### Abstract

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### 1 Introduction

In this paper, we examine the following nonlinear nonlocal problem:

where
*φ*, appearing in the nonlocal condition, we mention here four remarkable cases covered
by our general framework, *i.e.*:

•

•

•

•

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 [3], Aizicovici and McKibben [4], Aizicovici and Staicu [5], García-Falset [6], García-Falset 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 functional-differential equations when
the semigroup is compact and *φ* is convex and compact on a given ball. In [12], Fu and Ezzinbi studied neutral functional-differential equations with nonlocal conditions.
Benchohra and Ntouyas [13] 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 (
*A* to be monotone and the perturbation term to be multivalued, defined on
*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 [19] 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 [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 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.

### 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
*X* be a separable Banach space. Denote

Let
*x* to *A* is given by
*σ*-field of *X*. On

for all

It is well known that
*Z* is a Hausdorff topological space, a multifunction
*h*-continuous if it is continuous as a function from *Z* into

Let *Y*, *Z* be Hausdorff topological spaces and
*Y*. A USC multifunction has a closed graph in
*G* is locally compact (*i.e.*, for every
*U* of *y* such that
*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
*Y* as

Let

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
*V*. The system model considered here is based on this evolution triple. Let the embedding
be compact. Let
*H*. The norm in any Banach space *X* will be denoted by
*X*. Then the dual space of *X* is
*p*, *q* satisfying the above conditions, from reflexivity of *V* that both *X* and

Define
*X* and

**Lemma 2.1** (see [24])

*If**X**is 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

**Lemma 2.2** (see [25])

*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
*h*-continuous (*i.e.*, it is continuous from *X* to the metric space
*h* is a Hausdorff metric).

Let
*M* if there exists a function
*σ*-compact if there is a sequence
*M* and *σ*-compact. The following continuous selection theorem in the extreme point case is
due to Tolstonogov [26].

**Lemma 2.3** (see [26])

*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

where

**Definition 3.1** A function

where

We will need the following hypotheses on the data problem (3.1).

(H1)

(i)

(ii) for each
*that is*, there exists a constant
*t*) such that

for all

(iii) there exist a constant
*I*;

(iv) there exist

or

(H2)

(i)

(ii) for almost all

(iii) there exist a nonnegative function

where

(H3)

(i)
*I*;

(ii) there exists a continuous function

and

It is convenient to rewrite the system (3.1) as an operator equation in

It follows from Theorem 30.A of Zeidler [23] that the operator

for some constants

We define

where
*u*, *i.e.*,

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*.

*Proof* In the sequel we will show that *L* is maximal monotone. To prove this, suppose that

We have to show that
*i.e.*,
*u*, we choose

By the arbitrariness of *z*, one has that

Hence,

Choose a set of functions
*H* such that

**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.

*Step 1.* We claim that the equation

has only one solution.

Firstly, for every

has only one solution. By (H1) and (H3), it is easy to check that
*L* is a linear maximal monotone operator. Therefore,
*i.e.*,

By (3.5), we have

for all

By (H1)(ii), we have

Hence,

Invoking the Banach fixed point theorem, the operator *P* has only one fixed point
*i.e.*,

Therefore, we define

*Step 2.*

We only need to show that
*I* as

Let
*a priori* bound in

It follows that

By (H1)(iv),

or

with

Therefore,

or

with
*A*, *B*, we obtain that there exists an

Let
*F* and
*I*.

*Step 3.*

The closedness and decomposability of the values of
*I*. By the hypothesis (H2)(iii),

(see Hiai and Umegaki [27] Theorem 2.2). We will show that for every
*I* as

Therefore,

Since the embedding
*f*, we have

*Step 4.* We claim that the set

Let

By (H1)(iv) and (H3)(i), one has that

or

with

where

If

Thus, by virtue of the inequalities (3.12) and (3.13), we can find a constant
*A*, *B* and *f*, and the continuous embedding

for all

It follows from (3.14) that

Invoking the Leray-Schauder theorem, one has that there exists an
*i.e.*,

Let

By Step 3, we have that
*A* is hemicontinuous and monotone and *B* is a continuous linear operator, thus
*I*. Since

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)

(i)

(ii) for almost all

**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.

In this case, the multivalued Nemistsky operator
*H* and
*I*. Then by virtue of the hypothesis (H4)(i), for every

the last inclusion being a consequence of the hypothesis (H4)(ii). So,

Next, we show that

is closed. Let
*I*. Let

So, by the Dunford-Pettis theorem, we may assume that

then
*i.e.*,

We consider the following fixed point problem:

Recalling that

is bounded, as a proof of Theorem 3.1. Invoking the Leray-Schauder fixed point theorem,
one has that there exists an
*i.e.*,

Let

By Step 3,
*I*. Since

Hence, evidently *x* is a solution of (3.1). As in the proof of Theorem 3.1, we have that

Clearly,
*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

(H5)

(i)

(ii) for almost all
*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*.

*Proof* Since
*a priori* bound for
*I* for all

then
*i.e.*,
*W*, *W* is uniformly bounded in
*W*, we have

Therefore, the sequence
*H* for all
*m* being the Lebesgue measure on *R*). Since *A* is hemicontinuous and monotone and *B* is a continuous linear operator, thus
*I* and

Taking the inner product above with
*T*, one can see that

By the hypothesis (H1)(iii), it follows that

So, we can find

Using the integration by parts formula for functions in

By (4.5), we see that

So,
*I* for all

For the relation theorem of the problem (4.1), we need the following definition and hypotheses.

**Definition 4.1** A Carathéodory function

(H6) For each

and (H5) hold.

**Theorem 4.2***If hypotheses* (H1), (H3) *and* (H6) *hold*, *then*
*where the closure is taken in*

*Proof* Let
*I* such that

As before, let

Clearly, for every
*I*. So, we define the multifunction

We see that
*I*, and
*I* with

where

However,

Then

By

Hence, there exists a constant

as