Nonlocal evolution inclusions under weak conditions

We study evolution inclusions given by multivalued perturbations of m-dissipative operators with nonlocal initial conditions. We prove the existence of solutions. The commonly used Lipschitz hypothesis for the perturbations is weakened to one-sided Lipschitz ones. We prove an existence result for the multipoint problems that cover periodic and antiperiodic cases. We give examples to illustrate the applicability of our results.

Notice that problem (1.1) has a number of applications, since it describes many phenomena better than the local initial value problems. We refer to the recent interesting book [6], where (1.1) is comprehensively studied. See also [2], where the authors assume that X * is uniformly convex (as in the present paper) and A generates a compact semigroup. Nonautonomous A is also studied. In [21] the author considered the case where A is linear. In [16], it is assumed that A is of complete continuous type and generates a compact semigroup. The conditions on F(·, ·) and g(·) are mild, and the problem is studied in arbitrary (separable) Banach spaces. In fact the commonly assumptions used to prove the existence of solutions are either that A is of compact type (in particular, it generates a compact semigroup if X * is uniformly convex) and F satisfies some upper semicontinuity, or F is Lipschitz continuous. In [22] the authors assumed that X * is uniformly convex, and two cases are considered: one where A generates an equicontinuous semigroup and F(·, ·) is Lipschitz w.r.t. the Hausdorff measure of noncompactness, and the second is where A is m-dissipative and F(s, ·) is Lipschitz. In the recent paper [1], the existence of solutions of (1.1) is proved when F(s, ·) is Lipschitz with nonempty closed bounded values in general Banach spaces. The results of [1,22] are not applicable in the important case of periodic or antiperiodic boundary conditions, that is, g(y) = y(T) or g(y) = -y(T). This problem is especially studied in the present paper. The periodic or antiperiodic boundary conditions were studied in the literature when A generates a compact semigroup (see [6]).
In the present paper, we first prove the existence of solutions to problem (1.1), assuming that F(s, ·) is one-sided Lipschitz, which is weaker than the commonly used Lipschitz condition. Further, we relax the growth conditions on g(·) used in [22]. Our assumptions are weaker and more flexible. Notice also that even if F(s, ·) is Lipschitz, the one-sided Lipschitz constant (or function) is in general smaller (even negative) than the Lipschitz one.
Moreover, for the function g appearing in the nonlocal condition, we consider the particular instance g(y) = k i=1 α i y(t i ), where t 1 , t 2 , . . . , t k ∈ I are arbitrary but fixed, and k i=1 |α i | ≤ 1, which covers the remarkable periodic and antiperiodic cases. For this specific case of (1.1), we obtain an existence result under a one-sided Lipschitz condition with negative constant on F. Notice that there exist non-Lipschitz multifunctions, which are one-sided Lipschitz with respect to a negative constant.
To prove the existence of solutions to (1.1), we consider the corresponding local Cauchy problem ⎧ ⎨ ⎩ẏ (s) ∈ Ay(s) + F(s, y(s)), for which the existence of the solutions and some properties of the solution set are discussed in [5,9,10,12]. Here we provide estimates on dependence of the solution set of (1.2) on the initial conditions, which will be used to get our existence result for the nonlocal problem.
In the end of the paper, we give two examples to demonstrate the applicability of our results.

