Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays

We deal with the Cauchy problem associated with integro-differential inclusions of diffusion-wave type involving infinite delays. Based on the behavior of resolvent operator associated with the linear part, an explicit estimate for solutions will be established. As a consequence, the weak stability of zero solution is proved in case the resolvent operator is asymptotically stable.

1. Introduction. We consider the following problem Au(s)ds + F (t, u t ), t > 0, where the unknown function u takes values in a Banach space (X, · ), A is a closed, linear and unbounded operator, F is a multi-valued map which will be specified in Section 3, B is an admissible phase space that will be defined later.
Here α ∈ (1, 2) and u t stands for the history of the state function up to the time t, i.e. u t (s) = u(t + s), s ≤ 0. It is noted that the linear part of (1) can be rewritten as The last equation was studied in [15,16] in the case A = ∆ (the Laplacian) with the mention that it interpolates the heat equation (α = 1) and the wave equation (α = 2). In addition, the authors in [22,23,24] described the following equation x, u(t, x)), as a model for anomalous diffusion processes and wave propagation in viscoelastic materials. We also refer the reader to [1,6,8,9] for recent results on the existence of the so-called asymptotically almost automorphic, S-asymptotic ω-periodic solutions to diffusion-wave type equations.
In the recent paper [7], we studied (1) in the case of single-valued and obtained the existence of a unique decay solution under the assumption that F is Lipschitzian and the unknown function is subject to impulsive effect and nonlocal condition. Our problem in this paper involves the multi-valued nonlinearity, which derives from control systems with multi-valued feedback ( [21]), and various problems such as regularizing differential equations with discontinuous right-hand side ( [14]), converting differential variational inequalities ( [25]). The main aim of this work is to address, for the first time, weak stability of solutions to (1)- (2). We adopt the following concept of weakly asymptotic stability. Let Σ(ϕ) be the solution set of (1)- (2) with respect to the initial datum ϕ. Assume that 0 ∈ Σ(0), that is (1) admits zero solution. The zero solution of (1) is said to be weakly asymptotically stable if 1. It is stable, i.e. for every ε > 0 there exists δ > 0 such that if |ϕ| B < δ then |u t | B < ε for all u ∈ Σ(ϕ); 2. It is weakly attractive, i.e. for each ϕ ∈ B, there exists u ∈ Σ(ϕ) such that |u t | B → 0 as t → ∞.
Let us give a short description for our approach. By analyzing the behavior of αresolvent {S α (t)} t≥0 generated by the linear part, we construct appropriate solution spaces, on which the solution operator has a fixed point. To this end, we define a suitable measure of noncompactness (MNC) on the solution spaces, in which the solution operator is condensing. It is worth mentioning that the fixed point approach for studying stability of ordinary/functional differential equations was introduced by Burton and Furumochi in [4,5] as an alternative for the Lyapunov functional method. The rest of our paper is organized as follows. In Section 2, we recall the theory of α-resolvent for the linear part of (1) and the concept of phase spaces, using for differential systems with infinite delays. We also give the notion of MNC introduced in [21]. Especially, we define a new MNC on solution spaces C g ([0, ∞); X) for g being a non-decreasing function, which will be used to determine a compact criterion on these spaces. Section 3 deal with the case when the α-resolvent has an exponential growth. We will show that the problem (1)- (2), in this case, has an exponentially bounded solution. In section 4, we prove the weakly asymptotic stability of the zero solution when the α-resolvent is asymptotically stable. The last section is devoted to an application, in which a control problem governed by partial integro-differential equations (PDEs) and multi-valued feedback is considered.

Preliminaries.
2.1. Resolvent operators. Let L(X) be the space of bounded linear operators on X. We recall some notions and results on resolvent operators related to our problem.
Definition 2.1. Let A be a closed and linear operator with domain D(A) on a Banach space X. We say that A is the generator of an α-resolvent if there exists ω ∈ R and a strongly continuous function S α : R + → L(X) such that {λ α : Reλ > ω} ⊂ ρ(A) and It is known that, in the case α = 1, S α (·) = S 1 (·) is a C 0 -semigroup while if α = 2, we have a cosine family S 2 (·). By the Subordination Principle (see [3]), if A generates a β-resolvent with β > α then it also generates an α-resolvent. In particular, if A is the generator of a cosine family, there exists an α-resolvent generated by A with α ∈ (1, 2).
