Properties of the solutions set for a class of nonlinear evolution inclusions with nonlocal conditions
© Zhang et al.; licensee Springer. 2013
Received: 23 October 2012
Accepted: 18 January 2013
Published: 5 February 2013
In this paper, we consider the nonlocal problems for nonlinear first-order evolution inclusions in an evolution triple of spaces. Using techniques from multivalued analysis and fixed point theorems, we prove existence theorems of solutions for the cases of a convex and of a nonconvex valued perturbation term with nonlocal conditions. Also, we prove the existence of extremal solutions and a strong relaxation theorem. Some examples are presented to illustrate the results.
MSC:34B15, 34B16, 37J40.
where is a nonlinear map, is a bounded linear map, is a continuous map and is a multifunction to be given later. Concerning the function φ, appearing in the nonlocal condition, we mention here four remarkable cases covered by our general framework, i.e.:
, where are arbitrary, but fixed and .
Many authors have studied the nonlocal Cauchy problem because it has a better effect in the applications than the classical initial condition. We begin by mentioning some of the previous work done in the literature. As far as we know, this study was first considered by Byszewski. Byszewski and Lakshmikantham [1, 2] proved the existence and uniqueness of mild solutions for nonlocal semilinear differential equations when F is a single-valued function satisfying Lipschitz-type conditions. The fully nonlinear case was considered by Aizicovici and Lee , Aizicovici and McKibben , Aizicovici and Staicu , García-Falset , García-Falset and Reich , and Paicu and Vrabie . All these studies were motivated by the practical interests of such nonlocal Cauchy problems. For example, the diffusion of a gas through a thin transparent tube is described by a parabolic equation subjected to a nonlocal initial condition very close to the one mentioned above, see . For the nonlocal problems of evolution equations, in , Ntouyas and Tsamatos studied the case with compactness conditions. Subsequently, Byszewski and Akca  established the existence of a solution to functional-differential equations when the semigroup is compact and φ is convex and compact on a given ball. In , Fu and Ezzinbi studied neutral functional-differential equations with nonlocal conditions. Benchohra and Ntouyas  discussed second-order differential equations under compact conditions. For more details on the nonlocal problem, we refer to the papers of [14–18] and the references therein.
It is worth mentioning that many of these documents assume that a nonlocal function meets certain conditions of compactness and A is a strongly continuous semigroup of operators or accretive operators in studying the evolution equations or inclusions with nonlocal conditions. However, one may ask whether there are similar results without the assumption on the compactness or equicontinuity of the semigroup. This article will give a positive answer to this question. The works mentioned above mainly establish the existence of mild solutions for evolution equations or inclusions with nonlocal conditions. However, in the present paper, we consider the cases of a convex and of a nonconvex valued perturbation term in the evolution triple of spaces (). We assume the nonlinear time invariant operator A to be monotone and the perturbation term to be multivalued, defined on with values in (not in H). We will establish existence theorems of solutions for the cases of a convex and of a nonconvex valued perturbation term, which is new for nonlocal problems. Our approach will be based on the techniques and results of the theory of monotone operators, set-valued analysis and the Leray-Schauder fixed point theorem.
We pay attention to the existence of extreme solutions  that are not only the solutions of a system with a convexified right-hand side, but also they are solutions of the original system. We prove that, under appropriate hypotheses, such a solution set is dense and codense in the solution set of a system with a convexified right-hand side (‘bang-bang’ principle). Our results extend those of  and are similar to those of  in an infinite dimensional space. Furthermore, the process of our proofs is much shorter, and our conditions are more general. Finally, some examples are also given to illustrate the effectiveness of our results.
The paper is divided into five parts. In Section 2, we introduce some notations, definitions and needed results. In Section 3, we present some basic assumptions and main results, the proofs of the main results are given based on the Leray-Schauder alternative theorem. In Section 4, the existence of extremal solutions and a relaxation theorem are established. Finally, two examples are presented for our results in Section 5.
for all .
It is well known that is a complete metric space and is a closed subset of it. When Z is a Hausdorff topological space, a multifunction is said to be h-continuous if it is continuous as a function from Z into .