Preliminaries
We start this section by giving the notation and the main definitions used further in this paper. Also, we recall some known results, which will be used in the next sections.
For any nonempty closed bounded subset C of X and l ∈ X * , we denote by σ (l, C) = sup a∈C l, a the support function, where ·, · is the duality pairing. Denote by J(x) = {z ∈ X * ; z, x = |z| 2 = |x| 2 } the duality map. Since X * is uniformly convex, then J(·) is single-valued and uniformly continuous on the bounded sets (see, e.g., [3]). We denote by Ω J (r) = sup{|J(x) -J(y)|; |x -y| ≤ r, x, y ∈ X} its modulus of continuity. We define dist(x, A) = inf a∈A |x -a|, the distance from x ∈ X to A ⊂ X. The Hausdorff distance between two subsets A and B of X is defined by A multimap G : X ⇒ X is called hemicontinuous (upper hemicontinuous) if for every l ∈ X * , the support function σ (l, G(·)) is continuous (upper semicontinuous) as a realvalued function. A multifunction F : I × X ⇒ X is said to be almost upper hemicontinuous if for every ε > 0, there exists a compact I ε ⊂ I with meas(I \ I ε ) < ε such that F| I ε ×X is upper hemicontinuous.
It is well known that (2.1) has a unique solution.
We need the following theorem, which is a reformulation of [3], Theorem 4.1.
We state the standard assumptions of our paper.
(F1) F(·, ·) has nonempty convex weakly compact values, and there exists a Lebesgue integrable function λ(·) such that F(t, is measurable, and F(t, ·) is hemicontinuous. Regarding the existence of ε-solutions, we recall the following result proved in [10].
In the end of this section, we give the following lemma, which is a simplified version of [9], Lemma 1.

An existence result
In this section, we prove the first main result of this paper, namely the existence of solutions for the nonlocal problem (1.1). We add another assumption on F, much weaker than the Lipschitz continuity, called one-sided Lipschitz condition.
(F3) There exists a Lebesgue-integrable function L : I → R + such that, for all x, y ∈ X and t ∈ I, Then the nonlocal problem (1.1) has at least a solution.
To prove the theorem, we need some auxiliary results.
A version of this lemma was proved in [9], Lemma 2. Our assumptions are however stronger and allow us to obtain a more relevant result to the problems considered here.
By Lemma 2.6 there exists a (strongly) measurable f y (·) such that f y (t) ∈ F(t, y τ (τ )) and for a.e. t ∈ [t 0 , T]. Therefore, we get the following inequality Since On another hand, Hence It follows that for some positive constant c 2 . Clearly, We have used the fact that exp(2 t t 0 L(s) ds) is bounded. Applying now Zorn's lemma we get the existence of the δ-solution y(·) on the whole interval I.
The next result is a variant of the well-known lemma of Filippov and Plis.
For any positive number n, let f n (·) ∈ L 1 (I, X) with f n (t) ∈ F(t, x n (t) + ε n B) for a.e. t ∈ I be such that x n (·) is a solution ofẋ(t) ∈ Ax(t) + f n (t). Due to the growth condition (F1), the sequence (f n (·)) is integrally bounded and hence L 1 -weakly precompact. Passing to subsequences if necessary, we get that (f n (·)) converges L 1 -weakly to some function f (·) ∈ L 1 (I, X). By [5], Prop. 1, we get that z(·) is a solution of (1.2) with z(t 0 ) = x 0 .
Let n ∈ N be such that |x n (t)z(t)| < η/2 for any t ∈ I. We have that for any t ∈ I. Since η is arbitrary, we get the conclusion.
The following theorem is crucial in the proof of the main result.

Theorem 3.4 Assume (F1)-(F3).
For any x 0 , y 0 ∈ D(A), The proof is very similar to the proof of Corollary 1 in [9] and is omitted.

A multipoint problem
Our target now is to investigate inclusion (1.1) with a particular choice of the function g. More precisely, we consider the nonlocal problem ⎧ ⎨ ⎩ẏ (s) ∈ Ay(s) + F(s, y(s)), s ∈ I, where t 0 < t 1 < · · · < t k ≤ T are arbitrary but fixed, and α i ∈ R with k i=1 |α i | = κ ≤ 1.
Clearly, we can apply Theorem 3.1 to this problem in the case where κ < 1. However, this theorem is not applicable for κ = 1, since, in this case, (3.1) does not hold. We mention that the case κ = 1 includes periodic and antiperiodic boundary conditions, that is, y(t 0 ) = ±y(T).
We further provide an existence result the problem (4.1) that covers also the case κ = 1. To this aim, we assume the following stronger form of condition (F3).
(F3 ) There exists a positive constant m such that, for all x, y ∈ X and t ∈ I, To prove this theorem, we need the following lemma, which is used implicitly in [9] when the right-hand side is autonomous (see the proof of Theorem 4 therein).
Note that in [9], Lemma 2, we assumed that A generates an equicontinuous semigroup. This fact, however, is not used in the proof there. Now we are ready to prove the second main result of this paper.
Proof of Theorem 4.1 We will use the successive approximations method. We start with a point x 0 ∈ D(A) and let y 0 (·) be a solution of the local problem ⎧ ⎨ ⎩ẏ (t) ∈ Ay(t) + F(t, y(t)), For the existence of such a solution, see, for example, [9]. Consider now the problem ⎧ ⎨ ⎩ẏ (t) ∈ Ay(t) + F(t, y(t)), (4.5) Letᾱ =ᾱ(t 1 ) given by Lemma 4.2. Furthermore, there exists a solution y 1 (·) of (4.5) such that for all t ≥ t 1 .

