Existence of global solutions for some classes of integral equations
We study the existence of $L^p$ -solutions for a class of Hammerstein integral equations and neutral functional differential equations involving abstract Volterra operators. Using compactness-type conditions, we establish the global existence of solutions. In addition, a global existence result for...
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| author | Agarwal, P. Jabeen, T. Lupulescu, V. O’Regan, D. Агарвал, Р. П. Ябін, Т. Лупулеску, В. О'Реган, Д. |
| author_facet | Agarwal, P. Jabeen, T. Lupulescu, V. O’Regan, D. Агарвал, Р. П. Ябін, Т. Лупулеску, В. О'Реган, Д. |
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| datestamp_date | 2019-12-05T09:17:34Z |
| description | We study the existence of $L^p$ -solutions for a class of Hammerstein integral equations and neutral functional differential
equations involving abstract Volterra operators. Using compactness-type conditions, we establish the global existence of
solutions. In addition, a global existence result for a class of nonlinear Fredholm functional integral equations involving
abstract Volterra equations is given. |
| first_indexed | 2026-03-24T02:07:48Z |
| format | Article |
| fulltext |
UDC 517.9
T. Jabeen (Abdus Salam School Math. Sci., GC Univ., Lahore, Pakistan),
R. P. Agarwal (Texas A&M University-Kingvsille, Kingsville, USA),
V. Lupulescu (Constantin Brancusi Univ., Targu-Jiu, Romania),
D. O’Regan (School Math., Statistics and Appl. Math., Nat. Univ. Ireland, Galway, Ireland)
EXISTENCE OF GLOBAL SOLUTIONS FOR SOME CLASSES
OF INTEGRAL EQUATIONS
IСНУВАННЯ ГЛОБАЛЬНИХ РОЗВ’ЯЗКIВ ДЕЯКИХ КЛАСIВ
IНТЕГРАЛЬНИХ РIВНЯНЬ
We study the existence of Lp -solutions for a class of Hammerstein integral equations and neutral functional differential
equations involving abstract Volterra operators. Using compactness-type conditions, we establish the global existence of
solutions. In addition, a global existence result for a class of nonlinear Fredholm functional integral equations involving
abstract Volterra equations is given.
Вивчається iснування Lp -розв’язкiв для класу iнтегральних рiвнянь Гаммерштейна та нейтральних функцiональних
диференцiальних рiвнянь з абстрактними операторами Вольтерра. Iснування глобальних розв’язкiв встановлено за
допомогою умов типу компактностi. Крiм того, наведено результат про глобальне iснування розв’язку для класу
нелiнiйних функцiональних iнтегральних рiвнянь Фредгольма з абстрактними операторами Вольтерра.
1. Introduction. Many problems arising in modeling real world phenomena lead to mathematical
models described by nonlinear integral equations in abstract spaces. The theory of nonlinear integral
equations in abstract spaces, is a relatively old theory, but it is also current and has important
applications in physics, engineering and biology. The concept of abstract Volterra operator (or causal
operator), introduced by [47] and [46], plays an important role in physics and engineering [25, 42].
This concept arises naturally in classes of differential equations and integral equations such as ordinary
differential equations, integro-differential equations, differential equations with finite or infinite delay,
Volterra integral equations, neutral functional equations, and so on.
Let E be a real Banach space, Lp([0, a], E) be the space of all (classes of) strongly measurable
and Bochner integrable functions u : [0, a] \rightarrow E, and \scrL (E) the space of all bounded linear operators
from E into itself. In this paper, we consider the Hammerstein integral equation
u(t) = (\frakP u)(t) + \lambda
a\int
0
K(t, s)(Qu)(s)ds, a.e. t \in [0, a], (1.1)
and the Volterra – Hammerstein integral equation
u(t) = (\frakP u)(t) +
t\int
0
K(t, s)(Qu)(s)ds, a.e. t \in [0, a], (1.2)
where \frakP ,Q : Lp([0, a], E) \rightarrow Lp([0, a], E) are continuous abstract Volterra operators, K : [0, a] \times
\times [0, a] \rightarrow \scrL (E) is strongly measurable, \lambda \in \BbbR , and we provide conditions under which these
equations have solutions in Lp([0, a], E). In addition, under suitable conditions we establish the
existence of continuous solutions for the following nonlinear Fredholm functional-integral equation:
c\bigcirc T. JABEEN, R. P. AGARWAL, V. LUPULESCU, D. O’REGAN, 2018
130 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
EXISTENCE OF GLOBAL SOLUTIONS FOR SOME CLASSES OF INTEGRAL EQUATIONS 131
x(t) = x0(t) +
a\int
0
F (t, s, (Qx)(s)) ds, t \in [0, a],
where F (\cdot , \cdot , \cdot ) : [0, a]\times [0, a]\times Y \rightarrow X is a Carathéodory function, Q : C([0, a], X) \rightarrow L\infty ([0, a], Y )
is a continuous causal operator, x0(\cdot ) \in C([0, a], X), and X,Y are infinite dimensional spaces.
We recall that an operator Q : Lp([0, a], E) \rightarrow Lp([0, a], E) is called an abstract Volterra ope-
rators (or a causal operator) if, for each \tau \in [0, a) and for all u, v \in Lp([0, a], E) with u(t) = v(t)
for every t \in [0, \tau ], we have Qu(t) = Qv(t) for a.e. t \in [0, \tau ].
The study of differential equations involving abstract Volterra operators can be found in the
monographs [10, 19, 32, 40], and also in the papers [1, 2, 4, 11, 12, 14, 24, 34, 35, 37, 38, 41,
48, 50, 51]. The existence of Lp-solutions for different classes of differential equations and integral
equations were studied in [3, 6 – 9, 16, 26, 30, 31, 33, 36, 39, 43].
2. Preliminaries. Let E be a real Banach space endowed with the norm \| \cdot \| . If A is a nonempty
subset in E, then A, \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v} (A) and \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(A) denote the closure of A, the convex hull of A and
the closure of the convex hull of A, respectively. We denote by C([0, a], E) the Banach space of
continuous bounded functions from [0, a] into E endowed with the norm \| u(\cdot )\| = \mathrm{s}\mathrm{u}\mathrm{p}0\leq t\leq a \| u (t) \| .
The space of all (classes of) strongly measurable functions u : [0, a] \rightarrow E such that
\| u\| p :=
\left( a\int
0
\| u(t)\| p
\right) 1/p
<\infty
for 1 \leq p <\infty , will be denoted by Lp([0, a], E). Then Lp([0, a], E) is a Banach space with respect
to the norm \| u\| p. Also, we denote by L\infty ([0, a], E) the space of all (classes of) strongly measurable
functions u(\cdot ) : [0, a] \rightarrow E which are essentially bounded on [0, a]. Then L\infty ([0, a], E) is a Banach
space with respect to the norm
\| u\| \infty := \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
t\in [0,a]
\| u(t)\| = \mathrm{i}\mathrm{n}\mathrm{f}\{ M \geq 0; \| u(t)\| \leq M for a.e. t \in [0, a]\} .
We recall that, if 1 \leq p < q \leq \infty , then
Lq([0, a], E) \subset Lp([0, a], E)
and
\| u\| p \leq a1/p - 1/q\| u\| q for every u(\cdot ) \in Lq([0, a], E).
In the following, for a given p \geq 1, we shall denote by p\prime \geq 1 its conjugate; that is,
1
p
+
1
p\prime
= 1. We
denote the space of all bounded linear operators acting on a Banach space E by \scrL (E). Then \scrL (E)
is a Banach space with respect to the norm
\| T\| := \mathrm{i}\mathrm{n}\mathrm{f}\{ M \geq 0; \| Tu\| \leq M\| u\| for all u \in E\} , T \in \scrL (E).
We denote by \beta (A) the Hausdorff measure of non-compactness of a nonempty bounded set A \subset E,
and it is defined by [27]:
\beta (A) = \mathrm{i}\mathrm{n}\mathrm{f}\{ \varepsilon > 0; A admits a finite cover by balls of radius \leq \varepsilon \} .
