Abstract:
In this paper, we study the existence of solutions for first and second order impulsive neutral functional differential inclusions with variable times. Our main tool is a fixed point theorem due to Martelli for condensing multivalued maps.

Abstract:
In this paper we prove existence results for first order semilinear impulsive neutral functional differential inclusions under the mixed Lipschitz and Caratheodory conditions

Abstract:
We study the control systems governed by impulsive Riemann-Liouville fractional differential inclusions and their approximate controllability in Banach space. Firstly, we introduce the -mild solutions for the impulsive Riemann-Liouville fractional differential inclusions in Banach spaces. Secondly, by using the fractional power of operators and a fixed point theorem for multivalued maps, we establish sufficient conditions for the approximate controllability for a class of Riemann-Liouville fractional impulsive differential inclusions, which is a generalization and continuation of the recent results on this issue. At the end, we give an example to illustrate the application of the abstract results. 1. Introduction The concept of controllability plays an important part in the analysis and design of control systems. Since Kalman [1] first introduced its definition in 1963, controllability of the deterministic and stochastic dynamical control systems in finite-dimensional and infinite-dimensional spaces is well developed in different classes of approaches, and more details can be found in papers [2–4]. Some authors [5–7] have studied the exact controllability for nonlinear evolution systems by using the fixed point theorems. In [5–7], to prove the controllability results for fractional-order semilinear systems, the authors made an assumption that the semigroup associated with the linear part is compact. But if -semigroup is compact or the operator is compact, then the controllability operator is also compact and hence the inverse of it does not exist if the state space is infinite dimensional [8]. Thus, it is shown that the concept of exact controllability is difficult to be satisfied in infinite-dimensional space. Therefore, it is important to study the weaker concept of controllability, namely, approximate controllability for differential equations. In these years, several researchers [9–17] have studied it for control systems. In [13], Sakthivel et al. studied on the approximate controllability of semilinear fractional differential systems: where is Caputo’s fractional derivative of and is the infinitesimal generator of a -semigroup of bounded operators on the Hilbert space ; the control function is given in ; is a Hilbert space; is a bounded linear operator from to ; is a given function satisfying some assumptions and is an element of the Hilbert space . In [16], Sukavanam and Kumar researched approximate controllability of fractional-order semilinear delay systems: where ; is a closed linear operator with dense domain generating a -semigroup ; the

Abstract:
We firstly deal with the existence of mild solutions for nonlocal fractional impulsive semilinear differential inclusions involving Caputo derivative in Banach spaces in the case when the linear part is the infinitesimal generator of a semigroup not necessarily compact. Meanwhile, we prove the compactness property of the set of solutions. Secondly, we establish two cases of sufficient conditions for the controllability of the considered control problems. 1. Introduction During the past two decades, fractional differential equations and inclusions have gained considerable importance due to their applications in various fields, such as physics, mechanics, and engineering. For some of these applications, one can see [1–3] and the references therein. El-Sayed and Ibrahim [4] initiated the study of fractional multivalued differential inclusions. Recently, some basic theory for initial value problems for fractional differential equations and inclusions was discussed in [5–13]. The theory of impulsive differential equations and inclusions has been an object of interest because of its wide applications in physics, biology, engineering, medical fields, industry, and technology. The reason for this applicability arises from the fact that impulsive differential problems are an appropriate model for describing process which at certain moments change their state rapidly and which cannot be described using the classical differential problems. During the last ten years, impulsive differential inclusions with different conditions have been intensely studented by many mathematicians. At present, the foundations of the general theory of impulsive differential equations and inclusions are already laid, and many of them are investigated in details in the book of Benchohra et al. [14]. Moreover, a strong motivation for investigating the nonlocal Cauchy problems, which is a generalization for the classical Cauchy problems with initial condition, comes from physical problems. For example, it used to determine the unknown physical parameters in some inverse heat condition problems. The nonlocal condition can be applied in physics with better effect than the classical initial condition . For example, may be given by , where are given constants and . In the few past years, several papers have been devoted to study the existence of solutions for differential equations or inclusions with nonlocal conditions [15–17]. For impulsive differential equations or inclusions with nonlocal conditions of order we refer to [16, 17]. For impulsive differential equations or inclusions of

Abstract:
In paper the existence of solutions for first and second order impulsive neutral functional differential inclusions in Banach spaces is investigated. The results are obtained by using a fixed point theorem for condensing multivalued maps due to Martelli and the semigroup theory.

Abstract:
This paper investigates the existence of solutions for fractional-order neutral impulsive differential inclusions with nonlocal conditions. Utilizing the fractional calculus and fixed point theorem for multivalued maps, new sufficient conditions are derived for ensuring the existence of solutions. The obtained results improve and generalize some existed results. Finally, an illustrative example is given to show the effectiveness of theoretical results.

Abstract:
We study the existence of mild solutions for a class of impulsive fractional partial neutral stochastic integro-differential inclusions with state-dependent delay. We assume that the undelayed part generates a solution operator and transform it into an integral equation. Sufficient conditions for the existence of solutions are derived by using the nonlinear alternative of Leray-Schauder type for multivalued maps due to O'Regan and properties of the solution operator. An example is given to illustrate the theory.

Abstract:
In this paper, we consider a class of second-order evolution differential inclusions in Hilbert spaces. This paper deals with the approximate controllability for a class of second-order control systems. First, we establish a set of sufficient conditions for the approximate controllability for a class of second-order evolution differential inclusions in Hilbert spaces. We use Bohnenblust-Karlin's fixed point theorem to prove our main results. Further, we extend the result to study the approximate controllability concept with nonlocal conditions and extend the result to study the approximate controllability for impulsive control systems with nonlocal conditions. An example is also given to illustrate our main results.

Abstract:
In the paper, we study weak invariance of differential inclusions with non-fixed time impulses under compactness type assumptions. When the right-hand side is one sided Lipschitz an extension of the well known relaxation theorem is proved. In this case also necessary and sufficient condition for strong invariance of upper semi continuous systems are obtained. Some properties of the solution set of impulsive system (without constrains) in appropriate topology are investigated.