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.
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 .
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.
2.1. Multivalued Analysis
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]).
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.
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]).
Lemma 2.6 (see [32, Lemma 3.2]).
2.1.1. Closed Graphs
We recall the following two results; the first one is classical.
Lemma 2.9 (see [33, Proposition 1.2]).
is known as the set of selection functions.
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 ).
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 .
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.
The second one is due to Mazur, 1933.
Lemma 2.16 (Mazur's Lemma, ).
3. Resolvent Family
The following result is a direct consequence of ([16, Proposition 3.1 and Lemma 2.2]).
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. Existence Results
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.
The single-valued version may be stated as follows.
Proof of Theorem 4.3.
We have the following parts.
Part 1: Existence of Solutions
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
4.2. The Convex Case: An MNC Approach
It should be mentioned that these MNC satisfy all above-mentioned properties except regularity.
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]).
Lemma 4.10 (see [36, Theorem 5.2.2]).
Lemma 4.11 (see ).
Lemma 4.12 (see ).
4.2.1. Main Results
We have the following steps.
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.
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 .
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 following lemma is crucial in the proof of our existence theorem.
Lemma 4.20 (see, e.g., ).
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.
Let us introduce the following hypotheses:
Arguing as in Theorem 4.3, we can also prove the following result the proof of which is omitted.
5. Filippov's Theorem
5.1. Filippov's Theorem on a Bounded Interval
We will consider the following two assumptions.
6. The Relaxed Problem
Lemma 6.1 (see ).
Then our main contribution is the following.
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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