Appl. Math. Lett. Vol. 9, No. 1, pp. 1-8, 1996
Pergamon
Copyright@1996 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0893-9659/96 $15.00 + 0.00
0893-9659(95)00093-3
F i x e d - P o i n t T h e o r y for t h e S u m
of T w o Operators
D. O'REGAN
Department of Mathematics, University College Galway
Galway, Ireland
(Received and accepted August 1995)
A b s t r a c t - - W e extend a fixed-point theorem of Reinermann [1] for the sum of two operators. Our
results are then used to establish new existence results for differential equations in Banach spaces.
K e y w o r d s - - F i x e d points, Lipschitzian maps, Abstract spaces.
1. I N T R O D U C T I O N
This p a p e r uses two previously developed fixed-point theorems, one a nonlinear of Leray-Schauder
type [2-4] and the other a result of Furi-Pera t y p e by the a u t h o r [3,5] to establish existence results
for the sum of two operators F + G. In particular, if F is c o m p a c t and G is nonexpansive, we
establish some new results which extend a fixed-point result of R e i n e r m a n n [1]. These results are
t h e n used to establish new existence t h e o r y for the second-order b o u n d a r y value problem
y"+/(~,
y) = 0,
a.e. on [0, 1],
y(0) = y(1) = 0,
where f : [0, 1] x E -~ E is a C a r a t h 4 o d o r y function and E is a real B a n a c h space.
For the remainder of this section, we gather together some well-known ideas [2,4,6-8]. Let E be
a B a n a c h space and ftE the b o u n d e d subsets of E. T h e Kuratowskii measure of noncompactness
is the m a p a : f~E --~ [0, e~) defined by
o e ( X ) = i n f { e > O : X C _ t O i = 1X ~ a n d d i a m ( X i ) _ < e }
here, X c f ~ .
For convenience we recall some properties of ~. Let S, T E f~E. Then,
(i) c~(S) = 0 iff S is compact.
Oi) ~(s) = ~(s).
(iii) If S _c T, then a(S) < a(T).
(iv) c~(S U T) = max{or(S), c~(T)}.
(v)
~(~s) = I,qa(s),
,- c R .
(vi) ~(s + T) <_ a(S) + ~(T).
(vii) or(co(S)) = c~(S).
Let E1 and E2 be two B a n a c h spaces and let F : Y C E1 --* E2 be continuous and m a p
b o u n d e d sets into b o u n d e d sets. We call F an a-Lipschitzian m a p if there is a c o n s t a n t k .> 0
Typeset by Ajt/~-TEX
2
D. O'REGAN
with a ( F ( X ) ) < k a ( X ) for all bounded sets X _C Y. We call F a condensing map if F is
a-Lipschitzian with k = 1 and a ( F ( X ) ) < a ( X ) for all bounded sets X C_ Y with a ( X ) 7t O.
THEOREM 1 . 1 . [2--5]. Let E be a Banach space with C c E closed and convex. Assume U is a
relatively open subset of C with 0 E U, F ( U ) bounded and F : -U --* C a condensing map. Then
either,
(A1) F has a fixed point in U; or
(A2) there is a point u E OU and A • (0, 1) with u = AF(u).
REMARK. U and OU denote the closure of U in C and boundary of U in C, respectively.
THEOREM 1.2. [3,5]. Let E be a Banach space and Q a closed convex bounded subset of E
with 0 • int (Q). In addition, assume F : Q --* E is a condensing m a p with
oo is a sequence in OQ × [0,1] converging to (x, A) with
if {(xj, AJ)}j=l
x = AF(x) and 0 < A < 1, then A j F ( x j ) • Q for j sufficiently large
(I.i)
holding. Then, F has a fixed point.
REMARK. Theorem 1.2 was established by Furl and Pera [9] by different methods, in the case
when F : Q --* E is a compact map.
Also in this paper, the following well-known results will be used.
