Chapter 2
Asymptotic Derivatives and Asymptotic Scalar
Derivatives
Essentially this chapter has two parts. In the first part we present the notion
of the asymptotic derivative and some results related to this notion, and in the
second part we introduce the notion of the asymptotic scalar derivative. The
results presented in the first part are necessary for understanding the notions
that are given in the second part. It seems that the notion of an asymptotic
derivative was introduced by the Russian school, under the name of asymptotic
linearity. We found this notion in M. A. Krasnoselskii’s work and the reader is
referred to Krasnoselskii [1964a,b] and Krasnoselskii and Zabreiko [1984]. We
note that the main goal of this chapter is to present the notion of the asymptotic
scalar derivative and some of its applications. This chapter may be a stimulus
for new research in this subject.
2.1
Asymptotic Differentiability in Banach Spaces
Let (E, · ) and (F, · ) be Banach spaces. Let L(E, F ) be the Banach space
of linear continuous mappings, where the norm is L = supx=1 L(x),
for any L ∈ L(E, F ).
Definition 2.1 We say that a nonlinear mapping f : E → F is asymptotically linear, if there exists L ∈ L(E, F ) such that
f (x) − L(x)
= 0.
x
x→∞
lim
(2.1)
In this case we say that L is an asymptotic derivative of f .
Proposition 2.1 If f : E → F is asymptotically linear, then the mapping
L ∈ L(E, F ) that satisfies (2.1) is unique.
32
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Proof. Let f be asymptotically linear and let L1 , L2 ∈ L(E, F ) be two linear
mappings such that formula (2.1) is satisfied. We have
lim
x→∞
L1 (x) − L2 (x)
f (x) − L1 (x) f (x) − L2 (x)
≤ lim
+
.
x
x
x
x→∞
Then, for any ε > 0, there exists r > 0 such that for any x with x ≥ r, we
have
(L1 − L2 )(x)
< ε,
x
which implies
(
(
(
(
x
( < ε,
((L1 − L2 )
(
x (
for any x with x ≥ r. For any y ∈ S1 = {x ∈ E : x = 1} we consider
x = ρy with ρ > r and we have
(
(
(
x (
( < ε,
(
[(L1 − L2 )(y) = ((L1 − L2 )
x (
which implies L1 − L2 < ε and finally L1 − L2 = 0; that is, L1 = L2 . Remark 2.1 If f : E → F is asymptotically linear, then in this case we say
that the linear continuous mapping L used in Definition 2.1 is the asymptotic
derivative of f and we denote L = f∞ .
The following result due to M. A. Krasnoselskii is important in the study of
bifurcation problems [Amann, 1973, 1974b, 1976; Krasnoselskii, 1964a,b;
Krasnoselskii and Zabreiko, 1984]. We recall that a mapping f : E → F
is completely continuous, if it is continuous, and for any bounded set D ⊂ E,
we have that f (D) is relatively compact.
Theorem 2.2 Let f : E → F be a nonlinear mapping. If f is completely
continuous and asymptotically linear, then f∞ is completely continuous.
Proof. We use the fact that in a Banach space, a sequence is convergent if and
only if it is a Cauchy sequence.
Indeed, we assume that f∞ is not completely continuous. Then, we can
define a sequence {xn }n∈N ⊂ S(0, 1) = {x ∈ E : x = 1} such that
f∞ (xn )−f∞ (xm ) ≥ 3δ > 0, for any n and m such that n = m. Considering
formula (2.1) we deduce the existence of a real number r > 0 such that f (x)−
f∞ (x) < δx, for any x with x = r. Then, we have
f (rxn ) − f (rxm ) ≥ f∞ (rxn ) − f∞ (rxm ) − f (rxn ) − f∞ (rxn )
− f∞ (rxm ) − f (rxm ) > rf∞ (xn ) − f∞ (xm ) − 2δr,
33
2.1 Asymptotic Differentiability in Banach Spaces
which implies f (rxn ) − f (rxm ) ≥ δr for n = m and the compactness is
contradicted.
If a nonlinear mapping has an asymptotic derivative, the computation of this
derivative, generally cannot be so simple. We give now some examples.
(A) Let G ⊂ Rn be the closure of a bounded open set whose boundary is a
null set (i.e., G has a piecewise smooth boundary). Consider the following
Hammerstein mapping
)
K(t, s)f [s, ϕ(s)] d s,
A(ϕ)(t) =
G
where f : G × R → R and K : G × G → R. Suppose that the following
conditions are satisfied.
* *
(i) G G K 2 (t, s) d t d s < ∞.
(ii) The mapping f0 (ϕ)(s) = f [s, ϕ(s)], ϕ ∈ L2 is such that f0 : L2 → L2 .
+
(iii) |f (t, u) − u| ≤ nj=1 Sj (t)|u|1−pj + D(t), where t ∈ G; −∞ < u <
+ ∞; Sj (t) ∈ L2/pj , 0 < pj < 1, j = 1, 2, . . . , n, and D(t) ∈ L2 .
Consider the linear mapping
)
B(ϕ)(t) =
K(t, s)ϕ(s) d s.
G
In this case we have that A : L2 → L2 and B ∈ L(L2 , L2 ). Because
,) )
$2 -1/2
A(ϕ) − B(ϕ)
1
=
K(t, s)[f [s, ϕ(s)] − ϕ(s)] d s d t
ϕ
ϕ
G
G
.* *
/ ⎧ n )
$pj /2
2 (t, s) d t d s 1/2 ⎨
K
2/pj
G G
≤
Sj (t) d t
ϕ1−pj
⎩
ϕ
G
j=1
)
$ 1/2
D2 (t) d t
+
G
we deduce that
A(ϕ) − B(ϕ)
= 0;
ϕ
ϕ→∞
lim
that is, A∞ = B.
(B) Suppose that (E, · ) and (F, · ) are two particular Banach spaces of
functions defined on a particular subset (which can be as in Example (A)).
34
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Consider a function f (t, u) where t ∈ G and −∞ < u < +∞. Generally
we suppose that f satisfies Carathéodory conditions; that is, f is continuous with respect to u and measurable with respect to t. The operator
f ∗ (x)(t) = f [t, x(t)] is called the substitution mapping. Suppose that
f ∗ : E → F . If
f (t, u)
lim
t→∞
u
exists, we denote it g(t). In this case the asymptotic derivative of f ∗
must necessarily be of the form f∞ (h)(t) = g(t)h(t). In the next section
we present another interesting case when we can compute the asymptotic
derivative of a nonlinear mapping (when this derivative exists).
2.2
Hyers–Ulam Stability and Asymptotic Derivatives
The Hyers–Ulam stability of functional equations offers us the possibility to
compute the asymptotic derivative of an asymptotic differentiable mapping.
The notion of Hyers–Ulam stability of mappings has its origin in a problem defined by S. Ulam during a talk presented in 1940, at the mathematics club of the
University of Wisconsin, in which he discussed a number of unsolved problems.
This problem is related to the stability of homomorphisms. Given a group G1 ,
a metric group G2 with metric d(·, ·), and a real number ε > 0, does there exist
a δ > 0 such that if f : G1 → G2 satisfies d(f (xy), f (x)f (y)) < δ for all
x, y ∈ G1 , then a homeomorphism h : G1 → G2 exists with d(f (x), h(x)) < ε
for all x ∈ G1 ? In 1941, D. H. Hyers gave a positive answer to Ulam’s problem
for approximately additive mappings [Hyers, 1941]. He proved the following
result. If (E, · ) and (F, · ) are Banach spaces and f : E → F is a
mapping satisfying the condition
f (x + y) − f (x) − f (y) < ε
for all x, y ∈ E, then there is a unique additive mapping T satisfying
f (x) − T (x) ≤ ε.
Thus, the stability theory of Hyers and Ulam started. We note that for almost
three decades almost no progress was made on this problem, probably because
the theory of functional equations was not sufficiently developed at that time.
The Hyers–Ulam stability began to expand at the end of the seventies and now
there is extensive literature in the subject which forms the so-called Hyers–Ulam
stability theory. About this theory the reader is referred to the books by Czerwik
[1994, 2001], Hyers et al. [1998a] and Rassias [1978]. We present some results
from Hyers–Ulam stability theory related to the asymptotic differentiability.
In 1978, a generalized solution to Ulam’s problem for approximately linear
mappings was given by Th. M. Rassias (see [Rassias, 1978]). Let (E, · )
2.2 Hyers–Ulam Stability and Asymptotic Derivatives
35
and (F, · ) be Banach spaces. He considered a mapping f : E → F
satisfying the condition of continuity of f (tx) in t for each fixed x and such
that f (x + y) − f (x) − f (y) ≤ θ(xp + yp ), for any x, y ∈ E and that
T : E → F is the unique linear mapping satisfying
f (x) − T (x) ≤
2θ
xp .
2 − 2p
This result is valid also when p < 0 and when p > 1. The following definition
is due to G. Isac.
Definition 2.3 We say that a mapping f : E → F is ψ-additive if and only
if there exist θ > 0 and a function ψ : R+ → R+ such that
lim
t→∞
ψ(t)
=0
t
and
f (x + y) − f (x) − f (y) ≤ θ[ψ(x) + ψ(y)]
for all x, y ∈ E.
In 1991 Isac and Rassias proved the following result published in Isac and
Rassias [1993a].
Theorem 2.4 Let (E, · ) and (F, · ) be Banach spaces and f : E → F
a mapping such that f (tx) is continuous in t for each fixed x. If f is ψ-additive
and ψ satisfies
1. ψ(ts) ≤ ψ(t)ψ(s), for all t, s ∈ R+ ;
2. ψ(t) < t, for all t > 1;
then there exists a unique linear mapping T : E → F such that
$
2θ
ψ(x),
f (x) − T (x) ≤
2 − ψ(2)
for all x ∈ E1 .
Proof. We show that
( , n−1
(
ψ(2) $m
(
( f (2n x)
(
(
ψ(x)
( 2n − f (x)( ≤ θ
2
(2.2)
m=0
for any positive integer n, and for any x ∈ E. The proof of (2.2) follows by
induction on n. For n = 1 by ψ-additivity of f we have
f (2x) − 2f (x) ≤ 2θψ(x),
36
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
which implies
(
(
(
( f (2x)
( ≤ θψ(x).
(
−
f
(x)
(
( 2
Assume now that (2.2) holds for n and we want to prove it for the case n + 1.
Replacing x by 2x in (2.2) we obtain
( , n−1
(
ψ(2) $m
(
( f (2n 2x)
(
ψ(2x).
− f (2x)(
(≤ θ
( 2n
2
m=0
Because ψ(2x) ≤ ψ(2)ψ(x) we get
(
( , n−1 ψ(2) $m
( f (2n+1 x)
(
(
ψ(2)ψ(x).
− f (2x)(
(
(≤ θ
2n
2
(2.3)
m=0
Multiplying both sides of (2.3) by 1/2 we obtain
(
( , n ψ(2) $m
( f (2n+1 x) f (2x) (
(
(
ψ(x).
( 2n+1 − 2 ( ≤ θ
2
m=1
Now, using the triangle inequality we deduce
(
(
(
(
( 1 0 n+1 1
(
(
( 1 0 n+1 1 1
(
(
x) − f (x)(
x) − [f (2x)](
( 2n+1 f (2
(
( ≤ ( 2n+1 f (2
2
(
(
(
(1
(
+(
( 2 [f (2x)] − f (x)(
, n ψ(2) $m
ψ(x) + θψ(x)
≤
θ
2
m=1
,
= θψ(x) 1 +
$
n ψ(2) m
m=1
2
,
which proves (2.3). Thus,
,
(
(
$m n (
( 1 0 n+1 1
ψ(2)
2θψ(x)
(
.
≤
x) − f (x)(
( ≤ θψ(x) 1 +
( 2n+1 f (2
2
2 − ψ(2)
m=1
For m > n > 0 we have
(
(
(
( 1
1
m
n
( =
(
[f
(2
x)]
−
[f
(2
x)]
(
( 2m
2n
=
(
(
(
1 (
( 1 [f (2m x) − f (2n x)](
(
2n ( 2m−n
(
(
(
1
1 (
r
( [f (2 y) − f (y)]( ,
(
(
n
r
2 2
37
2.2 Hyers–Ulam Stability and Asymptotic Derivatives
where r = m − n and y = 2n x.