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
132 T. JABEEN, R. P. AGARWAL, V. LUPULESCU, D. O’REGAN
The Kuratowski measure of non-compactness of a nonempty bounded set A \subset E is defined
by [29]:
\alpha (A) = \mathrm{i}\mathrm{n}\mathrm{f}
\Bigl\{
\delta > 0; A can be expressed as the union of a finite number of sets
such that the diameter of each set does not exceed \delta
\Bigr\}
,
where the diameter of a bounded set A \subset E is defined by \mathrm{d}\mathrm{i}\mathrm{m}(A) = \mathrm{s}\mathrm{u}\mathrm{p}\{ \| x - y\| ;x, y \in A\} .
Let \gamma (\cdot ) be either \alpha (\cdot ) or \beta (\cdot ). If A,B are bounded subsets of E, then (see [5, 27]):
(1) \gamma (A) = 0 if and only if A is compact;
(2) \gamma (A) = \gamma (A) = \gamma (\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(A));
(3) \gamma (\lambda A) = | \lambda | \gamma (A) for every \lambda \in \BbbR ;
(4) \gamma (A) \leq \gamma (B) if A \subset B;
(5) \gamma (A+B) \leq \gamma (A) + \gamma (B);
(6) if T : E \rightarrow E is a bounded linear operator, then \gamma (TA) \leq \| T\| \gamma (A);
(7) if \{ An\} n\geq 1 is a decreasing sequence of bounded closed nonempty subsets of E and
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\gamma (An) = 0, then
\bigcap \infty
n=1
An is a nonempty and compact subset of E [29].
Remark 2.1. In general, for any bounded set A \subset E, one has \beta (A) \leq \alpha (A) \leq 2\beta (A) and both
inequalities can be strict. Also, for any bounded set A \subset E, we have that \gamma (A) \leq \mathrm{d}\mathrm{i}\mathrm{m}(A) and
\gamma (A) \leq 2d if \mathrm{s}\mathrm{u}\mathrm{p}x\in A \| x\| \leq d.
We recall the following lemma due to Heinz [21].
Lemma 2.1. Let \{ un(\cdot );n \geq 1\} be a sequence in L1([0, a], E) such that there exists m(\cdot ) \in
\in L1([0, a],\BbbR +) with \| un(t)\| \leq m(t) for each n \geq 1 and for a.e. t \in [0, a]. Then the function
t \mapsto \rightarrow \psi (t) := \gamma (\{ un(t);n \geq 1\} ) is integrable on [0, a] and, for each t \in [0, a], we have
(a) (Heinz [21])
\alpha
\left( \left\{
t\int
0
un(s)ds;n \geq 1
\right\}
\right) \leq 2
t\int
0
\psi (s)ds,
(b) (Kisielewicz [28], Lemma 2.2)
\beta
\left( \left\{
t\int
0
un(s)ds;n \geq 1
\right\}
\right) \leq
t\int
0
\psi (s)ds,
provided that E is a separable banach space.
In the following, we let \alpha p(\cdot ) denote the Kuratowski measures of noncompactness of sets in the
space Lp([0, a], E).
Lemma 2.2. Let 1 \leq p <\infty and let V \subset Lp([0, a], E) be a countable set such that there exists
m(\cdot ) \in L1([0, b],\BbbR +) with \| u(t)\| \leq m(t) for each u (\cdot ) \in A and for a.e. t \in [0, a].
(a) [43, 44] If
\mathrm{l}\mathrm{i}\mathrm{m}
h\rightarrow 0
\mathrm{s}\mathrm{u}\mathrm{p}
u\in A
a\int
0
\bigm\| \bigm\| u(t+ h) - u(t)
\bigm\| \bigm\| p dt = 0, (2.1)
then
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
EXISTENCE OF GLOBAL SOLUTIONS FOR SOME CLASSES OF INTEGRAL EQUATIONS 133
\alpha p(A) \leq 2
\left( a\int
0
[\alpha (V (t))]p dt
\right) 1/p
.
(b) [20] (Theorem 1.2.8) The set V is relatively compact in Lp([0, a], E) if and only if (2.1) is
satisfied and V (t) is relatively compact in E for a.e. t \in [0, a].
3. A global existence results for Hammerstein integral equations. Let p and q be real
numbers such that q > p \geq 1 and p
\biggl(
1 - 1
q
\biggr)
> 1. We also assume that
(H1) \frakP ,Q : Lp([0, a], E) \rightarrow Lp([0, a], E) are continuous operators such that there exist b(\cdot ), c(\cdot ) \in
\in Lp([0, a],\BbbR +) and d > 0 with
\| (\frakP u)(t)\| \leq b(t) and \| (Qu)(t)\| \leq c(t) + d\| u(t)\| for a.e. t \in [0, a]
and for every u(\cdot ) \in Lp([0, a], E);
(H2) K is a strongly measurable function from [0, a]\times [0, a] into \scrL (E) and
\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,a]
\left( a\int
0
\| K(t, s)\| qdt
\right) 1/q
:=M <\infty .
Lemma 3.1. If (H2) holds, then
\mathrm{l}\mathrm{i}\mathrm{m}
h\rightarrow 0
a\int
0
\left( a\int
0
\| K(t+ h, s) - K(t, s)\| qdt
\right) 1/q
ds = 0. (3.1)
Proof. For a.e. s \in [0, a], let us define the function \psi s(\cdot ) : [0, a] \rightarrow \scrL (E) by \psi s(t) = K(t, s),
t \in [0, a]. From (H2) it follows that \| \psi s(\cdot )\| q \in L\infty ([0, a],\BbbR +) and \| \psi s(\cdot )\| q \leq M < \infty for a.e.
s \in [0, a], so that \psi s(\cdot ) \in Lq([0, a],\scrL (E)) for a.e. s \in [0, a]. Let \{ hn\} n\geq 1 be a sequence of real
positive numbers such that hn \rightarrow 0 as n \rightarrow \infty , and t+ hn \in [0, a) for every t \in [0, a) and n \geq 1.
Also, for a.e. s \in [0, a], let
\theta n(s) :=
\left( a\int
0
\| \psi s(t+ hn) - \psi s(t)\| qdt
\right) 1/q
=
=
\left( a\int
0
\| K(t+ hn, s) - K(t, s)\| qdt
\right) 1/q
, n \geq 1.
Since \psi s(\cdot ) \in Lq([0, a],\scrL (E)) for a.e. s \in [0, a], then from the fact that translations of Lp functions
(1 \leq p <\infty ) are continuous in norm, we see that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\left( a\int
0
\| \psi s(t+ hn) - \psi s(t)\| qdt
\right) 1/q
= 0 for a.e. s \in [0, a],
so that \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \theta n(s) = 0 for a.e. s \in [0, a]. On the other hand, since (H2) implies
0 \leq \theta n(s) \leq \| \theta n\| \infty \leq 2M for a.e. s \in [0, a] and all n \geq 1,
then, by the Dominated Convergence Theorem, we have \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty
\int a
0
\theta n(s) ds = 0, so (3.1) is proved.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
134 T. JABEEN, R. P. AGARWAL, V. LUPULESCU, D. O’REGAN
Lemma 3.2. If (H2) holds, then the function \xi (\cdot ) : [0, a] \rightarrow \BbbR +, defined by
\xi (t) =
\left( a\int
0
\| K(t, s)\| q\prime ds
\right) 1/q\prime
for a.e. t \in [0, a], (3.2)
belongs to Lq([0, a],\BbbR +). Moreover, \| \xi \| q \leq Ma1/q
\prime
and \| \xi \| p \leq Ma1/p - 1/q+1/q\prime .
Proof. From (H2) and Tonelli’s theorem it is easy to see that the function \xi (\cdot ) : [0, a] \rightarrow \BbbR +
is measurable on [0, a]. Now, from q > p and p
\biggl(
1 - 1
q
\biggr)
> 1 it follows that q\prime < p < q; that is
q/q\prime > 1. Then, from (H2) and the integral version of Minkowski’s inequality, we have
a\int
0
\xi q(t)dt =
a\int
0
\left( a\int
0
\| K(t, s)\| q\prime ds
\right) q/q\prime
dt \leq
\leq
\left[ a\int
0
\left( a\int
0
\| K(t, s)\| qdt
\right) q\prime /q
ds
\right]
q/q\prime
\leq M qaq/q
\prime
,
so that \xi (\cdot ) \in Lq([0, a],\BbbR +) and \| \xi \| q \leq Ma1/q
\prime
. Since p < q, \| \xi \| p \leq a1/p - 1/q\| \xi \| q \leq
\leq Ma1/p - 1/q+1/q\prime .