THEOREM 1.3. (EBERLEIN SMULIAN). Suppose K is weakly dosed in a Banach space E. Then,
the following are equivMent:
(i) K is weakly compact,
(ii) K is weakly sequentially compact, i.e., any sequence in K has a subsequence which converges weakly.
THEOREM 1.4. [8,10]. Let E be a uniformly convex Banach space, ~ a dosed, bounded, convex
subset of E, and T : fl --* E a nonexpansive mapping. I f { u j } is a weakly convergent sequence
in f~ with weak limit uo and if (I - T ) ( u j ) converges strongly to an element Wo in E, then
(I - T)(uo) = wo (i.e., the operator I - T is demiclosed on f~).
2. F I X E D - P O I N T
THEORY
Theorems 1.1 and 1.2 will now be exploited to obtain a variety of fixed-point results. Our first
two results extend a theorem of Krasnoselskii [8].
THEOREM 2.1. Let U be an open set in a closed, convex set C of a Banach space E. Assume
0 E U, F ( U ) bounded and F : U --+ C is given by F = F1 ÷ F2, where F1 : U --~ E is continuous
and completely continuous (i.e., a-Lipschitzian with k = O) and F2 : U ---* E is a nonlinear
contraction (i.e., there exists a continuous nondeereasing function ¢ : [0, oo) --* [0, co) satisfying
¢(z) < z for z > 0, such that [IF2(x) - F2(y)I] <_ ¢(llx - Y[I) for all x , y E U). Then either,
(A1) F has a fixed point in U; or
(A2) there is a point u E OU and A E (0, 1) with u = AF(u).
PROOF. The result follows immediately from Theorem 1.1 once we show F : U --~ C is a condensing map. Let fi be a bounded set in U. Then,
a ( F ( ~ ) ) < a(F~(fi)) + a(F2(fl)) = a(F2(fl)),
(2.1)
since F1 : U --* C is completely continuous. We now claim
c~(F2(~)) <_ ¢(a(f~)).
(2.2)
Fixed-Point Theory
3
If this is true, t h e n (2.1) and (2.2) yield a ( F ( ~ ) ) < ¢ ( a ( f l ) ) , and so F : U -+ C is a condensing
map. Let e > 0 be given and suppose fl C_ Un=l~i with diam (fli) < a ( ~ ) + e. Now, F 2 ( ~ ) C_
tOn=lF2(~i) - un=IY/. If Wo, wl E Y / f o r some i, then there exists Xo, Xl E ~ i with F2(xo) = wo
and F 2 ( x l ) = wl, and so since ¢ is nondecreasing
IIF2(xo) - F2(xl)II ___¢(llx0 - xlll) _< ¢ ( ~ ( ~ ) + e).
Thus, diam (Y/) < ¢ ( a ( ~ ) + e) and so a ( F 2 ( ~ ) ) < ¢(c~(~) + e). Since e is arbitrary, t h e n (2.2)
follows.
|
THEOREM 2.2. L e t E be a B a n a c h space and Q a closed, convex, b o u n d e d subset o f E with
0 E i n t ( Q ) . I n addition, a s s u m e F : Q --+ E is given b y F = F1 + F 2 , w h e r e F 1 : Q --+ E
is continuous a n d c o m p a c t and F2 : Q --+ E is a nonlinear contraction (i.e., there exists a
continuous nondecreasing function ¢ : [0, oo) --~ [0, c~) satisfying ¢(z) < z for z > O, such that
HF2(x) - F2(~)II < ¢(llx - YlI) for ~1 x , y e Q). Suppose also (1.1) is satisfied. Then, F has a
fixed point.
PROOF. This follows immediately from T h e o r e m 1.2, since F : Q -* E is a condensing map.
|
We now obtain a variety of results for the sum of two operators F + G; here F is c o m p a c t
and G is nonexpansive. First, the case when E is a reflexive B a n a c h space is considered.
