Abstract
In this paper, we consider the nonlocal problems for nonlinear firstorder 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.
Keywords:
evolution inclusions; nonlocal conditions; LeraySchauder alternative theorem; extremal solutions1 Introduction
In this paper, we examine the following nonlinear nonlocal problem:
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 singlevalued function satisfying Lipschitztype conditions. The fully nonlinear case was considered by Aizicovici and Lee [3], Aizicovici and McKibben [4], Aizicovici and Staicu [5], GarcíaFalset [6], GarcíaFalset and Reich [7], and Paicu and Vrabie [8]. 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 [9]. For the nonlocal problems of evolution equations, in [10], Ntouyas and Tsamatos studied the case with compactness conditions. Subsequently, Byszewski and Akca [11] established the existence of a solution to functionaldifferential equations when the semigroup is compact and φ is convex and compact on a given ball. In [12], Fu and Ezzinbi studied neutral functionaldifferential equations with nonlocal conditions. Benchohra and Ntouyas [13] discussed secondorder differential equations under compact conditions. For more details on the nonlocal problem, we refer to the papers of [1418] 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, setvalued analysis and the LeraySchauder fixed point theorem.
We pay attention to the existence of extreme solutions [19] that are not only the solutions of a system with a convexified righthand 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 righthand side (‘bangbang’ principle). Our results extend those of [20] and are similar to those of [21] 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 LeraySchauder 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.
2 Preliminaries
In this section we recall some basic definitions and facts from multivalued analysis which we will need in what follows. For details, we refer to the books of Hu and Papageorgiou [22] and Zeidler [23]. Let , be the Lebesgue measurable space and X be a separable Banach space. Denote
Let , , then the distance from x to A is given by . A multifunction is said to be measurable if and only if, for every , the function is measurable. A multifunction is said to be graph measurable if with being the Borel σfield of X. On we can define a generalized metric, known in the literature as the ‘Hausdorff metric’, by setting
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 hcontinuous if it is continuous as a function from Z into .
Let Y, Z be Hausdorff topological spaces and . We say that is ‘upper semicontinuous (USC)’ (resp., ‘lower semicontinuous (LSC)’) if for all nonempty closed, (resp., ) is closed in Y. A USC multifunction has a closed graph in , while the converse is true if G is locally compact (i.e., for every , there exists a neighborhood U of y such that is compact in Z). A multifunction which is both USC and LSC is said to be ‘continuous’. If Y, Z are both metric spaces, then the above definition of LSC is equivalent to saying that for all , is upper semicontinuous as valued function. Also, lower semicontinuity is equivalent to saying that if in Y as , then
Let . By , we denote the LebesgueBochner 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 [23], 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 [23]). So, every element in has a representative in . Since the embedding is compact, the embedding is also compact (see Problem 23.13 of [23]). 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 [24])
IfXis a Banach space, is nonempty, closed and convex withandis an upper semicontinuous multifunction which maps bounded sets into relatively compact sets, then one of the following statements are true:
(ii) thehas a fixed point, i.e., there existssuch 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 BressanColombo continuous selection theorem.
Lemma 2.2 (see [25])
LetXbe a separable metric space and letbe a lower semicontinuous multifunction with closed decomposable values. ThenFhas 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 hcontinuous (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 σcompact if there is a sequence of compact subsets such that . Let be such that is dense in M and σcompact. The following continuous selection theorem in the extreme point case is due to Tolstonogov [26].
Lemma 2.3 (see [26])
Let the multifunctionbe Carathéodory type and integrably bounded. Then there exists a continuous functionsuch that for almost all, if , then, and if, then.
3 Main results
Let , consider the following evolution inclusions:
where is a nonlinear map, is a bounded linear map, is a continuous map and is a multifunction satisfying some conditions mentioned later.
Definition 3.1 A function is called a solution to the problem (3.1) iff
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
for all , and the map is continuous on for all ;
(iii) 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
or
(H2) is a multifunction such that
(iii) there exist a nonnegative function and a constant such that
(H3)
(i) is a bounded linear selfadjoint operator such that for all , a.e. on I;
(ii) there exists a continuous function such that
It is convenient to rewrite the system (3.1) as an operator equation in . For , we get
It follows from Theorem 30.A of Zeidler [23] that the operator is bounded, monotone, hemicontinuous and coercive. By using the same technique, one can show that the operator is bounded and satisfies
We define
where stands for the generalized derivative of u, i.e.,
For the proofs of main results, we need the following lemma.
Lemma 3.1Letbe an evolution triple and let, whereand. Then the linear operatordefined by (3.2) is maximal monotone.
Proof In the sequel we will show that L is maximal monotone. To prove this, suppose that and
We have to show that and , i.e., . Due to the arbitrariness of u, we choose , where , and . Then , so . From , we obtain that
By the arbitrariness of z, one has that
Hence, . Since , then . It remains to show that . Using the integration by parts formula for functions in (see Zeidler [23], Proposition 23.23), we obtain from (3.2) that
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 3.1If 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.
Step 1. We claim that the equation
has only one solution.
Firstly, for every , and , we claim that the equation
has only one solution. By (H1) and (H3), it is easy to check that is bounded, monotone, hemicontinuous and coercive. Moreover, by Lemma 3.1, L is a linear maximal monotone operator. Therefore, , i.e., is surjective (see [23], p.868]). The uniqueness is clear. Hence, for the Cauchy problem (3.5) has a unique . By , then the operator is defined as follows:
By (3.5), we have
for all . Take an inner product over (3.5) with , then
By (H1)(ii), we have
Hence,
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 onetoone 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.
Let be a bound set, for any , there is a priori bound in for the possible solution of (3.4). Then
It follows that
By (H1)(iv),
or
Therefore,
or
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.
The closedness and decomposability of the values of are easy to check. For the nonemptiness, note that if , by the hypothesis (H2)(i), is graph measurable, so we apply Aumann’s selection theorem and obtain a measurable map such that a.e. on I. By the hypothesis (H2)(iii), . Thus, for every , . To prove the lower semicontinuity of , we only need to show that every , is a USC valued function. Note that
(see Hiai and Umegaki [27] Theorem 2.2). We will show that for every , the superlevel set is closed in . Let and assume that in . By passing to a subsequence if necessary, we may assume that a.e. on I as . By (H2)(ii), is an upper semicontinuous valued function. So, via Fatou’s lemma, we have
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.
Let , then we have . Note that
By (H1)(iv) and (H3)(i), one has that
or
with . By using the integration by parts formula, we have
where . By (3.9), (3.10) and (3.11), if , then we have
Thus, by virtue of the inequalities (3.12) and (3.13), we can find a constant such that for all . From the boundedness of operators A, B and f, and the continuous embedding , we obtain , and for some constants , , and all . Therefore,
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.
Invoking the LeraySchauder theorem, one has that there exists an such that , i.e., is a solution of the following problem:
Let and . For every , there exists an which is a solution of the following equations:
By Step 3, we have that is uniformly bounded. By the boundedness of the sequence , it follows that the sequence is uniformly bounded and passing to subsequence if necessary, we may assume that in . Evidently, and in . Since the embedding is compact, then in . Hence, from the hypothesis (H2)(ii), we obtain and . Since the operator A is hemicontinuous and monotone and B is a continuous linear operator, thus , in as . Therefore, we obtain , a.e. on I. Since in and is continuous, then we have
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:
(H4) is a multifunction such that
(ii) for almost all , has a closed graph; and (H2)(iii) hold.
Theorem 3.2If 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.
In this case, the multivalued Nemistsky operator has nonempty closed, convex values in and is USC from into furnished with the weak topology (denoted by ). The closedness and convexity of the values of are clear. To prove the nonemptiness, let and be a sequence of step functions such that in H and a.e. on I. Then by virtue of the hypothesis (H4)(i), for every , admits a measurable selector . From the hypothesis (H4)(iii), we have that , so is uniformly integrable. So, by the DunfordPettis theorem, and by passing to a subsequence if necessary, we may assume that weakly in . Then from Theorem 3.1 in [28], we have
the last inclusion being a consequence of the hypothesis (H4)(ii). So, , which means that is nonempty.
Next, we show that is USC from into . Let Ξ be a nonempty and weakly closed subset of . Obviously, it is sufficient to show that the set
is closed. Let and assume in . Passing to a subsequence, we can get that a.e. on I. Let , . Then by virtue of the hypothesis (H4)(iii), we have
So, by the DunfordPettis theorem, we may assume that in . As before, we have
then , i.e., is closed in . This proves the upper semicontinuity of from into .
We consider the following fixed point problem:
Recalling that is completely continuous, we see that is USC and maps bounded sets into relatively compact sets. We easily check that
is bounded, as a proof of Theorem 3.1. Invoking the LeraySchauder fixed point theorem, one has that there exists an such that , i.e., is a solution of the following problem:
Let and . For every , there exists an which is a solution of the following problem:
By Step 3, is uniformly bounded. By the boundedness of the sequence , it follows that the sequence is uniformly bounded and, passing to subsequence if necessary, we may assume that in . Thus, , in as . Evidently, there exists , by virtue of the hypothesis (H4)(iv), we have that , so is uniformly integrable. So, by the DunfordPettis theorem and by passing to a subsequence if necessary, we may assume that weakly in . Therefore, we obtain , a.e. on I. Since in and is continuous, then we have
Hence, evidently x is a solution of (3.1). As in the proof of Theorem 3.1, we have that , for some . So, is uniformly bounded. So, by the DunfordPettis theorem and by passing to a subsequence if necessary, we may assume that weakly in . As before, we have
4 Relaxation theorem
Now, we prove the existence of extremal solutions and a strong relaxation theorem. Consider the extremal problem of the following evolution inclusion:
where denotes the extremal point set of . We need the following hypothesis:
(H5) is a multifunction such that
(ii) for almost all , is hcontinuous; and (H2)(iii) holds.
Theorem 4.1If hypotheses (H1), (H3) and (H5) hold, then the problem (4.1) has at least one solution.