(
(
(
( 1
1
m
n
( ≤
(
[f
(2
x)]
−
[f
(2
x)]
(
( 2m
2n
=
≤
$
1
2ψ(y)
θ
2n 2 − ψ(2)
$
1
2ψ(y)
θ
2n 2 − ψ(2)
$
1
2ψ(2n )ψ(x)
θ
2n
2 − ψ(2)
ψ(2)
2
≤
But because
lim
n→∞
we have that
2
1
2n
ψ(2)
2
$n $
2ψ(x)
.
θ
2 − ψ(2)
$n
= 0,
3
[f (2 x)]
n
n∈N
is a Cauchy sequence. Set
f (2n x)
,
n→∞
2n
for all x ∈ E. The mapping x → T (x) is additive. Indeed, we have
T (x) = lim
f [2n (x + y)] − f (2n x) − f (2n y) ≤ θ[ψ(2n x) + ψ(2n y)]
= θ[ψ(2n nx) + ψ(2n y)]
≤ θψ(2n )[ψ(x) + ψ(y)],
which implies that
1
f [2n (x + y)] − f (2n x) − f (2n y)
2n
$
$
ψ(2) n
ψ(2n )
θ[ψ(x) + ψ(y)] ≤
θ[ψ(x) + ψ(y)].
≤
2n
2
However,
$
ψ(2) n
= 0,
lim
n→∞
2
thus
1
lim
f [2n (x + y)] − f (2n x) − f (2n y) = 0.
n→∞ 2n
38
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Therefore,
T (x + y) = T (x) + T (y),
(2.4)
for all x, y ∈ E. Because of (2.4) it follows that T (rx) = rT (x) for any
rational number r, which implies that T (ax) = aT (x) for any real value of a.
Hence, T is a linear mapping. From
(
(
(
( f (2n x)
ψ(x)
(
(
( 2n − f (x)( ≤ 2θ 2 − ψ(2) ,
taking the limit as n → ∞ we obtain
T (x) − f (x) ≤
2θψ(x)
.
2 − ψ(2)
(2.5)
We claim that T is the unique such linear mapping. Suppose that there exists
another one, denoted g : E → F satisfying
f (x) − g(x) ≤
2θ1 ψ1 (x)
.
2 − ψ1 (2)
(2.6)
From (2.5) and (2.6) we get
T (x) − g(x) ≤ T (x) − f (x) + f (x) − g(x)
≤
2θψ(x) 2θ1 ψ1 (x)
+
.
2 − ψ(2)
2 − ψ1 (2)
Then,
(
(
(
(1
1
(
T (x) − g(x) = ( T (nx) − g(nx)(
(
n
n
$
$ $
$
ψ(n) 2θψ(x)
ψ1 (n) 2θ1 ψ1 (x)
≤
+
,
n
2 − ψ(2)
n
2 − ψ1 (2)
for every positive integer n > 1. However,
ψ(n)
ψ1 (n)
= 0 = lim
.
n→∞ n
n→∞
n
lim
Therefore, T (x) = g(x) for all x ∈ E.
The mapping T defined by Theorem 2.4 has some remarkable properties.
(A) If f (S) is bounded, where S = {x ∈ E : x = 1}, in particular if f
is completely continuous, then T is continuous. Indeed, this is the conse-
2.2 Hyers–Ulam Stability and Asymptotic Derivatives
39
quence of the inequalities
T (x) ≤ f (x) + T (x) − f (x)
≤ f (x) +
2θ
ψ(x)
2 − ψ(2)
≤ f (x) +
2θ
ψ(1),
2 − ψ(2)
for all x ∈ S.
(B) When, the linear mapping T defined by Theorem 2.4 is continuous, in particular when f (S) is bounded or f is completely continuous, we have that
f is asymptotically linear and f∞ = T . Indeed, we have
lim
x→+∞
2θ
f (x) − T (x)
ψ(x)
≤
lim
= 0.
x
2 − ψ(2) x→+∞ x
The class of functions ψ : R+ → R+ , which satisfies conditions asked in
Theorem 2.4; that is,
ψ(t)
= 0;
t
(i1 ) ψ(ts) ≤ ψ(t)ψ(s), for all t, s ∈ R+ ;
(i2 ) ψ(t) < t, for all t > 1;
(i0 )
lim
t→+∞
is not empty. In this case we can cite the following functions.
(1) ψ(t) = tp , with p ∈ [0, 1[;
2
0 if t = 0,
(2) ψ(t) =
tp if t > 0, where p < 0.
Now, we show that it is possible to enlarge the class of functions ψ such
that the conclusion of Theorem 2.4 remains valid. Let F(ψ) be the set of all
functions ψ : R+ → R+ satisfying conditions (i0 ), (i1 ), and (i2 ). Let P(ψ)
be the convex cone (for the definition of a convex cone see the first section of
Chapter 4) generated by the set F(ψ) (i.e., the smallest convex cone containing
this set). We remark that a function ψ ∈ P(ψ) satisfies the assumption (i0 )
but generally does not satisfy the assumptions (i1 ) and (i2 ). However, we show
that Theorem 2.4 remains valid for ψ-additive functions with ψ ∈ P(ψ). The
following result is a consequence of the main result proved in Gavruta [1994].
Lemma 2.5 If φ : E × E → [0, +∞[ is a mapping such that
φ0 (x, y) =
∞
k=0
2−k ψ(2k x, 2k y) < +∞,
40
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
for all x, y ∈ E and f : E → F is a continuous mapping such that
f (x + y) − f (x) − f (y) ≤ φ(x, y),
for all x, y ∈ E, then there exists a unique linear mapping T : E → F such
that
1
f (x) − T (x) ≤ φ0 (x, x),
2
for all x ∈ E. Moreover
f (2n x)
,
n→∞
2n
T (x) = lim
for all x ∈ E.
A consequence of Lemma 2.5 is the following result which is a generalization
of Theorem 2.4.
Theorem 2.6 Let f : E → F be a continuous mapping and ψ ∈ P(ψ),
such that
m
ai ψi ,
ψ=
i=1
where for each i, ai > 0 and ψi ∈ F(ψ). If f is ψ-additive, then there exists a
unique linear mapping T : E → F such that
f (x) − T (x) ≤ 2θM ψ(x),
for any x ∈ E, where
2
M = max
3
1
: i = 1, 2, · · · , m
2 − ψi (2)
and
f (2n x)
,
n→∞
2n
T (x) = lim
for any x ∈ E. Moreover,
ψ(x)
= 0.
x→∞ x
lim
Proof. We consider the function
Φ(x, y) = θ[ψ(x) + ψ(y)],
for any x, y ∈ E, where θ is the constant used in the ψ-additivity assumption,
and we apply Lemma 2.5. To do this first we must show that
Φ0 (x, y) =
∞
k=0
2−k Φ(2k x, 2k y)
41
2.2 Hyers–Ulam Stability and Asymptotic Derivatives
is convergent for any x, y ∈ E. Indeed, we have
Φ0 (x, y) = θ
∞
m
2−k
i=1
k=0
m
= θ
ai
∞
i=1
≤ θ
i=1
because the series
+m
ai
2
ψi (2 x) +
+∞
k
2
4
k=0
ψi (2)
2
$
∞ ψi (2) k
2
$
∞ ψi (2) k
k=0
ai ψi (2k y)
m
ai
i=1
k=0
k=0
and
−k
$
∞ ψi (2) k
i=1 ai
i=1
k=0
,m
+
ai ψi (2k x) +
m
2
∞
−k
2
ψi (2 y)
k
k=0
ψi (x)
5k
3
ψi (y) < ∞,
ψi (x)
ψi (y)
are convergent. Applying Lemma 2.5 we have that T is well defined by
f (2n x)
,
n→∞
2n
T (x) = lim
for any x ∈ E. Because f is continuous, we have that T is not only additive as
in Gavruta [1994] but it is linear too. We have
1
f (x) − T (x) ≤ Φ0 (x, x),
2
for any x ∈ E. Now, we evaluate Φ0 (x, x). We have
Φ0 (x, x) = 2θ
,m
i=1
= 4θ
+m
ai
$
∞ ψi (2) k
k=0
2
i=1 ai ψi (x)
ψi (x)
1
,
2 − ψi (2)
42
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
which implies
f (x) − T (x) ≤ 2θ
m
ai ψi (x)
i=1
≤ 2θM
m
1
2 − ψi (2)
ai ψi (x) = 2θM ψ(x),
i=1
2
where
M = max
3
1
: i = 1, 2, . . . , m .
2 − ψi (2)
Because for any i = 1, 2, . . . , m, ψi satisfies assumption (i0 ), we have that
ψi (x)
= 0,
x
x→∞
lim
which implies that
ψi (x)
= 0.
x
x→∞
lim
Remark 2.2
1. If f : E → F is continuous, ψ-additive with ψ ∈ P(ψ) and f (S) is
bounded, then in this case we also have that f∞ = T .
2. Theorem 2.6 is significant, because the class of ψ-additive mappings with
ψ ∈ P(ψ) is strictly larger than the class of mappings defined in Theorem
2.4. In this sense we remark the following results.
(a) If f : E → F is a ψ-additive mapping with ψ ∈ P(ψ) and L ∈
L(E, F ), then L + f is a ψ-additive mapping with respect to the same
function ψ.
(b) If f : E1 → E2 is a ψ-additive mapping with ψ ∈ P(ψ) and L ∈
L(E2 , E3 ), then L ◦ f is a ψ-additive mapping from E1 into E3 with
respect to the same function ψ and the constant θ replaced by θL. We
note that E1 , E2 , and E3 are Banach spaces.
(c) If f1 , f2 : E → E are mappings such that f1 is ψ1 additive and f2 is ψ2
additive, then for every a1 , a2 ∈ R+ \{0}, we have that a1 f1 + a2 f2 is
a ψ-additive mapping where ψ = ψ1 + ψ2 and θ = max{a1 θ1 , a2 θ2 }.
More results about ψ-additivity and its generalizations are given in Czerwik
[1994, 2001], Gavruta [1994], Hyers et al. [1998a,b], Isac and Rassias [1993a,b,
1994]. It is interesting to note that Theorem 2.4 was recently proved again by
43
2.2 Hyers–Ulam Stability and Asymptotic Derivatives
V. Radu using the fixed point theory [Radu, 2003]. Perhaps Radu’s method will
open a new research direction in the Hyers–Ulam stability of mappings.
Remark 2.3 We note that the constant
2θ
2 − ψ(2)
used in Theorem 2 given in Isac and Rassias [1993b] must be the constant 2θM
computed in Theorem 2.6, or the constant
2θ
2 − ψ(2)
must be
2θ
,
2 − ψi0 (2)
where
1
= M = max
2 − ψi0
2
1
2 − ψi (2)
3m
.
i=1
We remarked above that under the assumptions of Theorem 2.6, if f is continuous and f (S) is bounded, we have that
f (2n x)
,
n→∞
2n
f∞ (x) = lim
for any x ∈ E. Conversely, if f has an asymptotic derivative, namely,
f∞ = T ∈ L(E, F ),
that is, if
f (x) − T (x)
,
n→∞
x
lim
then at any point x ∈ E, we have that
f (2n x)
,
n→∞
2n
f (x) = lim
for any x ∈ E. Indeed, if x ∈ E\{0}, then we have that 2n x → ∞ as
n → ∞ and
f (2n x) − T (2n x)
n→∞
2n x
0 = lim
f (2n x) − T (2n x)
n→∞
2n x
(
( n
(
(2 x
1
lim (
− T (x)(
=
(.
(
n
x n→∞ 2
=
lim
44
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Therefore,
f (2n x)
,
n→∞
2n
for any x ∈ E, because this formula is also true for x = 0.
T (x) = lim
We recall the following classical result due to Krasnoselskii.
