Funkcialaj Ekvacioj, 30 (1987) I9-29
An Existence Theorem of Functional Differential Equations
With Infinite Delay in a Banach Space
By
Jong Son SHIN
(Korea University, Japan)
1. Introduction
.
: ¥ ¥ ¥
, $ 0<a¥leqq¥infty$ ,
Let be a Banach space with norm
be defined by $x_{t}(¥theta)=x(t+¥theta)$ ,
,
then for any $¥in(-¥infty, ¥sigma+a)$ we let : (
$0]$ .
In this paper we are concerned with an abstract Cauchy problem
$E$
$1¥mathrm{f}$
$|¥cdot|_{E}$
$t$
$¥mathrm{x}_{t}$
$-¥infty$
$(- infty,
$x$
sigma+a) rightarrow E$
$¥mathrm{O}]¥rightarrow E$
$¥theta¥in(-¥infty,$
(1.1)
$¥frac{dx}{dt}=f(f, ¥chi_{t})$
(1.2)
$x_{¥sigma}=¥phi¥in¥ovalbox{¥tt¥small REJECT}$
$ t>¥sigma$
,
is an abstract phase
and
is a -valued mapping defined on [ ,
space which is a semi-normed space satisfying suitable axioms introduced by
Hale and Kato [3].
The purpose of this paper is to study the local existence of solutions for
the Cauchy problem (1.1)?(1.2). It is well known that the compactness condition
which is described by means of a-measure of noncompactness introduced by
K. Kuratowski is useful in showing the existence of solutions of ordinary differential equations in a Banach space ([2, 4, 7, 9, 10]). In this paper the local
existence of solutions for the Cauchy problem (1.1)?(1.2) will be shown by using
a property (Theorem 2.1) of a-measure of noncompactness which is concerned
. Our result (Theorem 3.1) on the local existence of
with the phase space
solutions is an extension of the main result obtained in [2] (also refer to [10]).
Finally, as an application of Theorem 3.1, we shall show the existence of solutions
for some integro-differential equation.
where
$f$
$¥mathrm{E}$
$¥sigma$
$¥infty)¥times¥ovalbox{¥tt¥small REJECT}$
$¥ovalbox{¥tt¥small REJECT}$
$¥ovalbox{¥tt¥small REJECT}$
2. Phase space
$¥mathscr{B}$
and properties of
α-measure
of noncompactness
introduced by Hale and
First of all, we shall explain the phase space
be a
Kato [3]. Let $R^{-}=(-¥infty, 0], R^{+}=[0, ¥infty)$ , $R=(-¥infty, ¥infty)$ and let
be a linear space of functions
. Further let
Banach space with norm
having the following axioms;
into with a semi-morm
mapping
$¥mathscr{B}$
$E$
$|¥cdot|_{E}$
$R^{-}$
$E$
$¥mathscr{B}$
$|¥cdot|_{g}$
20
Jong Son SHIN
If :
and
$x$
$(¥mathrm{B}_{1})$
then
$¥mathrm{x}_{t}¥in¥ovalbox{¥tt¥small REJECT}$
$(-¥infty, ¥sigma+a)¥rightarrow E$
$¥mathrm{x}_{t}$
$a>0$ ,
,
is continuous in
There exist functions
$(¥mathrm{B}_{2})$
$ t¥in$
is continuous on [ ,
$¥sigma$
$[¥sigma,$
$¥sigma+a)$
$x_{¥sigma}¥in¥ovalbox{¥tt¥small REJECT}$
,
.
and
$K(t)>0$
and
$¥sigma+a)$
with the following
$M(t)¥geqq 0$
properties ;
.
is continuous for ¥
$M(t)$ is locally bounded on
.
having the properties in
For every function
it holds that
i)
$t in R^{+}$
$K(t)$
$R^{+}$
$¥mathrm{i}¥mathrm{i})$
$x$
$¥mathrm{i}¥mathrm{i}¥mathrm{i})$
$|x_{t}|_{g}¥leqq K(t-¥sigma)¥sup$
$(¥mathrm{B}_{3})$
$(¥mathrm{B}_{4})$
$¥{|x(s)|_{E}:
¥sigma¥leqq s¥leqq t¥}+M(t-¥sigma)|x_{¥sigma}|_{¥ovalbox{¥tt¥small REJECT}}$
There exists a constant $L>0$ such that
The quotient space
and
$(B_{1})$
$¥hat{¥ovalbox{¥tt¥small REJECT}}:=¥ovalbox{¥tt¥small REJECT}/|¥cdot|_{¥ovalbox{¥tt¥small REJECT}}$
$|¥phi(0)|_{E}¥leqq L|¥phi|_{g}$
$t$
$¥in$
$[¥sigma,$
$¥sigma+a)$
,
.
for all
$¥phi¥in¥ovalbox{¥tt¥small REJECT}$
.
is a Banach space.
.
satisfying all axioms
See [3] for examples of the phase space
mapping
the set of all functions
, we denote by
and
For
$x(t)$
.
on
is
continuous
and
, to such that
,
,
(
Next, we shall consider the properties of Kuratowski’s measure of noncompacrness (for brevity, a-measure). Let be a bounded subset of a Banach
of is defined as
. The a-measure
space Ywith norm
$(¥mathrm{B}_{1})-(¥mathrm{B}_{4})$
$¥ovalbox{¥tt¥small REJECT}$
$¥sigma¥in R$
$¥phi¥in¥ovalbox{¥tt¥small REJECT}$
$-¥infty$
$¥sigma+¥delta]$
$X_{¥sigma}^{¥phi}[¥delta]$
$E$
$¥delta>¥mathrm{C}$
$x$
$[¥sigma, ¥sigma+¥delta]$
$ x_{¥sigma}=¥phi$
$¥Omega$
$¥alpha(¥Omega)$
$|¥cdot|_{Y}$
$¥Omega$
{ $ d>0:¥Omega$ has a finite cover of diameter
$¥alpha(¥Omega)=¥inf$
$<d$ }.
