Impulsive Functional Differential Inclusions Research Articles

On the solution of T−controllable abstract fractional differential equations with impulsive effects

In this research article, we delimitate the definition of mild solution for abstract fractional differential equations with state-dependent delay (AFDEw/SDD) of order α ∈(1,2) with impulsive effects and compare the solution to the second-order impulsive differential equations. Further, we obtain sufficient conditions of the existence of mild solution for instantaneous and non-instantaneous impulsive fractional functional differential inclusions with state-dependent delay (IFDIw/SDD) using the multi-valued fixed point theory and operator techniques. Furthermore, we study the trajectory controllability (T−controllability) of the AFDEw/SDD. At last, we present some examples to illustrate the sufficient conditions involving partial and ordinary derivatives.

Mild solutions for nonlinear impulsive functional differential inclusions with nonlocal conditions

Mild solutions for nonlinear impulsive functional differential inclusions with nonlocal conditions

Existence, uniqueness and stability of fractional impulsive functional differential inclusions

In the paper, we discuss necessary and sufficient conditions to obtain the existence, uniqueness and stability of solutions of fractional impulsive functional differential equations towards the \(\psi \)-Liouville–Caputo fractional derivative, through fixed point theorem, Arzela–Ascoli theorem and multivalued analysis theory.

Existence for Impulsive Semilinear Functional Differential Inclusions

In this paper we investigate the existence of solutions for first-order impulsive semilinear functional differential inclusions in Banach spaces. Sufficient condition for the existence is obtained with the well-known Covitz and Nadler’s fixed point theorem for multivalued contractions.

Approximate controllability for abstract semilinear impulsive functional differential inclusions based on Hausdorff product measures

A second order semilinear impulsive functional differential inclusion in a separable Hilbert space is considered. Without imposing hypotheses of the compactness on the cosine families of operators, some sufficient conditions of approximate controllability are formulated in the case where the multivalued nonlinearity of the inclusion is a completely continuous map dominated by a function. By the use of resolvents of controllability Gramian operators and developing appropriate computing techniques for the Hausdorff product measures of noncompactness, the results of approximate controllability for position and velocity are derived. An example is also given to illustrate the application of the obtained results.

Controllability for noninstantaneous impulsive semilinear functional differential inclusions without compactness

The first part of this paper considers the controllability for a functional semilinear differential inclusion governed by a family of operators {A(t):t∈[0,b]} generating an evolution operator in a Banach space in the presence of noninstantaneous impulse effects. In the second part of this paper we study the controllability for a fractional noninstantaneous impulsive semilinear differential inclusion with delay, where the linear part is an infinitesimal generator of a C0−semigroup. Using a weakly convergent criterion in the space of piecewise continuous functions and weak topology theory (for weak sequentially closed graph operators) we establish sufficient conditions to guarantee controllability results. Examples are given to illustrate the abstract results.

Impulsive functional delay differential inclusions of fractional order at variable times

We are concerned with some sufficient conditions for the existence of solutions of a class of initial value problems for impulsive fractional differential inclusions with functional delay at variable moments.

Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay

In this paper, we prove the existence of mild solutions for a first-order impulsive semilinear stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay. We consider the cases in which the right hand side is convex or nonconvex valued. The results are obtained by using two different fixed point theorems for multivalued mappings. Copyright © 2015 John Wiley & Sons, Ltd.

On The Exact Controllability of Impulsive Fractional Semilinear Functional Differential Inclusions

AbstractIn this paper, we deal with the exact controllability for impulsive fractional semilinear functional differential inclusions in Hilbert spaces. Under suitable assumptions, the exact controllability of the control systems are formulated and proved. The main tools in our study rely on the fixed point theorem for multivalued maps due to Dhage associated with evolution systems.

Erratum to: Structure of the solution set to impulsive functional differential inclusions on the half-line

The aim of the paper is twofold. At first, it is an erratum to: Gabor G., Grudzka A.: Structure of the solution set to impulsive functional differential inclusions on the half-line. NoDEA 19, 609–627 (2012). We fill a little gap in a proof of Theorem 3.5 by a change of a compactness type condition, and give a discussion on assumptions sufficient to obtain an Rδ-structure of solution sets. Simultaneously, we improve the main theorems by the use of modified proof-arguments. Namely, the assumptions on impulse maps can be weakened.