Theorem 2.7 Let f : E → E be a mapping such that for a ρ > 0 sufficiently
large, the mapping f has a Frechét derivative denoted by f (x), at any element
x with x > ρ, and
lim f (x) − T = 0,
x→∞
where T ∈ L(E, F ), then f∞ = T .
Proof. This result is a particular case of Theorem 3.3 proved in the Russian
edition of Krasnoselskii [1964a].
The following result is inspired by Theorem 2.7. The Frechét derivative is
replaced by a ψ-additive mapping.
Theorem 2.8 Let f : E → F be a mapping, g : E → F a continuous
mapping such that g(S) is bounded, and ψ1 a mapping which satisfies condition
(i0 ). If the following two assumptions are satisfied,
1. there exist two constants, ρ > 0 and M1 > 0 such that
f (x) − g(x) ≤ M1 ψ1 (x),
for any x ∈ E with x > ρ,
2. g is ψ2 -additive, with ψ2 ∈ P(ψ),
then f is an asymptotically linear mapping and
g(2n x)
f (2n x)
=
lim
n→∞
n→∞
2n
2n
f∞ (x) = lim
for any x ∈ E.
Proof. Because g is ψ2 -additive with ψ2 ∈ P(ψ), we apply Theorem 2.7
and we obtain a constant M2 > 0 and a continuous linear mapping T : E → F
such that
g(x) − T (x) ≤ 2θM2 ψ2 (x),
for every x ∈ E. We know that
g(2n x)
,
n→∞
2n
T (x) = lim
2.3 Asymptotic Differentiability Along a Convex Cone in a Banach Space
45
for any x ∈ E. We have
f (x) − T (x) ≤ f (x) − g(x) + g(x) − T (x)
≤ M1 ψ1 (x) + 2θM2 ψ2 (x),
for all x ∈ E, which implies
f (x) − T (x)
2θM2 ψ2 (x)
≤ lim M1 ψ1 (x)x + lim
= 0.
x
x
x→∞
x→∞
Remark 2.4 Krasnoselskii in [1964a] considers the following definition of
the asymptotic derivative of a mapping f : E → F . We say that a linear
mapping f∞ ∈ L(E, F ) is the asymptotic derivative of f if
f (x) − f∞ (x)
= 0.
n→∞ x≥R
x
lim sup
Many interesting applications of this notion are given in Krasnoselskii in [1964a].
2.3
Asymptotic Differentiability Along a Convex Cone in a
Banach Space
Let (E, · ) be a Banach space and K ⊂ E a closed pointed convex cone (for a
definition see the first section of Chapter 4). We say that K is a generating cone
if E = K − K. If K has a nonempty interior then K is generating. Indeed, let
K̊ be the interior of K. Let v0 ∈ K̊ be an arbitrary element. For any x ∈ E,
there exists λ ∈]0, 1[ such that
y = λv0 + (1 − λ)x ∈ K.
If
ρ=
1−λ
λ
and
z=
1
y,
λ
then we have
z = v0 + ρx ∈ K,
which implies
x = u − v,
where
u=
1
z∈K
ρ
46
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
and
1
v0 ∈ K.
ρ
Let (F, · ) be another Banach space and f : K → F a mapping.
v=
Definition 2.9 We say that f is asymptotically linear along the cone K if
there exists T ∈ L(E, F ) such that
lim
x→∞
x∈K
f (x) − T (x)
= 0.
x
In this case we say that T is an asymptotic derivative of f with respect to K or
a derivative at infinity with respect to K.
If K is a generating cone in E and T ∈ L(E, F ) is an asymptotic derivative,
∞ . Now, we
then T is unique. In this case we denote the linear mapping T by fK
suppose that F = E and K ⊂ E is a generating, closed pointed convex cone.
We say that a linear mapping T : E → E is positive if T (K) ⊆ K. Similarly,
we say that a general mapping f : E → E is positive if f (K) ⊆ K.
Proposition 2.2 If f : E → E is a positive and asymptotically linear
∞ is a positive linear mapping.
mapping, along the cone K, then fK
∞ (x ) ∈
Proof. Suppose that there exists x∗ ∈ K such that fK
∗ / K. Without
restriction we may suppose that x∗ = 1. By the formula used in Definition
2.9 we have
f (ax∗ )
∞
= fK
(x∗ ),
lim
a→∞
a
∞ (x ) ∈ K and we have a contradiction.
and by the closedness of K we have fK
∗
∞ (K) ⊆ K; that is, f ∞ is positive.
Therefore, fK
K
Lemma 2.10 If (E, · ) is a Banach space ordered by a pointed generating
closed convex cone K ⊂ E, then there exists a constant M > 0 such that for
any element x ∈ E, there exist u, v ∈ K such that x = u−v and u ≤ M x,
v ≤ M x.
Proof. A proof of this result is given on p. 102 of the Russian edition of
Krasnoselskii [1964a].
We recall that a mapping f : E → E is completely continuous with respect
to K if f is continuous and for any bounded set D ⊂ K, we have that f (D) is
relatively compact. A mapping can be completely continuous with respect to
K but not completely continuous with respect to the space E.
Theorem 2.11 Let (E, · ) be a Banach space ordered by a generating,
closed pointed convex cone K ⊂ E. Let f : E → E be a completely continuous
2.3 Asymptotic Differentiability Along a Convex Cone in a Banach Space
47
mapping with respect to K. If f is asymptotically linear along the cone K,
∞ is a linear completely continuous mapping.
then fK
Proof. Because K is a generating cone, then by Lemma 2.10, there exists
a constant M0 > 0 such that every x ∈ E has a decomposition of the form
x = u − v, with u, v ∈ K and such that u + v ≤ M x. Considering
∞ maps B(0, 1) ∩ K into
this fact it is easily seen that it suffices to show that fK
a compact set. Suppose that this is not true. In this case we can suppose that
there exists ε > 0 and a sequence {xn }n∈N ⊂ S(0, 1) ∩ K such that
∞
(xn − xm ) > 3ε
fK
for n = m. Let α > 0 be a real number such that for all x ∈ K with x = α,
∞
(x) < εx
f (x) − fK
∞ ). Then for n = m we have
(we used the definition of fK
∞
(xn − xm )
f (αxn ) − f (αxm ) ≥ αfK
∞
∞
(αxm ) ≥ αε,
− f (αxn ) − fK (αxn ) − f (αxm ) − fK
which contradicts the compactness of f on bounded subsets of K.
It is well known that the asymptotic derivative of a nonlinear mapping, with
respect to a Banach space or with respect to a closed convex cone has many
interesting applications to the study of bifurcation problems or to the study
of fixed points. About this subject the reader is referred to Amann [1973,
1974a,b, 1976], Cac and Gatica [1979], Krasnoselskii [1964a,b], Krasnoselskii
and Zabreiko [1984], Talman [1973] among others. We also note that the notion of asymptotic derivative inspired some ideas developed in Mininni [1977].
∞ ) the spectral radius
Now, we cite the following result. We denote by ρ(fK
∞
of fK .
Theorem 2.12 (Krasnoselskii) Let (E, ·) be a Banach space ordered
by a generating, closed pointed convex cone. Let f : E → E be a positive
∞ ) < 1,
completely continuous mapping. If f is asymptotically linear and ρ(fK
then f has a fixed point in K.
Proof. A proof of this result can be found in Amann [1974a] and Krasnoselskii
[1964a]. Several authors generalized this result, but in this chapter we give another generalization following another point of view and using the asymptotic
scalar derivatives.
Let (H, ·, ·) be a Hilbert space, · the norm generated by ·, ·, and
f : H → H. We again use the notion of ψ-additivity (Definition 2.3).
48
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Theorem 2.13 Suppose that f (tx) is continuous in t for each fixed x. If f
is ψ-additive and ψ satisfies
1. ψ(ts) ≤ ψ(t)ψ(s), for all t, s ∈ R+ ;
2. ψ(t) < t, for all t > 1;
then there exists a linear mapping T : H → H such that
|f (x) − T (x), x| ≤
2θψ(x)x
,
2 − ψ(2)
(2.7)
for all x ∈ H. S is another linear mapping satisfying (2.7) iff T − S is
skew-adjoint.
Proof. By Theorem 2.4 there exists a unique linear mapping T such that
f (x) − T (x) ≤
2θψ(x)
,
2 − ψ(2)
(2.8)
for all x ∈ H. Moreover, we have T (x) = limn→∞ (f (2n x)/2n ), for all
x ∈ H. Hence, by using the Cauchy inequality in (2.8), we obtain (2.7).
Suppose that S is another linear mapping satisfying (2.7). Hence,
|T (x) − S(x), x| ≤ |T (x) − f (x), x| + |f (x) − S(x), x|
4θψ(x)x
.
≤
2 − ψ(2)
Then,
1
1
T (nx) − S(nx), x |T (x) − S(x), x| = n
n
ψ(n) 4θψ(x)x
.
≤
n
2 − ψ(2)
Because limn→∞ (ψ(n)/n) = 0, we obtain that T (x) − S(x), x = 0. Thus,
T − S is skew-adjoint. Conversely, if T − S is skew-adjoint, then T (x) −
S(x), x = 0. Hence,
|f (x) − S(x), x| ≤ |f (x) − T (x), x| + |T (x) − S(x), x|
2θψ(x)x
.
= |f (x) − T (x), x| ≤
2 − ψ(2)
2.4 Asymptotic Differentiability in Locally Convex Spaces
2.4
49
Asymptotic Differentiability in Locally Convex Spaces
First we recall the following definition of a locally convex space. Let E be a real
vector space. We suppose that in E is defined a family of seminorm {| · |α }α∈A
which generates a topology τ such that E endowed with this topology is a
locally convex topological vector space; that is, the collection of sets {{x :
|x|α ≤ λ} : α ∈ A and λ is a positive real number} is a base for a filter of
neighbourhoods of zero in E. About the family {| · |α }α∈A of seminorms we
suppose satisfied the following properties.
(i) (∀x ∈ E)(x = 0)(∃α0 ∈ A)(|x|α0 = 0),
(ii) (∀α1 , α2 ∈ A)(∃α ∈ A)(| · |α1 , | · |α2 ≤ | · |α ).
We note that the topology defined on E by the family {|·|α }α∈A of seminorms is
a Hausdorff topology. We denote this locally convex space by (E, {| · |α }α∈A ).
Let (E, {| · |α }α∈A ) and (F, {| · |β }β∈B ) be two locally convex spaces and
f : E → F a linear mapping. We know (see [Marinescu, 1963]) that f
is continuous if and only if there exists a function ψ : B → A such that
|x|ψ(β) = 0 implies |f (x)|β = 0 and
|f |β,ψ(β) :=
|f (x)|β
< ∞,
|x|ψ(β) =0 |x|ψ(β)
sup
for every β ∈ B.
If L(E, F ) is the vector space of linear continuous mappings from E into
F , then L(E, F ) is the pseudo-topological union of spaces Lψ (E, F ), where
Lψ (E, F ) = {f : E → F | f is linear and |f |β,ψ(β) < +∞, for every β ∈ B};
that is,
L(E, F ) =
Lψ (E, F ),
ψ∈F (B,A)
where
F(B, A) = {ψ : B → A}.
For this result the reader is referred to Marinescu [1963]. Let K ⊂ E be a closed
pointed convex cone. We suppose that K is total in E; that is, K − K = E.
Definition 2.14 We say that a mapping f : K → F is asymptotically
linear along the cone K if there exist a function ψ : B → A and a linear
continuous mapping f∞ ∈ Lψ (E, F ) such that
lim
x∈K
|x|ψ(β)
|f (x) − f∞ (x)|β
=0
|x|ψ(β)
50
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
for any β ∈ B.
Because the cone K is total in E, the mapping f∞ is unique. We say that f∞
is the asymptotic derivative of f along the cone K. Let (E, {| · |α }α∈A ) be an
arbitrary locally convex space. For every α ∈ A and every subset Ω ⊂ E we
define the measures of noncompactness:
γα (Ω) = inf{d > 0 : Ω can be covered by a finite number of sets of
· α -diameter≤ d},
χα (Ω) = inf{r > 0 : Ω can be covered by a finite number of · α -balls of
· α -radius≤ d}.