From the definition we can get some properties of a-measure immediately,
see [6, p. 17-18].
Let A and be bounded subsets of Y.
$ A=¥sup$
, where
(i)
.
(Bi)
if
, where A? denotes the closure of A.
(iii)
(vi)
if and only if A? compact.
.
, where
,
(v)
¥
¥
, where $A+B=$
,
(vi) ¥ ¥
$¥{x+y:x¥in ¥mathrm{A}, y¥in B¥}$ .
(Conv A), where Conv A is the convex hull of A.
(vii)
is
a family of nonempty bounded subsets of such that
(viii) If
, then
is nonempty and compact.
for $ n=1,2,¥cdots$ , and
Lemma 2.1.
$B$
$¥alpha(¥mathrm{A})¥leqq¥alpha(B)$
$¥{|x-y|_{¥mathrm{Y}} :
$¥mathrm{d}¥mathrm{i}¥mathrm{a}$
$¥alpha(A)¥leqq ¥mathrm{d}¥mathrm{i}¥mathrm{a}¥mathrm{A}$
x, y¥in ¥mathrm{A}¥}$
$¥mathrm{A}¥subset B$
$¥alpha(¥mathrm{A})=¥alpha(¥overline{A})$
$is$
$¥alpha(¥mathrm{A})=0$
$a(¥lambda ¥mathrm{A})=|¥lambda|a(¥mathrm{A})$
$ alpha( mathrm{A} cup B)= max$
$¥lambda¥in R$
$¥lambda ¥mathrm{A}=$
$¥{¥alpha(¥mathrm{A}), ¥alpha(B)¥}$
$¥{¥lambda ¥mathrm{x}:x ¥in ¥mathrm{A}¥}$
$¥alpha(¥mathrm{A}+B)¥leqq¥alpha(¥mathrm{A})+a(B)$
$¥alpha(A)=¥alpha$
$¥mathrm{Y}$
$¥{¥mathrm{A}_{n}¥}$
$A_{n}$
$¥lim_{n¥rightarrow¥infty}¥alpha(¥mathrm{A}_{n})=0$
$¥mathrm{A}_{n+1}¥subseteqq$
$¥bigcap_{n=1}^{¥infty}¥overline{A}_{n}$
Yand of radius
will be denoted by $K(x,
Then open ball centered at
$(y,$
$B)<r$ } for ¥
: dist
Note that
Define $K(B, r)=¥bigcup_{x¥in B}K(x, r)=$ {
$x$
$¥in$
$r¥geqq 0$
(2. 1)
r)$ .
$B subset Y$
$y¥in ¥mathrm{Y}$
$K(B, r)=B+K(0, r)=B+rK(0,1)$ ,
where 0 is the zero element of Y.
Put $ d(¥mathrm{A}, B)=¥inf$ { : A
$r$
$¥subset K(B,$
$r)$
} and
$D(¥mathrm{A}, B)=$
FDE with
$¥max¥{d(¥mathrm{A}, B), d(B, ¥mathrm{A})¥}$
Infinite Delay in a Banach space
, where A and
$B$
21
are subsets of Y. Then from (v), (vi)
¥
. Let
be the
¥
¥
¥
and (2. 1) it follows that ¥ ¥
set of all continuous functions defined on
$C([a, b], 7)$ we define
$| alpha( mathrm{A})- alpha(B)| leqq alpha(K(0,1))D(¥mathrm{A}, B)$
$¥int_{a}^{t}H(s)ds$
1) If
mathrm{Y})$
with values in Y.
for
$=¥{¥int_{a}^{t}x(s)ds:x¥in H¥}$
where $H(s)=¥{x(s)¥in ¥mathrm{Y}:x¥in H¥}$ .
needed.
Lemma 2.2.
$[a, b]$
$C([a, b],
$t¥in[a, b]$
For
$ H¥subset$
,
To our purpose the following lemma will be
$H¥subset C([a, b], ¥mathrm{Y})$
is a bounded set, then
$|¥alpha¥{H(t)¥}-¥alpha¥{H(s)¥}|¥leqq¥alpha(K(0,1))D(H(t), H(s))$
$¥leqq¥alpha(K(0,1))¥omega(H, |t-s|)$
where
$¥omega(H, r)=¥sup$ $¥{|x(t)-x(s)|_{Y} :
2) If
i)
$¥mathrm{i}¥mathrm{i})$
$H¥subset C([a, b], ¥mathrm{Y})$
$¥alpha(H)=¥sup_{a¥leqq t¥leqq b}¥alpha¥{H(t)¥}$
,
x¥in H, |t-s|<r¥}$ .
is a bounded, equicontinuous set, then
.
for
$¥alpha¥{¥int_{a}^{t}H(s)ds¥}¥leqq¥int_{a}^{t}¥alpha¥{H(s)¥}ds$
$t¥in[a, b]$
.
For a proof refer to [1, p. 69, 6, p. 20].
