Cauchy Problems for Functional Evolution Inclusions Involving Accretive Operators

We study the existence and stability of solutions for a class of nonlinear functional evolution inclusions involving accretive operators. Our approach is employing the fixed point theory for multivalued maps and using estimates via the Hausdorff measure of noncompactness.


Introduction
Let X be a Banach space.We consider the following problem u (t) + Au(t) f (t), t ∈ J := [0, T ], (1.1) where the state function u takes values in X, A is an m-accretive operator on X and F is a multivalued function defined on J × X × C([−h, 0]; X).It is known that system (1.1)- (1.3) is an abstract model of many problem involving retarded differential equations and inclusions.In the case when A is a linear operator and F is single-valued, there has been a great literature devoted to studying the global existence and asymptotic behavior of solutions.Regarding the latter objective, one of the most important and interesting problems is studying the stability of solutions to (1.1)- (1.3).For the stability theory for functional differential equations, see for instance the monographs of Driver [7], Halanay [9] and Hale [10].Since the uniqueness for (1.1)-(1.3) is unavailable, the stability for this problem is a quite large subject.In the present paper, we will touch only the initial data dependence of the solution set and the exponential stability of the zero solution of problem (1.1)-(1.3)after proving its global solvability.
Nowadays, the evolution inclusions associated with m-accretive operators and nonlinear perturbations are getting more attractive.There are many works studying such problems with/without delays and subject to standard/nonlocal initial conditions.Let us quote in this note some significant results in [2,12,15,17], among others.In most cases, the authors of the mentioned papers assume that −A generates a compact semigroup.This assumption is then utilized to prove some compactness properties of the solution operator (whose fixed points are desired solutions).The reader is also referred to [5,6] for some generalized cases of undelayed evolution problems with accretive operators.Precisely, in [5] some range conditions were imposed on A instead of m-accretive property while in [6], in dealing with reaction-diffusion systems, reaction terms (nonlinearities) were supposed to be merely measurable.In this paper, by using the techniques presented by Bothe in [4], we treat (1.1)- (1.3) in the case that the semigroup generated by −A is equicontinuous only.The latter case, in particular, makes sense when A is in the form of the subdifferential of a proper, convex and lower semicontinuous functional Φ so that the level set H R = {x ∈ X : ||x|| 2 + Φ(x) ≤ R} is not compact (see, e.g.[16]).To deal with the case of a noncompact semigroup, we impose a regular condition on the multivalued nonlinearity F expressed by measures of noncompactness (MNCs) in order to employ the technique of MNC estimates.Under this condition, we first prove that (1.1)-(1.3) is globally solvable for any T > 0 in Section 3. It is worth noting that, our existence result, in part, extends the one obtained by Bothe [4].In Section 4, since the solution set is compact, we show that it depends semicontinuously on the initial data.Furthermore, under some additional assumptions, we prove that the zero solution of (1.1)-(1.3) is exponentially stable by using Halanay's inequality.In comparison with [4], the retarded case in our problem needs more sophisticated MNC estimates.The last section is an application of obtained results for a concrete problem, namely, the doubly nonlinear boundary problem with delays.

Preliminaries
Let (X, .) be a Banach space and P(X) the collection of all nonempty subsets of X.For x, y ∈ X, h ∈ R\{0}, the following product exists and satisfies (see [3]) Here and in the sequel, we write (x, y) ∈ A if x ∈ D(A) and y ∈ Ax.An accretive operator A is said to be m-accretive if R(I + λA) = X for all λ > 0. If, in addition, A − ωI is accretive for ω ∈ R, we say that A is ω-m-accretive.
Consider the Cauchy problem where f ∈ L 1 (J; X) and u 0 ∈ D(A) given.A function u : for each s, t ∈ J, s ≤ t.
Denote by {S(t)} t≥0 the semigroup generated by −A, that is S(t) : D(A) → D(A), S(t)u 0 = u(t, u 0 , 0) being the integral solution of (2.1)-(2.2) with respect to f = 0.The semigroup {S(t)} t≥0 is said to be compact if S(t) is a compact operator for each t > 0. It is called equicontinuous if for each 0 < a < b, S(•)D is an equicontinuous set in C([a, b]; X) for any bounded set D ⊂ X.
We also denote by W the solution map for (2.1)-(2.2) with respect to f for fixed u 0 .That is If Ω ⊂ L 1 (J; X) such that for all f ∈ Ω, f (t) ≤ ν(t) for a.e.t ∈ J, where ν ∈ L 1 (J) then we say that Ω is integrably bounded.
Let B(X) be the collection of all bounded subsets of X.The following function defined on B(X), χ(D) = inf{ : D has a finite -net}, is called the Hausdorff measure of noncompactness (MNC) on X.
Lemma 2.2.Let A be an m-accretive operator on X such that −A generates an equicontinuous semigroup.Then we have where χ is the Hausdorff MNC on X.In addition, if Let E be a Banach space and Y a metric space.Definition 2.1.A multivalued map (multimap) F : Y → P(E) is said to be: The following facts will be used.
Lemma 2.3 ([11, Theorem 1.1.12]).Let X and Y be metric spaces and G : X → P(Y ) a closed quasi-compact multimap with compact values.Then G is u.s.c.

Lemma 2.4 ([4, Proposition 2])
. Let E be a Banach space and Ω be a nonempty subset of another Banach space.Assume that G : Ω → P(E) is a multimap with weakly compact and convex values.Then G is weakly u.s.c iff {x n } ⊂ Ω with , up to a subsequence.