THEOREM 2.3. L e t U be a b o u n d e d open convex set in a reflexive Banach space E. A s s u m e
0 E U and F : U -* E is given by F = F1 + F2, where F1 : U -+ E is continuous and c o m p a c t
and F2 : U -~ E is a nonexpansive m a p (i.e., IiF2(x) - F2(y)[[ <_ Itx - y[[ for all x , y E U). In
addition, suppose I - F : U --~ E is demic]osed on U. T h e n either,
(A1) F has a fixed point in U; or
(A2) there is a point u E OU and A E (0, 1) with u = AF(u).
REMARK. A m a p p i n g G : ~ C E --+ E is called demiclosed on f~ if for every sequence {Xn} E fl
with x~ --~ x and G ( x n ) --+ y as n --+ 0% we have t h a t x E t2 and G ( x ) = y; here ~ denotes weak
convergence.
PROOF. Assume (A2) does not hold. Consider for each n E {2, 3 , . . . }, the m a p p i n g
S~ =
1-
F : U + E.
(2.3)
Notice (1 - 1 / n ) F 2 is a contraction and (1 - 1 / n ) F 1 is continuous and compact. A p p l y Theorem 2.1 to S~. If there exists A E (0, 1) and u E OU with u = AS,~(u), then,
u=A(1-1)
F(u)=7?F(u),
where 0 < r / = A ( 1 - - l n )
<1,
whieh is a contradiction since (A2) is assumed not to hold. Consequently, for each n E {1, 2 . . . . },
we have (from T h e o r e m 2.1) t h a t S~ has a fixed point u~ E U.
A s t a n d a r d result in funetional analysis (if E is a reflexive B a n a c h space, t h e n any n o r m
b o u n d e d sequence in E has a weakly convergent subsequence) implies (since U is b o u n d e d ) t h a t
there exists a subsequence S of integers and a u E U (notice U is strongly closed and convex so
weakly closed) with
un ~ u,
as n --+ oo in S.
Notice also since un = (1 - 1 / n ) F ( u n ) , we have
II(I - F)(un)ll = -1 HFl(u,-,) + F2(u,~)l] < _1 {llF~(u~)ll + IIF2(u.) - F2(0)ll + I[F2(0)II}
n
n
1
< - {l[&(u,)ll + Ilu~ll + IIF2(0)II},
n
4
D. O'REGAN
so
( I - F ) ( u n ) --~ O,
as n --* oo in S.
Since I - F is demiclosed on U, we have t h a t ( I - F ) ( u ) = 0 i.e.; u = F ( u ) .
I
THEOREM 2.4. Let U be a bounded open convex set in a uniformly convex Banach space E .
A s s u m e 0 E U and F : U --~ E is given by F = F1 + F2, where F1 : U --~ E is continuous and
c o m p a c t and F2 : U --~ E is a nonexpansive map. In addition, s u p p o s e F1 : U --* E is s t r o n g l y
continuous. T h e n either,
(A1) F has a fixed p o i n t in U; or
(A2) there is a p o i n t u E OU and A E (0, 1) with u = AF(u).
REMARK. F1 : U ~ E is said to be strongly continuous [1,8,11] if xn -* x implies Fl(x,~)
FI(x); here x,~, x E U.
PROOF. One could use Theorem 2.3. However, we will prove it directly. Assume (A2) does not
occur. As in Theorem 2.3, Sn (given by (2.3)) has a fixed-point u,~ E U, and there exists a
subsequence S of integers and a u E U with u,~ --~ u as n --* oo in S. Also,
[1(-]'--p2)(un) --_Pl(U)[I :
-1V2(un)-}- F l ( u n ) - Fl(U ) -- XFl(Un )
1
< - {llu,~l[ + IIF2(O)I[ + IlY~(u,,)l[} + [IF~(u,,) - Fx(u)ll,
n
so since F1 is strongly continuous, we have ( I - F2)(Un) ~ FI(U) as n ~ c~ in S. T h e o r e m 1.4
now implies F l ( u ) = ( I - F2)(u).