Let $X^{¥sigma}((-¥infty, ¥sigma+a), E)$ , $ 0<a¥leqq¥infty$ , be a set of functions mapping $(-¥infty$ ,
$¥sigma+a)$ into $E$ such that
lies in
and $x(t)$ is continuous in on [ , $¥sigma+a)$ . To
state the theorem below, we use the following notations:
$x$
$¥ovalbox{¥tt¥small REJECT}$
$t$
$x_{¥sigma}$
$¥sigma$
$X(t)=¥{x(t)¥in E:x¥in X^{¥sigma}((-¥infty, ¥sigma+a), E)¥}$
$X_{t}=¥{x_{t}¥in¥ovalbox{¥tt¥small REJECT}:x¥in X^{¥sigma}((-¥infty, ¥sigma+a), E)¥}$
and
$X[¥sigma, t]=¥{x| [¥sigma, t]:x¥in X^{¥sigma}((-¥infty, ¥sigma+a), E)¥}$
,
where
is the restriction of to
. For a subset
) and $x|$
of , we put
, where is the equivalence class containing .
Now, we get the main theorem in Section 2, which is useful for the study of
the existence of solutions for the Cauchy problem (1.1)?(1.2).
$t¥in[¥sigma,$
$¥ovalbox{¥tt¥small REJECT}$
$¥sigma+a$
$[¥sigma, t]$
$¥hat{Z}=¥{¥hat{x}¥in¥hat{¥ovalbox{¥tt¥small REJECT}} :
Theorem 2.1.
bounded sets in
holds;
$¥ovalbox{¥tt¥small REJECT}$
(2.2)
Let
and
$¥ovalbox{¥tt¥small REJECT}$
x ¥in Z¥}$
$x$
$Z$
$[¥sigma, t]$
$¥hat{x}$
$x$
satisfy the axioms
. If
and $X[¥sigma, t]$ are
, respectively, then the following inequality
$(¥mathrm{B}_{1})-(¥mathrm{B}_{4})$
$X_{¥sigma}$
$C([¥sigma, t], E)$
$¥alpha(¥hat{X}_{t})¥leqq K(t-¥sigma)¥alpha(X[¥sigma, t])+M(t-¥sigma)¥alpha(¥hat{X}_{¥sigma})$
.
22
Son Son SHIN
In particular, if $X[¥sigma, t]$ is an equicontinuous subset
above inequality (2.2) becomes
(2.5)
of
$C([¥sigma, t], E)$
$a(¥hat{X}_{t})¥leqq K(t-¥sigma)¥sup_{¥sigma¥leqq s¥leqq t}¥alpha(X(s))+M(t-¥sigma)¥alpha(¥hat{X}_{¥sigma})$
, then the
.
From the definition of a-measure it follows that for any
¥
¥
¥
, $ i=1,¥cdots$ , $m$ , and bounded sets
exist bounded sets
$ j=1,¥cdots$ ,
, such that
Proof.
$¥epsilon>0$
$X^{i}[ sigma, t] subset X[ sigma, t]$
, there
$X_{¥sigma}^{i}¥subset X_{¥sigma}$
,
$n$
$ i=1,¥cdots$
(2.4)
,
$m$
,
$¥left¥{¥begin{array}{l}¥mathrm{d}¥mathrm{i}¥mathrm{a}X^{i}[¥sigma,t]¥leqq¥alpha(X[¥sigma,t])+¥epsilon/2K^{*}(t-¥sigma),¥¥m¥¥X[¥sigma,t]=¥cup X^{i}[¥sigma,t]¥end{array}¥right.i=1$
and
$ j=1,¥cdots$
(2.5)
where
Put
, ,
$n$
$¥left¥{¥begin{array}{l}¥mathrm{d}¥mathrm{i}¥mathrm{a}X_{¥sigma}^{j}¥leqq¥alpha(¥hat{X}_{¥sigma})+¥epsilon/2M^{*}(t-¥sigma),¥¥X_{¥sigma}=¥cup nX_{¥sigma}^{j},¥end{array}¥right.j=1$
$K^{*}(t)=¥sup_{0¥leqq s¥leqq t}K(s)$
$X_{t}^{i,j}=$
{
and
$x_{t}¥in X_{t}$
:
$M^{*}(t)=¥sup_{0¥leqq s¥leqq t}M(s)$
$x|$
[ ,
$¥sigma$
$t]¥in X^{i}$
[ ,
$¥sigma$
$t]$
.
and
$x_{¥sigma}¥in X_{¥sigma}^{j}$
}
Then we have
(2.6)
$X_{t}=¥bigcup_{i=1}^{m}¥bigcup_{j=1}^{n}X_{t}^{i,j}$
By (2.4) and 2.5) we obtain, for
$x_{t}$
,
$y_{t}¥in X_{t}^{i,j}$
,
$|¥chi_{t}-y_{t}|_{¥ovalbox{¥tt¥small REJECT}}¥leqq K(t-¥sigma)¥sup_{¥sigma¥leqq s¥leqq t}|x(s)-y(s)|_{E}+M(t-¥sigma)|x_{¥sigma}-y_{¥sigma}|_{¥ovalbox{¥tt¥small REJECT}}$
$¥leqq K(t-¥sigma)¥mathrm{d}¥mathrm{i}¥mathrm{a}$
$¥mathrm{X}¥{¥mathrm{a}$
,
$ t]+M(t-¥sigma)¥mathrm{d}¥mathrm{i}¥mathrm{a}X_{¥sigma}^{j}$
$¥leqq K(t-¥sigma)¥alpha(X[¥sigma, t])+M(t-¥sigma)¥alpha(¥hat{X}_{¥sigma})+¥epsilon$
and hence, by (2.6)
.
$¥alpha(¥hat{X}_{t})¥leqq K(t-¥sigma)¥alpha(X[¥sigma, t])+M(t-¥sigma)¥alpha(¥hat{X}_{¥sigma})+¥epsilon$
Letting
$¥epsilon¥rightarrow 0$
,
we have
$¥alpha(¥hat{X}_{t})¥leqq K(t-¥sigma)¥alpha(X[¥sigma, t])+M(t-¥sigma)¥alpha(¥hat{X}_{¥sigma})$
.
Furthermore, the inequality (2.3) immediately follows from the property 2)
in Lemma 2.2, and so, the proof is complete.
FDE with
Define a linear operator
Infinite
$S(t):¥ovalbox{¥tt¥small REJECT}¥rightarrow¥ovalbox{¥tt¥small REJECT}$
$¥mathrm{r}$
(2.7)
Delay in a Banach space
,
$t¥geqq 0$
$¥phi(t+¥theta)$
$[S(t)¥phi](¥theta)=¥{¥phi(0)$
23
, by
if
$t+¥theta<0$
if
$t+¥theta¥geqq 0$
.
Then a linear operator
is induced by the relation
for
. This operator plays an essential role in studying of linear functional
differential equations with infinite delay (refer to [8]). Corollary 2.1 below is
an immediate consequence of Theorem 2. 1.
$¥hat{S}(t)¥hat{¥phi}=(S(t)¥phi)^{¥wedge}$
$¥hat{S}(t):¥hat{¥ovalbox{¥tt¥small REJECT}}¥rightarrow¥hat{¥ovalbox{¥tt¥small REJECT}}$
$¥hat{¥phi}¥in¥hat{¥ovalbox{¥tt¥small REJECT}}$
Corollary 2.1. If the space
then the following results hold.
1)
2) If
$¥ovalbox{¥tt¥small REJECT}$
of
consists
$R^{n_{-}}$
valued mappings,
for every
$ n<¥infty$
,
bounded set
.
is a relatively compact subset of
and $X[¥sigma, t]$ is a bounded and
equicontinuous subset of $C([¥sigma, t], R^{n})$ , then
is a relatively compact subset
$¥alpha(¥hat{S}(t-¥sigma)B)¥leqq M(t-¥sigma)¥alpha(B)$
$B¥subset¥hat{¥ovalbox{¥tt¥small REJECT}}$
$¥hat{X}_{¥sigma}$
$¥hat{¥ovalbox{¥tt¥small REJECT}}$
$¥hat{X}_{t}$
of
$¥hat{¥ovalbox{¥tt¥small REJECT}}$
.
3. Existence of solutions
We shall prove the existence of solutions for the Cauchy problem (1.1)?(1.2)
by using Theorem 2. 1.
Theorem 3.1.
1)
$f$
:
Let
$¥ovalbox{¥tt¥small REJECT}$
satisfy the axioms
,
$[¥sigma, ¥sigma+a]¥times K(¥phi, r)¥rightarrow E$
for
. Assume that
, is uniformly continuous and
$(¥mathrm{B}_{1})-(¥mathrm{B}_{4})$
$K(¥phi, r)¥subset¥ovalbox{¥tt¥small REJECT}$
all ¥ ¥ $[¥sigma, ¥sigma+a]¥times K(¥phi, r)$ .
2) There exists a function $W$ : ( , $¥sigma+a]¥times[0,¥mathit{2}r]R^{+}such$ that $¥alpha(f(t, B))¥leqq$
and all sets $B¥subset K(¥phi, r)$ .
for .
3) $W$ satisfies the following properties;
(i) $W(t, s)$ is Lebesgue measurable in for fixed , and continuous in
$|f(t, ¥psi)|_{E}¥leqq H$
$(t,
psi) in$
$¥sigma$
$W(t, ¥alpha(¥hat{B}))$
$¥mathrm{a}.¥mathrm{a}$
$ t¥in$
$(¥sigma, ¥sigma+a)$
$t$
for fixed
$t$
$s$
.
(ii) $W(t, s)$ is nondecreasing in for fixed
], it holds that
(iii) For any
$s$
$t¥in(¥sigma,$
$t$
.
$¥sigma+a$
$¥int_{¥sigma+}^{t}W(s, K(s-¥sigma)v(s))ds<¥infty$
,
whenever $K(t-¥sigma)v(t)¥in[0,¥mathit{2}r]$ for all
, where
$v(t)$ is the continuous
function with the condition;
$ t¥in$
(3. 1)
$[¥sigma, ¥sigma+a]$
$¥lim_{t¥rightarrow¥sigma¥dagger}¥frac{v(t)}{t-¥sigma}=v(¥sigma)=0$
(iv)
$s$
is the unique
integral inequality
$v(t)¥equiv 0$
function
$K(t)$
is as in
$(¥mathrm{B}_{2})$
and
.
satisfying the condition (3.1) and the
24
Jong Son SHIN
(3.2)
$v(t)¥leqq¥int_{¥sigma+}^{t}W(s, K(s-¥sigma)v(s))ds$
Then there is a number ,
has a solution defined on $(-¥infty,$
$¥delta$
Proof.
, such that the Cauchy problem (1.1)?(1.2)
$0<¥delta¥leqq a$
$¥sigma+¥delta]$
.
.
be the operator defined by (2.7). Clearly,
,
Let
, $¥sigma+b]$ , $0<b¥leqq a$ , if
is a solution of the Cauchy problem (1.1)?(1.2) on (
,
defined by $y(t+¥theta)=x(t+¥sigma+¥theta)-[S(t)¥phi](¥theta)$ , $t¥in[0, b]$ ,
and only if
satisfies
$S(t):¥ovalbox{¥tt¥small REJECT}¥rightarrow¥ovalbox{¥tt¥small REJECT}$
$t¥geqq 0$
$-¥infty$
$x$
$¥theta¥in R^{-}$
$y$
$tt¥in[0, b]¥in R^{-}$
$¥sim v(t)=¥{¥int_{0}^{t}0,f(s+¥sigma, S(s)¥phi+y_{s})ds$
.
