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.
Keywords:evolution inclusions; nonlocal conditions; Leray-Schauder alternative theorem; extremal solutions
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 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.
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  and Zeidler . 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 σ-field of X. On we can define a generalized metric, known in the literature as the ‘Hausdorff metric’, by setting
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 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 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 )
IfXis a Banach space, is nonempty, closed and convex with and is an upper semicontinuous multifunction which maps bounded sets into relatively compact sets, then one of the following statements are true:
(i) the set is unbounded;
(ii) 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 )
LetXbe a separable metric space and let be 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 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
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).
(H1) is an operator such that
(i) is measurable;
(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
(H2) is a multifunction such that
(i) is graph measurable;
(ii) for almost all , is LSC;
(iii) there exist a nonnegative function and a constant such that
(i) is a bounded linear self-adjoint 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  that the operator is bounded, monotone, hemicontinuous and coercive. By using the same technique, one can show that the operator is bounded and satisfies
for some constants and all .
where stands for the generalized derivative of u, i.e.,
For the proofs of main results, we need the following lemma.
Lemma 3.1Let be an evolution triple and let , where and . Then the linear operator defined 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 , 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 , 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
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.
Let be a bound set, for any , there is a priori bound in for the possible solution of (3.4). Then
It follows that
with . But
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  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
with . By using the integration by parts formula, we have
where . By (3.9), (3.10) and (3.11), if , then we have
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,
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.
Invoking the Leray-Schauder 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
(i) is graph measurable;
(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 Dunford-Pettis theorem, and by passing to a subsequence if necessary, we may assume that weakly in . Then from Theorem 3.1 in , 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 Dunford-Pettis 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 Leray-Schauder 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 Dunford-Pettis 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 Dunford-Pettis theorem and by passing to a subsequence if necessary, we may assume that weakly in . As before, we have
Clearly, , then . Thus, S is weakly compact in . □
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
(i) is graph measurable;
(ii) for almost all , is h-continuous; 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 Dunford-Pettis 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
So, we can find such that
Using the integration by parts formula for functions in (see Zeidler , 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  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 II-Theorem 8.31 of  (in , 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
By in and in , we have that
Hence, there exists a constant , one has that
as . It follows that