I
We now extend Theorem 2.3.
THEOREM 2.5. L e t U be an open subset o# a Banach space E and U a w e a k l y c o m p a c t subset
o r E . Assume 0 E U and F : U --* E is given by F = F1 + F2, where F1 : U --* E is continuous
and c o m p a c t and F2 : U -* E is a nonexpansive map. In addition, s u p p o s e I - F : U --* E is
demiclosed on U. T h e n either,
(A1) F has a fixed p o i n t in U; or
(A2) there is a p o i n t u E OU and A E (0, 1) with u = AF(u).
PROOF. Assume (A2) does not occur. As in Theorem 2.3, Sn (given by (2.3)) has a fixed-point
u,~ E U for each n E {1, 2 , . . . }. Notice U is bounded [12] since U is weakly compact. Now the
Eberlein Smulian theorem guarantees a subsequence S of integers and a u E U with un --~ u as
n -~ oo in S. Essentially, the same reasoning as in Theorem 2.3 establishes the result.
I
We now formulate a nonlinear alternative for a-Lipschitzian maps with k = 1 (see Section 1).
This generalizes the above results.
THEOREM 2.6. L e t U be an open subset o f a closed, convex set C in a B a n a c h space E . A s s u m e
0 E U and F : U --~ C is a a-Lipschitzian m a p with k = 1. Also, suppose F ( U ) is b o u n d e d and
( I - F ) ( U ) is closed. T h e n either,
(A1) F has a fixed p o i n t in U; or
(A2) there is a p o i n t u E OU and A E (0, 1) with u = AF(u).
REMARK. A result of this type was proved in [10, p. 230] for the case when U is bounded.
PROOF. Assume (A2) does not hold. Consider for each n E {1, 2 , . . . }, the condensing m a p Sn
(given by (2.3)). As in Theorem 2.3, Sn has a fixed point un E U. Notice also since un =
(1 - 1 / n ) F ( u n ) , we have t h a t un - F ( u ~ ) = - ( 1 / n ) F ( u n ) and so un - F ( u n ) --* 0 as n --+ oa
(since F ( U ) is bounded). Consequently, 0 E ( I -- F ) ( U ) , since ( I - F ) ( U ) is closed. Thus, there
exists u E U with 0 -- ( I - F)(u).
I
REMARK. For examples of operators with ( I - F ) ( U ) closed, see [8].
Fixed-Point Theory
5
Finally, we o b t a i n some new fixed-point results for the s u m of two o p e r a t o r s F + G, where F
is c o m p a c t and G is nonexpansive.
THEOREM 2.7. L e t E be a reflexive Banach space and Q a closed, convex, b o u n d e d subset o f E
with 0 E int (Q). In addition, assume F : Q ~ E is given by F = F 1 + F 2 , where F1 : Q --* E is
continuous and c o m p a c t and F2 : Q --~ E is a nonexpansive map. Also, suppose I - F : Q -~ E
is demielosed on Q and that (1.1) holds. Then, F has a fixed point.
PROOF. For each n • {2, 3 , . . . }, consider Sn given by (2.3). Let { x j , Aj}~_ 1 be a sequence in
OQ × [0, 1] converging to (x,A) with x = ASh(x) and 0 < A < 1. Then,
for j sufficiently large,
AjSn(Zj)=Aj(I-1)F(xj)=-~jF(xj)CQ,
since F satisfies (1.1) (note #j = Aj (1 - 1 / n ) is a sequence in [0,1] with pj --* A ( 1 - - 1 / n ) -#, 0 < # < 1 and x = ASh(x) = A (1 - l / n ) F ( x n ) ) . A p p l y T h e o r e m 2.2 to Sn to deduce t h a t S~
has a fixed-point u~ • Q.