,
Therefore the Cauchy problem (1.1)?(1.2) is equivalent to the existence of a
fixed point of the operator $T$ defined in the following manner. From the
continuity of
and the axiom
we can choose a number , ¥ ¥ ,
such that $|S(t)¥phi-¥phi|_{g}<r/2$ for ¥ ¥
and $K^{*}(¥delta)H¥delta<r/2$ , where $K^{*}(t)=$
. Let $BC((-¥infty, ¥delta], E)$ be the Banach space of all continuous and
bounded functions from (
into
,
with the $¥sup$ -norm. Define a set
as follows
$ t¥rightarrow S(t)¥phi$
$¥delta$
$(¥mathrm{B}_{2})$
$t in[0,
$0< delta leqq a$
delta]$
$¥sup_{0¥leqq s¥leqq t}K(s)$
$-¥infty$
$¥mathrm{Y}^{0}=¥{y$
$¥delta]$
$E$
$¥in BC((-¥infty, ¥delta], E):y(t)¥equiv 0$
for
$¥mathrm{Y}^{0}$
$t¥in R^{-}$
and
$|y(t)-y(¥tau)|¥leqq H|t-¥tau|$
for ,
$t$
Then
is a closed bounded convex subset of the Banach space
Now, we define operators and as follows; For
,
$¥mathrm{Y}^{0}$
$T$
$F$
$(Fy)(t)=f(t+¥sigma, S(t)¥phi+y_{t})$
$¥tau¥in[0, ¥delta]¥}$
.
$BC((-¥infty, ¥delta], E)$
.
$y¥in ¥mathrm{Y}^{0}$
for
$t¥in[0, ¥delta]$
and
$(Ty)(t)=¥{¥int_{0}^{t}0(Fy)(s)ds$
$¥mathrm{f}¥mathrm{f}¥mathrm{o}¥mathrm{r}¥mathrm{o}¥mathrm{r}tt¥in[0, ¥delta]¥in R^{-}$
.
First of all, we shall prove that the operator
is a continuous mapping
from
into
. Since $|(Ty)(t)-(Ty)(¥tau)|_{E}¥leqq H|t-¥tau|$ for ,
,
and
it is obvious that
. Suppose a sequence
of
tends to a
as
. Then using the Lebesgue’s dominated convergence theorem for vectorvalued functions, we see that the right hand side of the inequality
$T$
$¥mathrm{Y}^{0}$
$¥mathrm{Y}^{0}$
$t$
$T¥mathrm{Y}^{0}¥subset ¥mathrm{Y}^{0}$
$¥{y^{n}¥}$
$¥tau¥in[0, ¥delta]$
$¥mathrm{Y}^{0}$
$ n¥rightarrow¥infty$
$¥sup_{0¥leqq ¥mathrm{r}¥leqq¥delta}|(Ty^{n})(t)-(Ty^{0})(t)|_{E}¥leqq¥int_{0}^{¥delta}|(Fy^{n})(s)-(Fy^{0})(s)|_{E}ds$
$y¥in ¥mathrm{Y}^{0}$
$y^{0}$
Infinite Delay in a Banach space
FDE with
25
. This implies that the operator $T$ is a continuous mapping
tends to 0 as
.
into
from
for ¥ ¥
, $n=0,1$ , 2, .
and
Conv
Set
converges to a
Then, using the same argument as in [2], we can see that
.
as
satisfies
and
uniformly on
¥
¥
$V(t)
¥
equiv
0$
¥
¥
¥
¥
.
, where
for
Next, we shall show that
is equicontinuous and uniformly
From the condition 1) in the theorem, the set
bounded: from the condition 2) in the theorem it follows that
. Since $V^{n+1}(t)=a$ ((Conv
) $=$
, by using the property 2) in Lemma 2.2, we get
$ n¥rightarrow¥infty$
$¥mathrm{Y}^{0}$
$¥mathrm{Y}^{0}$
$¥mathrm{Y}^{n+1}=$
$V^{n}(t)=a(¥mathrm{Y}^{n}(t))$
$(T¥mathrm{Y}^{n})$
$t in[0,
delta]$
$¥cdots$
$¥{V^{n}(t)¥}$
$[0, ¥delta]$
$V^{¥infty}(t)$
$t¥rightarrow 0+$
$V^{¥infty}(t)/t¥rightarrow 0$
$V^{¥infty}(t)$
$t in[0,
delta]$
$V(t)= sup_{0 leqq s leqq t}V^{ infty}(s)$
$F¥mathrm{Y}^{0}$
$¥alpha¥{(F¥mathrm{Y}^{n})(s)¥}¥leqq$
$W(s+¥sigma, ¥alpha(¥hat{S}(s)¥hat{¥phi}+¥hat{¥mathrm{Y}}_{s}^{n}))¥leqq W(s+¥sigma, ¥alpha(¥hat{¥mathrm{Y}}_{¥mathrm{s}}^{n}))$
$T¥mathrm{Y}^{n})(t)$
$¥alpha(T¥mathrm{Y}^{n}(t))$
$V^{n+1}(t)=¥alpha¥{¥int_{0}^{t}(F¥mathrm{Y}^{n})(s)ds¥}$
$¥leqq¥int_{0}^{t}¥alpha¥{(F¥mathrm{Y}^{n})(s)¥}ds$
$¥leqq¥int_{0+}^{t}W(s+¥sigma, ¥alpha¥{¥mathrm{Y}_{s}^{n}¥})ds$
.
Moreover, by using Theorem 2.1, we have
for all $t¥in[0, ¥delta]$ . From this inequality and properties (ii) and (iii) of
we have
$¥alpha(¥hat{¥mathrm{Y}}_{t}^{n})¥leqq K(t)¥sup_{0¥leqq s¥leqq t}¥alpha(¥mathrm{Y}^{n}(s))¥leqq r$
$V^{n+1}(t)¥leqq¥int_{0+}^{t}W(s+¥sigma, K(s)¥sup_{0¥leqq¥tau¥leqq s}V^{n}(¥tau))ds$
Since
conclude that
$¥lim_{n¥rightarrow¥infty}V^{n}(t)=V^{¥infty}(t)$
uniformly on
$[0, ¥delta]$
and
$W(t, s)$
,
.
$V^{n}(t)¥leqq V^{1}(t)$
, we can easily
$V^{¥infty}(t)¥leqq¥int_{0+}^{t}W(s+¥sigma, K(s)¥sup_{0¥leqq¥tau¥leqq s}V^{¥infty}(¥tau))ds$
and hence
$V(t)¥leqq¥int_{0+}^{t}W(s+¥sigma, K(s)V(s))ds$
.
by the property (iv) of $W(t, s)$ .
Thus we obtain $V(t)¥equiv 0$ , that is,
Finally, we shall prove the existence of fixed points of the operator $T$. Since
. Therefore
as
, it follows that
is a nonempty, convex and compact subset of $BC((-¥infty, ¥delta], E)$
and $T$ is
Since
because of the property (viii) in Lemma 2.1.
by Schauder’s fixed point theorem. This
continuous, $T$ has a fixed point in
completes the proof.
$V^{¥infty}(t)¥equiv 0$
$¥alpha(¥mathrm{Y}^{n})=¥sup¥alpha(¥mathrm{Y}^{n}(t))=¥sup V^{n}(t)$
$¥alpha(¥mathrm{Y}^{n})¥rightarrow 0$
$ n¥rightarrow¥infty$
$¥mathrm{Y}^{¥infty}=¥bigcap_{l}^{¥infty},=1¥overline{¥mathrm{Y}}^{n}$
$T¥mathrm{Y}^{¥infty}¥subset ¥mathrm{Y}^{¥infty}$
$¥mathrm{Y}^{¥infty}$
Corollary 3.1.
Assume that
$W(t, K(t-¥sigma)v(t))¥rightarrow 0$
as
$t¥rightarrow¥sigma+$
, where
$v(t)$
is
26
Jong Son SHIN
any continuous function satisfying the condition (3.1). Then the condition (iv)
in Theorem 3.1 can be replaced by the following condition;
¥
is the only absolutely continuous function satisfying the con(v)
dition (3. 1) and the scalar differential equation
$z(t) equiv 0$
(3.3)
$¥frac{dz}{dt}=W(t, K(t-¥sigma)z(t))$
Proof.
In view of (3.2), define
$u(t)$
,
$¥mathrm{a}.¥mathrm{e}$
.
$ t¥in$
$u(t)$
.
as
$u(t)=¥int_{¥sigma+}^{t}W(s, K(s-¥sigma)v(s))ds$
Since
$¥sigma+a]$
$(¥sigma,$
.
is the absolutely continuous function, we have
$¥frac{du}{dt}=W(t, K(t-¥sigma)v(t))$
Clearly, it follows from (3.2) that
differential inequality
,
$v(t)¥leqq u(t)$
$¥mathrm{a}.¥mathrm{e}$
on
.
$ t¥in$
$(¥sigma,$
$¥sigma+a]$
$[¥sigma, ¥sigma+a]$
$¥frac{du}{dt}¥leqq W(t, K(t-¥sigma)u(t))$
.
.
Thus we obtain the
,
because $W(t, s)$ is nondecreasing in . Furthermore, by the assumption, for
any
we can choose a
so that $ 0¥leqq W(t, K(t-¥sigma)v(t))<¥epsilon$ if ¥ ¥ ¥ .
Thus, if
, then
$s$
$¥epsilon>0$
$¥delta(¥epsilon)>0$
$t$
$ sigma<t< sigma+ delta$
$¥in(¥sigma, ¥sigma+¥delta)$
,
$ 0¥leqq¥frac{u(t)}{t-¥sigma}=¥frac{1}{t-¥sigma}¥int_{¥sigma+}^{t}W(s, K(s-¥sigma)v(s))ds<¥frac{1}{t-¥sigma}¥int_{¥sigma}^{t}¥epsilon ds=¥epsilon$
¥
¥
which yields that ¥ ¥
. Therefore, by the hypothesis (v) and
¥
the comparison theorem (see [5, Lemma 2.3]) we obtain
, and hence
¥
. This completes the proof.
$ lim_{t rightarrow sigma+}u(t)/(t- sigma)=0$
$u(t) equiv 0$
$v(t) equiv 0$
Example 3.1.
in Corollary 3.1.
Let
$¥sigma=¥dot{0}$
.
Then the function
$W(t, s)=s/K(t)t$
is admissible
Remark 3.1. Let $E=R^{n}$ . Then it is clear from the proof of Theorem 3.1
that we may take the continuous function in place of the uniformly continuous
functions as /. Moreover, from the boundedness
it follows that $¥alpha(f(t, B))¥equiv 0$
$B
¥
subset
K(
¥
phi,
r)$
for every bounded set
. Thus the function $W(t, s)¥equiv 0$ is admissible
in Theorem 3.1. This implies that the condition 3) is quite a trivial condition
for equations on
.