Existence results for third-order impulsive functional differential inclusions with multiplier p(t)

We study the existence of solutions of third-order impulsive functional differential inclusions with multiplier p(t). Two new results are obtained by using a suitable fixed-point theorem combined with the multivalued-analysis theory.

Existence and controllability results for some impulsive partial functional differential inclusion

In this work, we prove the existence of mild and extremal mild solutions for first-order semilinear non densely defined impulsive functional differential inclusions in separable Banach spaces with local and nonlocal conditions. Firstly, we show the existence of mild and extremal solutions. Secondly, we study the controllability of a semilinear functional differential inclusions.

Existence results on random impulsive semilinear functional differential inclusions with delays

This article presents the result on existence of mild solutions for random impulsive semilinear functional dierential inclusions under sucient conditions. The results are obtained by using the Martelli xed point theorem and the xed point theorem due to Covitz and Nadler.

Existence and solution sets of impulsive functional differential inclusions with multiple delay

In this paper, we present some existence results of solutions and study the topological structure of solution sets for the following first-order impulsive neutral functional differential inclusions with initial condition: \[ \begin{cases}\frac{d}{dt}[y(t)-g(t,y_t)] \in F(t,y_t) + \sum_{i=1}^{n_*} y(t-Ti), & a.e.\, t \in J\setminus\{t_1,...,t_m\} y(t_k^+)-y(t_k^-)=I_k(y(t_k^-)), & k=1,...,m, y(t)=\phi(t), & t \in [-r,0],\end{cases} \] where \(J:=[0,b]\) and \(0=t_0\lt t_1 \lt ...\lt t_m\lt t_{m+1}=b\) (\(m \in \mathbb{N}^*\)), \(F\) is a set-valued map and \(g\) is single map. The functions \(I_k\) characterize the jump of the solutions at impulse points \(t_k\) (\(k=1,...,m\)). Our existence result relies on a nonlinear alternative for compact u.s.c. maps. Then, we present some existence results and investigate the compactness of solution sets, some regularity of operator solutions and absolute retract (in short AR). The continuousdependence of solutions on parameters in the convex case is also examined. Applications to a problem from control theory are provided.

Structure of the solution set to impulsive functional differential inclusions on the half-line

A topological structure of the set of solutions to impulsive functional differential inclusions on the half-line is investigated. It is shown that the solution set is nonempty, compact and, moreover, an Rδ-set. It is proved on compact intervals and then, using the inverse limit method, obtained on the half-line.

Existence results for [formula omitted]th-order impulsive functional differential inclusions

In this paper, we study the existence of solutions for m th-order impulsive functional differential inclusions. Three new results are obtained by using the fixed point theorem and multivalued analysis theory. Finally, an example is given to illustrate our main results.

Approximate controllability for neutral impulsive differential inclusions with nonlocal conditions

In this paper, we study the approximate controllability of semilinear impulsive functional differential inclusions with nonlocal conditions. Analytic semigroup theory and -norm arguments are employ...

Impulsive semilinear neutral functional differential inclusions with multivalued jumps

In this paper we establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some densely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces. We rely on a fixed point theorem for the sum of completely continuous and contraction operators.

Controllability for impulsive semilinear functional differential inclusions with a non-compact evolution operator

We study a controllability problem for a system governed by a semilinear functional differential inclusion in a Banach space in the presence of impulse effects and delay. Assuming a regularity of the multivalued non-linearity in terms of the Hausdorff measure of noncompactness we do not require the compactness of the evolution operator generated by the linear part of inclusion. We find existence results for mild solutions of this problem under various growth conditions on the nonlinear part and on the jump functions. As example, we consider the controllability of an impulsive system governed by a wave equation with delayed feedback.

Nonlinear impulsive partial functional differential inclusions with state-dependent delay and multivalued jumps

In this paper, we shall establish sufficient conditions for the existence of integral solutions for some nondensely defined impulsive semilinear functional differential inclusions with state-dependent delay in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators.