On solvability of the impulsive Cauchy problem for integro-differential inclusions with non-densely defined operators

We prove the existence of at least one integrated solution to an impulsive Cauchy problem for an integro-differential inclusion in a Banach space with a non-densely defined operator. Since we look for integrated solution we do not need to assume that A is a Hille Yosida operator. We exploit a technique based on the measure of weak non-compactness which allows us to avoid any hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term. As the main tool in the proof of our existence result, we are using the Glicksberg–Ky Fan theorem on a fixed point for a multivalued map on a compact convex subset of a locally convex topological vector space. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

We prove the existence of at least one integrated solution to an impulsive Cauchy problem for an integro-differential inclusion in a Banach space with a non-densely defined operator. Since we look for integrated solution we do not need to assume that A is a Hille Yosida operator. We exploit a technique based on the measure of weak non-compactness which allows us to avoid any hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term. As the main tool in the proof of our existence result, we are using the Glicksberg-Ky Fan theorem on a fixed point for a multivalued map on a compact convex subset of a locally convex topological vector space.
This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

Introduction
In this paper, we study the following impulsive Cauchy problem for an integro-differential inclusion in a Banach space with a non-densely defined operator: t 0 x(s) ds a.e. t ∈ [0, b], t = t k , k = 1, . . . , N x(t + k ) = x(t k ) + c k , k = 1, . . . , N x(0) = x 0 ∈ E. (1.1) Here E is a weakly compactly generated Banach space, 1 A : D(A) ⊂ E → E is the generator of an integrated semigroup, F : [0, b] × E E is a multivalued map (multimap for short). We require that 0 < t 1 < t 2 < · · · < t N < b and c k ∈ E, k = 1, . . . , N are given elements.
The space where solutions to (1.1) naturally lie is the space PC ([0, b]; E) of all piece-wise continuous functions x : [0, b] → E with discontinuity points at t = t k , k = 1, . . . , N such that all values x(t + k ) = lim s→t + k x(s) and x(t − k ) = lim s→t − k x(s) are finite and x(t k ) = x(t − k ) for all k. In many situations the domain of the operator A may be non-dense in the Banach space E. It comes, for example, from restrictions on the space on which the operator A is defined (periodic continuous functions, Hölder continuous functions, etc.) or from boundary conditions (e.g. the space of smooth functions vanishing on the boundary of a domain is not dense in the space of continuous functions). Thus, several types of differential equations, such as delay differential equations, age-structure models in population dynamics, evolution equations with nonlinear boundary conditions, can be written as semilinear Cauchy differential or integro-differential problems with non-dense domain, see Da Prato & Sinestrari [1], Thieme [2] and the recent papers by Magal et al. [3] or by Yang [4].
In order to better describe natural phenomena it is useful to consider non-necessarily continuous propagation of the studied process, allowing that the model is subjected to shortterm perturbations in time, the so-called impulses. For instance, in the periodic treatment of some diseases, impulses may correspond to administration of a drug treatment; in environmental sciences, impulses may correspond to seasonal changes or harvesting; in economics, impulses may correspond to abrupt changes of prices.
Multivalued nonlinearity is related to models where the parameters are known up to some degree of uncertainty.