We consider the set of functions
C = {f : A → [0, +∞]}
ordered by the ordering
f, g ∈ C, f ≤ g if and only if f (α) ≤ g(α) for any α ∈ A.
We consider also the following functions: γ : 2E → C defined by
Ω → (γ(Ω))(α) = γα (Ω),
for any α ∈ A. χ : 2E → C defined by
Ω → (χ(Ω))(α) = χα (Ω),
for any α ∈ A. Because the functions γ and χ have similar properties we denote
by Φ or the function γ or the function χ. In nonlinear analysis it is known that
the function Φ has the following properties.
(1) Φ(A ∪ B) ≤ max{Φ(A), Φ(B)}, for any A, B ∈ 2E .
(2) Φ(A) = 0 if and only if A is a totally bounded set. (We recall that a set
A is totally bounded if for each 0-neighbourhood U there exists a finite
subset A0 ⊂ A such that A ⊂ A0 + U . Because a locally convex space is
Hausdorff, a subset A is totally bounded if and only if it is precompact.)
(3) Φ(co(A)) = Φ(A), where co(A) is the closed convex hull of A.
(4) A ⊆ B implies Φ(A) ≤ Φ(B).
(5) For any λ ∈ R+ and any α ∈ A we have Φ(λA) = λΦ(A).
(6) If Bα is the open ball of | · |α -radius = 1, then Φ(Bα ) ≤ 2.
2.4 Asymptotic Differentiability in Locally Convex Spaces
51
(7) For any α ∈ A we have Φ(A + B) ≤ Φ(A) + Φ(B).
For the proof of properties (1)–(7) the author is referred to references [1], [4],
[12–15] and [17] cited in Isac [1982]. Let E and F be locally convex spaces
such that the family of seminorms for each space is denoted by the same set
A; that is, (E, { · α }α∈A ) and (F, { · β }β∈A ). Therefore, for both spaces
we have the same set C. We note that we have this situation in particular when
E = F or when E and F are Frechét spaces. We denote by ΦE (resp., ΦF ) the
function defined above considering the function γ (resp., χ). Let D be a subset
of E (supposed to be a nonempty set). We can have D = E.
Definition 2.15 We say that a mapping f : D → F is an (α∗ , Φ)-contraction
if
ΦF (f (Q)) ≤ α∗ ΦE (Q),
for any nonempty bounded set Q ⊂ D, where α∗ is a function from A into R+ .
Remark 2.5 Because ΦE (Q) : A → [0, +∞] the inequality used in Definition 2.15 means
ΦF (f (Q))(α) ≤ α∗ (α)ΦE (Q)(α).
The following result is given with respect to a total closed convex cone, but we
have a similar proof when the cone is the space itself. This result is due to Isac.
Theorem 2.16 Let E and F be locally convex spaces such that the family of
seminorms for each space is indexed by the same set A; that is, (E, {·α }α∈A )
and (F, {·β }β∈A ). Let K ⊂ E be a total closed convex cone and f : K → F
a (α∗ , Φ)-contraction mapping. If f is asymptotically linear along the cone K,
then f∞ |K is an (α∗ , Φ)-contraction.
Proof. We denote u = f∞ . Let α ∈ A be an arbitrary element and A ⊂ K a
bounded subset such that there exists ρ > 0 with the property that |x|ψ(α) ≥ ρ,
for any x ∈ A (the function ψ is given by Definition 2.15). Let σ be a positive
real number such that
σ > sup{|x|ψ(α) : x ∈ A},
and let ε > 0 be arbitrary. We denote r = f − u. Because f is asymptotically
linear along the cone K, there exists δ > 0 such that for any x ∈ K with the
property |x|ψ(α) ≥ δ we have
|r(x)|α ≤
ε
.
2σ
If we denote BαF = {x ∈ F : |x|α < 1}, then for any λ ≥ (δ/ρ) we have
r(λA) ⊂
λε
,
BαF
52
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
because
|λx|ψ(α) = λ|x|ψ(α) ≥
δ
xψ(α) ≥ δ.
ρ
Therefore, we have
u(λA) ⊂ f (λA) − r(λA) ⊂ f (λA) +
λε F
B ,
2 α
which implies (denoting by γ E (resp., γ F ) the measure of noncompactness on
E ( resp., on F ))
λε F F
γ (B )
2 α α
≤ α∗ (α)γαE (λA) + λε = λ(α∗ (α)γαE (A) + ε).
λγαF (u(A)) = γαF (u(λA)) ≤ γαF (f (λA)) +
Because ε > 0 is arbitrary we have
γαF (u(A)) ≤ α∗ (α)γαF (A).
(2.9)
Now, we suppose that A ⊂ K is an arbitrary nonempty bounded set and ε > 0
is an arbitrary real number. Because u = f∞ ∈ Lψ (E, F ), where ψ : A → A
is used for the continuity of U and for the asymptotic linearity of f we have,
|u|α,ψ(α) =
|u(x)|α
< ∞,
|x|ψ(α) =0 |x|ψ(α)
sup
and if |x|ψ(α) = 0, then |u(x)|α = 0. For the properties of |u|α,ψ(α) see
Marinescu [1963]. First, we suppose that |u|α,ψ(α) = 0 and we take
ρ=
ε
2|u|α,ψ(α)
.
E
We define A1 = A ∩ ρBψ(α)
and A2 = A\A1 . In this case we have
ε
u(A1 ) ⊂ BαF .
2
(2.10)
E
If |u|α,ψ(α) = 0, we take A1 = A ∩ Bψ(α)
and A2 = A\A1 and considering
the definition of |u|α,ψ(α) and the continuity of u we obtain again the formula
(2.10). In both situations we have
γαF (u(A1 )) ≤ ε.
From the first part of the proof we deduce
γαF (u(A2 )) ≤ α∗ (α)γαE (A2 ) ≤ α∗ (α)γαE (A).
2.4 Asymptotic Differentiability in Locally Convex Spaces
53
Finally, we obtain
γαF (u(A)) = γαF (u(A1 ) ∪ u(A2 )) ≤ max{γαF (u(A1 )), γαF (u(A2 ))}
≤ max{ε, α∗ (α)γαE (A)}.
Because ε > 0 is arbitrary, we obtain formula (2.9) for an arbitrary bounded
set A ⊂ K. Now, if we pass to the function Φ we have
ΦF (u|K (A)) ≤ α∗ ΦE (A),
because the same proof is valid if we replace for any α ∈ A the measure of
noncompactness γα by χα . The proof of the theorem is complete.
We recall that A. Granas defined the notion of quasi-bounded mapping
Granas [1962]. Let (E, · ) be a Banach space and f : E → E a mapping. We say that f is a quasi-bounded mapping if
lim sup
x→∞
f (x)
f (x)
= inf sup
< ∞.
ρ>0 x≥ρ x
x
If f is quasi-bounded, then the real number
|f |qb = lim sup
x→∞
f (x)
x
is called the quasi-norm of f . Any bounded linear mapping L : E → E is
quasi-bounded and |L|qb = L. If f is a nonlinear mapping such that ∃M > 0
with the property
f (x) ≤ M x,
for any x ∈ E, then f is quasi-bounded. The notion of quasi-bounded mapping
has interesting applications in fixed point theory. Now, we generalize this
notion to locally convex spaces. Let (E, {| · α }α∈A ) and (F, {| · β }β∈B ) be
two totally convex spaces and f : E → F a mapping.
Definition 2.17 We say that f is quasi-bounded if there exists a function
ψ : B → A such that the numbers
,
|f (x)|β
sup
f β,ψ(β) = inf
0<ρ<∞ |x|ψ(β)≥ρ |x|ψ(β)
are finite for any β ∈ B.
The following results are consequences of Definition 2.17.
Proposition 2.3 If the mapping f : E → F is quasi-bounded, then there
exists a function ψ : B → A such that for any ε > 0 there exists ρ > 0 with
the property that for any x ∈ E with |x|ψ(β) > ρ, we have
|f (x)|β ≤ (f β,ψ(β) + ε)|x|ψ(β) .
54
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Proposition 2.4 If there exists a function ψ : B → A and for each β ∈ B
there exist two constants k1,ψ(β) ≥ 0 and k2,ψ(β) ≥ 0 such that
|f (x)|β ≤ k1,ψ(β) |x|ψ(β) + k2,ψ(β) ,
for any x ∈ E, then f is a quasi-bounded mapping and f β,ψ(β) ≤ k1,ψ(β) .
We recall that if T : E → F is a linear mapping, then T is continuous if for
any β ∈ B there exists α ∈ A and a constant Mβ,α ≥ 0 such that
|T (x)|β ≤ Mβ,α |x|α ,
for any x ∈ E. For any T ∈ L(E, F ), α ∈ A, and β ∈ B we define
|T |β,α = sup |T (x)|β .
|x|α ≤1
We observe that |T |β,α can be zero for T = 0 and it can be +∞. We have that
T ∈ L(E, F ) is continuous if and only if there exists a function ψ : B → A
such that
T ∈ Lψ (E, F ) = {f : E → F : f is linear and |f |β,ψ(β) < +∞ for any
β ∈ B}.
Proposition 2.5 If f : E → F is asymptotically linear along the cone
K = E, then f is quasi-bounded and
f β,ψ(β) = |f∞ |β,ψ(β) ,
where ψ is the function used in the definition of asymptotic linearity and in the
continuity of f∞ .
In 1973, Louis A. Talman, presented in his PhD thesis (Graduate School
of the University of Kansas) another approach of asymptotical differentiability
along a closed convex cone in a locally convex space [Talman, 1973]. Now
we present his approach and some of his results. Let (E, {| · |α }α∈A ) be an
arbitrary (Hausdorff) locally convex space. Consider again the power set 2E
and the set C(A) = C = {f : A → [0, +∞]} ordered by f ≤ g if and only if
f (α) ≤ g(α), for any α ∈ A.
Definition 2.18 We say that a function Ψ : 2E → C(A) is a measure of
noncompactness on E if for every A, B ∈ 2E , for every λ ∈ R, and for every
λ ∈ A the following properties are satisfied.
(1) Ψ(A)(α) < +∞ if A is bounded.
(2) Ψ(A) ≡ 0 if and only if A is precompact.
2.4 Asymptotic Differentiability in Locally Convex Spaces
55
(3) Ψ(A ∪ B) ≤ max(Ψ(A), Ψ(B)).
(4) Ψ(λA) = |λ|ψ(A).
(5) Ψ(A + B) ≤ Ψ(A) + Ψ(B).
(6) Ψ(A) = Ψ(clE A) = Ψ(conv A), where clE A is the closure of A with
respect to E and conv A is the convex hull of A.
(7) There is a convex balanced neighbourhood of zero, Uα , in E such that
Ψ(Uα ) = 1 and such that if Ψ(A)(α) ≤ ρ < ∞, then for every δ > 0
there is a finite set {x1 , x2 , . . . , xn } ⊆ E with the property that
A⊆
n
[xk + (ρ + δ)Uα ].
k=1
From property (3) we deduce that a measure of noncompactness is monotone;
that is, if A ⊆ B, then Ψ(A) ≤ Ψ(B). Also, a consequence of properties (2)
and (5) is the fact that a measure of noncompactness is translation invariant.
Indeed, we have
Ψ(A) = Ψ((x0 + A) − x0 ) ≤ Ψ(x0 + A) + Ψ(−x0 ) = Ψ(x0 + A)
≤ Ψ(x0 ) + Ψ(A) = Ψ(A),
so that Ψ(x0 + A) = Ψ(A). For more information and results about measures
of noncompactness the reader is referred to Sadovskii [1968] and Banas and
Goebel [1980]. The notion of noncompactness defined above includes the two
most commonly used measures of noncompactness, namely, γ (the Kuratowski
measure of noncompactness) and χ (the Hausdorff measure of noncompactness)
defined in this section of this chapter.
Let Ψ : 2E → C(A) be a measure of noncompactness, M ⊂ E a nonempty
set, and f : M → E a mapping.