As in T h e o r e m 2.3, there exists a subsequence S of integers and a u • Q with un ~ u as
n --+ cx~ in S. Essentially, the same reasoning as in T h e o r e m 2.3 establishes the result.
|
T h e next result generalizes a t h e o r e m of R e i n e r m a n n [1].
THEOREM 2.8. Let E be a uniformly convex Banach space and Q a closed, convex, b o u n d e d
subset o f F with 0 • int (Q). In addition, assume F : Q --+ E is given by F = F1 + F2, where
F1 : Q --~ E is continuous and c o m p a c t and F2 : Q -~ E is a nonexpansive map. Also, s u p p o s e
F1 : Q ---+ E is strongly continuous and that (1.1) holds. Then, F has a fixed point.
|
PROOF. Fix n • {1, 2 , . . . }. Essentially, the same reasoning as in T h e o r e m 2.7 implies t h a t S~
(given by (2.3)) has a fixed point un • Q. Also, there exists a subsequence S of integers and a
u • Q with u~ ~ u as n --~ cx~ in S. In addition (as in T h e o r e m 2.4), ( I - F2)(un) converges
strongly to Fl(U). T h e o r e m 1.4 now implies (I - F2)(u) = Fx(u).
|
A generalization of T h e o r e m 2.8 is the following fixed-point result.
THEOREM 2.9. Let E be a Banach space and Q a dosed, convex, b o u n d e d subset o f E with
0 • int (Q). In addition, assume F : Q -+ E is a a-Lipschitzian m a p with k = 1. Also, s u p p o s e
(I - F ) ( Q ) is d o s e d and that (1.1) holds. Then, F has a fixed point.
PROOF. Fix n • {1, 2 , . . . }. Essentially, the same reasoning as in T h e o r e m 2.7 implies t h a t S~
(given by (2.3)) has a fixed point un • Q. Also (as in T h e o r e m 2.6), un - F ( u n ) --~ 0 as n --* oc,
and hence, 0 • (I - F ) ( Q ) .
|
3. D I F F E R E N T I A L E Q U A T I O N S IN B A N A C H S P A C E S
In this section, we obtain a new existence result for the second-order b o u n d a r y value p r o b l e m
y" + , f ( t , y) = 0,
y(0) = y(1) = 0,
a.e. on [0, 1],
(3.1)
where E is a real B a n a c h space (with n o r m ][. [[) and # _> 0 is a p a r a m e t e r . P r o b l e m s of the above
form have been discussed extensively in the literature; see [4,6,7,13-17] and their references. We
denote by C([0, 1], E ) the space of continuous functions u : [0, 1] --+ E. Let u : [0, 1] -+ E be a
m e a s u r a b l e function. B y f ~ u(t) dt, we m e a n the Bochner integral of u, assuming it exists. We
define the Sobolev class W1,1([0, 1], E ) to be the space of continuous functions u such t h a t there
exists v • LI([0, 1], E ) with u ( t ) - u(O) =
v ( s ) d s for all t • [0, 1]. Notice if u • W1'1([0, 1], E ) ,
f~
t h e n u is differentiable a.e. on [0, 1], u' • nl([0, 1], E ) and u(t) - u(O) -- fo u ' ( s ) d s for t • [0, 1].
Also, if E is a reflexive B a n a c h space, u • W1'1([0, 1], E) iff u is absolutely continuous. Now,
u • W2'l([0, 1],E) if u, u' • W l ' l ( [ 0 , 1 ] , E).
6
D.
O'REGAN
B y a solution to (3.1), we mean a function u E W2'I([0,1],E) t h a t satisfies the differential
equation in (3.1) a.e. on [0, 1] and the stated boundary data.
A function g : [0, 1] x E ~ E is a Carath6odory function if
(i) the map t ~-* g(t, z) is measurable for each z E E;
(ii) the map z H g(t, z) is continuous for almost all t E [0, 1];
(iii) for each r > 0, there exists hr E LI[0,1] such t h a t I[z[[ < r implies [[g(t,z)[[ <_ hr(t) for
almost all t E [0, 1].
