- Research Article
- Open Access
Some Results for Integral Inclusions of Volterra Type in Banach Spaces
© R. P. Agarwal et al. 2010
- Received: 29 July 2010
- Accepted: 29 November 2010
- Published: 6 December 2010
We first present several existence results and compactness of solutions set for the following Volterra type integral inclusions of the form: , where , is the infinitesimal generator of an integral resolvent family on a separable Banach space , and is a set-valued map. Then the Filippov's theorem and a Filippov-Ważewski result are proved.
- Banach Space
- Mild Solution
- Fractional Differential Equation
- Separable Banach Space
- Measurable Selection
In the past few years, several papers have been devoted to the study of integral equations on real compact intervals under different conditions on the kernel (see, e.g., [1–4]) and references therein. However very few results are available for integral inclusions on compact intervals, see [5–7]. Topological structure of the solution set of integral inclusions of Volterra type is studied in .
where and is the generator of an integral resolvent family defined on a complex Banach space , and is a multivalued map.
is the Riemann-Liouville kernel. In this case (1.1) and (1.2) can be represented in the form of fractional differential equations and inclusions or abstract fractional differential equations and inclusions. Also in the case where , and is a Rieman-Liouville kernel, (1.1) and (1.2) can be represented in the form of fractional differential equations and inclusions, see for instants [25–27].
Our goal in this paper is to complement and extend some recent results to the case of infinite-dimensional spaces; moreover the right-hand side nonlinearity may be either convex or nonconvex. Some auxiliary results from multivalued analysis, resolvent family theory, and so forth, are gathered together in Sections 2 and 3. In the first part of this work, we prove some existence results based on the nonlinear alternative of Leray-Schauder type (in the convex case), on Bressan-Colombo selection theorem and on the Covitz combined the nonlinear alternative of Leray-Schauder type for single-valued operators, and Covitz-Nadler fixed point theorem for contraction multivalued maps in a generalized metric space (in the nonconvex case). Some topological ingredients including some notions of measure of noncompactness are recalled and employed to prove the compactness of the solution set in Section 4.2. Section 5 is concerned with Filippov's theorem for the problem (1.1). In Section 6, we discuss the relaxed problem, namely, the density of the solution set of problem (1.1) in that of the convexified problem.
Denote by , closed}, bounded}, convex}, compact .
2.1. Multivalued Analysis
Let and be two metric spaces and be a multivalued map. A single-valued map is said to be a selection of and we write whenever for every .
is called upper semicontinuous (u.s.c. for short) on if for each the set is a nonempty, closed subset of , and if for each open set of containing , there exists an open neighborhood of such that . That is, if the set is closed for any closed set in . Equivalently, is u.s.c. if the set is open for any open set in .
The following two results are easily deduced from the limit properties.
Lemma 2.1 (see, e.g., [29, Theorem 1.4.13]).
Lemma 2.2 (see, e.g., [29, Lemma 1.1.9]).
where refers to the closure of the convex hull of .
is said to be completely continuous if it is u.s.c. and, for every bounded subset , is relatively compact, that is, there exists a relatively compact set such that . is compact if is relatively compact. It is called locally compact if, for each , there exists such that is relatively compact. is quasicompact if, for each subset , is relatively compact.
A multivalued map is said measurable provided for every open , the set is Lebesgue measurable.
is Lebesgue measurable.
The following two lemmas are needed in this paper. The first one is the celebrated Kuratowski-Ryll-Nardzewski selection theorem.
Lemma 2.5 (see [31, Theorem 19.7]).
Let be a separable metric space and a measurable multivalued map with nonempty closed values. Then has a measurable selection.
Lemma 2.6 (see [32, Lemma 3.2]).
2.1.1. Closed Graphs
We denote the graph of to be the set .
is closed if is a closed subset of , that is, for every sequences and , if , as with , then .
We recall the following two results; the first one is classical.
Lemma 2.9 (see [33, Proposition 1.2]).
If is u.s.c., then is a closed subset of . Conversely, if is locally compact and has nonempty compact values and a closed graph, then it is u.s.c.
If is quasicompact and has a closed graph, then is u.s.c.
A multivalued map is called a Carathéodory function if
(a)the function is measurable for each ;
(b)for a.e. , the map is upper semicontinuous.
is known as the set of selection functions.
For each , the set is closed whenever has closed values. It is convex if and only if is convex for .
From  (see also  when is finite-dimensional), we know that is nonempty if and only if the mapping belongs to . It is bounded if and only if the mapping belongs to ; this particularly holds true when is -Carathéodory. For the sake of completeness, we refer also to Theorem 1.3.5 in  which states that contains a measurable selection whenever is measurable and is a Carathéodory function.
Lemma 2.13 (see ).
has a closed graph in .