Definition 2.19 We say that f is a k-Ψ-contraction if there is a function
k : A → [0, 1[ such that
Ψ(f (B))(α) ≤ k(α)Ψ(β)(α),
for every bounded set B ⊆ M and for every α ∈ A.
It is known that there exists an extensive literature concerning k-Ψ-contractions
in Banach spaces. For locally convex spaces we cite Talman [1973].
Let K ⊂ E be a closed pointed convex cone. We recall that K is total in E
if E = K − K and K is generating if E = K − K.
56
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Definition 2.20 We say that K is sharp (or locally bounded) if there is a
neighbourhood U of zero in E such that K ⊂ ∂E U is bounded and nonempty.
(We denote by ∂E U the boundary of U with respect to E.)
In any Banach space any closed pointed convex cone is sharp. In a locally
convex space we can show that if U is a neighbourhood of zero for which
K ∩∂E U is bounded, then K ∩U is also bounded. Talman has given an example
of a sharp cone in a non-normed vector space, which is also a generating cone.
Now we present his example.
Let (E, ·) be a Banach space, E ∗ its topological dual, and suppose that the
σ(E, E ∗ ) topology on E (the weak topology) is not a norm topology. We select
a nonzero element x∗ ∈ E ∗ and choose x0 ∈ E with the property x∗ , x0 = 1.
We define
K = {x ∈ E : x − x∗ , xx0 ≤ x∗ , x}.
(We note that · is the norm on E and ·, · is the bilinear form which defines the
duality between E ∗ and E.) The set K is a convex cone. Indeed, if x, y ∈ K,
then
(x + y) − x∗ , x + yx0 ≤ x − x∗ , xx0 + y − x∗ , yx0 ≤ x∗ , x + x∗ , y = x∗ , x + y,
which implies that x + y ∈ K.
Obviously, if λ ≥ 0 and x ∈ K, then λx ∈ K. We also remark that
K = {0} because x0 ∈ K. Because K is convex and closed in the norm
topology, K is closed for σ(E, E ∗ ). Now, we show that K is pointed; that is,
K ∩ (−K) = {0}. Indeed, if x ∈ K and −x ∈ K, then we have
0 ≤ ± x − x∗ , ±xx0 ≤ x∗ , ±x,
which implies that x∗ , x = 0. Thus,
0 = x − x∗ , xx0 = x,
and x = 0. Hence, K is a pointed convex cone. Let
U = {x ∈ E : |x∗ , x| ≤ 1.
Then, U is a neighbourhood of zero for σ(E, E ∗ ) and we can show that
σ
U = {x ∈ E : |x∗ , x| = 1}
∂E
σ U means the boundary mapping for the weak topology
(here of course ∂E
∗
σ(E, E )). Hence,
σ
U = {x ∈ E : |x∗ , x| = 1 and x − x∗ ≤ 1},
K ∩ ∂E
2.4 Asymptotic Differentiability in Locally Convex Spaces
57
which is clearly bounded for the norm topology and therefore is bounded for
σ(E, E ∗ ). Finally, we show that K is a generating cone in E. Indeed, let x ∈ E
be an arbitrary element. We must show that x ∈ K − K, and we must assume
/ K, we
that x ∈
/ K. Put α = x∗ , x, and let β = x − αx0 . Because x ∈
have β − α > 0. Let y = x + (β − α)x0 . Then, we have
y − x∗ , yx0 = x + (β − α)x0 − x∗ , x + (β − α)x0 = x + (β − α)x0 − x∗ , xx0 − x∗ , (β − α)x0 x0 = x + (β − α)x0 − αx0 − (β − α)x0 = x − αx0 = β = α + (β − α)
= x∗ , x + (β − α)x∗ , x0 = x∗ , x + (β − α)x0 = x∗ , y.
Therefore, y ∈ K. Because β − α > 0 and x0 ∈ K, we know that (β − α)x0 ∈
K. Then, x = y − (β − α)x ∈ K − K and we have that E = K − K.
Let (E, {pα }α∈A ) be a Hausdorff locally convex space and K ⊂ E a closed
pointed convex cone. We denote by L(E, E) the vector space of linear continuous mappings from E into E.
Definition 2.21 We say that a mapping f : K → E is Hyers–Lang
asymptotically linear (HLAL) along K if there is a continuous linear mapping D∞ f : E → E (i.e., D∞ f ∈ L(E, E)) such that for any α, β ∈ A there
exist γ ∈ A and constants cα , cβ > 0 such that the following properties are
satisfied.
1. pα (x) ≤ cα pγ (x), for any x ∈ K.
2. For any ε > 0 there exists M > 0 with the property that x ∈ K and
pγ (x) ≥ M imply
pβ [f (x) − D∞ f (x)] ≤ εcβ pγ (x).
Remark 2.6 In Definition 2.21 we can suppress the constants cα and cβ . We
can do this if we denote again by pγ the seminorm cpγ , where c = max{cα , cβ }.
Now, we recall a well known notion in the theory of locally convex spaces.
Let A ⊂ E be a nonempty subset. We say that A is balanced (circled) if
λA ⊆ A, whenever |λ| ≤ 1 and we say that A is radial (absorbing), if for each
x ∈ E there is an ε > 0 such that tx ∈ A for t ∈ [0, ε]. If B ⊂ E is a radial,
balanced, and convex set, then the nonnegative real function
x → pB (x) = inf{λ > 0 : x ∈ λB}
is called the Minkowski functional associated with B. In this case we can show
that pB is a seminorm. We recall that a subset A ⊂ E is bounded if for each
0-neighbourhood U ∈ E, there exists λ ∈ R such that A ⊂ λU . If B ⊂ E is
radial, balanced, convex, and bounded, then in this case pB is a norm on E.
58
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
We denote by EB the normed space obtained by equipping the linear subspace
of E that is spanned by B with the norm arising from the Minkowski functional
pB . If q is any continuous seminorm on E, we let Eq denote the quotient space
E/ ker(q) equipped with the canonical norm induced by q. (We recall that
ker(q) = {x ∈ E : q(x) = 0}.) We denote by πq : E → Eq the quotient
mapping. If u : E → E is a continuous linear mapping, then πq ◦ u|EB :
EB → Eq is continuous.
If q = pα , we denote the space Epα by Eα . Let B be a basis consisting of
closed, balanced, absorbing, convex sets for the bornology on E.
Definition 2.22 We say that a mapping f : K → E is B-asymptotically
linear (BAL) along K if there is a continuous linear mapping D∞ f : E → E
such that for every α ∈ A and for every B ∈ B, πα ◦ D∞ f |EB : EB → Eα is
the asymptotic derivative (in the sense of normed spaces along K ∩ EB of the
mapping πα ◦ f |EB : EB → Eα ).
Remark 2.7 The notion of B-asymptotic linearity is highly sensitive to the
selection of B.
Proposition 2.6 Let (E, {pα }α∈A ) be a Hausdorff locally convex space
and K ⊂ E a total closed pointed convex cone. If f : K ⊂ E is HLAL
along K with HL-asymptotic derivative D∞ f and for some basis B for the
bornology of E, f is BAL along K with B-asymptotic derivative D∞ f , then
D∞ f = D∞ f .
Proof. Let x0 ∈ K be an arbitrary element such that x0 = 0. Let B ∈ B be
such that x0 ∈ B and let α ∈ A be arbitrary. Then, for every λ > 0 we have
pα (D∞ f (λx0 ) − D∞ f (λx0 ))
pB (λx0 )
pα (D∞ f (λx0 ) − f (λx0 )) + pα (f (λx0 ) − D∞ f (λx0 ))
.
≤
pB (λx0 )
(2.11)
Let ε > 0 be given. Because f is HLAL, there is a β ∈ A such that pβ (x0 ) = 0,
and such that λ > 0 sufficiently large implies that
εpB (x0 )
pα (f (λx0 ) − D∞ f (λx0 ))
<
.
pβ (λx0 )
2pβ (x0 )
(We note that pB (x0 ) = 0, because otherwise λx0 ∈ B for all λ > 0, which
is impossible because B is bounded.) It now follows that, for large λ > 0, we
have
pB (x0 )pα (f (λx0 ) − D∞ f (λx0 ))
ε
pα (f (λx0 ) − D∞ f (λx0 ))
=
< .
pB (λx0 )
pB (λx0 )pB (x0 )
2
2.4 Asymptotic Differentiability in Locally Convex Spaces
59
On the other hand f is BAL, so that when λ > 0 is large, we have
ε
pα (D∞ f (λx0 ) − f (λx0 ))
< .
pB (λx0 )
2
Combining the last two inequalities with 2.4, we obtain , for λ > 0 sufficiently
large,
pα (D∞ f (λx0 ) − D∞ f (λx0 ))
< ε.
pB (λx0 )
Inasmuch as D∞ f and D∞ f are both linear this is equivalent to
pα (D∞ f (x0 ) − D∞ f (x0 )) < εpB (x0 ).
It follows that
D∞ f (x0 ) − D∞ f (x0 ) ∈ ker pα ,
and because α ∈ A was arbitrary and E is Hausdorff, we must have
D∞ f (x0 ) = D∞ f (x0 ).
Taking into account the totality of K, the proof is complete.
Remark 2.8 In Talman [1973] are given several examples to show that in
general neither of the implications BAL =⇒ HLAL and HLAL =⇒ BAL is
true, but if some special conditions are satisfied we have an interesting relation
between BAL and HLAL along K.
A basis B for the bornology of a locally convex space E is said to be strict
if each B ∈ B has the property that for every bounded set D ⊂ E, B absorbs
D ∩ EB . It seems that there is no topological condition which guarantees that
such a basis exists. There exist spaces in which there is no such basis [Talman,
1973].
Theorem 2.23 Let (E, {pα }α∈A ) be a locally convex space whose bornology admits a strict basis B and let K be a sharp cone in E. Then, f : K → E
is BAL along K if and only if f is HLAL along K.
Proof. Assume that f is BAL along K. Let α, β ∈ A be given. Find θ ∈ A
so that Bθ ∩ K is bounded, where Bθ is the open unit pθ -ball. Choose γ ∈ A
such that max{pα , pθ } ≤ pγ . Then Bγ ∩ K ⊆ B for some B ∈ B and pβ ≤ pγ
on K. But Bγ is a neighbourhood of zero, so Bγ absorbs B, and this means
that there is a k > 0 such that pγ ≤ kpβ on EB , which contains K.
If ε > 0 is given, we find M so that whenever x ∈ K and pB (x) ≥ M ,
we have pβ (f (x) − D∞ f (x)) ≤ εpB (x). If x ∈ K and pγ (x) ≥ kM , then
pB (x) ≥ M , so that
pβ (f (x) − D∞ f (x)) ≤ εpB (x) ≤ εpγ (x).
60
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
It follows that f is HLAL along K.
Conversely, assume that f is HLAL along K, and let α ∈ A, β ∈ B be
given. We take β so that pβ ∩ K is bounded. Find γ so that pβ ≤ pγ , and
for every ε > 0 there is an M such that x ∈ K and pγ (x) ≥ M imply that
pα (f (x) − D∞ f (x)) < εpγ (x). Because B is strict, there is a k1 > 0 such that
pB ≤ k1 pγ on K (because K is sharp and Bγ ∩ K is bounded).
Bγ is a neighbourhood of zero, and thus there is a k2 > 0 such that pγ ≤ k2 pB
on EB . Let ε > 0 be given. Find M so that pγ (x) ≥ M implies that
pα (f (x) − D∞ f (x)) ≤
ε
pγ (x).
k2
Then pB (x) ≥ k1 M implies that pγ (x) ≥ M , which in turns implies that
pα (f (x) − D∞ f (x)) ≤
ε
pγ (x) ≤ εpB (x).
k2
Hence, f is BAL along K.
Proposition 2.7 If f : K → K is HLAL along K, then D∞ f (K) ⊆ K.
/ K. We
Proof. We suppose that there is an h ∈ K such that D∞ f (h) ∈
∗
select a positive x∗ ∈ E such that
x∗ , D∞ f (h) < 0.
We denote
μ = x∗ , D∞ f (h)
and we define
φ(t) = x∗ , f (th).