Existence result
Let us introduce the notations for all (x, y) ∈ A and s, t ∈ J, s ≤ t.
We now define the multioperator F : C ϕ → P(C ϕ ) as follows In order to prove the existence result for problem (1.1)-(1.3),we make use of the following fixed point theorem (see e.g., [8]). is weakly u.s.c for fixed t; for all bounded subsets B ⊂ D A , C ⊂ C h .By using the same arguments as in [4, Theorem 1], one gets the following results.Proposition 3.2.Let the hypotheses (A), (F)(1) and (F)(2) hold.Then the following assertions hold: (1) If X * is uniformly convex then the multioperator F is well-defined, that is P F (u) = ∅ for each u ∈ C ϕ .In addition, P F : C(J; X) → P(L 1 (J; X)) is weakly u.s.c with weakly compact and convex values; (2) The multioperator F has closed contractible values.
We are in a position to state the main result of this section.Proof.Let {S(t)} t≥0 be the semigroup generated by −A and v(t) = S(t)ϕ(0).
where ψ is the solution of the integral equation It is clear that M 0 is a closed convex subset of C ϕ .We first show that F(M 0 ) ⊂ M 0 .Indeed, taking u ∈ M 0 and w ∈ F(u), there exists f ∈ P F (u) such that Noting that Thus w ∈ M 0 .Set here the notation conv stands for the closure of convex hull of a subset in C ϕ .
We see that M k is closed, convex and It follows that M k+1 is equicontinuous for all k ≥ 0. Thus M is equicontinuous as well.In order to apply the Arzelà-Ascoli theorem, we have to prove that M(t) is compact for each t ≥ 0. This will be done if we show that µ k (t) = χ(M k (t)) → 0 as k → ∞.
To verify the last claim, we make use of the fact that (see, e.g.[1]), for Ω ⊂ X, > 0, there exists a sequence ω n ⊂ Ω such that χ(Ω) ≤ 2χ({ω n }) + .Taking EJQTDE, 2013 No. 75, p. 6 {u j } ⊂ M k+1 such that µ k+1 (t) ≤ 2χ({u j (t)}) + , one can choose a sequence v j ∈ M k , f j ∈ P F (v j ) such that u j = W (f j ).Obviously, χ({v j (τ )}), thanks to (F)(3).Hence, by Lemma 2.2, we obtain thanks to (3.3).The last inequality implies Since is arbitrary, we have Observing that the right term of the last inequality is non-decreasing in t, we can write where ν ∞ (t) = lim k→∞ ν k (t) for t ∈ J. Taking into account that M k (0) = {ϕ(0)}, one has µ k (0) = 0 and then ν k (0) = 0 for all k ∈ N.This leads to ν ∞ (0) = 0. Therefore, (3.4) deduces that ν ∞ (t) = 0 for all t ∈ J. Now, since 0 So we have M(t) is compact as desired.Now, consider F : M → P(M).To apply the fixed point principle given by Lemma 3.1, it remains to show that F is u.s.c.By Lemma 2.3, this is the case if Then, by the definition of F, one can take f n ∈ P F (u n ) such that v n = W (f n ).Since P F is weakly u.s.c with weakly compact and convex values (Proposition 3.2), one obtains f n f * ∈ P F (u * ), up to a subsequence (Lemma 2.4).By virtue of Lemma 2.2, we have v * = W (f * ), and thus v * ∈ F(u * ), which completes the proof.
Remark 3.1.In fact, the fixed point set of F is compact.Indeed, let Ω = Fix(F), then Ω ⊂ F(Ω).Assume that {u j } ⊂ Ω, then one can choose f j ∈ P F (u j ) such that u j = W (f j ).By using similar estimates as in the proof of Theorem 3.3 for {u j }, we obtain that {u j } is relatively compact.
On the other hand, if −A generates a compact semigroup on X, then one can drop assumption (F)(3) due to the compactness of W . Indeed, since the subsets M k , k ≥ 1 in the latter proof are compact, the set M is compact as well and we are able to obtain the conclusion of the Theorem easily.

Stability Results
The aim of this section is twofold.We first show that the solution set of (1.1)-( 1.3) semicontinuously depends on the initial data.Then, under some additional conditions, we assert that the zero solution of (1.1)-(1.3) is exponentially stable in the sense of Lyapunov.Let Σ : C h → P(C(J; X)) Obviously, Theorem 4.1.Under assumptions (A) and (F), the solution map Σ defined by (4.1) is u.s.c.
By using Halanay's inequality, one obtains where is the solution of the equation ω − a * = + b * e h .
Remark 4.1.In the case when A is a linear operator such that −A generates an exponentially stable semigroup and F depends on the time and the history state only, i.e.F = F (t, u t ), our condition in (F * ) that a * + b * < ω reduces to the condition b * < ω.This is exactly the result by Travis and Webb [14].

Application
Let Ω be a bounded set in R n with smooth boundary ∂Ω.We consider the doubly nonlinear boundary value problem: where λ is a positive number and ϕ, ψ : R → R are such that EJQTDE, 2013 No. 75, p. 10 • ϕ is proper, convex, lower semicontinuous, ϕ(0) = 0; • ψ is a continuous and convex function and there is C > 0 such that 0 ≤ ψ(s) ≤ C(s 2 + 1), s ∈ R.

Lemma 3 . 1 .
Let E be a Banach space and D ⊂ E be a nonempty compact convex subset.If the multivalued map F : D → P(D) is u.s.c with closed contractible values, then F has a fixed point.Concerning operator A and function F in problem (1.1)-(1.3),we assume that: (A) The operator A is an m-accretive operator such that −A generates an equicontinuous semigroup.(F) The multivalued function F : R + × D A × C h → P c (X) is such that (1) F (•, x, y) has a strongly measurable selection for fixed x, y and F (t, •, •)

Theorem 3 . 3 .
Let the hypotheses (A)and (F) hold.If X * is uniformly convex then problem (1.1)-(1.3)has at least one integral solution for all initial data ϕ ∈ C h .