It is well known that if A is a Hille-Yosida operator and it is densely defined (i.e. D(A) = E), it generates a C 0 -semigroup, say {T(t)} t≥0 . Thus, given an initial datum in E, the semilinear Cauchy problem where f : [0, b] × E → E, has been extensively studied, under several regularity assumptions on A and f , see, e.g., Pazy [5]. In particular, the integration of the equation in (1.2) yields the variation of constants formula When A is non-densely defined (i.e. D(A) = E) the variation of constants formula may be not well-defined. Therefore, the well-posedness of the problem may be recovered by integrating the equation twice to introducing integrated semigroups (see [6] semigroup {V(t)} t≥0 , then the function t → V(t)x is differentiable for each x ∈ D(A) and, moreover, the derivative {V (t)} t≥0 is a C 0 -semigroup on D(A), thus the integral solution to (1.2) given by Da Prato and Sinestrari (the so-called mild solutions) is equivalent to If A is neither a Hille-Yosida operator, the Cauchy problem (1.2) may not have an integral solution in the sense of Da Prato and Sinestrari. Thus, one can introduce the definition of integrated solution given by Thieme [2, definition 6.4] for linear equations, i.e. when f (t, x(t)) = f (t) for every t ∈ [0, b], which is obtained by integrating (1.2) twice. Thieme [2, theorem 6.5], proved that the integrated solution is equivalent to the variation of constants formula After the pioneering work of Da Prato and Sinestrari, the existence of mild solutions of (1.2) with a non-densely defined operator A was extensively investigated. For instance, Kellerman & Hieber [7] and Thieme [2], assuming A a Hille-Yosida operator, studied the semilinear Cauchy problem respectively with a bounded and a Lipschitz perturbation of the closed linear operator A. Liu et al. [8], imposing an additional condition to assure the existence of mild solutions of the Cauchy problem, extended Thieme's results [2] to the case when the operator A is not Hille-Yosida. Semilinear equations with non-densely defined linear part and impulses were considered subject to several problems: controllability, delay, non-local initial conditions, evolution operators, fractional equations, neutral equations, integro-differential equations, see for instance [9][10][11][12][13][14][15][16][17] and references therein.
We considered a rather simple problem compared to the ones cited; the reason is that our focus is to weaken the assumptions as much as possible both on the solution and on the terms of the equations.
We prove an existence result for an integrated solution to (1.1), see definition 3.8. As far as we know this is the first time that integrated solutions, related to operators A which do not satisfy the Hille-Yosida condition, are introduced for nonlinear differential problems with impulses. We stress that in order to find an integrated solution it is sufficient to assume that A generates an integrated semigroup and not to pose the stronger assumption that it is a Hille-Yosida operator. Moreover, if A is a Hille-Yosida operator defined in a reflexive Banach space E, by [18, corollary 2.19] it has a dense domain. Thus, it is a contradiction to consider a non-densely defined Hille Yosida operator in a reflexive Banach space. Since we assume only that A is the generator of an integrated semigroup, unlike [9][10][11][12][13][14][15][16][17] we can consider also a reflexive Banach space E.
By means of the measure of weak non-compactness, we exploit a technique based on the weak topology, developed in [19] and extended to non-necessarily reflexive Banach spaces in [20]. Thus, we avoid hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term F. In particular, in contrast to [12][13][14][15]17], this approach allows us to treat a class of nonlinear multimaps F which are not necessarily compact-valued. In fact, even for single valued nonlinearities, assuming that the nonlinear term satisfies a regularity condition in terms of measure of weak non-compactness is a much weaker assumptions than assuming the same condition with respect to strong measure of non-compactness (see remark 3.1). For example the first condition can be easily verified in a reflexive Banach space, where bounded sets are weakly relatively compact ones. Moreover, unlike [9,11,16], we can handle also linear part A generating integrated semigroups whose derivative is a non-compact semigroup.
The paper is organized as follows. In §2 we recall some known results on the integrated semigroups, as well as on the measure of non-compactness, and on Eberlein Smulian theory, we also state the Glicksberg-Ky Fan fixed point theorem we are going to use to prove our main result. Section 3 is divided in four subsections: in §3a we state the problem we are going to study, in §3b we show how the integrated solution to (1.1) is obtained, in §3c we construct a solution operator whose fixed points are integrated solutions to (1.1); finally, in §3d we prove the existence of at least one integrated solution to (1.1).

Preliminaries
In the whole paper (E, · ) stands for a Banach space. We use the notation B r (E) to denote the closed ball in E centred at 0 with radius r. In the whole paper, without generating misunderstanding, we denote by Thus, the above characterization of weakly convergent sequences holds also for the space Now we recall some definitions and properties of the integrated semigroups, we refer to [2,7] for more details on this topic.

Definition 2.1 ([7, definition 1.1]).
An integrated semigroup is a family {V(t)} t≥0 of bounded linear operators on E with the following properties:

Definition 2.4 ([7, p. 163]). Let
Then for λ > ω, λI X − A is invertible and for the resolvent R(λ, A) the following relation holds: We recall now some properties of the generator of an integrated semigroup that we will use in the sequel.

Lemma 2.7 ([2, lemma 3.5]). Let A be the generator of an integrated semigroup {V(t)} t≥0 then
We recall now the definition of a measure of non-compactness, describe its main properties and consider some relevant examples. Further information and all the proofs can be found, for instance, in [22].
As an example of a MNC obeying all the above properties we can consider the Hausdorff MNC χ E , defined as The aim of this paper is to relax the compactness requirements as far as possible. Thus, we consider the Banach space E endowed with the weak topology and to prove our main result we use the De Blasi measure of weak non-compactness introduced in [23] which for Ω ⊆ E is defined as Remark 2.9. The De Blasi measure of weak non-compactness is monotone, non-singular, algebraically semiadditive and regular. Moreover, for every linear and bounded operator L : For weakly compactly generated E, we can use the following result which is a special case of [24, theorem 2.8]: Theorem 2.10. Let E be a weakly compactly generated Banach space and Ω be a positive measure space with a finite measure. Then for every sequence of functions g n : Ω → E the functions β(g n (·)) are measurable in Ω, and where β denotes the De Blasi measure of weak non-compactness.
In order to define a measure of weak noncompactness in the space PC ([0, b]; E), we consider the following characteristic. The proof is the same as the one in [20, proposition 3.1], we present it here for the reader's convenience.
Proof. Since β is monotone, it is sufficient to prove that β L (conv M) ≤ β L (M) for every M ⊆ PC([0, b]; E), i.e., that, given γ < β L (convM) arbitrary, we get β L (M) > γ . By definition of β L , there exist a sequence {g n } n ⊆ convM and t 0 ∈ [0, b] such that β({g n (t 0 )} n )e −Lt 0 > γ . Since convM is a convex set, then by the Hahn-Banach theorem, its closure in the weak topology coincides with its closure in the norm topology. Therefore, since {g n } n is a separable subset of convM, according to [25, proposition 3.55], there exists a sequence {h n } n ⊆ M such that {g n } n ⊆ conv{h n } n . And the conclusion follows from To prove our main result we exploit the Glicksberg-Ky Fan fixed point theorem [26,27]. We recall the Krein-Smulian theorem.

Impulsive differential problem (a) Statement of the problem
As stated in the introduction we study the following problem: where β is the De Blasi measure of weak noncompactness in E.

Remark 3.1.
In the case of a reflexive Banach space E, since a bounded set is a weakly relatively compact set, from (F3) it follows that β(F(t, Ω)) = β(Ω) = 0 for every bounded Ω ⊂ E, thus assumption (F4) is automatically true.

(b) Definition of a solution
In order to define the concept of solution to problem (3.1) at first we consider the following linear Cauchy problem associated to a semilinear differential inclusion with non-densely defined operator: To provide the existence of such a solution one must impose some smooth conditions on x 0 , e.g.
x 0 ∈ D(A), and on f , either some regularity assumptions such as f (t) ∈ D(A) for every t. Without smoothness assumptions, it is necessary to consider solutions of (3.2) in a generalized sense. The first attempt in this direction is the definition of mild solution given by Da Prato & Sinestrari [1], which is obtained by integrating (3.2) in time and assuming x 0 ∈ D(A).

Proposition 3.3 ([30, theorem 3.1 page 67]). The following assertions are equivalent
(i) x is the mild solution of (3.2) (ii) If we want to relax smoothness conditions even more, we can introduce the definition of integrated solution given by Thieme [2, definition 6.4], which is obtained by integrating (3.2) twice.

Definition 3.4. A continuous function
Notice that the integrated solution y : [0, b] → E, is the integral function of the mild solution x : [0, b] → E given by Da Prato and Sinestrari, namely y(t) = t 0 x(s) ds. Thus, integrating (3.3) we get the following result.

Proposition 3.5 ([2, theorem 6.5]). y is an integrated solution of (3.2) if and only if
(3.5) Now consider the following linear Cauchy problem with impulses: In order to define the integrated solution of the impulsive linear problem (3.6), we first show a formula for the mild solution of a Cauchy problem with starting point a > 0.

Proposition 3.6. Given a > 0, x is the mild solution of x (t) ∈ Ax(t) + f (t) in [a, b] if and only if
(3.7) Proof. By (3.4) Substituting this value in (3.7) and exploiting the semigroup properties of V we get, for every t ≥ a, and the proof is completed.
The following proposition establishes a formula for the mild solution of (3.6). We give here a direct proof. For an alternative proof see [11]. Proposition 3.7. If x is a mild solution of problem (3.6) then Proof. Let x be a mild solution of (3.6). Then, according to proposition 3.3, for every t ∈ [0, t 1 ], By (3.7), recalling the semigroup properties of V , for t ∈ (t 1 , t 2 ] we get By induction we obtain that, for every t ∈ [0, b], Let x 0 ∈ D(A) and x : [0, b] → D(A) be an absolutely continuous function satisfying (3.1). Integrating the equation in (3.1) we get the following integral equation with f (s) ∈ F(s, s 0 x(r) dr) for a.e. s ∈ [0, b]. As stated above, this integral equation (3.9) is equivalent to with f (s) ∈ F(s, s 0 x(r) dr) = F(s, y(s)) for a.e. s ∈ [0, b]. This approach motivates the following definition.