If t ≥ 0, then th ∈ K, so that f (th) ∈ K and φ(t) ≥ 0. But for t > 0,
$
f (th) − D∞ f (th)
φ(t) = t x∗ ,
+μ ,
t
and the function
x → |x∗ , x|
is a continuous semi-norm on E. Hence, there is an α ∈ A such that
|x∗ , x| ≤ pα (x)
(modulo a multiplicative constant, which we ignore). Because E is Hausdorff,
there is a β ∈ A with K\ ker pβ = ∅. Select γ ∈ A so that pβ ≤ pγ and so that
for every ε > 0 there is an M > 0 such that t ≥ 0 and pγ (th) ≥ M imply that
pα (f (th) − D∞ f (th)) ≤ εpγ (th).
2.4 Asymptotic Differentiability in Locally Convex Spaces
61
If we now choose ε so that
0<ε<−
−μ
2pγ (h)
and find M accordingly we see that when
t≥
M
pγ (h)
we must have
x∗ , f (th) − D∞ f (th) ≤ pα f (th) − D∞ f (th)
t
t
μ
1
≤ εpγ (th) = εpγ (h) < − .
t
2
But then, for such t we would have
$
μt
f (th) − D∞ f (th)
φ(t) = t x∗ ,
+μ <
< 0,
t
2
which is not possible because φ(t) ≥ 0. This contradiction completes the proof.
Proposition 2.8 If f : K → K is BAL along K for some basis B for the
bornology of E, then D∞ f (K) ⊆ K.
Proof. With D∞ f in place of D∞ f , we proceed as in the proof of Proposition
2.7 up through the definition of the function φ : R → R. Select a B ∈ B so
that {h} ⊆ H. Because x → |x∗ , x| is a continuous semi-norm on E, there
is an M > 0 such that x ∈ EB ∩ K and pB (x) ≥ M imply that
|x∗ , f (x) − D∞ f (x)| < −
But then when
t>
μ pB (x)
.
2 pB (h)
M
,
pB (h)
we have pB (th) > M , so that, for such t,
μ
1 μ pB (th)
1
|x∗ , f (th) − D∞ f (th)| < −
=−
t
t 2 pB (h)
2
and again φ(t) < 0. The proof is complete.
62
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Theorem 2.24 Let (E, {pα }α∈A ) be a Hausdorff locally convex space,
Ψ : 2E → C(A),
a measure of noncompactness on E, and K ⊂ E a closed pointed convex cone.
Let f : K → E be a k-Ψ-contraction (k : A → [0, 1]). If f is HLAL along
K, then D∞ f |K is a k-Ψ-contraction.
Proof. Let α ∈ A. From Property (7) cited in Definition 2.18, there is a
neighbourhood Uα of zero in E such that Ψ(Uα )(α) = 1. Choose β ∈ A so
that Bβ ⊆ Uα . Because f is HLAL along K, we can find γ ∈ A so that
(i) K\ ker pγ = ∅ (because E is Hausdorff).
(ii) For every ε > 0 there is M > 0 such that
pβ (f (x) − D∞ f (x)) ≤ εpγ (x),
whenever pγ (x) ≥ M and x ∈ K.
Let S ⊂ K be bounded, and suppose for the moment that pγ ≥ 1 for every
x ∈ S. Let ε > 0 be given, and put
σ = sup{pγ (x) : x ∈ S}.
Find M > 0 so that we have
pβ (f (x) − D∞ f (x)) ≤
ε
pγ (x),
σ
for every x ∈ K with pγ (x) ≥ M . Let λ be a real number such that λ ≥ M .
Then, if x ∈ λS (i.e., x = λxs for some xs ∈ S), we have
pγ (x) = pγ (λxs ) = λpγ (xs ) ≥ M pγ (xs ) ≥ M,
and therefore
pβ (f (x) − D∞ f (x)) ≤
ε
ε
pγ (x) ≤ λpγ (xs ) ≤ ελ.
σ
σ
We deduce that
f (λS) − D∞ f (λS) ⊆ ελBβ ⊆ ελUα
and it follows that
D∞ f (λS) ⊆ f (λS) − [f (λS) − D∞ f (λS)] ⊆ f (λS) + ελUα ,
which implies
λΨ(D∞ f (S))(α) = Ψ(D∞ f (λS))(α) ≤ Ψ(f (λS) + ελUα )(α)
≤ Ψ(f (λS))(α) + Ψ(ελUα )(α) ≤ k(α)Ψ(λS)(α) + ελ
= λ(k(α)Ψ(S)(α) + ε).
2.4 Asymptotic Differentiability in Locally Convex Spaces
63
Dividing through by λ and letting ε go to zero, we obtain
Ψ(D∞ f (S))(α) ≤ k(α)Ψ(S)(α).
Now, if S is an arbitrary bounded subset of K, let
δ = 1 + sup{pα (x) : x ∈ S}.
Because K\ ker pγ = ∅, we can find x0 ∈ K such that pγ (x0 ) = δ. Considering S = x0 + S, we find for x = x0 + xs ∈ S that
pγ (x) = pγ (x0 + xs ) ≥ pγ (x0 ) − pγ (xs ) = δ − pγ (xs )
≥ δ − (δ − 1) = 1.
But we have just seen above that
Ψ(D∞ f (S ))(α) ≤ k(α)Ψ(S )(α).
We have
Ψ(D∞ f (S ))(α) = Ψ(D∞ f (x0 + S))(α) = Ψ(D∞ f (x0 ) + D∞ f (S))(α)
= Ψ(D∞ f (S))(α),
and
Ψ(S )(α) = Ψ(x0 + S)(α) = Ψ(S)(α),
which imply that the proof is complete.
We have a similar result for the B-asymptotic linearity.
Theorem 2.25 Let (E, {pα }α∈A ) be a Hausdorff locally convex space
Ψ : 2E → C(A)
a measure of noncompactness on E and K ⊂ E a closed pointed convex cone.
Let f : K → E be a k-Ψ-contraction (k : A → [0, 1]). If f is BAL along K
for some basis B of the bornology on E, then D∞ |K is a k-Ψ-contraction.
Proof. Let α ∈ A and let Uα be a convex neighbourhood of zero with
Ψ(Uα )(α) = 1. Select β ∈ A so that Bβ ⊂ Uα . If S ⊂ K is bounded,
find B ∈ B such that S ⊂ EB and S ⊂ μB for some μ > 0. If ε > 0 is
given, we assume for the moment that pB (x) ≥ 1 for every x ∈ S. We then
find M > 0 so that x ∈ K and pB (x) ≤ M imply that
pB (f (x) − D∞ f (x)) ≤ εpB (x).
Let λ be a real number such that λ ≥ M . When x ∈ λS, we have pB (x) ≥ M
so that
pB (f (x) − D∞ f (x)) ≤ εpB (x).
64
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Hence, we have
f (λS) − D∞ f (λS) ⊂ ελBβ ,
which implies that
D∞ f (λS) ⊆ f (λS) − [f (λS) − D∞ f (λS)] ⊆ f (λS) + ελBβ .
Thus, exactly as in the previous argument, we have
Ψ(D∞ f (S))(α) ≤ k(α)Ψ(S)(α).
Now, if S ⊂ K is an arbitrary bounded set, note that we may assume that
EB ∩ K = {0} (we need only to choose x0 ∈ K, x0 = 0, and require
S ∪ {x0 } ⊂ EB , rather than S ⊂ EB ). Moreover, S ⊂ μB for some μ > 0
means that S is bounded for the norm pB on EB . We can repeat the translation
argument used in the proof of Theorem 2.24, using pB in place of pγ . The proof
is complete. Remark 2.9 The condition that f is a k-Ψ-contraction in Theorem 2.24 and
in Theorem 2.25 cannot be relaxed to the condition that f is Ψ-condensing.
Now we cite, without proof, a fixed point theorem in locally convex spaces
due to Talman which is based on the notion of an asymptotic derivative along
a cone.
Theorem 2.26 Let (E, {pα }α∈A ) be a Hausdorff, quasi-complete locally
convex space. Let K ⊂ E be a sharp total positive cone and let f : K → K be
a continuous k-Ψ-contraction which is HL-asymptotically linear (respectively,
B-asymptotically linear for some basis B for the bornology on E). If D∞ f
(respectively, D∞ f ) does not have any positive eigenvector belonging to an
eigenvalue which is greater than or equal to one, then f has a fixed point in K.
Proof. A proof of this theorem is in Talman [1973] and it is based on several
intermediate results and on the topological index.
Remark 2.10 We note that Theorem 2.26 is a generalization of Krasnoselskii’s fixed point theorem.
2.5
The Asymptotic Scalar Differentiability
Inspired by the notion of scalar derivatives Isac introduced in 1999 the notion
of the asymptotic scalar derivative [Isac, 1999c]. In this section we present this
notion and some relations with the scalar derivative.
Let (E, · ) be an arbitrary real Banach space. We say that a semi-inner
product (in Lumer’s sense) is defined on E, if to any x, y ∈ E there corresponds
a real number denoted by [x, y] satisfying the following properties.
65
2.5 The Asymptotic Scalar Differentiability
(s1 ) [x + y, z] = [x, z] + [y, z].
(s2 ) [λx, y] = λ[x, y], for x, y, z ∈ E, λ ∈ R.
(s3 ) [x, x] > 0 for x = 0.
(s4 ) |[x, y]|2 ≤ [x, x][y, y].
It is known [Giles, 1967; Lumer, 1961] that a semi-inner product space is a
normed linear space with the norm xs = [x, x]1/2 and that every Banach
space can be endowed with a semi-inner product (and in general in infinitely
many different ways, but a Hilbert space in a unique way).
Obviously if (H, ·, ·) is a Hilbert space, the inner product ·, · is the unique
semi-inner product in Lumer’s sense on H, [Giles, 1967; Lumer, 1961].
We note that it is possible to define a semi-inner product such that [x, x] =
x2 (where · is the norm given in E). In this case we say that the semi-inner
product is compatible with the norm · . By the proof of Theorem 1 [Giles,
1967] this semi-inner product can be defined to have the homogeneity property:
(s5 ) [x, λy] = λ[x, y], for x, y ∈ E, λ ∈ R.
Throughout this chapter we suppose that all semi-inner products compatible
with the norm satisfy (s5 ).
The following definition is an extension of Example 5.1, p.169 of [do Carmo,
1992].
Definition 2.27 The mapping
i : E\{0} → E\{0}; i(x) =
x
[x, x]
is called the inversion (of pole 0) with respect to [·, ·].
It is easy to see that i is one to one and i−1 = i. Indeed, because
i(x)s =
1
,
xs
by the definition of i we have
i(i(x)) =
i(x)
= x2s i(x) = x.
i(x)2s
Hence i is a global homeomorphism of E\{0} which can be viewed as a global
nonlinear coordinate transformation in E.
Let A ⊆ E such that 0 ∈ A and A\{0} is an invariant set of the inversion i
with respect to [·, ·]; that is, i(A\{0}) = A\{0} and f : A → E. Examples of
invariant sets of the inversion i with respect to [·, ·] are:
66
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
1. F \{0} where F is a linear subspace of E (in particular F can be the whole
E)
2. K\{0} where K ⊆ E is a convex cone
Now we define the inversion (of pole 0) with respect to [·, ·] of the mapping f .
Definition 2.28 The inversion (of pole 0) with respect to [·, ·] of the mapping
f is the mapping I(f ) : A → E defined by:
2
[x, x](f ◦ i)(x) if x = 0,
I(f )(x) =
0
if x = 0.
Proposition 2.9 The inversion of mappings I with respect to [·, ·] is a oneto-one mapping on the set of mappings {f | f : A → E; f (0) = 0} and
I −1 = I; that is, I(I(f )) = f .
Proof. By definition I(I(f ))(0) = 0. Hence, I(I(f ))(0) = f (0). If x =
0, then I(I(f ))(x) = x2s I(f )(i(x)) = x2s i(x)2s f (i(i(x))) = f (x).