THEOREM 3.1. Assume f : [0, 1] x E --* E has the decomposition f ( t , u ) = f l ( t , u ) + f2(t,u)
with f l and f2 Carathdodory functions such that the following conditions are satisfied:
for each bounded set f~ C_ C([0, 1],E), and for each t E [0, 1], the set
/'
{ /o'
(l-t)
sfl(s,y(s))ds+t
is relatively compact,
(1-s) fa(s,y(s))ds: yEf~
}
(3.2)
there exists q E LI[0, 1], with q > 0 a.e. on [0, 1], and a continuous nondecreasing
(3.3)
function ¢ : [0, c~) --* [0, co) satisfying ¢(z) < z for z > 0 such that
N f 2 ( t , Ul)
--
f2(t, u2)ll _< q ( t ) ¢ ( l l u l
# sup
[0,1]
(
(l-t)
- u211) for a.e. t E [0,1] and a n Ux, u2 C E,
1'
sq(s)ds+t
/'
(1-s)q(s)ds
)
< 1,
(3.4)
and
there exists a conthauous nondecreasing function ¢ : [0, oo) ---* [0, co),
with ¢(u) > 0 for u > O, and a function 71 E L 1[0, 1], with
r / > 0 a.e. on [0, 1], with [If(t,u)ll <
n(t)C(llull)
(3.5)
on (0,1) x E.
Let
Qo=sup
[0,1]
(
and let #o satisfy
/o'
(l-t)
sv(s)ds+t
/'
(1-s)rl(s)ds
sup( c ) >
(o,oo) tzo Qo ¢(c)
)
,
(3.6)
1.
IfO < # < #o, then (3.1) has a solution.
REMARK. The supremum in (3.6) is allowed to be infinite.
PROOF. Fix # < #0. Let Mo > 0 satisfy
Mo
Qo¢(M0)
>1.
(3.7)
Let y be a solution to
y"+~,f(t,y)=0
y(0)=y(1)=0,
a.e. o n [0,1],
(3.s)~
for 0 < A < 1. Then, for t E [0, 1], we have
y(t)=A#
(
(l-t)
/o'
sf(s,y(s))ds+t
S'
(1-s)f(s,y(s))ds
)
,
Fixed-PointTheory
7
and so (3.5) yields
(
[y(t)[ -<#
/:
(i-t)
(1-s),7(s)¢([ly(s){{)ds
s~?(s)¢(Hy(s)ll)ds+t
)
< ~ ¢({{y{{o) Qo,
where [}Y}{0-- suP[o,ll
Ily(s){{.
Consequently,
[[Y}[o
< 1.
(3.9)
~ Qo'C'(llyl}o) -
Let
}[U}]o<
U = {u • C([0,1],E) :
Mo}.
Solving (3.1) is equivalent to finding a fixed point of N : C([0, 1], E) -+ C([0, 1], E) given by
y(t)=#
(
(l-t)
/0
sf(s,y(s))ds+t
fl
(1-s)
f(s,y(s))ds
)
=Nly(t)+N2y(t),
where
Niy(t) = #
( (1 - t) ~0t s fi(s, y(s)) ds + t fl
(1 - s ) f i ( s , y(s)) ds
)
,
i = 1, 2.
Since fl and f2 are Carath6odory, Ni and N2 are well defined and continuous. Condition (3.2)
together with the Arzela-Ascoli theorem implies that N1 is completely continuous. Consequently,
Ni : U --+ C([0, 1], E) is compact.
Also, for u , v E U and t E [0, 1], we have from (3.3) that
[ £
IlN2u(t) - N2u(t){{ -< # (1 - t)
+ t
s q ( s ) ¢ ( N u ( s ) - v(s){]) ds
(1 - s ) q ( s ) ¢ ( l l u ( s )
<_#¢([[u-v{[o) sup
[0,1l
- v(s)ll)ds
(l-t)
sq(s)ds+t
( 1 - s ) q(s)ds
.