Theorem 3.1. Let conditions (H1), (H2) be satisfied. Suppose that there exist k1 \in [0, 1) and
k2 > 0 such that
\alpha ((\frakP A)(t)) \leq k1\alpha (A(t)) and \alpha ((QA)(t)) \leq k2\alpha (A(t)) (3.3)
for t \in [0, a] and for each bounded subset A \subset Lp([0, a], E).
Then there exists a positive number \lambda 0 such that for every \lambda \in R with | \lambda | < \lambda 0, the integral
equation (1.1) has at least one solution in Lp([0, a], E).
Proof. First, we show that each solution of (1.1) is a priori bounded in Lp([0, a], E). Indeed,
since
\| u(t)\| \leq b(t) + | \lambda |
a\int
0
\| K(t, s)\| \| (Qu)(s)\| ds, t \in [0, a],
then, using the Minkowski’s inequality and the integral version of Minkowski inequality, we obtain
\| u\| p \leq
\left( a\int
0
| b(t)| pdt
\right) 1/p
+ | \lambda |
\left[ a\int
0
\left( a\int
0
\| K(t, s)\| \| (Qu)(s)\| ds
\right) p
dt
\right] 1/p
\leq
\leq \| b\| p + | \lambda |
a\int
0
\left[ a\int
0
[\| K(t, s)\| \| (Qu)(s)\| ]p dt
\right] 1/p
ds \leq
\leq \| b\| p + | \lambda |
a\int
0
\| (Qu)(s)\|
\left( a\int
0
\| K(t, s)\| pdt
\right) 1/p
ds.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
EXISTENCE OF GLOBAL SOLUTIONS FOR SOME CLASSES OF INTEGRAL EQUATIONS 135
Since q > p then, using (H1), (H2) and Hölder’s inequality, we get
a\int
0
\| (Qu)(s)\|
\left( a\int
0
\| K(t, s)\| pdt
\right) 1/p
ds \leq
\leq a1/p - 1/q
a\int
0
\| (Qu)(s)\|
\left( a\int
0
\| K(t, s)\| qdt
\right) 1/q
ds \leq
\leq Ma1/p - 1/q
a\int
0
\| (Qu)(s)\| ds \leq Ma1/p - 1/qa1/p
\prime
\left( a\int
0
\| (Qu)(s)\| pds
\right) 1/p
\leq
\leq Ma1/q
\prime
(\| c\| p + d\| u\| p) ,
so that
\| u(\cdot )\| p \leq \| b(\cdot )\| p + | \lambda | Ma1/q
\prime
(\| c(\cdot )\| p + d\| u(\cdot )\| p) .
Put
\lambda 0 := \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
1
dMa1/q\prime
,
1 - k1
2k2a1/p - 1/q\| \xi \| p
\biggr\}
,
where the function \xi (\cdot ) is defined in (3.2). Then for each | \lambda | < \lambda 0, we have \| u\| p \leq r, where
r := \gamma (1 - \rho ) - 1, \rho := | \lambda | dMa1/q
\prime
< 1 and \gamma := \| b\| p + | \lambda | Ma1/q
\prime \| c\| p, so that u bounded in
Lp([0, a], E). Moreover, we remark that \| Qu\| p \leq \| c\| p + dr if \| u\| p \leq r. We also notice that
\| u(t)\| \leq b(t) + | \lambda | a1/p - 1/q(\| c\| p + dr)\xi (t) for a.e. t \in [0, a];
that is, for every u \in B, we have
\| u(t)\| \leq \varphi (t) for a.e. t \in [0, a], (3.4)
where \varphi (t) = b(t) + | \lambda | a1/p - 1/q(\| c\| p + dr)\xi (t), t \in [0, a] and B := \{ u(\cdot ) \in Lp([0, a], E); \| u\| p \leq
\leq r\} . Moreover, from Lemma 3.2 it follows that \varphi (\cdot ) \in Lp([0, a],\BbbR +), and
\| \varphi \| p \leq \| b\| p + | \lambda | Ma2(1/p - 1/q)+1/q\prime (\| c\| p + dr). (3.5)
Now, define the operator T : Lp([0, a], E) \rightarrow Lp([0, a], E) by
(Tu) (t) = (\frakP u)(t) + \lambda
a\int
0
K(t, s)(Qu)(s)ds, t \in [0, a]. (3.6)
As above, we can show that
\| (Tu)(t)\| \leq \varphi (t) for a.e. t \in [0, a],
and
\| Tu\| p \leq \| b\| p + | \lambda | Ma1/q
\prime
(\| c\| p + d\| u\| p) ,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
136 T. JABEEN, R. P. AGARWAL, V. LUPULESCU, D. O’REGAN
for every u(\cdot ) \in Lp([0, a], E), so that T is well defined. Moreover, it is easy to see that T(B) \subset B;
that is, T is an operator from B into itself. Next, we show that T is a continuous operator. For this,
let \{ un(\cdot )\} n\geq 1 be a convergent sequence in Lp([0, a], E) such that un(\cdot ) \rightarrow u(\cdot ) as n\rightarrow \infty . Since
\| (Tun) (t) - (Tu) (t)\| \leq \| (\frakP un) (t) - (\frakP u) (t)\| +
+| \lambda |
a\int
0
\| K(t, s)\| \| (Qun)(s) - (Qu)(s)\| ds
for every t \in [0, a], then using Minkowski’s inequality we have
\| Tun - Tu\| p \leq
\left( a\int
0
\| (\frakP un) (t) - (\frakP u) (t)\| pdt
\right) 1/p
+
+| \lambda |
\left[ a\int
0
\left( a\int
0
\| K(t, s)\| \| (Qun)(s) - (Qu)(s)\| ds
\right) p
dt
\right] 1/p
. (3.7)
Now, using (H2) and the integral version of Minkowski inequality, we obtain\left[ a\int
0
\left( a\int
0
\| K(t, s)\| \| (Qun)(s) - (Qu)(s)\| ds
\right) p
dt
\right] 1/p
\leq
\leq
a\int
0
\left[ a\int
0
[\| K(t, s)\| \| (Qun)(s) - (Qu)(s)\| ]p dt
\right] 1/p
ds \leq
\leq
a\int
0
\| (Qun)(s) - (Qu)(s)\|
\left( a\int
0
\| K(t, s)\| pdt
\right) 1/p
ds \leq
\leq a1/p - 1/q
a\int
0
\| (Qun)(s) - (Qu)(s)\|
\left( a\int
0
\| K(t, s)\| qdt
\right) 1/q
ds \leq
\leq Ma1/p - 1/qa1/p
\prime
\left( a\int
0
\| (Qun)(s) - (Qu)(s)\| pds
\right) 1/p
=
=Ma1/q
\prime \| Qun - Qu\| p,
so that (3.7) become
\| Tun - Tu\| p \leq \| \frakP un - \frakP u\| p +M | \lambda | a1/q\prime \| Qun - Qu\| p.
Since \frakP and Q are continuous operators, from the above inequality it follows that \| Tun - Tu\| p \rightarrow 0
as n\rightarrow \infty , and so T is a continuous operator. In the next step, we will show that
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EXISTENCE OF GLOBAL SOLUTIONS FOR SOME CLASSES OF INTEGRAL EQUATIONS 137
\mathrm{l}\mathrm{i}\mathrm{m}
h\rightarrow 0
\mathrm{s}\mathrm{u}\mathrm{p}
u\in B
a\int
0
\bigm\| \bigm\| (Tu)(t+ h) - (Tu)(t)
\bigm\| \bigm\| pdt = 0. (3.8)
If t \in [0, a] and t+ h \in [0, a], then for every u(\cdot ) \in B we have\bigm\| \bigm\| (Tu)(t+ h) - (Tu)(t)
\bigm\| \bigm\| \leq
\bigm\| \bigm\| (Pu)(t+ h) - (Pu)(t)
\bigm\| \bigm\| +
+| \lambda |
a\int
0
\bigm\| \bigm\| K(t+ h, s) - K(t, s)
\bigm\| \bigm\| \| (Qu)(s)\| ds.