Thus, I(I(f ))(x) = f (x) for all x ∈ A. Therefore, I(I(f )) = f . Remark 2.11 We note that the inversion of mappings with respect to [·, ·] is
linear and has the following properties.
1. If T ∈ L(E, E) and j : A → E is the embedding of A into E, then
I(T ◦ j) = T ◦ j.
2. If the semi-inner product is compatible with the norm of E and
x → +∞, then i(x) → 0.
Now, we introduce the notion of a scalar derivative with respect to a semiinner product [·, ·].
Let (E, · ) be an arbitrary real Banach space and [·, ·] a semi-inner product
6⊆E
on E. Let G ⊆ E be a set which contains at least one nonisolated point, G
6
6
such that G ⊆ G, f : G → E and x0 a nonisolated point of G. The following
definition is an extension of Definition 2.2 [Nemeth, 1992].
Definition 2.29 The limit
inf
f #,G (x0 ) = lim
x→x
0
x∈G
[f (x) − f (x0 ), x − x0 ]
x − x0 2s
is called the lower scalar derivative of f at x0 along G with respect to [·, ·].
Taking lim sup in place of lim inf, we can define the upper scalar derivative
#,G
f
(x0 ) of f at x0 along G with respect to [·, ·] similarly.
2.5 The Asymptotic Scalar Differentiability
67
6 then without confusion, we can say, for short, lower
Remark 2.12 If G = G,
scalar derivative and upper scalar derivative instead of lower scalar derivative
along G and upper scalar derivative along G, respectively. In this case, we
omit G from the superscript of the corresponding notations.
Proposition 2.10 Suppose that [·, ·] is compatible with the norm · . Let
K ⊆ E be an unbounded set such that 0 ∈ K and K\{0} is an invariant set
of the inversion i with respect to [·, ·]. Let g : E → E. Then we have
lim inf
x→∞
x∈K
[g(x), x]
= I(g)#,K (0).
x2
Proof. Because K ⊆ E is unbounded and K\{0} is an invariant set of i, 0
is a nonisolated point of K. Hence, I(g)#,K (0) is well defined. Consider the
global nonlinear coordinate transformation y = i(x). Then x = i(y) and we
have
[g(x), x]
lim inf
= lim inf [I(g)(y), i(y)],
y→0
x→∞
x2
y∈K
x∈K
from where, by using the definition of the lower scalar derivative along a set,
the assertion of the lemma follows easily.
Remark 2.13 Obviously, if the Banach space (E, · ) is a Hilbert space
(H, ·, ·), in Definition 2.29 and Proposition 2.10 we replace the semi-inner
product [·, ·] by the inner product ·, · defined on H.
Let (E, · ) be an arbitrary Banach space, [·, ·] a semi-inner product on E,
and K ⊂ E an unbounded set .
The following definition is an extension of the notion of an asymptotic scalar
derivative given on Hilbert space by Isac [Isac, 1999c].
Let f : K → E be an arbitrary mapping.
Definition 2.30 We say that T ∈ L(E, E) is an asymptotic scalar derivative of f along K, with respect to the semi-inner product [·, ·] if
lim sup
x→∞
x∈K
[f (x) − T (x), x]
≤ 0.
x2s
(∞). For the next results
The mapping of Definition 2.30 is denoted fs,K
we suppose that 0 ∈ K and K\{0} is an invariant set of the inversion i with
respect to [·, ·].
Remark 2.14 If the semi-inner product [·, ·] is compatible with the norm ·,
then in Definitions 2.29 and 2.30 we can replace x − x0 2s by x − x0 2 and
x2s by x2 , respectively.
68
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Proposition 2.11 If T is an asymptotic scalar derivative of f with respect
to the semi-inner product [·, ·], then for any c > 0 the mapping T + cI is also
an asymptotic scalar derivative of f with respect to [·, ·].
Proof. This proposition is a consequence of Definition 2.30.
Theorem 2.31 If [·, ·] is a semi-inner product compatible with the norm ·,
then T ∈ L(E) is an asymptotic scalar derivative of f with respect to [·, ·] if and
#
only if the upper scalar derivative of h in 0 is nonpositive (i.e., h (0) ≤ 0),
where h : K → E is defined by h = I(f − T ◦ j) = I(f ) − T ◦ j, and
j : K → E is the embedding of K into E.
Proof. We suppose that T ∈ L(E) is an asymptotic scalar derivative of f
#
with respect to the semi-inner product [·, ·] and prove that h (0) ≤ 0. The
converse implication can be proved similarly. Indeed, because T ∈ L(E) is an
asymptotic scalar derivative of f with respect to [·, ·], we have that
lim sup [f (x) − T (x), i(x)] ≤ 0.
(2.12)
x→+∞
x∈K
Consider the global nonlinear coordinate transformation y = i(x) given by the
global diffeomorphism i. Because K is unbounded and K\{0} is invariant
under i, 0 is a nonisolated point of K. Then, x = i(y) and by (2.12),
lim sup[(f ◦ i)(y) − (T ◦ j ◦ i)(y), y] ≤ 0.
y→0
y∈K
Hence,
lim sup[I(f )(y) − I(T ◦ j)(y), i(y)] ≤ 0.
y→0
y∈K
Thus, by the definition of the upper scalar derivative with respect to [·, ·] we
#
have h (0) ≤ 0.
Corollary 2.32 If the semi-inner product [·, ·] is compatible with the norm
· , then 0 is an asymptotic scalar derivative of f with respect to [·, ·] if and
#
only if I(f ) (0) ≤ 0.
The following theorem shows the surprising fact that if [·, ·] is compatible
with the norm · , then every f whose inversion has a finite upper scalar
derivative with respect to [·, ·] at 0 is asymptotically scalarly differentiable with
respect to [·, ·].
Theorem 2.33 If [·, ·] is a semi-inner product compatible with the norm · #
and I(f ) (0) < +∞, then f is asymptotically scalarly differentiable with
69
2.5 The Asymptotic Scalar Differentiability
#
respect to [·, ·] and T = I(f ) (0)I is an asymptotic scalar derivative of f
with respect to [·, ·], where I : E → E is the identity mapping (the asymptotic
scalar differentiability is along the unbounded set K).
#
#
Proof. Indeed, h (0) = 0, where h = I(f )−T ◦j = I(f )−I(f ) (0)(I ◦j).
Hence, the result follows by using Theorem 2.31.
Remark 2.15 If the semi-inner product [·, ·] is compatible with the norm
#
· and I(f ) (0) < +∞, then every mapping cI is an asymptotic scalar
#
derivative with respect to [·, ·], where c ≥ I(f ) (0).
If the Banach space (E, · ) is in particular a Hilbert space and the norm
· is the norm defined by the inner product ·, · given on the vector space H,
then Definition 2.30 has the following form.
Definition 2.34 Let (H, ·, ·) be a Hilbert space and K ⊂ H an unbounded set. We say that T ∈ L(H, H) is an asymptotic scalar derivative
of f : K → H, along K if
lim sup
x→∞
x∈K
f (x) − T (x), x
≤ 0.
x2
Now, we consider a more general situation. Let (E, · ) be a Banach space,
E ∗ the topological dual of E, E, E ∗ a duality between E and E ∗ with respect
to a bilinear functional on E × E ∗ , denoted ·, · and satisfying the separation
axioms. Let K ⊆ E be an unbounded set K̃ ⊂ E such that K ⊆ K̃ and
f : K̃ → E ∗ be a mapping.
Definition 2.35 We say that T ∈ L(E, E ∗ ) is an asymptotic scalar derivative of f along K if
lim sup
x→+∞
x∈K
x, f (x) − T (x)
≤ 0.
x2
(∞). If K = K̃ we can
The mapping used in Definition 2.35 is denoted fs,K
say asymptotic scalar derivative for short instead of asymptotic scalar derivative
along K.
Remark 2.16 If in Definitions 2.30, 2.34, and 2.35 we have that K = E,
K = H, and K = E, respectively, we say that T is the asymptotic scalar
derivative with respect to the space E, H, E, respectively.
Let (E, · ) be a Banach space and [·, ·] a semi-inner product (in Lumer’s
sense) and let · s be the norm defined by this semi-inner product. Let K ⊂ E
be a closed convex cone and f : E → E.
70
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Definition 2.36 We say that T ∈ L(E, E) is an asymptotic derivative of f
along K if
f (x) − T (x)s
= 0.
lim
xs →+∞
xs
x∈K
Proposition 2.12 If T ∈ L(E, E) is an asymptotic derivative of f along
K, then T is an asymptotic scalar derivative along K.
Proof. The proposition is a consequence of the relation
lim sup
xs →+∞
x∈K
[f (x) − T (x), x]
f (x) − T (x)s xs
≤ lim sup
2
xs
x2s
xs →+∞
x∈K
=
lim
xs →+∞
x∈K
f (x) − T (x)s
= 0.
xs
Remark 2.17 Let (H, ·, ·) be a Hilbert space. By the definition of the
asymptotic scalar derivative, it follows easily that if U is an asymptotic scalar
derivative of f and g : H → H satisfies the relation
g(x), x ≤ 0,
(2.13)
for all x ∈ H, then U is also an asymptotic scalar derivative of f + g. Particularly, for any skew-adjoint mapping Z, the mapping U is an asymptotic scalar
derivative of f + Z, or equivalently U + Z is an asymptotic scalar derivative
of f . Moreover, for any P continuous linear positive semi-definite mapping,
U + P is also an asymptotic scalar derivative of f . An example for a nonlinear
mapping g satisfying (2.13) is g : R3 → R3 ;
g(u, v, w) = (−u + vw, −v + uw, −w − 2uv).
It would be interesting to study the properties of mappings satisfying the condition (2.13). Of course, 0 is an asymptotic scalar derivative of these mappings.
Remark 2.18 We have already shown that every asymptotic derivative of
f is an asymptotic scalar derivative of f . However, the converse is not true.
Indeed, it can be easily checked that if f : R3 → R3 ,
f (u, v, w) = (vw, uw, −2uv),
then 0 is an asymptotic scalar derivative of f but it is not an asymptotic derivative of f .
71
2.6 Some Applications
Remark 2.19 Every continuous mapping S satisfying (2.7) is an asymptotic
scalar derivative of f . Indeed, we have
2θ
f (x) − T (x), x
ψ(x)
lim
= 0.
≤
2
x
2 − ψ(2) x→+∞ x
x→+∞
lim sup
Let K ⊂ E be a closed convex cone and f : K → E a mapping. Let
R : H → K be a retraction. We have the following result.
Proposition 2.13 If the retraction R is a ρ-Lipschitz mapping with respect
to the norm · s and T ∈ L(E, E) is an asymptotic derivative of f along K
such that T (K) ⊆ K, then T is an asymptotic scalar derivative of R ◦ f along
K.
Proof. Indeed, we have
lim
xs →+∞
x∈K
≤
lim
xs →+∞
x∈K
[R(f (x)) − T (x), x]
=
x2s
R(f (x)) − R(T (x))s xs
≤
x2s
lim
[R(f (x)) − R(T (x)), x]
x2s
lim
ρf (x) − T (x)s xs
x2s
xs →+∞
x∈K
xs →+∞
x∈K
= 0.
Remark 2.20 We note that Proposition 2.13 has interesting applications to
the study of nonlinear complementarity problems in Hilbert spaces. In this case
the retraction R is the projection onto a closed convex cone.
2.6
Some Applications
We present in this section some applications to the study of fixed points of nonlinear mappings and also to the study of nonlinear complementarity problems.
First, we give an interesting variant of Krasnoselskii’s fixed point theorem
(Theorem 2.7). We need to introduce a notation.
Let (H, ·, ·) be a Hilbert space, K ⊂ H a closed convex cone, K ∗ its dual
cone, and f, g : K → H two mappings. The relation h ≤K ∗ g means that
g(x) − f (x) ∈ K ∗ (the dual of the cone K), for all x ∈ K. In this case we
have in particular h(x), x ≤ g(x), x for all x ∈ K.