In what follows, for J ⊂ R we denote by L 1 (J; X) the space of functions defined on J, taking values in X, which are integrable in the sense of Bochner. If X = R then we write L 1 (J) for brevity. In addition, by C(J; X) we mean the space of continuous functions v : J → X.
Given T > 0, consider the operator W : L 1 (0, T ; X) → C([0, T ]; X) given by A subset Ω ⊂ L 1 (0, T ; X) is said to be integrably bounded if there exists a function ν ∈ L 1 (0, T ) such that for all f ∈ Ω,  2.2. Phase spaces. In this work, we will deploy the axiomatic definition of the phase space B introduced by Hale and Kato in [18]. The space B is a linear space of functions mapping (−∞, 0] into X endowed with a seminorm | · | B and satisfying the following fundamental axioms. If a function y : is continuous, M is locally bounded, and they are independent of y.
In this work, we need an additional assumption on B: We give here some examples of phase spaces. For more examples, we refer to the book by Hino, Mukarami and Naito [20]. The first one is given by where γ is a positive number. This phase space satisfies (B1)-(B3) with and it is a Banach space with the norm Considering another typical example, suppose that 1 ≤ p < +∞, 0 ≤ r < +∞ and g : (−∞, −r] → R is a nonnegative, Borel measurable function on (−∞, −r). Let CL p g is a class of functions ϕ : (−∞, 0] → X such that ϕ is continuous on [−r, 0] and g(·) ϕ(·) p ∈ L 1 (−∞, −r). A seminorm in CL p g is given by Assume further that −r s g(θ)dθ < +∞, for every s ∈ (−∞, −r) and (8) where G : (−∞, 0] → R + is locally bounded. We know from [20] that if (8)-(9) hold true, then CL p g satisfies (B1)-(B3). Moreover, one can take 2.3. Measures of compactness and condensing multi-valued maps. Let E be a Banach space. Denote We will use the following definition of measure of noncompactness (see, e.g. [21]).
An important example of MNC satisfying all properties, is the Hausdorff MNC χ(·), which is defined as follows We now define two useful MNCs on C([0, T ]; X). For given L > 0 and D ⊂ C([0, T ]; X), put According to [ and D is equicontinuous. Hence D is relatively compact by the Arzelà-Ascoli theorem. Consider the following function space endowed with the norm is a continuous and nondecreasing function.
Then it is easily seen that C g (R + ; X) is a Banach space. We now define an MNC on this space. We make use of the restriction operator π T : where ω T and mod T is given by (12) and (13), respectively. Then one can check that χ ∞ , d ∞ and χ * are monotone, nonsingular MNCs on C g (R + ; X). The following lemma tests the compactness in C g (R + ; X).
Proof. Let > 0. Since d ∞ (Ω) = 0, one can choose T > 0 such that Let {u n } be a sequence in Ω. Then Accordingly, Combining (18)- (19), one gets for all n, m ≥ N ( ). Therefore {u n } is a Cauchy sequence in C g (R + ; X). The proof is complete.
We are now in a position to recall a basic estimate based on the Hausdorff MNC. Then We now recall some notions of set-valued analysis and fixed point theory for condensing multi-valued maps. Let Y be a metric space. Let β be a monotone nonsingular MNC in E. The application of the topological degree theory for condensing maps (see, e.g. [21]) yields the following fixed point principle. 3. Exponentially bounded solutions. Concerning problem (1)-(2), we give the following assumptions: (A) The operator A is sectorial of type (ω, θ) with ω ≥ 0 and 0 ≤ θ < π(1 − α/2) so that the α-resolvent S α (·) generated by A is norm continuous. (1) for any ψ ∈ B the multimap F (·, ψ) : R + → Kv(X) admits a locally strongly measurable selector, i.e. for each T > 0 one can find a strongly measurable function f : (3) there exist nonnegative functions m, p such that m, p g ∈ L 1 (R + ), and for every ψ ∈ B we have for a.e. t ∈ R + ; (4) there exists a function k ∈ L ∞ (R + ) such that, for every bounded set In view of (3), by (A) we get So in this section, we consider the space C g (R + ; X) with g(t) = e βt for a number β > ω 1 α . By this setting, we have For ϕ ∈ B, we define as a closed subset of C g (R + ; X). For any v ∈ C g,ϕ , we define the function v[ϕ] : For v ∈ C g,ϕ put for a.e. t ∈ R + }. Using the assumption (F)(1)-(F)(3) and the arguments as in [17], one gets P F (v) = ∅ for each v ∈ C g,ϕ , that is the multimap P F is well-defined.