$¥mathrm{o}¥mathrm{f}/$
$R^{n}$
4. An applictaion
In this section, as an application of Theorem 3.1, we shall show the existence
FDE with
27
in a Banach space
Infinite Delay
of solutions for some integro-differential equation.
$¥mathrm{s}¥mathrm{i}¥mathrm{s}¥mathrm{t}¥mathrm{i}¥mathrm{n}¥mathrm{L}¥mathrm{e}¥mathrm{t}¥mathrm{g}¥mathrm{o}¥mathrm{f}$
$1]¥mathrm{c}¥mathrm{o}¥mathrm{n}¥mathrm{t}¥mathrm{i}¥mathrm{n}¥mathrm{u}¥mathrm{o}¥mathrm{u}¥mathrm{s}¥mathrm{f}¥mathrm{u}¥mathrm{n}¥mathrm{c}¥mathrm{t}¥mathrm{i}¥mathrm{o}¥mathrm{n}¥mathrm{s}¥mathrm{m}¥mathrm{a}¥mathrm{p}¥mathrm{p}¥mathrm{i}¥mathrm{n}¥mathrm{g}¥overline{I}¥mathrm{i}¥mathrm{n}¥mathrm{t}¥mathrm{o}R¥mathrm{w}¥mathrm{i}¥mathrm{t}¥mathrm{h}¥overline{I}=[0,¥mathrm{l}]¥mathrm{a}¥mathrm{n}¥mathrm{d}¥mathrm{d}¥mathrm{e}¥mathrm{n}¥mathrm{o}¥mathrm{t}¥mathrm{e}¥mathrm{b}¥mathrm{y}C=C(¥overline{I},R)¥mathrm{a}$
$I¥mathrm{a}11¥mathrm{t}¥mathrm{h}¥mathrm{e}=(0,$
$¥mathrm{B}¥mathrm{t}¥mathrm{h}¥mathrm{e}¥sup- ¥mathrm{n}¥mathrm{o}¥mathrm{r}¥mathrm{m}|¥cdot|_{C}¥mathrm{a}¥mathrm{n}¥mathrm{a}¥mathrm{c}¥mathrm{h}¥mathrm{s}¥mathrm{p}¥mathrm{a}¥mathrm{c}¥mathrm{e}¥mathrm{c}¥mathrm{o}¥mathrm{n}-$
.
As a phase space
$¥ovalbox{¥tt¥small REJECT}=¥{¥psi:$
(
$-¥infty$
we take the following space;
$¥ovalbox{¥tt¥small REJECT}$
,
$¥mathrm{O}]¥rightarrow C$
: measurable on (
$-¥infty$
,
$-r]$ ,
continuous on
$[-r, 0]$
where
$ 0¥leqq r<¥infty$
and
$|¥psi|_{¥ovalbox{¥tt¥small REJECT}}<¥infty$
},
and
.
$|¥psi|_{¥ovalbox{¥tt¥small REJECT}}=¥sup_{-r¥leqq¥theta¥leqq 0}|¥psi(¥theta)|_{C}+¥int_{-¥infty}^{0}e^{¥theta}|¥psi(¥theta)|_{C}d¥theta$
Clearly, the phase space
$(¥mathrm{B}_{2})$
$¥ovalbox{¥tt¥small REJECT}$
satisfies the axioms
$(¥mathrm{B}_{1})-(¥mathrm{B}_{4})$
and
$K(t)=2-e^{-t}$
in
(see [3]).
We shall consider the existence of solutions for the integro-differential
equation;
(4. 1)
$¥frac{¥partial U(t,x)}{¥partial t}=¥frac{1}{2}¥int_{0}^{1}dy¥int_{-¥infty}^{0}$
tA(θ,x)
$¥sin¥frac{U(t+¥theta,y)}{t^{2}}d¥theta$
,
$+¥frac{1}{2}B(t, x)U(t-r, ¥mathrm{x})+¥int_{0}^{1}G(t, ¥mathrm{x}, y)H(t, U(t-r, .v))dy$
$0¥leqq t¥leqq 1$
,
$0¥leqq ¥mathrm{x}¥leqq 1$
,
under the initial condition
(4.2)
for
$U(t, x)=¥phi^{0}(t, x)$
$(t, x)¥in(-¥infty,$
0]
$¥times¥overline{I}$
and
,
$¥phi^{0}¥in¥ovalbox{¥tt¥small REJECT}$
.
To see the existence of solutions of Eq. (4. 1) under the initial condition (4.2),
we make the following hypotheses;
is the continuous function such that A
A: (
,
.
on $(-¥infty,$
is the continuous function such that $|B(¥theta, x)|¥leqq 1$ on .
are the continuous function.
and
$(¥mathrm{H}_{1})$
$-¥infty$
$¥mathrm{O}]¥times¥overline{I}¥rightarrow R$
$|$
$(¥theta, x)|¥leqq$
$0]¥times¥overline{I}$
$e^{¥theta}$
$(¥mathrm{H}_{2})$
$(¥mathrm{H}_{3})$
$B:¥overline{I}^{2}¥rightarrow R$
$G:¥overline{I}^{3}¥rightarrow R$
Theorem 4.1.
exists a solution,
condition (4.2).
Proof.