This, together with (3.4), implies
{{N2u - N2v{}o < ¢({{u - v{{o),
so N2 : U -+ C([0, 1], E) is a nonlinear contraction.
If condition (A2) of Theorem 2.1 holds, then there exists A c (0, 1) and y c OU with y = A N y .
Then, y is a solution of (3.8)~ with HyII0 = M0. Now, (3.9) implies
Mo
_<1,
# Q0 ¢(Mo)
which contradicts (3.7). Hence, N has a fixed point in U by Theorem 2.1.
|
REMARKS.
(i) In (3.3), we can replace [[f2(t, u l ) - f 2 ( t , u2)ll -< q(t)¢(llul -u21[) for a.e. t e [0,1], and
all u l , u 2 E E with [[f2(t, u l ) - f2(t, u2)][ ___~ q(t)¢([[ux - u211) for a.e. t e [0,1] and all
u l , u 2 e E satisfying [[ul[[, Hush _< Mo; here M0 is as defined in (3.7).
(ii) Notice in the proof of Theorem 3.1, we only showed that any solution of (3.8)~ satisfies
IlYl[o ¢ Mo. We do n o t claim that any solution of (3.8)x satisfies [lY[10 -< Mo.
8
D. O'REGAN
REFERENCES
1. J. l~einermann, Fixpunkts~tze vom Krasnoselski-type, Math. Z. 119, 339-344 (1971).
2. J. Dugundji and A. Granas, Fixed point theory, Monografie Mat., PWN, Warsaw, (1982).
3. D. O'Regan, A fixed point theorem for condensing operators and applications to Hammerstein integral
equations in Banach spaces, Computers Math. Applic. 30 (9), 39-49 (1995).
4. R. Precup, On the topological transversality principle, Jour. Nonlinear Anal. 20, 1-9 (1993).
5. D. O'Regan, Some fixed point theorems for concentrative mappings between locally convex linear topological
spaces, Jour. Nonlinear Anal. (to appear).
6. J. Banas and K. Goebel, Measures of Noneompactness in Banach Spaces, Marcel Dekker, New York, (1980).
7. l=t.H. Martin, Jr., Nonlinear Operators and Differential Equations in Banach Spaces, John Wiley and Sons,
New York, (1976).
8. E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. I, Springer-Verlag, (1990).
9. M. Furi and P. Pera, A continuation method on locally convex spaces and applications to ordinary differential
equations on noncompact intervals, Ann. Polon. Math. 47, 331-346 (1987).
10. F.E. Browder, Nonlinear operators and nonlinear equations of evolutions in Banach spaces, In Proc. Symp.
Pure Math, Vol. 18, Part II, Amer. Math. Soc., Providence, RI, (1976).
11. S. F u ~ , Remarks on monotone operators, Comment. Math. Univ. Carolinae 11, 435-448 (1970).
12. N. Dunford and J.T. Schwartz, Linear Operators, Part I, Interscience, New York, (1967).
13. M. Frigon and J.W. Lee, Existence principles for Carath~odory differential equations in Banach spaces, Top.
Methods in Nonlinear Anal. 1, 91-106 (1993).
14. M. Frigon and D. O'Regan, Existence results for initial value problems in Banach spaces, Diferential
Equations and Dynamical Systems 2, 41-48 (1994).
15. V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press,
New York, (1981).
16. J.W. Lee and D. O'Regan, Existence results for differential equations in Banach spaces, Comment. Math.
Univ. Carolinae 34, 239-251 (1993).
17. J. Mawhin, Two point boundary value problems for nonlinear second order differential equations in Hilbert
spaces, Tohohu Math. Jour. 32, 225-233 (1980).
18. J.A. Gatica and W.A. Kirk, Fixed point theorems for contractive mappings with applications to nonexpansive
and pseudo-contractive mappings, Rocky Mount. J. Math. 4, 69-79 (1974).