By the arguments in [7], we adopt the following definition of solution to (1)-(2).
Definition 3.1. Given ϕ ∈ B. A function u : R → X is said to be an integral solution of problem (1)- (2) if there exists f ∈ P F (u| R + ) such that Now we consider the multi-valued operator F : C g,ϕ → P(C g,ϕ ), It is clear that if v is a fixed point of F then v[ϕ] is an integral solution to (1)-(2).
Thanks to the formulation of the operator W given by (4), F can be rewritten as To determine the closedness of F, we need the following property for P F .
Considering the MNC ω T defined in (12), we choose L > 0 such that 4 sup  Proof. Let Ω ⊂ C g,ϕ be a bounded set. We will show that if χ * (F(Ω)) ≥ χ * (Ω) then Ω is relatively compact. Put Ω T = π T (Ω) and Θ T = π T (F(Ω)). Then by the formulation of F we have thanks to Proposition 2. By using (F)(4) one gets thanks to the fact that Ω[ϕ](s + θ) = {ϕ(s + θ)} for θ < −s, which is a singleton. Using the last inequality in (23) we get Hence We are in a position to estimate d ∞ (F(Ω)). Let z ∈ F(Ω), then we can take v ∈ Ω and f ∈ P F (v) such that

WEAK STABILITY FOR INTEGRO-DIFFERENTIAL INCLUSIONS 3647
Let v g ≤ R. So For s ≥ 0 we have where Since Sα(t) → 0 as t → ∞, one can find D α , T 2 > 0 such that Hence it follows from (27)-(30) that, for all t ≥ T 1 + T 2 We have shown that for any > 0, there exist C, T > 0 such that for all z ∈ F(Ω). This implies
We formulate the main result of this section as follows.
Proof. We will show that the solution operator F has a fixed point in C g,ϕ . Obviously F(v) is convex for each v ∈ C g,ϕ , due to the convexity of P F (v). On the other hand, by the arguments in the proof of Lemma 3.4 we get Hence F(v) is a relatively compact set. Since F is closed, F(v) is compact. That is, F has compact, convex values. Taking into account Theorem 2.8, Lemma 3.3 and 3.4, it suffices to prove that F(B R ) ⊂ B R for some R > 0, here B R is the closed ball with center at origin and radius R. Assume to the contrary that for each n ∈ N there exists v n ∈ C g,ϕ with v n g ≤ n such that z n g > n for some z n ∈ F(v n ). Take f n ∈ P F (v n ) such that Then using the same estimates as in the proof of Lemma 3.4, we have where which is uniformly bounded, thanks to (20) and the fact that m, p g ∈ L 1 (R + ). It follows from (33) that Passing to the limit in the last inequality as n → ∞, we get a contradiction with (32). The proof is complete.
4. Weak stability result. In this section, we consider the case when the operator A is sectorial of type (ω, θ) with ω < 0 and 0 ≤ θ < π(1 − α/2), i.e. the α-resolvent S α (·) is asymptotically stable: One observes that, by choosing g ≡ 1 and proceeding as in the previous section, we can prove the existence of attracting solutions to (1)-(2) and then infer the weakly asymptotic stability of zero solution. Precisely, we consider the solution operator F on the space In this circumstance, (A), (B) and (F) are replaced by the following assumptions.
is satisfied, then the zero solution of the problem (1)-(2) is weakly asymptotically stable.
We first verify that this solution is stable, i.e. for ε > 0 there exists δ > 0 such that for |ϕ| B < δ, we have |z t | B < ε, whenever z ∈ Σ(ϕ). Indeed, for z ∈ Σ(ϕ) we can take f ∈ P F (z| R + ) such that Then by (F')(3) and (B3)-(B4) we get where So one has, for all t ≥ 0 where K ∞ = sup t≥0 K(t) and M ∞ = sup t≥0 M (t). This ensures the stability of the zero solution.
It remains to show that the zero solution is weakly attractive. Taking z ∈ Σ(ϕ) such that z(t) = o(1) as t → ∞, we show that |z t | B → 0 as t → ∞. Deploying (B3) again, but now with σ = t 2 , we have thanks to (38). Let ε > 0, then there is T > 0 such that z(r) < ε and M (r) < ε for all r ≥ T , thanks to (B'). Therefore, inequality (39) implies that, for all t ≥ 2T where The proof is complete.