Using Minkowski’s inequality, we obtain
J :=
\left( a\int
0
\| (Tu)(t+ h) - (Tu)(t)\| pdt
\right) 1/p
\leq
\left( a\int
0
\| (\frakP u)(t+ h) - (\frakP u)(t)\| pdt
\right) 1/p
+
+| \lambda |
\left[ a\int
0
\left( a\int
0
\| K(t+ h, s) - K(t, s)\| \| (Qu)(s)\| ds
\right) p
dt
\right] 1/p
= J1 + J2. (3.9)
Since \frakP u \in Lp([0, a], E), then from the fact that translations of Lp-functions (1 \leq p < \infty ) are
continuous in norm, we see that J1 \rightarrow 0 as h\rightarrow 0.
Next, using the integral version of Minkowski inequality, we get
J2 \leq
a\int
0
\left( a\int
0
[\| K(t+ h, s) - K(t, s)\| \| (Qu)(s)\| ]p dt
\right) 1/p
ds =
=
a\int
0
\| (Qu)(s)\|
\left( a\int
0
\| K(t+ h, s) - K(t, s)\| pdt
\right) 1/p
ds \leq
\leq a1/p - 1/q
a\int
0
\| (Qu)(s)\|
\left( a\int
0
\| K(t+ h, s) - K(t, s)\| qdt
\right) 1/q
ds \leq
\leq a1/p - 1/q
\left( a\int
0
\| (Qu)(s)\| q\prime ds
\right) 1/q\prime \left[ a\int
0
\left( a\int
0
\| K(t+ h, s) - K(t, s)\| qdt
\right) ds
\right] 1/q
\leq
\leq a1/q
\prime - 1/q\| Qu\| p
\left[ a\int
0
\left( a\int
0
\| K(t+ h, s) - K(t, s)\| qdt
\right) ds
\right] 1/q
,
so that
Jq
2 \leq aq/q
\prime - 1 (\| c\| p + dr)q
a\int
0
\left( a\int
0
\| K(t+ h, s) - K(t, s)\| qdt
\right) ds.
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138 T. JABEEN, R. P. AGARWAL, V. LUPULESCU, D. O’REGAN
Then, from Lemma 3.1, we have J2 \rightarrow 0 as h\rightarrow 0. Therefore, from (3.9) it follows that
Jp =
a\int
0
\| (Tu)(t+ h) - (Tu)(t)\| pdt\rightarrow 0 as h\rightarrow 0,
uniformly with respect to u \in B, so that (3.8) is proved. Next, let A be a countable subset of B
such that A \subset \mathrm{c}\mathrm{o}((TA) \cup \{ 0\} ). We will use the compactness criteria from Lemma 2.2 to show that
A is a relatively compact set in Lp([0, a], E). First, from (3.8) we have
\mathrm{l}\mathrm{i}\mathrm{m}
h\rightarrow 0
\mathrm{s}\mathrm{u}\mathrm{p}
u\in A
a\int
0
\| u(t+ h) - u(t)\| p dt = 0. (3.10)
Since A is a bounded set in Lp([0, a], E) then, from (3.10) and Lemma 2.2, we have
\alpha p(A) \leq 2
\left( a\int
0
[\alpha (A(t))]p dt
\right) 1/p
. (3.11)
On the other hand, using the properties of the Kuratowski measures of noncompactness and (3.3), we
have
\alpha (A(t)) \leq \alpha (\mathrm{c}\mathrm{o}((TA) (t) \cup \{ 0\} )) = \alpha ((TA) (t)) \leq
\leq \alpha
\left( (\frakP A)(t) + \lambda
a\int
0
K(t, s)(QA)(s)ds
\right) \leq
\leq \alpha ((\frakP A)(t)) + | \lambda | \alpha
\left( a\int
0
K(t, s)(QA)(s)ds
\right) \leq
\leq k1\alpha (A(t)) + | \lambda | \alpha
\left( a\int
0
K(t, s)(QA)(s)ds
\right) . (3.12)
Next, for each u(\cdot ) \in A, the function s \mapsto \rightarrow \| K(t, s)(Qu)(s)\| is measurable on [0, t] for a.e. t \in [0, a].
From (3.4) it follows that\bigm\| \bigm\| K(t, s)(Qu)(s)
\bigm\| \bigm\| \leq \| K(t, s)\|
\bigl(
c(s) + d\| u(t)\|
\bigr)
\leq \| K(t, s)\| (c(s) + d\varphi (t)) ,
and consequently
a\int
0
\| K(t, s)(Qu)(s)\| ds \leq
\left( a\int
0
\| K(t, s)\| q\prime ds
\right) 1/q\prime \left( a\int
0
(c(s) + d\varphi (t))q ds
\right) 1/q
\leq
\leq a1/p - 1/q
\bigl(
\| c\| p + d\| \varphi \| p
\bigr)
\xi (t),
so that s \mapsto \rightarrow
\bigm\| \bigm\| K(t, s)(Qu)(s)
\bigm\| \bigm\| belong to L1([0, a],\BbbR +) for a.e. t \in [0, a]. Hence, from Lemma 3.1,
Hölder’s inequality and Lemma 3.2, we have
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EXISTENCE OF GLOBAL SOLUTIONS FOR SOME CLASSES OF INTEGRAL EQUATIONS 139
\alpha
\left( a\int
0
K(t, s)(QA)(s)ds
\right) \leq 2
a\int
0
\alpha (K(t, s)(QA)(s)) ds \leq
\leq 2k2
a\int
0
\| K(t, s)\| \alpha ((A)(s)) ds \leq
\leq 2k2
\left( a\int
0
\| K(t, s)\| q\prime ds
\right) 1/q\prime \left( a\int
0
[\alpha ((A)(s))]q ds
\right) 1/q
\leq
\leq 2k2a
1/p - 1/q
\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
\xi (t), (3.13)
so that, from (3.12) and Lemma 2.2, we obtain\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
\leq k1
\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
+
+2k2| \lambda | a1/p - 1/q\| \xi \| p
\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
\leq
\leq (k1 + 2| \lambda | k2a1/p - 1/q\| \xi \| p)
\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
. (3.14)
Since k1 + 2| \lambda | k2a1/p - 1/q\| \xi \| p < 1, from the last inequality we obtain\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
= 0
and thus, from (3.11) it follows that \alpha p(A) = 0; that is, A is a relatively compact set in Lp([0, a], E).
Summarizing, we have shown that T : B \rightarrow B is a continuous operator with the property that for a
countable subset A of B such that A \subset \mathrm{c}\mathrm{o}
\bigl(
(TA)\cup \{ 0\}
\bigr)
we have that A is relatively compact. Since
B is a closed and convex set in Lp([0, a], E) then, by the Mönch fixed point theorem, it follows that
there exists u(\cdot ) \in B such that u = Tu; that is, the integral equation (1.1) has a least one solution
u(\cdot ) \in B.
Theorem 3.1 is proved.
Remark 3.1. Suppose that \lambda = 1 and the conditions (H1), (H2) are satisfied. If (3.3) holds for
some k1, k2 \geq 0 with k1 +2k2a
1/p - 1/q\| \xi \| p < 1, then from the above proof it is easy to see that the
integral equation (1.1) has at least one solution in Lp([0, a], E).
Theorem 3.2. Let conditions (H1), (H2) be satisfied and suppose that (3.3) holds for some
k1, k2 \geq 0 with k1+2k2a
1/p - 1/q\| \xi \| p < 1. Then the integral equation (1.2) has at least one solution
in Lp([0, a], E).