Let . By , we denote the Lebesgue-Bochner space equipped with the norm , . A set is said to be ‘decomposable’ if for every and for every measurable, we have .
Let H be a real separable Hilbert space, V be a dense subspace of H having structure of a reflexive Banach space, with the continuous embedding , where is the topological dual space of V. The system model considered here is based on this evolution triple. Let the embedding be compact. Let denote the pairing of an element and an element . If , then , where is the inner product on H. The norm in any Banach space X will be denoted by . Let be such that . We denote by X. Then the dual space of X is and is denoted by . For p, q satisfying the above conditions, from reflexivity of V that both X and are reflexive Banach spaces (see Zeidler , p.411]).
Define , where the derivative in this definition should be understood in the sense of distribution. Furnished with the norm , the space becomes a Banach space which is clearly reflexive and separable. Moreover, embeds into continuously (see Proposition 23.23 of ). So, every element in has a representative in . Since the embedding is compact, the embedding is also compact (see Problem 23.13 of ). The pairing between X and is denoted by . By ‘⇀’ we denote the weakly convergence. The following lemmas are still needed in the proof of our main theorems.
Lemma 2.1 (see )
the set is unbounded;
the has a fixed point, i.e., there exists such that .
Let X be a Banach space and let be the Banach space of all functions which are Bochner integrable. denotes the collection of nonempty decomposable subsets of . Now, let us state the Bressan-Colombo continuous selection theorem.
Lemma 2.2 (see )
Let X be a separable metric space and let be a lower semicontinuous multifunction with closed decomposable values. Then F has a continuous selection.
Let X be a separable Banach space and be the Banach space of all continuous functions. A multifunction is said to be Carathéodory type if for every , is measurable, and for almost all , is h-continuous (i.e., it is continuous from X to the metric space , where h is a Hausdorff metric).
Let , a multifunction is called integrably bounded on M if there exists a function such that for almost all , . A nonempty subset is called σ-compact if there is a sequence of compact subsets such that . Let be such that is dense in M and σ-compact. The following continuous selection theorem in the extreme point case is due to Tolstonogov .
Lemma 2.3 (see )
Let the multifunction be Carathéodory type and integrably bounded. Then there exists a continuous function such that for almost all , if , then , and if , then .
3 Main results
where is a nonlinear map, is a bounded linear map, is a continuous map and is a multifunction satisfying some conditions mentioned later.
where , for all and almost all .
We will need the following hypotheses on the data problem (3.1).
- (ii)for each , the operator is uniformly monotone and hemicontinuous, that is, there exists a constant (independent of t) such that
there exist a constant , a nonnegative function and a nondecreasing continuous function such that for all , a.e. on I;
- (iv)there exist , , such that
is graph measurable;
for almost all , is LSC;
- (iii)there exist a nonnegative function and a constant such that
is a bounded linear self-adjoint operator such that for all , a.e. on I;
- (ii)there exists a continuous function such that
for some constants and all .
For the proofs of main results, we need the following lemma.
Lemma 3.1 Let be an evolution triple and let , where and . Then the linear operator defined by (3.2) is maximal monotone.
Choose a set of functions in H such that as . For , let , then . By (3.3), we have as . Hence, . This completes the proof. □
Theorem 3.1 If hypotheses (H1), (H2) and (H3) hold, the problem (3.1) has at least one solution.
Proof The process of proof is divided into four parts.
has only one solution.
Invoking the Banach fixed point theorem, the operator P has only one fixed point , i.e., is the uniform solution of (3.4).
Therefore, we define as and . By Step 1, we have is one-to-one and surjective, and so is well defined.
Step 2. is completely continuous.
We only need to show that is continuous and maps a bounded set into a relatively compact set. We claim that is continuous. In fact, let such that as . From (H1)(ii) and (H3), we infer that , , , a.e. I as . Obviously, . Therefore, is continuous and is continuous.
with . So, there exists an such that . Because of the boundedness of operators A, B, we obtain that there exists an such that . Hence, for some constant . Therefore, we have is bounded in . But is compactly embedded in . Therefore, is relatively compact in .