For further readings and details on multivalued analysis, we refer to the books by Andres and Górniewicz , Aubin and Cellina , Aubin and Frankowska , Deimling , Górniewicz , Hu and Papageorgiou , Kamenskii et al. , and Tolstonogov .
2.2. Semicompactness in
the image sequence is relatively compact in for a.e. .
We recall two fundamental results. The first one follows from the Dunford-Pettis theorem (see [36, Proposition 4.2.1]). This result is of particular importance if is reflexive in which case (a) implies (b) in Definition 2.14.
Every semicompact sequence is weakly compact in .
The second one is due to Mazur, 1933.
Lemma 2.16 (Mazur's Lemma, ).
Let be a normed space and be a sequence weakly converging to a limit . Then there exists a sequence of convex combinations with for and , which converges strongly to .
if the integral is absolutely convergent for . In order to defined the mild solution of the problems (1.1) we recall the following definition.
In this case, is called the integral resolvent family generated by .
The following result is a direct consequence of ([16, Proposition 3.1 and Lemma 2.2]).
is strongly continuous for and ;
and for all ;
In particular, .
The uniqueness of resolvent is well known (see Prüss ).
(see [22, Theorem 2.1]). For example, the case corresponds to the generator of a -semigroup and actually corresponds to the generator of a sine family; see . A characterization of generators of integral resolvent families, analogous to the Hille-Yosida Theorem for -semigroups, can be directly deduced from [22, Theorem 3.4]. More information on the -semigroups and sine families can be found in [41–43].
4.1. Mild Solutions
In order to define mild solutions for problem (1.1), we proof the following auxiliary lemma.
This lemma leads us to the definition of a mild solution of the problem (1.1).
Consider the following assumptions.
(B1) The operator solution is compact for .
(B3) For every , is uniformly continuous.
In all the sequel we assume that is exponentially bounded. Our first main existence result is the following.
Assume is a Carathéodory map satisfying or . Then problem (1.1) has at least one solution. If further is a reflexive space, then the solution set is compact in .
has at least one fixed point,
the set is unbounded.
The single-valued version may be stated as follows.
either there is a point and with ,
or has a fixed point in .
Proof of Theorem 4.3.
We have the following parts.
Part 1: Existence of Solutions
Notice that the set is nonempty (see Remark 2.12,(b)). Since, for each , the nonlinearity takes convex values, the selection set is convex and therefore has convex values.
The right-hand side tends to zero as since is uniformly continuous.
which tends to 0 as . Therefore, there are precompact sets arbitrarily close to the set . This set is then precompact in .
Step 2 ( has a closed graph).
Hence , proving our claim. Lemma 2.10 implies that is u.s.c.
Step 3 (a priori bounds on solutions).
and consider the operator . From the choice of , there is no such that for some . As a consequence of the Leray-Schauder nonlinear alternative (Lemma 4.4), we deduce that has a fixed point in which is a mild solution of problem (1.1).
Part 2: Compactness of the Solution Set
This with (4.33) imply that . Since has closed, convex values, we deduce that , for a.e. , as claimed. Hence which yields that is closed, hence compact in .
4.2. The Convex Case: An MNC Approach
for every .
monotone if , implies .
nonsingular if for every .
invariant with respect to the union with compact sets if for every relatively compact set , and .
real if and for every bounded .
semiadditive if for every .
regular if the condition is equivalent to the relative compactness of .
Recall that a bounded set has a finite -net if there exits a finite subset such that where is a closed ball in .
It should be mentioned that these MNC satisfy all above-mentioned properties except regularity.
implies the relative compactness of .
Some important results on fixed point theory with MNCs are recalled hereafter (see, e.g.,  for the proofs and further details). The first one is a compactness criterion.
Lemma 4.9 (see [36, Theorem 5.1.1]).
Let be an abstract operator satisfying the following conditions:
(S2) is weakly-strongly sequentially continuous on compact subsets: for any compact and any sequence such that for a.e. , the weak convergence implies the strong convergence as .
Then for every semicompact sequence , the image sequence is relatively compact in .
Lemma 4.10 (see [36, Theorem 5.2.2]).
Let an operator satisfy conditions together with the following:
where is the Hausdorff MNC.
where is the constant in .
The next result is concerned with the nonlinear alternative for -condensing u.s.c. multivalued maps.
Lemma 4.11 (see ).
for all and . Then .
Lemma 4.12 (see ).
Let be a closed subset of a Banach space and is a closed -condensing multivalued map where is a monotone MNC on . If the fixed point set Fix is bounded, then it is compact.
4.2.1. Main Results
Let be a Carathéodory multivalued map which satisfies Lipschitz conditions with respect to the Hausdorf MNC.
(B4) There exists such that for every bounded in ,
Under conditions and , the operator is closed and , for every where is as defined in the proof of Theorem 4.3.
We have the following steps.
Step 1 ( is closed).
As a consequence, , as claimed.
Step 2 ( has compact, convex values).