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140 T. JABEEN, R. P. AGARWAL, V. LUPULESCU, D. O’REGAN
Proof. If we put
K\ast (t, s) :=
\left\{ K(t, s) if 0 \leq s \leq t \leq a,
0 otherwise,
and \lambda = 1, then the integral equation (1.2) is equivalent to
u(t) = (\frakP u)(t) +
a\int
0
K\ast (t, s)(Qu)(s)ds a.e. t \in [0, a]. (3.15)
Since K satisfies (H2), it follows that K\ast is a strongly measurable function from [0, a]\times [0, a] into
\scrL (E),
\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
s\in [0,a]
\left( a\int
0
\| K\ast (t, s)\| qdt
\right) 1/q
:=M <\infty ,
and
\mathrm{l}\mathrm{i}\mathrm{m}
h\rightarrow 0
a\int
0
\left( a\int
0
\| K\ast (t+ h, s) - K\ast (t, s)\| qdt
\right) 1/q
ds = 0.
Also, it is easy to check that the function \xi \ast : [0, a] \rightarrow \BbbR +, defined by
\xi \ast (t) =
\left( a\int
0
\| K\ast (t, s)\| q\prime ds
\right) 1/q\prime
for a.e. t \in [0, a],
belongs to Lq([0, a],\BbbR +), \| \xi \ast \| q \leq Ma1/q
\prime
and \| \xi \ast \| p \leq Ma1/p - 1/q+1/q\prime . Then, by Remark 3.1, it
follows that the integral equation (3.15) has at least one solution in Lp([0, a], E), so that the integral
equation (1.2) has at least one solution in Lp([0, a], E).
Theorem 3.2 is proved.
Remark 3.2. Suppose that there exist m0 > 0, k2 > 0 such that
\alpha ((\frakP A)(t)) \leq m0
\left( t\int
0
[\alpha (A(s))]p ds
\right) 1/p
and \alpha ((QA)(t)) \leq k2\alpha (A(t)) (3.16)
for t \in [0, a] and for each bounded subset A \subset Lp([0, a], E). We notice that if there exists m1 > 0
such that
\alpha ((\frakP A)(t)) \leq m1
t\int
0
\alpha (A(s))ds, t \in [0, a],
then
\alpha ((\frakP A)(t)) \leq m1a
1/p\prime
\left( t\int
0
[\alpha (A(s))]p ds
\right) 1/p
,
so that \frakP satisfies (3.16) with m0 := m1a
1/p\prime . Now, let A be a countable subset of B such that
A \subset \mathrm{c}\mathrm{o}((TA) \cup \{ 0\} ), where T is defined by (3.6) and B := \{ u \in Lp([0, a], E); \| u\| p \leq r\} . Then
(3.12) becomes
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EXISTENCE OF GLOBAL SOLUTIONS FOR SOME CLASSES OF INTEGRAL EQUATIONS 141
\alpha (A(t)) \leq m0
\left( t\int
0
[\alpha (A(s))]p ds
\right) 1/p
+ | \lambda | \alpha
\left( a\int
0
K(t, s)(QA)(s)ds
\right) (3.17)
for all t \in [0, a]. Then, by (3.13), (3.14) and (3.17), we obtain\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
\leq m0
\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
+
+2k2| \lambda | a1/p - 1/q\| \xi \| p
\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
\leq
\leq (m0 + 2| \lambda | k2a1/p - 1/q\| \xi \| p)
\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
.
If m0 + 2| \lambda | k2a1/p - 1/q\| \xi \| p < 1, then the last inequality implies\left( a\int
0
[\alpha ((A)(s))]p ds
\right) 1/p
= 0.
Therefore, under conditions (H1), (H2) the result of Theorem 3.1 remains true if (3.16) holds for some
m0 > 0. Consequently, the result of Theorem 3.2 remains also true if (3.16) holds for some m0 > 0
with m0 + 2k2a
1/p - 1/q\| \xi \| p < 1.
4. Neutral functional differential equation. The aim of this section is to apply Theorem 3.2
to a class of neutral functional differential equations involving abstract Volterra equations. Some
interesting results about neutral differential equations can be found in [17, 18, 22, 23]. In the
following, we consider the neutral functional differential equation
d
dt
[u(t) - (\frakC u)(t)] = (Qu)(t) for a.e. t \in [0, a], (4.1)
together the initial conditions u(0) = u0, where \frakC ,Q : Lp([0, a], E) \rightarrow Lp([0, a], E) are continuous
causal operators such that (\frakC u)(0) = \theta for every u(\cdot ) \in Lp([0, a], E).
A function u(\cdot ) \in Lp([0, a], E) is said to be a solution of (4.1) with initial condition u(0) = u0
if t \mapsto \rightarrow u(t) - (\frakC u)(t) is an absolutely continuous function and satisfies (4.1) for a.e. t \in [0, a]. Note
that u(\cdot ) itself may not be differentiable on the interval of existence. It is easy to see that if u(\cdot ) is a
solution of equation (4.1), then it satisfies the integral equation
u(t) = (\frakP u)(t) +
t\int
0
(Qu)(s) ds for a.e. t \in [0, a], (4.2)
where (\frakP u)(t) := u0 + (\frakC u)(t), t \in [0, a]. Conversely, if u(\cdot ) \in Lp([0, a], E) satisfies the integral
equation (4.2), then u(\cdot ) is a solution of equation (4.1) with initial value u(0) = u0. Let condition (H1)
be satisfied and suppose that there exists k1 \in [0, 1) and k2 > 0 such that \frakC and Q satisfy (3.3).
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142 T. JABEEN, R. P. AGARWAL, V. LUPULESCU, D. O’REGAN
Taking K(t, s) = I for all (t, s) \in \Delta := \{ (t, s); 0 \leq s \leq t \leq a\} , by Theorem 3.2 it follows that
the integral equation (4.2) has at least one solution in Lp([0, a], E). A similar result was obtained by
Corduneanu [10] (Section 6.4) in the finite dimensional case. For instance, the above result can be
applied to the neutral functional differential equation
d
dt
\left[ u(t) - t\int
0
K(t, s)u(s)ds
\right] = g(t, u(t)) for a.e. t \in [0, a], (4.3)
with the initial conditions u(0) = u0, where K : \Delta \rightarrow \scrL (E) satisfies (H2) and g : [0, a]\times E \rightarrow E is
a Carathéodory function; that is,
(a) g(t, \cdot ) \in C(E,E) for each t \in [0, a];
(b) g(\cdot , u) is strongly measurable for each u \in E;
(c) There exist mg(\cdot ) \in Lp([0, a],\BbbR +) and d \geq 0 such that
\| g(t, u)\| \leq mg(t) + d\| u\| for every t \in [0, a] and u \in E.
Also, we assume that the following condition holds:
(H3) t \mapsto \rightarrow u(t) -
\int t
0
K(t, s)u(s) ds is an absolutely continuous function on [0, a].
Now, it is easy to see that if u(\cdot ) is a solution of equation (4.3), then it satisfies the following integral
equation:
u(t) = (\frakP u)(t) +
t\int
0
(Qu)(s) ds for a.e. t \in [0, a], (4.4)
where
(\frakP u)(t) := u0 +
t\int
0
K(t, s)u(s)ds, t \in [0, a],
is a Volterra operator and
(Qu)(t) := g(t, u(t)), t \in [0, a],
is the Nemitskii operator. Conversely, if u(\cdot ) \in Lp([0, a], E) satisfies the integral equation (4.4), then
u (\cdot ) is a solution of equation (4.3) with initial value u(0) = u0.
Theorem 4.1. Suppose that K : \Delta \rightarrow \scrL (E) satisfies (H2) and g(\cdot , \cdot ) : [0, a] \times E \rightarrow E is a
Carathéodory function such that there exists k2 > 0 such that Ma1/p
\prime +1/q\prime + 2k2a
1/p - 1/q\| \xi \| p < 1
and
\alpha (g(t, A)) \leq k2\alpha (A) (4.5)
for t \in [0, a] and for each bounded subset A \subset E. If (H3) hold, then the neutral functional differential
equation (4.3) has at least one solution in Lp([0, a], E) satisfying the initial condition u(0) = u0.