Definition 2.37 We say that a mapping f : K → H is scalarly compact,
if for any sequence {xn }n∈N ⊂ K weakly convergent to an element x∗ ∈ K,
there exists a subsequence {xnk }k∈N such that
lim supxnk − x∗ , f (xnk ) ≤ 0.
k→+∞
72
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
Examples
(1) Any completely continuous mapping is scalarly compact.
(2) Given a mapping f : K → H, if there exists a completely continuous
mapping h : K → H such that
y, f (x) ≤ |y, h(x)|
for any x, y ∈ K, then f is scalarly compact .
Theorem 2.38 Let (H, ·, ·) be a Hilbert space, K ⊂ H a pointed closed
convex cone, and f : K → K a mapping.
If the following assumptions are satisfied,
(i) f is demicontinuous;
(ii) f is scalarly compact;
(iii) there exists an asymptotic scalarly differentiable mapping f0 : K → H
(∞) < 1;
such that f ≤K ∗ f0 and f0s
then f has a fixed point in K.
Proof. We use the notion of a nonlinear complementarity problem defined
in (2.21) and the notion of a variational inequality defined in (3.9). We define
h = I − f , where I is the identity mapping. From the complementarity
theory we know that f has a fixed point in K if and only if the nonlinear
complementarity problem NCP(h, K) has a solution.
For every m ∈ N we define the set
Km = {x ∈ K : x ≤ m}
and we observe that Km is closed, convex, weakly closed, and K = ∪∞
m=1 Km .
Obviously, any set Km is bounded.
First, we show that for every m ∈ N the variational inequality VI(I −f, Km )
∗ ∈ K . Indeed, let m ∈ N arbitrary and denote by Λ the
has a solution ym
m
family of all finite-dimensional subspaces of H ordered by inclusion. Consider
the mapping h(x) = x − f (x) for all x ∈ K and define Km (E) = Km ∩ E
for each E ∈ Λ.
For each E ∈ Λ we set
AE = {y ∈ Km : h(y), x − y ≥ 0 for all x ∈ Km (E)}
and we have that AE is nonempty. Indeed, the solution set of the problem
VI(h, Km (E)) is a subset of AE , but the solution set of VI(h, Km (E)) is nonempty. To see this, we consider the mappings j : E → H and j ∗ : H ∗ → E ∗ ,
73
2.6 Some Applications
where j is the inclusion and j ∗ is the adjoint of j. The mapping j ∗ ◦ h ◦ j :
Km (E) → E ∗ is continuous and
j ∗ ◦ h ◦ j(y), x − y = h(j(y)), j(x − y) = h(y), x − y,
for all x, y ∈ Km (E). Applying the classical Hartman–Stampacchia theorem
to the set Km (E) and to the mapping j ∗ ◦ h ◦ j we obtain that the problem
VI(h, Km (E)) has at least a solution.
For every E ⊂ Λ we denote ĀσE the weak closure of AE . We have that
∩E∈Λ ĀσE is nonempty. Indeed, let ĀσE1 , ĀE2 , . . . , ĀσEn be a finite subfamily of
the family {ĀσE }E ∈ Λ. Let M be the finite-dimensional subspace generated
by E1 , E2 , . . . , En .
Because Ek ⊆ M for all k ∈ {1, 2, . . . , n}, we have that Km (Ek ) ⊆
Em (M ) for all k ∈ {1, 2, . . . , n}. Therefore, AM ⊆ AEK for all k ∈
{1, 2, . . . , n}, which implies that ∩k=1 ĀσEK is nonempty. The weak compactσ
∗ ∈ ∩
ness of Km implies that ∩E∈Λ ĀσE = ∅. Let ym
E∈Λ ĀE be arbitrary and
let x ∈ Km be any element of this set. There exists some E ∈ Λ such that
∗ ∈ E. Because y ∗ ∈ Āσ , there exists a sequence {y }
x, ym
n n∈N ⊂ AE such
m
E
∗
that {yn }n∈N is weakly convergent to ym (we applied S̆mulian’s theorem). We
have
∗
− yn ≥ 0,
h(yn ), ym
and
h(yn ), x − yn ≥ 0,
or
∗
∗
≤ f (yn ), yn − ym
,
yn , yn − ym
(2.14)
yn , x − yn ≤ f (yn ), x − yn .
(2.15)
and
From (2.14) and assumption (ii) we have that {yn } has a subsequence, denoted again {yn }, such that
∗
≤ 0,
lim supyn , yn − ym
(2.16)
n→∞
which implies
∗ 2
∗
∗
= lim supyn − ym
, yn − ym
0 ≤ lim sup yn − ym
n→∞
n→∞
∗
∗
∗
+ lim sup[−ym
, yn − ym
] ≤ 0.
≤ lim supyn , yn − ym
n→∞
n→∞
∗ . Because f is demicontinuWe deduce that {yn } is strongly convergent to ym
∗ ).
ous, we have that {f (yn )}n∈N is weakly convergent to f (ym
74
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
From (2.15) we have
∗
∗
∗
ym
− f (ym
), x − ym
≥0
∗ is a solution of VI(I − f, K ). (We note that to
for any x ∈ Km ; that is, ym
m
obtain the last inequality we also used the following fact, “If {un } is weakly
convergent to an element u∗ and {vn } is strongly convergent to an element v∗ ,
then limn→∞ un , vn = u∗ , v∗ .”)
Now, we pass to the second part of the proof. In the first part we proved that
for every m ∈ N the problem VI(I − f, Km ) has a solution ym ; that is,
ym − f (ym ), x − ym ≥ 0, for all x ∈ Km .
(2.17)
Taking x = 0 in (2.17), we obtain
ym , ym ≤ f (ym ), ym .
(2.18)
The sequence {ym }m∈N is bounded. Indeed, if this is false, we may assume
that ym → +∞ as m → +∞, which implies (using (2.18) and assumption
(iii))
1=
f (ym ), ym ym , ym ≤
lim
2
ym ym 2
ym →+∞
f0 (ym ), ym ≤ lim sup
ym 2
ym →+∞
(∞)(y ), y (∞)(y ), y f0 (ym ) − f0s
f0s
m
m
m
m
+
lim
sup
2
2
ym ym ym →+∞
ym →∞
≤ lim sup
≤ lim sup
ym 2
(∞)y 2
f0s
m
= f0s
(∞) < 1.
ym 2
We have a contradiction and therefore {ym }m∈N is a bounded sequence. By
the reflexivity of H and the weak closedness of K we have that there exists
a subsequence {ymk }k∈N of the sequence {ym }m∈N , weakly convergent to
y0 ∈ K. For all x ∈ K, there exists an m0 ∈ N such that y0 and x are in Km0 .
Thus, for all m ≥ 0 we have y0 , x ∈ Km . We have
ym − f (ym ), y0 − ym ≥ 0.
(2.19)
ym − f (ym ), x − ym ≥ 0
(2.20)
and
Using inequality (2.19) and the scalar compactness of f (i.e., assumption (ii))
we have that there exists a subsequence {ymk }k∈N of the sequence {ym }m∈N
such that
lim supymk , ymk − y0 ≤ lim supf (ymk ), ymk − y0 ≤ 0
k→∞
k→∞
75
2.6 Some Applications
which implies that {ymk }k∈N is strongly convergent to y0 , as we can see considering the following inequalities,
0 ≤ lim sup ymk − y0 2 = lim supymk − y0 , ymk − y0 k→∞
k→∞
≤ lim supymk , ymk − y0 + lim sup[−y0 , ymk − y0 ≤ 0.
k→∞
k→∞
Considering (2.20) for all mk ≥ m0 we have
ymk − f (ymk ), x − ymk ≥ 0.
Computing the limit in the last inequality we obtain
y0 − f (y0 ), x − y0 ≥ 0
for any x ∈ K. Therefore, f (y0 ) = y0 and the proof is complete. Corollary 2.39 Let (H, ·, ·) be a Hilbert space, K ⊂ H a pointed closed
convex cone, and f : K → K a mapping. If the following assumptions are
satisfied,
(1) f is demicontinuous;
(2) f is scalarly compact;
(3) f has an asymptotic scalar derivative fs (∞) and fs (∞) < 1;
then f has a fixed point in K
Corollary 2.40 Let (H, ·, ·) be a Hilbert space and K ⊂ H a generating
closed pointed convex cone. Let f : K → K be a completely continuous
mapping. If f is asymptotically linear and f (∞) < 1, then f has a fixed
point in K.
Corollary 2.41 Let (H, ·, ·) be a Hilbert space and K ⊂ H a generating
closed pointed convex cone. Let f : K → K be a completely continuous
mapping. If there exists an asymptotically linear mapping f0 : K → K such
that f ≤K ∗ f0 and fs < 1, then f has a fixed point in K.
Let (H, ·, ·) be a Hilbert space, K ⊂ H a closed pointed convex cone, K ∗
its dual cone, and f : H → H a mapping. We consider the general nonlinear
complementarity problem
2
find x0 ∈ K such that
(2.21)
NCP(f, K) :
f (x0 ) ∈ K ∗ and x0 , f (x0 ) = 0.
We say that f is a completely continuous field if f has a representation of the
form f (x) = x − T (x), for any x ∈ H, where T : H → H is a completely
76
2 Asymptotic Derivatives and Asymptotic Scalar Derivatives
continuous mapping . Also we say that f is an asymptotically differentiable
field with respect to K if f has a representation of the form f (x) = x − T (x),
along K.
for any x ∈ H, where T : H → H has an asymptotic derivative T∞
We have the following result related to the NCP(f, K) problem.
Theorem 2.42 Let (H, ·, ·) be a Hilbert space, K ⊂ H a generating
closed pointed convex cone, and f : H → H a mapping. The mapping f is
supposed to be a completely continuous and asymptotically differentiable field
< 1 and T (K) ⊆ K,
of the form f (x) = x − T (x) for any x ∈ H. If T∞
∞
then the problem NCP(f, K) has a solution.
Proof. From the complementarity theory it is known that the problem
NCP(f, K) has a solution if and only if the mapping
Φ(x) = PK (x − f (x)) = PK (T (x))
has a fixed point. Obviously, Φ(K) ≤ K and Φ is a completely continuous
mapping. Therefore, Φ is demicontinuous and scalarly compact. Because
(x) = P (T (x)), consequently
T∞
K ∞
lim
x→∞
x∈K
(x)
(x))
PK (T (x)) − T∞
PK (T (x)) − PK (T∞
≤ lim
x→∞
x
x
x∈K
≤ lim
x→∞
x∈K
(x))
T (x) − (T∞
= 0.
x
is also an asymptotic derivative of the mapping Φ, which
We have that T∞
. Because the assumptions of Theorem 2.38 are
implies that Φs (∞) = T∞
satisfied our theorem is proved.
Remarks
(K) ⊆ K is satisfied if T (K) ⊆ K (see [Krasnoselskii,
1. The assumption T∞
1964a]).
2. Theorem 2.42 is applicable to complementarity problems defined by completely continuous fields of the form f (x) = x − T (x), where T is an
integral operator. It is known that many nonlinear integral operators (as for
example, Hammerstein operators or Urysohn mappings are asymptotically
differentiable [Krasnoselskii, 1964b].
3. Theorem 2.42 is also applicable to complementarity problems NCP(f, K),
where f has a representation of the form f (x) = αx − T (x), where α is a
positive real number and T : H → H is a completely continuous mapping.
In this case, in the proof of Theorem 2.42 we consider the mapping
1
1
T (x) .
Ψ(x) = ΦK x − f (x) = PK
α
α
2.6 Some Applications
77
< α.
In this case we must ask to have T∞
Another interesting application of Theorem 2.38 to complementarity problems is when the cone K is an isotone projection cone and K is self-adjoint;
that is, K = K ∗ . In this case if f (x) = x − T (x) and there exists a mapping
T0 such that T0 : H → H and T (x) ≤K T0 (x) for any x ∈ H, we have that
PK (T (x)) ≤K PK (T0 (x)). If T0 has an asymptotic derivative (T0 )∞ such that
(T0 )∞ (K) ⊆ K, and the mapping f is a completely continuous field, then by
Theorem 2.38 we have that the problem NCP(f, K) has a solution if in addition
(T0 )∞ < 1.
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