Proof Since , as in the proof of Theorem 3.1, we obtain a priori bound for . We know that there exists , such that and for all . Let , . We may assume that a.e. on I for all . By Theorem 3.1, let , , then is well defined. So, let
then is a compact convex subset in . Obviously, is convex. We only need to show the compactness. Let , then there exists such that , i.e., . By the definition of W, W is uniformly bounded in . By the DunfordPettis theorem, passing to a subsequence if necessary, we may assume that in for some . From the definition of W, we have
Therefore, the sequence is bounded. Because of the compactness of the embedding , we have that the sequence is relatively compact. So, by passing to a subsequence if necessary, we may assume that in . Moreover, by the boundedness of the sequence , it follows that the sequence is uniformly bounded and, passing to subsequence if necessary, we may assume that in . Since the embedding is continuous and is compact, it follows that in and in . Hence, in H for all , (m being the Lebesgue measure on R). Since A is hemicontinuous and monotone and B is a continuous linear operator, thus , in and as , we obtain a.e. on I and . Note that
Taking the inner product above with and integrating from 0 to T, one can see that
By the hypothesis (H1)(iii), it follows that
Using the integration by parts formula for functions in (see Zeidler [26], Proposition 23.23), for any , we have
By (4.5), we see that
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 .
(H6) For each , there exists a Kamke function such that
and (H5) hold.
Theorem 4.2If hypotheses (H1), (H3) and (H6) hold, then, where the closure is taken in.
Proof Let , then there exist and a.e. on I such that
As before, let , then is a compact convex subset in . For every , we define the multifunction
Clearly, for every , , and it is graph measurable. On applying Aumann’s selection theorem, we get a measurable function such that almost everywhere on I. So, we define the multifunction
We see that has nonempty and decomposable values. It follows from Theorem 3 of [29] that is LSC. Therefore, is LSC and has closed and decomposable values. So, we apply Lemma 2.2 to get a continuous map such that for all . Invoking IITheorem 8.31 of [22] (in [22], p.260]), we can find a continuous map such that almost everywhere on I, and for all . Now, let and set , . Note that a.e. on I with , so we have in . We consider the following problem:
where . We see that is a compact operator and by the Schauder fixed point theorem, we obtain a solution of (4.1). We see that the sequence is uniformly bounded. So, by passing to a subsequence if necessary, we may assume that in . From the proof of Theorem 4.1, we know that in and . Note that . So, we have that
However,
Then
Hence, there exists a constant , one has that
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 . □
5 Examples
As an application of the previous results, we introduce two examples. Let Ω be a bounded domain in with smooth boundary ∂Ω, , . Firstly, consider the following nonlinear evolution equation with a discontinuous righthand side:
The pLaplacian arises in many applications such as Finsler geometry and nonNewtonian fluids. In [30], Liu showed the existence of antiperiodic solutions to the problem (5.1) where is continuous.
Since is not continuous, the problem (5.1) need not have solutions. To obtain an existence theorem for (5.1), we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points of . So, we introduce the functions and defined by
and
Set
Then, instead of (5.1), we study the following multivalued nonlinear evolution inclusion:
The hypotheses on the data of this problem (5.1) are the following:
(H7)
(i) () are Nemitskymeasurable, i.e., for all measurable, () is measurable;
(ii) there exists , , such that
In this case, the evolution triple is , and . From the Sobolev embedding theorem, we see that all embeddings are compact. Let us define the following operator on V:
By the monotone property of pLaplacian, it is easy to verify that A satisfies our hypothesis (H1). Let be defined by
The hypothesis (H7) implies that (H4) is satisfied. Note that is lower semicontinuous, is upper semicontinuous, and so is USC (see [22], 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 5.1If the hypothesis (H7) holds, then the problem (5.1) has a nonempty set of solutionssuch that.
Secondly, we present an example of a quasilinear distributed parameter control system, with a priori feedback (i.e., state dependent control constraint set). So, let and be a bounded domain with boundary Γ. Let , , . We consider the following control system:
The hypotheses on the data (5.3) are the following:
(i) is measurable on for every , is continuous on for all almost all ;
(ii) with a nonnegative function and for almost all ;
(H9) The function satisfies the following:
(iii) for almost all and all , we have
(H10) is a multifunction such that
(ii) for all , is hcontinuous;
(iii) for almost all and all , , with .
Let , , . Then is an evolution triple with compact embeddings. Let , be the operators defined by
Evidently, using the hypothesis (H8) , it is straightforward to check that A, B, φ satisfy hypotheses (H1), (H3). Also, let be defined by
Using hypotheses (H9) and (H10), it is straightforward to check that F satisfies the hypothesis (H5).
Rewrite the problem (5.3) in the following equivalent evolution inclusion form:
It is easy to get the following theorem by applying Theorem 4.1 to the problem (5.3).
Theorem 5.2If hypotheses (H8)(H10) hold, then the problem (5.3) has one solutionwith.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
JZ and YC carried out the main part of this manuscript. CY participated in the discussion and corrected the main theorem. FC provided all examples for our results. All authors read and approved the final manuscript.
Acknowledgements
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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