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EXISTENCE OF GLOBAL SOLUTIONS FOR SOME CLASSES OF INTEGRAL EQUATIONS 143
Proof. From (H2) and Theorem 9.5.1 in [15] it follows that \frakP is a continuous operator from
Lp([0, a], E) into itself. If V is a bounded countable set in Lp([0, a], E), then we have
\alpha ((\frakP V )(t)) \leq \alpha
\left( t\int
0
K(t, s)V (s)ds
\right) \leq
t\int
0
\alpha (K(t, s)V (s)) ds \leq
\leq
t\int
0
\| K(t, s)\| \alpha (V (s)) ds \leq
\leq
\left( t\int
0
\| K(t, s)\| qds
\right) 1/q \left( t\int
0
[\alpha (V (s))]q
\prime
ds
\right) 1/q\prime
\leq
\leq Ma1/q
\prime - 1/p
\left( t\int
0
[\alpha (V (s))]p ds
\right) 1/p
,
so that \frakP satisfies (3.8). Also, by (c) it follows that the Nemitskii operator Q is a continuous
operator from Lp([0, a], E) into itself. Next, using (4.5), for any bounded and countable set in
Lp([0, a], E) we have \alpha ((QV )(t)) = \alpha (g(t, V (t))) \leq k2\alpha
\bigl(
V (t)
\bigr)
for t \in [0, a], so that Q also
satisfies (3.3). Consequently, (H1), (H2) and (3.3) are satisfied so that, by Remark 3.2, the neutral
functional differential equation (4.3) has at least one solution in Lp([0, a], E) satisfying the initial
condition u(0) = u0.
Theorem 4.1 is proved
5. A global existence result for nonlinear Fredholm functional integral equations. In this
section we obtain a result on the global existence of solutions for a nonlinear Fredholm functional
integral equation involving an abstract Volterra operator. A similar result was obtained by Warga
[49] ([Theorem II.5.1]) in the finite dimensional case. If X, Y are given real separable Banach
spaces, we denote by C(Y,X) the Banach space of all continuous and bounded functions from Y
into X endowed with the norm \| f(\cdot )\| C(Y,X) = \mathrm{s}\mathrm{u}\mathrm{p}y\in Y \| f(y)\| . We shall identify two functions
g(\cdot , \cdot ), h(\cdot , \cdot ) : [0, a]\times Y \rightarrow X if g(t, \cdot ) = h(t, \cdot ) a.e. on[0, a], and we will denote by \Omega := \Omega ([0, a]\times
\times Y,X) the vector space of (equivalence classes of) all functions g(\cdot , \cdot ) : [0, a] \times Y \rightarrow X such
that:
(c1) g(t, \cdot ) \in C(Y,X) for each t \in [0, a];
(c2) g(\cdot , y) is strongly measurable for each y \in Y ;
(c3) there exists a function mg(\cdot ) \in Lp([0, a],\BbbR +) such that
\| g(t, \cdot )\| C(Y,X) \leq mg(t) for every t \in [0, a].
An element of \Omega is called a Carathéodory function.
Remark 5.1. It is easy to see that the function t \mapsto \rightarrow \| g(t, \cdot )\| C(Y,X) is Lebesgue integrable on
[0, a] for every g(\cdot , \cdot ) \in \Omega . Moreover, the function g \mapsto \rightarrow \| g\| \Omega : \Omega \rightarrow \BbbR +, given by
\| g\| \Omega :=
a\int
0
\| g(t, \cdot )\| C(Y,X) dt
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144 T. JABEEN, R. P. AGARWAL, V. LUPULESCU, D. O’REGAN
is a norm on \Omega . Also, for every g(\cdot , \cdot ) \in \Omega and for every strongly measurable functions y(\cdot ) :
[0, a] \rightarrow Y, the function t \mapsto \rightarrow g(t, y(t)) is Bochner integrable on [0, a].
In the following, if F (\cdot ) \in C([0, a],\Omega ) is a given function, then we will write F (t, s, y) instead
of F (t)(s, y) for (s, y) \in [0, a]\times Y.
Consider the nonlinear Fredholm functional-integral equation
u(t) = u0(t) +
a\int
0
F (t, s, (Qu)(s))ds, t \in [0, a], (5.1)
where F (\cdot ) \in C([0, a],\Omega ), Q : C([0, a], X) \rightarrow L\infty ([0, a], Y ), and u0(\cdot ) \in C([0, a], X) are assumed
to satisfy the following assumptions:
(A1) Q : C([0, a], X) \rightarrow L\infty ([0, a], Y ) is continuous and there exists b > 0 and 0 < c < 1 such
that
\| Qu\| \infty \leq b(1 + \| u(\cdot )\| )c, u(\cdot ) \in C([0, a], X);
(A2) there exist 0 < d < 1 and an integrable function h(\cdot , \cdot ) : [0, a]\times [0, a] \rightarrow \BbbR + such that
\gamma := \mathrm{s}\mathrm{u}\mathrm{p}
0\leq t\leq a
a\int
0
h(t, s)ds <\infty
and
\| F (t, s, y)\| \leq h(t, s)(1 + \| y\| )d for t, s \in [0, a] and y \in Y ;
(A3) there exist k, k0 > 0 and \psi (\cdot ) \in L1([0, a],\BbbR +) such that
\beta (F (t, s, B)) \leq k\beta (B)
for all t, s \in [0, a] and any bounded set B \subset Y, and
\beta ((QV )(t)) \leq k0\beta
\bigl(
V (t)
\bigr)
for every t \in [0, a] and every bounded set V \subset C([0, a], X).
Theorem 5.1. If assumptions (A1) – (A3) are satisfied, u0(\cdot ) \in C([0, a], X) and kk0 < 1, then
the integral equation (5.1) has at least one solution in C([0, a], X).
Proof. Since F (t, \cdot , \cdot ) \in \Omega and the function s \mapsto \rightarrow (Qu)(s) is strongly measurable on [0, a]
for each u(\cdot ) \in C([0, a], X), by Remark 5.1 it follows that the function s \mapsto \rightarrow F (t, s, (Qu)(s)) is
Bochner integrable on [0, a] for every t \in [0, a], so that the operator
(Ku)(t) := u0(t) +
a\int
0
F (t, s, (Qu)(s))ds, t \in [0, a],
is well defined for every u(\cdot ) \in C([0, a], X). Since 0 < c, d < 1, it is easy to check that, for a given
r \geq \mathrm{m}\mathrm{a}\mathrm{x}\{ 1, \gamma (1 + 2b)c\} , we have \gamma
\bigl[
1 + b(1 + r)d
\bigr] c \leq r0. Let
Wr :=
\bigl\{
u(\cdot ) \in C([0, a], X); \| u(\cdot )\| \leq r
\bigr\}
,
where r := r + \| u0(\cdot )\| . First, we remark that, for every u(\cdot ) \in Wr, we obtain
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
EXISTENCE OF GLOBAL SOLUTIONS FOR SOME CLASSES OF INTEGRAL EQUATIONS 145
\| (Qu)(s)\| \leq b(1 + \| u(s)\| )c \leq r0 := b(1 + r)c
for a.e. s \in [0, a], so that QWr \subset \{ y(\cdot ) \in L\infty ([0, a], Y ); \| y(\cdot )\| \leq r0\} . From (A1) and (A2) it follows
that, for each u(\cdot ) \in Wr, we get
\| (Ku)(t)\| \leq \| u0(\cdot )\| +
a\int
0
\| F (t, s, (Qu)(s))\| ds \leq
\leq \| u0(\cdot )\| +
a\int
0
h(t, s)(1 + \| (Qu)(s)\| )dds \leq
\leq \| u0(\cdot )\| + (1 + \| Qu\| \infty )d \mathrm{s}\mathrm{u}\mathrm{p}
0\leq t\leq a
a\int
0
h(t, s)ds \leq
\leq \| u0(\cdot )\| + \gamma
\Bigl[
1 + b(1 + r)d
\Bigr] c
\leq \| u0(\cdot )\| + r = r,
so that Ku \in Wr for every u(\cdot ) \in Wr. Since Wr is bounded and KWr \subset Wr, KWr is also bounded.