Let be a multivalued Nemitsky operator corresponding to F and was defined by a.e. on I.
Step 3. has nonempty, closed, decomposable values and is LSC.
Therefore, and this proves the LSC of . By Lemma 2.2, we obtain a continuous map such that . To finish our proof, we need to solve the fixed point problem: .
Since the embedding is compact, the embedding is compact. That is, in whenever in . By using the above relation and the continuity of f, we have in whenever in . So, is compact.
Step 4. We claim that the set is bounded.
for all .
It follows from (3.14) that for some constant . Hence, Γ is a bounded subset of . So, Γ is a bounded subset of since the embedding is compact.
Hence, x is a solution of (3.1). The proof is completed. □
Next, we consider the convex case, the assumption on F is as follows:
is graph measurable;
for almost all , has a closed graph; and (H2)(iii) hold.
Theorem 3.2 If hypotheses (H1), (H3) and (H4) hold, the problem (3.1) has at least one solution; moreover, the solution set is weakly compact in .
Proof The proof is as that of Theorem 3.1. So, we only present those particular points where the two proofs differ.
the last inclusion being a consequence of the hypothesis (H4)(ii). So, , which means that is nonempty.
then , i.e., is closed in . This proves the upper semicontinuity of from into .
Clearly, , then . Thus, S is weakly compact in . □
4 Relaxation theorem
where denotes the extremal point set of . We need the following hypothesis:
is graph measurable;
for almost all , is h-continuous; and (H2)(iii) holds.
Theorem 4.1 If hypotheses (H1), (H3) and (H5) hold, then the problem (4.1) has at least one solution.
So, in . Since with , we conclude that is compact. From Lemma 2.3, we can find a continuous map such that a.e. on I for all . Then is a compact operator. On applying the Schauder fixed point theorem, there exists an such that . This is a solution of (4.1), and so in . □
For the relation theorem of the problem (4.1), we need the following definition and hypotheses.
Definition 4.1 A Carathéodory function is said to be a Kamke function if it is integrally bounded on the bounded sets, and the unique solution of the differential equation , is .
and (H5) hold.
Theorem 4.2 If hypotheses (H1), (H3) and (H6) hold, then , where the closure is taken in .
Hence, , where and . By (4.7), then . Let , we have . Therefore, , i.e., and , and so . Also, S is closed in (see the proof of Theorem 3.2), thus . □
The p-Laplacian arises in many applications such as Finsler geometry and non-Newtonian fluids. In , Liu showed the existence of anti-periodic solutions to the problem (5.1) where is continuous.
The hypotheses on the data of this problem (5.1) are the following:
() are Nemitsky-measurable, i.e., for all measurable, () is measurable;
- (ii)there exists , , such that
The hypothesis (H7) implies that (H4) is satisfied. Note that is lower semicontinuous, is upper semicontinuous, and so is USC (see , Example 2.8, p.371]). Let , it is easy to check that φ satisfies our hypothesis (H3)(ii). Then, we rewrite equivalently (5.1) as (3.1) , with A and F as above. Finally, we can apply Theorem 3.2 to the problem (5.1) and obtain the following.
Theorem 5.1 If the hypothesis (H7) holds, then the problem (5.1) has a nonempty set of solutions such that .
The hypotheses on the data (5.3) are the following:
is measurable on for every , is continuous on for all almost all ;
with a nonnegative function and for almost all ;
for almost all ;
for all .
for all , is measurable;
for all , is continuous;
- (iii)for almost all and all , we have
with , .
for all , is measurable;
for all , is h-continuous;
for almost all and all , , with .
for all . Let .
Using hypotheses (H9) and (H10), it is straightforward to check that F satisfies the hypothesis (H5).
It is easy to get the following theorem by applying Theorem 4.1 to the problem (5.3).
Theorem 5.2 If hypotheses (H8)-(H10) hold, then the problem (5.3) has one solution with .
The authors are in debt to the anonymous referees whose comments helped them to improve the final version of this paper. This work is partially supported by the National Natural Science Foundation of China (No. 11172036, 11171350, 10902125) and the Natural Science Foundation of Jilin Province Grants 201115133.
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