(c) Solution operator
In this section we shall prove the existence of an integrated solution of (1.1). We first justify the well posedness of the superposition operator P F : For this purpose we need the following proposition which is a special case of [20, theorem 4.4]. Hence {q n } n is a bounded sequence and q n (t) q(t) for every t ∈ [0, b]. By (F1), there exists a sequence of functions {f n } n such that f n (t) ∈ F(t, q n (t)) for a.a. t ∈ [0, b] and f n : [0, b] → E is measurable for any n ∈ N. Then by proposition 3.9 we get the existence of a subsequence, still denoted as the sequence, weakly converging to f ∈ P F (q) in L 1 ([0, b]; E) and the proposition is proved.
Consider now the operator Γ : : Under the above assumptions, the operator Γ is well defined, i.e. for every q ∈ PC([0, b]; E), Γ (q) = ∅. Clearly if y is a fixed point of Γ then there exists f ∈ P F (y) such that i.e. y is an integrated solution of (1.1). Therefore, we study the solvability of (1.
is a linear and bounded operator.
Proof. The linearity follows by the linearity of the operators V(t), t ∈ [0, b] and of the integral operator. Moreover from (A), being {V(t)} t≥0 exponentially bounded, we have ; E) satisfying y n ∈ Γ (q n ) for all n and q n q, y n y in PC([0, b]; E); we will prove that y ∈ Γ (q).
The weak convergence of {q n } n to q in PC([0, b]; E) implies its boundedness and the weak convergence of {q n (t)} n to q(t) for a.a. t ∈ [0, b]. Moreover the fact that y n ∈ Γ (q n ) means that there exists a sequence {f n } n , f n ∈ P F (q n ) for every n, such that for every t ∈ [0, b], Hence, by proposition 3.9 we have the existence of a subsequence, denoted as the sequence, and a function f ∈ P F (q) such that f n f in L 1 ([0, b]; E).
By proposition 3.11 we have that G(f n ) G( f ). Thus, we have implying, for the uniqueness of the weak limit in E, that y 0 (t) = y(t) for all t ∈ [0, b], i.e. that y ∈ Γ (q).
Proposition 3.13. The multioperator Γ has convex and closed values.
Proof. Taking a fixed q ∈ PC([0, b]; E), since F is convex valued, from the linearity of the operator G it follows that the set Γ (q) is convex. The closedness of Γ (q) follows from proposition 3.12. there exists n ∈ N such that Γ (Q n ) ⊂ Q n , where for n ∈ N, Proof. Assume to the contrary, that there exist two sequences {q n } n and {y n } n such that q n ∈ Q n , y n ∈ Γ (q n ) and y n / ∈ Q n for all n ∈ N. By the definition of Γ , there exists a sequence {f n } n , f n ∈ P F (q n ), Moreover, q n ∈ Q n implies, by (F3 ), that f n (t) ≤ ϕ n (t) for a.a. t ∈ [0, b], hence f n 1 ≤ ϕ n 1 . Consequently, According to (3.12), there exists a subsequence, still denoted as the sequence, such that lim n→∞ 1 n b 0 |ϕ n (s)| ds = 0, (3.14) thus it is possible to choose n such that, for every n ≥ n, getting a contradiction with y n / ∈ Q n for every n.

(d) Existence result
We are now ready to state our main result. By proposition 3.14 Q n ∈ A, hence A = ∅. Moreover 0 ∈ K for every K ∈ A. Therefore 0 ∈ K∈A K := Q, so Q is non-empty. The set Q is closed and convex and Q is bounded, because Q ⊂ Q n . Moreover for every K ∈ A we have  obtaining β L (Q) = 0. By remark 2.12 and corollary 2.15, since Q is closed, Q(t) is weakly compact for every t ∈ [0, b] and so also P F (Q)(t) is weakly compact by (F4). Moreover, by (F3) , P F (Q) is also bounded and uniformly integrable and so again by the Dunford Pettis theorem we get that P F (Q) is weakly compact. Since G is linear and bounded we obtain that Γ (Q) is weakly compact. Therefore, the restriction of the operator Γ to the set Q is a weakly upper semicontinuous map, mapping Q onto itself. Finally, from (3.15) and theorem 2.16 we get that Q is weakly compact and the existence of a fixed point of Γ follows from theorem 2.13.
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