Now, we show that K is a continuous operator on Wr. For this, let \{ un(\cdot )\} n\geq 1 be a sequence in Wr
converging to some u(\cdot ) \in Wr. Then by (A1) we have that \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty (Qun)(s) = (Qu)(s) for a.e.
s \in [0, a]. Also, since F (t, \cdot , \cdot ) \in \Omega ,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
F (t, s, (Qun)(s)) = F (t, s, (Qu)(s))
and \bigm\| \bigm\| F (t, s, (Qun)(s))\bigm\| \bigm\| \leq \mathrm{s}\mathrm{u}\mathrm{p}
\| y\| \leq r0
\| F (t, s, y\| \leq (1 + r0)
dh(t, s)
for each t \in [0, a] and for a.e. s \in [0, a], by the Lebesgue dominated convergence theorem we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
a\int
0
F (t, s, (Qun)(s)ds =
a\int
0
F (t, s, (Qu)(s)ds
for each t \in [0, a]. Consequently, K is a continuous operator. Next, for every t, s \in [0, a] and every
u(\cdot ) \in Wr, we have
\bigm\| \bigm\| (Ku)(t) - (Ku)(s)
\bigm\| \bigm\| \leq
a\int
0
\| F (t, \tau , (Qu)(\tau )) - F (s, \tau , (Qu)(\tau ))\| d\tau \leq
\leq
a\int
0
\mathrm{s}\mathrm{u}\mathrm{p}
y\in Y
\bigm\| \bigm\| F (t, \tau , y) - F (s, \tau , y)
\bigm\| \bigm\| d\tau =
=
a\int
0
\bigm\| \bigm\| F (t, \tau , \cdot ) - F (s, \tau , \cdot )
\bigm\| \bigm\|
C(Y,X)
d\tau =
\bigm\| \bigm\| F (t, \cdot , \cdot ) - F (s, \cdot , \cdot )
\bigm\| \bigm\|
\Omega
.
Since F (\cdot ) \in C([0, a],\Omega ), for every \varepsilon > 0, there exists \delta = \delta (\varepsilon ) > 0 such that
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
146 T. JABEEN, R. P. AGARWAL, V. LUPULESCU, D. O’REGAN\bigm\| \bigm\| (Ku)(t) - (Ku)(s)
\bigm\| \bigm\| \leq
\bigm\| \bigm\| F (t, \cdot , \cdot ) - F (s, \cdot , \cdot )
\bigm\| \bigm\|
\Omega
\leq \varepsilon
for all t, s \in [0, a] with | t - s| \leq \delta and for every u(\cdot ) \in Wr, so that KWr is equicontinuous. Next,
put W0 :=Wr and define Wn+1 = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(KWn), n = 0, 1, 2, . . . .
Now, from KW0 \subset W0, it follows that
W1 = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(KW0) \subset \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}(W0) =W0,
and thus, W1 \subset C([0, a], X) is bounded, closed, convex and equicontinuous. By Mathematical
induction it is easy to see that Wn+1 \subset Wn and Wn \subset C([0, a], X) are bounded, closed, convex
and equicontinuous for n = 0, 1, 2, . . . . Next, since C([0, a], X) is separable, then for each n =
= 0, 1, 2, . . . , there exists a countable set V n = \{ vnk ; k = 1, 2, . . .\} \subset C([0, a], X) such that
V n =Wn. Then, by Lemma 3.1, the properties of the measure of noncompactness and (A3), we have
\beta (Wn+1(t)) = \beta
\bigl(
\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}((KWn)(t))
\bigr)
= \beta
\bigl(
(KWn)(t)
\bigr)
= \beta
\bigl(
(KV n)(t)
\bigr)
\leq
\leq \beta
\left( a\int
0
F (t, s, (QV n)(s))ds
\right) \leq
\leq k
a\int
0
\beta
\bigl(
(QV n)(s)
\bigr)
ds \leq kk0
a\int
0
\beta
\bigl(
(V n(s)
\bigr)
ds,
that is,
\beta
\bigl(
Wn+1(t)
\bigr)
\leq kk0
a\int
0
\beta (Wn(s)) ds, t \in [0, b].
From a finite number of steps, we obtain
\beta (Wn(t)) \leq (kk0)
n
a\int
0
\beta (W0(s)) ds, t \in [0, b], n \geq 1.
From Wn+1 \subset Wn, n = 0, 1, 2, . . . , and property (4) of the measure of noncompactness, it follows
that, for each t \in [0, b], the sequence \{ \beta (Wn(t))\} n\geq 0 is bounded and decreasing. Hence, there exists
h(t) := \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \beta (Wn(t)) , t \in [0, b]. Taking n\rightarrow \infty on both sides of the last inequality we get
h(t) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\beta (Wn(t)) \leq \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
(kk0)
n
a\int
0
\beta (W0(s)) ds = 0, t \in [0, b],
and thus, h(t) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \beta (Wn(t)) = 0, t \in [0, b]. Since Wn, n = 0, 1, . . . , are bounded and
equicontiuous, it follows that \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \beta c (Wn) = 0. By property (7) of the measure of noncompact-
ness, it follows that W :=
\bigcap \infty
n=0
Wn is a compact set of C([0, a], X) and KW \subset W. Consequently,
by the Schauder fixed point theorem, it follows that the operator K has at least one fixed point
u(\cdot ) \in W, which is a solution of (5.1).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
EXISTENCE OF GLOBAL SOLUTIONS FOR SOME CLASSES OF INTEGRAL EQUATIONS 147
References
1. Agarwal R. P., Zhou Y., Wang J. R., Luo X. Fractional functional differential equations with causal operators in
Banach spaces // Math. and Comput. Modelling. – 2011. – 54. – P. 1440 – 1452.
2. Agarwal R. P., Arshad S., Lupulescu V., O’Regan D. Evolution equations with causal operators // Different. Equat.
and Appl. – 2015. – 7, № 1. – P. 15 – 26.
3. Agarwal R. P., O’Regan D. Infinite interval problems for differential, differential and integral equations. – New York:
Kluwer Acad. Publ., 2001.
4. Ahangar R. Nonanticipating dynamical model and optimal control // Appl. Math. Lett. – 1989. – 2. – P. 15 – 18.
5. Akmerov R., Kamenskii M., Potapov A., Sadovskii B. Measures of noncompactness and condensing operators. – Basel:
Birkhäuser, 1992.
6. Banaś J. Integrable solutions of Hammerstein and Urysohn integral equations // J. Austral. Math. Soc. Ser. A. – 1989. –
46. – P. 61 – 68.
7. Barton T. A., Purnaras I. K. Lp -solutions of singular integro-differential equations // J. Math. Anal. and Appl. –
2012. – 386. – P. 830 – 841.
8. Barton T. A., Zhang B. Lp -solutions of fractional differential equations // Nonlinear Stud. – 2012. – 19. – P. 161.
9. Bedivan D. M., O’Regan D. The set of solutions for abstract Volterra equations in Lp([0, a], Rm) // Appl. Math.
Lett. – 1999. – 12. – P. 7 – 11.
10. Corduneanu C. Functional equations with causal operators. – London; New York: Taylor and Francis, 2002.
11. Corduneanu C. A modified LQ-optimal control problem for causal functional differential equations // Nonlinear Dyn.
and Syst. Theory. – 2004. – 4. – P. 139 – 144.
12. Corduneanu C., Mahdavi M. Neutral functional equations with causal operators on a semi-axis // Nonlinear Dyn. and
Syst. Theory. – 2008. – 8. – P. 339 – 348.
13. Darwish M. A., El-Bary A. A. Existence of fractional integral equation with hysteresis // Appl. Math. and Comput. –
2006. – 176. – P. 684 – 687.
14. Drici Z., McRae F. A., Devi J. V. Differential equations with causal operators in a Banach space // Nonlinear Anal.:
Theory, Methods and Appl. – 2005. – 62. – P. 301 – 313.
15. Edwards R. E. Functional analysis – theory and applications. – New York: Holt, Rinehart and Winston, Inc., 1965.
16. Emmanuele G. About the existance of integrable solution of a functional-integral equation // Rev. Mat. Univ. Complut.
Madrid. – 1991. – 4, № 1.
17. Gil’ M. I. Explicit stability conditions for neutral type vector functional differential equations. A survey // Surv. Math.
and Appl. – 2014. – 9. – P. 1 – 54.