For
$¥overline{I}^{2}$
$H:¥overline{I}¥times R¥rightarrow R$
Let all the hypotheses
defined on
$(t, ¥phi)¥in¥overline{I}¥times¥ovalbox{¥tt¥small REJECT}$
$f_{1}(t, ¥phi)(x)=¥frac{1}{2}¥int_{0}^{1}dy¥int_{-¥infty}^{0}$
where
$[0, d]$
$g(t, s)=t¥sin(s/t^{2})$
for
$(¥mathrm{H}_{1})-(¥mathrm{H}_{3})$
for some
$d>0$ ,
of
be
$Eq$
.
satisfied.
Then there
under
the initial
(4.1)
, set
A $(¥theta, x)g(t,¥phi(¥theta, y))d¥theta+¥frac{1}{2}B(t.
$(t, s)¥in¥overline{I}¥times R$
,
x)¥phi(-r, x)$
,
Jong Son SHIN
28
$f_{2}(t, ¥phi)(x)=¥int_{0}^{1}G(t, x, y)H(t, ¥phi(-r, y))dy$
and
$f(t, ¥phi)(x)=f_{1}(t, ¥phi)(x)+f_{2}(t, ¥phi)(x)$
.
, is bounded and
,
Then it is easy to show that
uniformly continuous.
¥
and
for ¥
Next, we shall show the inequality
,
, we can obtain, for
and
. By the hypotheses
$K(¥phi^{0}, a)$ ,
$f:¥overline{I}¥times K(¥phi^{0}, a)¥rightarrow C$
$K(¥phi^{0}, a)¥subset¥ovalbox{¥tt¥small REJECT}$
$¥alpha(f(t, B))¥leqq¥alpha(¥hat{B})/2t$
$t¥in I$
$B subset K( phi^{0}, a)$
$(t, ¥phi)$
$(¥mathrm{H}_{2})$
$(¥mathrm{H}_{1})$
$(t, ¥psi)¥in I¥times$
$|f_{1}(t, ¥phi)(x)-f_{1}(t, ¥psi)(x)|$
$¥leqq¥frac{1}{2}¥{¥int_{0}^{1}dy¥int_{-¥infty}^{0}e^{¥theta}|g(t, ¥phi(¥theta, y))-g(t, ¥Psi(¥theta, y))|d¥theta+|¥phi(-r, x)-¥psi(-r, x)|¥}$
$¥leqq¥frac{1}{2t}¥{¥int_{0}^{1}dy¥int_{-¥infty}^{0}e^{¥theta}|¥phi(¥theta, y)-¥psi(¥theta, y)|d¥theta+|¥phi(-r, x)-¥psi(-r, x)|¥}$
$¥leqq¥frac{1}{2¥mathrm{t}}|¥phi-¥psi|_{¥ovalbox{¥tt¥small REJECT}}$
.
that
Moreover, it follows from the hypothesis
$B
¥
subset
K(
¥
phi^{0},
a)$
for
and
compact operator. Thus we have,
$(¥mathrm{H}_{3})$
$f_{2}$
:
$¥overline{I}¥times K(¥phi^{0}, a)¥rightarrow C$
$t¥in I$
is a
,
$¥alpha(f(t, B))¥leqq¥alpha(f_{1}(t, B))+¥alpha(f_{2}(t, B))$
$¥leqq¥alpha(f_{1}(t, B))$
$¥leqq¥frac{¥alpha(¥hat{B})}{2t}$
Since
$K(t)=2-e^{-t}$
in
$(¥mathrm{B}_{2})$
, we have
$¥frac{1}{2t}(2-e^{-t})v(t)¥rightarrow 0$
as
$t¥rightarrow 0+$
,
where $v(t)$ is any continuous function satisfying the condition (3.1). Thus, by
using the Nagumo’s uniqueness theorem, the condition (v) in Corollary 3.1 is
satisfied. This implies that all the hypotheses of Theorem 3.1 are satisfied, which
completes the proof.
References
[1] Banas, J. and Goebel, K., Mesures of noncompactness in Banach spaces, Marcal Dekker
Inc. New York and Basel, 1980.
[2] Goebel, K. and Rzymowski, W., An existence theorem for the equations $x^{¥prime}=f(t, x)$
FDE with
Infinite Delay
in a Banach space
29
in Banach space, Bull. Acad. Polon. Sci., Ser. Math. Astronom. Phys., 18 (1970), 367-370.
[3] Hale, J. K. and Kato, J., Phase space for retarded equations with infinite delay,
Funkcial Ekvac., 21 (1978), 11-41.
[4] Hu, S. C., Ordinary differential equations involving perturbations in Banach spaces,
Nonlinear Analysis, 7 (1983), 933-940.
[5] Kato, S., Some remarks on nonlinear ordinary differential equations in Banach spaces,
Hokkaido Math. J., 6 (1975), 205-226.
[6] Lakshmikantham, V. and Leela, S., Differential and Integral Inequalities, Vol. 1.
Academic Press, New York, 1969.
[7]?, Nonlinear Differential Equations in Abstract Space, Pergamon Press, 1981.
[8] Naito, T., On linear autonomous retarded equations with an abstract phase space for
infinite delay, J. Differential Equations, 33 (1979), 74-91.
[9] Sugiyama, S., On integral-like operator equations in a Banach space, Mem. School
of Sci. Waseda Univ., 47 (1983), 47-59.
[10] Rzymowski, W., On the existence of solution the equation $x^{¥prime}=f(t, x)$ in a Banach
space, Bull. Acad. Polon. Sci., Ser. Math. Astronom. Phys., 19 (1971), 295-299.
nuna adreso:
Department of Mathematics
Korea University
Ogawa, Kodaira, Tokyo
Japan
(Ricevita la 4-an de februaro, 1985)
(Reviziita la 6-an de septembre, 1985)
(Reviziita la 11-an novembro, 1985)