The convexity of follows immediately by the convexity of the values of . To prove the compactness of the values of , let for some and . Then there exists satisfying (4.47). Arguing again as in Step 1, we prove that is semicompact and converges weakly to some limit , a.e. hence passing to the limit in (4.47), tends to some limit in the closed set with satisfying (4.49). Therefore the set is sequentially compact, hence compact.
Under the conditions and , the operator is u.s.c.
As a consequence, is u.s.c.
We are now in position to prove our second existence result in the convex case.
then the set of solutions for problem (1.1) is nonempty and compact.
where is the collection of all countable subsets of . Then the MNC is monotone, regular and nonsingular (see [36, Example 2.1.4]).
Hence (4.60) implies that . To show that , i.e, the set is equicontinuous, we proceed as in the proof of Theorem 4.3 Step 1 Part (b). It follows that which implies, by (4.61), that . We have proved that is relatively compact. Hence is u.s.c. and -condensing, where is as in the proof of Theorem 4.3. From the choice of , there is no such that for some . As a consequence of the nonlinear alternative of Leray-Schauder type for condensing maps (Lemma 4.11), we deduce that has a fixed point in , which is a solution to problem (1.1). Finally, since is bounded, by Lemma 4.12, is further compact.
4.3. The Nonconvex Case
In this section, we present a second existence result for problem (1.1) when the multivalued nonlinearity is not necessarily convex. In the proof, we will make use of the nonlinear alternative of Leray-Schauder type  combined with a selection theorem due to Bressan and Colombo  for lower semicontinuous multivalued maps with decomposable values. The main ingredients are presented hereafter. We first start with some definitions (see, e.g., ). Consider a topological space and a family of subsets of .
is called measurable if belongs to the -algebra generated by all sets of the form where is Lebesgue measurable in and is Borel measurable in .
where stands for the characteristic function of the set .
Let be a multivalued map with nonempty closed values. Assign to the multivalued operator defined by . The operator is called the Nemyts'kiĭ operator associated to .
Let be a multivalued map with nonempty compact values. We say that is of lower semicontinuous type (l.s.c. type) if its associated Nemyts'kiĭ operator is lower semicontinuous and has nonempty closed and decomposable values.
Next, we state a classical selection theorem due to Bressan and Colombo.
Let be a separable metric space and let be a Banach space. Then every l.s.c. multivalued operator , with closed decomposable values has a continuous selection, that is, there exists a continuous single-valued function such that for every .
Let us introduce the following hypothesis.
the mapping is measurable;
the mapping is lower semicontinuous for a.e. .
The following lemma is crucial in the proof of our existence theorem.
Lemma 4.20 (see, e.g., ).
Let be an integrably bounded multivalued map satisfying . Then is of lower semicontinuous type.
Suppose that the hypotheses or and are satisfied. Then problem (1.1) has at least one solution.
From the choice of , there is no such that for in . As a consequence of the nonlinear alternative of the Leray-Schauder type (Lemma 4.5), we deduce that has a fixed point which is a solution of problem (4.68), hence a solution to the problem (1.1).
4.4. A Further Result
where and . Then is a metric space and is a generalized metric space (see ). In particular, satisfies the triangle inequality.
a contraction if it is -Lipschitz with .
Let be a complete metric space. If is a contraction, then .
Let us introduce the following hypotheses:
(A1) ; is measurable for each ;
Let Assumptions be satisfied. Then problem (1.1) has at least one solution.
proving that .
is the Bielecki-type norm on . So, is a contraction and thus, by Lemma 4.23, has a fixed point , which is a mild solution to (1.1).
Arguing as in Theorem 4.3, we can also prove the following result the proof of which is omitted.
Let be a reflexive Banach space. Suppose that all conditions of Theorem 4.24 are satisfied and . Then the solution set of problem (1.1) is nonempty and compact.
5.1. Filippov's Theorem on a Bounded Interval
We will consider the following two assumptions.
for all is measurable,
the map is integrable.
where refers to the closure of the convex hull of the set . We will need the following auxiliary results in order to prove our main relaxation theorem.
Lemma 6.1 (see ).
With being a reflexive Banach space, the following hypotheses will be assumed in this section:
The function satisfies
(a)for all the map is measurable,
(b)the map is integrable bounded (i.e., there exists such that .
Then our main contribution is the following.
Notice that the multivalued map also satisfies .
Part 1 ( )
To show that satisfies the assumptions of Lemma 4.23, the proof will be given in two steps.
proving that .
Then is a contraction and hence, by Lemma 4.23, has a fixed point , which is solution to problem (6.2).
Since is arbitrary, , showing the density relation .
This paper was completed when the second and forth authors visited the Department of Mathematical Analysis of the University of Santiago de Compostela. The authors would like to thank the department for its hospitality and support. The authors are grateful to the referee for carefully reading the paper.
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