18. Gil’ M. I. Stability of neutral functional differential equations. – Paris: Atlantis Press, 2014.
19. Gripenberg G., Londen S. O., Staffans O. Volterra integral and functional equations. – Cambridge Univ. Press, 1990.
20. Guo D., Lakshmikantham V., Liu X. Nonlinear integral equations in abstract spaces. – Dordrecht: Kluwer Acad. Publ.,
1996.
21. Heinz H. P. On the behaviour of measures of noncompactness with respect to differentiation and integration of vector
valued-functions // Nonlinear Anal. – 1983. – 7. – P. 1351 – 1371.
22. Hernández E., Balachandran K. Existance results for abstract degenrate neutral functional differential equations //
Bull. Austral. Math. Soc. – 2010. – 81. – P. 329 – 342.
23. Hernández E., Perri M., Prokopczyk A., Hernández E., Pierri M., Prokopczyk A. On a class of abstract neutral
functional differential equations // Nonlinear Anal. – 2011. – 74. – P. 3633 – 3643.
24. Hernández E., O’Regan D., Ben M. A. On a new class of abstract integral equations and applications // Appl. Math.
and Comput. – 2012. – 219, № 4. – P. 2271 – 2277.
25. Ilchmann A., Ryan E. P., Sangwin C. J. Systems of controlled functional differential equations and adaptive tracking //
SIAM J. Control and Optim. – 2002. – 40, № 6. – P. 1746 – 1764.
26. Isaia F. On a nonlinear integral equation without compactness // Acta Math. Univ. Comenian. – 2006. – 75, № 2. –
P. 233 – 240.
27. Kamenskii M., Obukhovskii V., Zecca P. Condensing multivalued maps and semilinear differential inclusions in
Banach spaces. – Berlin; New York: Walter de Gruyter, 2001.
28. Kisielewicz M. Multivalued differential equations in separable Banach spaces // J. Optim. Theory and Appl. – 1982. –
37, № 2. – P. 231 – 249.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
148 T. JABEEN, R. P. AGARWAL, V. LUPULESCU, D. O’REGAN
29. Kuratowski C. Sur les espaces complets // Fund. Math. – 1930. – 51. – P. 301 – 309.
30. Kwapisz M. Remarks on the existance and uniqueness of solutions of volterra functional equations in Lp spaces // J.
Integral Equat. and Appl. – 1991. – 3, № 3.
31. Kwapisz M. Bielecki’s method, existence and uniqueness results for Volterra integral equations in Lp space // J.
Math. Anal. and Appl. – 1991. – 154. – P. 403 – 416.
32. Lakshmikantham V., Leela S., Drici Z., McRae F. A. Theory of causal differential equations // Atlantis Stud. Math.
Eng. and Sci. – 2010. – Vol. 5.
33. Liang J., Yan S.-H., Agarwal R. P., Huang T.-W. Integral solution of a class of nonlinear integral equations // Appl.
Math. and Comput. – 2013. – 219. – P. 4950 – 4957.
34. Lupulescu V. Causal functional differential equations in Banach spaces // Nonlinear Anal. – 2008. – 69. – P. 4787 –
4795.
35. Lupulescu V. On a class of functional differential equations in Banach spaces // Electron. J. Qual. Theory Different.
Equat. – 2010. – 64. – P. 1 – 17.
36. Mamrilla D. On Lp -solutions of nth order nonlinear differential equations // Čas. pěstov. mat. – 1988. – 113. –
P. 363 – 368.
37. Martynyuk A. A., Martynyuk-Chernienko Yu. A. Analysis of the set of trajectories of nonlinear dynamics: equations
with causal robust operator // Different. Equat. – 2015. – 51, № 1. – P. 11 – 22.
38. Obukhovskii V., Zecca P. On certain classes of functional inclusions with causal operators in Banach spaces //
Nonlinear Anal.: Theory, Methods and Appl. – 2011. – 74. – P. 2765 – 2777.
39. Olszowy L. A family of measures of noncompactness in the space L1
loc(R+) and its application to some nonlinear
Volterra integral equation // Mediterr. J. Math. – 2014. – 11. – P. 687 – 701.
40. Prüss J. Evolutionary integral equations and applications. – Basel: Birkhäuser, 1993.
41. O’Regan D., Precup R. Existence criteria for integral equations in Banach spaces // J. Inequal. and Appl. – 2001. –
6. – P. 77 – 97.
42. Ryan E. P., Sangwin C. J. Controlled functional differential equations and adaptive tracking // Systems Control Lett. –
2002. – 47. – P. 365 – 374.
43. Szufla S. Existence for Lp -solutions of integral equations in Banach spaces // Publ. Inst. Math. – 1986. – 54. –
P. 99 – 105.
44. Szufla S. Appendix to the paper An existence theorem for the Urysohn integral equation in Banach spaces // Comment.
Math. Univ. Carolin. – 1984. – 25. – P. 763 – 764.
45. Taggart R. J. Evolution equations and vector-valued Lp spaces: PhD thesis. – New South Wales Univ., 2004.
46. Tikhonov A. N. Functional Volterra-type equations and their applications to certain problems of mathematical physics //
Bull. Mosk. Gos. Univ. Sekt. A. – 1938. – 1, № 8. – P. 1 – 25.
47. Tonelli L. Sulle equazioni funzionali di Volterra // Bull. Calcutta Math. Soc. – 1930. – 20. – P. 31 – 48.
48. Wang F. A fixed point theorem for nonautonomous type superposition operators and integrable solutions of a general
nonlinear functional integral equation // J. Inequal. and Appl. – 2014. – 2014.
49. Warga J. Optimal control of differential and functional equations. – New York: Acad. Press, 1972.
50. Zhu H. On a nonlinear integral equation with contractive perturbation / Adv. Difference Equat. – 2011. – 2011. –
Article ID 154742. – 10 p., doi:10.1155/2011/154742.
51. Zhukovskii E. S., Alves M. J. Abstract Volterra operators // Russ. Math. – 2008. – 52. – P. 1 – 14.
Received 26.11.17
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|
| id | umjimathkievua-article-1546 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:07:48Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ec/d30dee12cb4693d1c1ac17965ad18eec.pdf |
| spelling | umjimathkievua-article-15462019-12-05T09:17:34Z Existence of global solutions for some classes of integral equations Існування глобальних розв’язкiв деяких класiв iнтегральних рiвнянь Agarwal, P. Jabeen, T. Lupulescu, V. O’Regan, D. Агарвал, Р. П. Ябін, Т. Лупулеску, В. О'Реган, Д. We study the existence of $L^p$ -solutions for a class of Hammerstein integral equations and neutral functional differential equations involving abstract Volterra operators. Using compactness-type conditions, we establish the global existence of solutions. In addition, a global existence result for a class of nonlinear Fredholm functional integral equations involving abstract Volterra equations is given. Вивчається iснування $L^p$ -розв’язкiв для класу iнтегральних рiвнянь Гаммерштейна та нейтральних функцiональних диференцiальних рiвнянь з абстрактними операторами Вольтерра. Iснування глобальних розв’язкiв встановлено за допомогою умов типу компактностi. Крiм того, наведено результат про глобальне iснування розв’язку для класу нелiнiйних функцiональних iнтегральних рiвнянь Фредгольма з абстрактними операторами Вольтерра. Institute of Mathematics, NAS of Ukraine 2018-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1546 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 1 (2018); 130-148 Український математичний журнал; Том 70 № 1 (2018); 130-148 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1546/528 Copyright (c) 2018 Agarwal P.; Jabeen T.; Lupulescu V.; O’Regan D. |
| spellingShingle | Agarwal, P. Jabeen, T. Lupulescu, V. O’Regan, D. Агарвал, Р. П. Ябін, Т. Лупулеску, В. О'Реган, Д. Existence of global solutions for some classes of integral equations |
| title | Existence of global solutions for some classes
of integral equations |
| title_alt | Існування глобальних розв’язкiв деяких класiв
iнтегральних рiвнянь |
| title_full | Existence of global solutions for some classes
of integral equations |
| title_fullStr | Existence of global solutions for some classes
of integral equations |
| title_full_unstemmed | Existence of global solutions for some classes
of integral equations |
| title_short | Existence of global solutions for some classes
of integral equations |
| title_sort | existence of global solutions for some classes
of integral equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1546 |
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