Remark 4.3
Using more carefully the estimations, we can prove the conclusion of Theorem 4.1 when κ > 1 and k i=1 e -m(t i -t 0 ) |α i | < 1.

Examples
In this section we give two examples to apply the abstract results to partial differential inclusions. The first one, inspired by [13], Section 5, illustrates the applicability of Theorem 3.1.
We suppose that the following hypotheses are satisfied.
Example 5.3 Let Ω, ϕ, and ψ be as in the previous example, and let G : [0, T] × R 2 ⇒ R 2 be a given multifunction. We consider the following system: We suppose that the multifunction G satisfies the following conditions: (G1) there exist a(·), b(·) ∈ L 1 (0, T) such that G(t, z) ≤ a(t) + b(t)|z| for all (t, z) ∈ [0, T] × X; (G2) G is almost upper semicontinuous with nonempty closed convex values; (G3) G(t, ·) is one-sided Lipschitz with negative constant. Under these hypotheses, it is easy to prove that (F1), (F2), and (F3 ) hold. Then, due to Theorem 4.1, we obtain the following result.

Concluding remarks
In this paper, we investigate the nonlocal problem (1.1) and prove two existence results.
The first one extends Theorem 4.1 of [22] in several directions. We recall that, in [22], the authors established an existence result for the nonlocal differential inclusion (1.1) assuming that X is separable with uniformly convex dual, F(·, x) is measurable, F(t, ·) is Lipschitz with the Lipschitz function p(·) ∈ L 1 (I, R + ), and K + T t 0 p(s) ds < 1, (6.1) where K is the Lipschitz constant of g(·). Our condition (3.1) is weaker than (6.1), as can be seen from [1], Lemma 2.7. In fact, this condition (3.1) was used in [1] to prove an existence result for the nonlocal problem (1.1) in general Banach spaces under the assumption that F(t, ·) is Lipschitz continuous. Clearly, the one-sided Lipschitz condition assumed in this paper is much weaker than the Lipschitz one, and, moreover, if F(t, ·) is p(t) Lipschitz, then it is L(t) one-sided Lipschitz with L(t) ≤ p(t). We further give two simple examples of maps that are one-sided Lipschitz but not Lipschitz. We also refer the reader to [11], where the advantages of the one-sided Lipschitz condition are shown.
Example 6.1 Let X be a Hilbert space. We define the map Clearly, f (·) is continuous and one-sided Lipschitz with the constant 0, but it is not Lipschitz.
Another example is in X = L 3 (Ω), where Ω ⊂ R n is a bounded domain. The dual space is X * = L 3/2 (Ω), and the duality map is for a.e. ω ∈ Ω and for all x ∈ L 3 (Ω). The map F : L 3 (Ω) ⇒ L 3 (Ω) given by is upper semicontinuous at 0 and one-sided Lipschitz with the constant -1, but it is not Lipschitz, and even discontinuous at 0.
The second main result of this paper is devoted to the so-called multipoint problem. Note that this problem cannot be studied under the assumptions of Theorem 3.1. Such a kind of problems is studied in the literature under compactness-type assumptions or under stronger assumptions on the right-hand side. We refer the reader to [6]. Another approach is assuming that F(t, ·) is m-Lipschitz, A is m-dissipative, and, moreover, A + λI is dissipative with m < λ. We can see that our assumptions are more clear than the assumptions in [6] (although their results are applicable in more general Banach spaces).