Impulsive Differential Inclusions Research Articles

Existence results for impulsive differential inclusions with nonlocal conditions

In this paper, we shall establish sufficient conditions for the existence of mild solutions for nonlocal impulsive differential inclusions. On the basis of the fixed point theorems for multivalued maps and the technique of approximate solutions, new results are obtained. Examples are also provided to illustrate our results.

Existence of random impulsive abstract neutral non-autonomous differential inclusions with delays

This article presents the results on existence of mild solutions of random impulsive neutral functional differential inclusions under sufficient conditions. The results are obtained by using the Dhage’s fixed point theorem.

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...

The partial averaging of fuzzy impulsive differential inclusions

In this paper the substantiation of the method of partial averaging for fuzzy impulsive differential inclusions is considered. Such variants of the averaging method also lead to the simplification of the initial inclusion and happen to be useful when the average of some mappings does not exist or their presence in the system does not complicate its research. These results generalize the results of [24] for impulsive differential inclusions with Hukuhara derivative and of [18] for fuzzy impulsive differential equations.

Overview of V.A. Plotnikov’s research on averaging of differential inclusions

In this review, we will first look in detail at Plotnikov’s results on the substantiation of full and partial schemes of averaging of differential inclusions of the standard form on finite and infinite intervals. Then we will consider the algorithms where it is not possible to find the average, but there is a possibility to find its estimation from below and from above. Such an approach is also used when the average can be found only approximately. This situation is common for differential inclusions with fast and slow variables. In the end, we will present the results on the substantiation of the full and partial averaging methods for impulsive differential inclusions on finite and infinite intervals.

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.

Some results on impulsive boundary value problem for fractional differential inclusions

This paper deals with impulsive fractional differential inclusions with a fractional order multi-point boundary condition and with fractional order impulses. By use of multi-valued analysis and topological fixed point theory, we present some existence results under both convexity and nonconvexity conditions on the multi-valued right-hand side. The compactness of the solutions set and continuous version of Filippov’s theorem are also investigated.

Fractional order impulsive partial hyperbolic differential inclusions with variable times

Fractional order impulsive partial hyperbolic differential inclusions with variable times

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.

Controllability of Neutral Impulsive Differential Inclusions with Non-Local Conditions

In this short article, we have studied the controllability result for neutral impulsive differential inclusions with nonlocal conditions by using the fixed point theorem for condensing multi-valued map due to Martelli [1]. The system considered here follows the P.D.E involving spatial partial derivatives with α-norms.

Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay

This paper deals with the controllability of a class of impulsive neutral stochastic functional differential inclusions with infinite delay in an abstract space. Sufficient conditions for the controllability are derived with the help of the fixed point theorem for discontinuous multi-valued operators due to Dhage. An example is provided to illustrate the obtained theory.

Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces

In this paper, we consider the existence of solutions as well as the topological and geometric structure of solution sets for first-order impulsive differential inclusions in some Fréchet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, A R and R δ ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals.

Discrete Approximation of Impulsive Differential Inclusions

The article deals with the approximation of the solution set and the reachable sets of an impulsive differential inclusion with variable times of impulses. It is strongly connected to [11] and is its continuation. We achieve order of convergence 1 for the Euler approximation under Lipschitz assumptions on the set-valued right-hand side and on the functions describing the jump surfaces and jumps themselves. Another criterion prevents the beating phenomena and generalizes available conditions. Several test examples illustrate the conditions and the practical evaluation of the jump conditions.

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.

First-order periodic impulsive semilinear differential inclusions: Existence and structure of solution sets

In this paper, we present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive semilinear differential inclusions with periodic boundary conditions: { ( y ′ − A y ) ( t ) ∈ F ( t , y ( t ) ) , a.e. t ∈ J ∖ { t 1 , … , t m } , y ( t k + ) − y ( t k − ) = I k ( y ( t k − ) ) , k = 1 , … , m , y ( 0 ) = y ( b ) where J = [ 0 , b ] and 0 = t 0 < t 1 < ⋯ < t m < t m + 1 = b ( m ∈ N ∗ ) A is the infinitesimal generator of a C 0 -semigroup T on a separable Banach space E and F is a multi-valued map. The functions I k characterize the jump of the solutions at impulse points t k ( k = 1 , … , m ). We will have to distinguish between the cases when either or neither 1 lies in the resolvent of T ( b ) . Accordingly, the problem is either formulated as a fixed point problem for an integral operator or for a Poincaré translation operator. Our existence results rely on fixed point theory and on a new nonlinear alternative for compact u.s.c. maps respectively. Then, we present some existence results and investigate the topological structure of the solution set. A continuous version of Filippov’s theorem is provided and the continuous dependence of solutions on parameters in both convex and nonconvex cases are examined.

On impulsive functional differential inclusions with Hille–Yosida operators in Banach spaces

We consider a semilinear functional differential inclusion with infinite delay and impulse characteristics in a Banach space assuming that its linear part is a non-densely defined Hille–Yosida operator. We assume that the multivalued nonlinearity of upper Carathèodory or almost lower semicontinuous type satisfies a regularity condition expressed in terms of the measures of noncompactness. We apply the theory of integrated semigroups and the theory of condensing multivalued maps to obtain local and global existence results. The application to an optimization problem for an impulsive feedback control system is given.

Extrapolation method and some nondensely defined impulsive semilinear neutral partial functional differential inclusions

In this paper, we use the extrapolation method combined with a fixed-point theorem for the sum of completely continuous and contraction operators, to establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some classes of nondensely defined impulsive semilinear neutral functional differential inclusions in separable Banach spaces with infinite delay.

The existence of viable trajectories in state-dependent impulsive systems

Suitable topological tools for studying the existence of viable and periodic viable trajectories in state-dependent impulsive differential equations and inclusions are provided. Some results based on the Ważewski retract method are applied. The techniques presented in the paper are illustrated with several examples.

Fixed points of multivalued operators in ordered metric spaces with applications

Abstract The purpose of this paper is to present some new (hybrid) fixed point theorems involving multivalued operators which satisfy weakly generalized contractive conditions in an ordered complete metric space. An example of the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is given to illustrate the usability of our results.

Impulsive partial hyperbolic differential inclusions of fractional order

AbstractIn this paper we investigate the existence of solutions of a class of partial impulsive hyperbolic differential inclusions involving the Caputo fractional derivative. Our main tools are appropriate fixed point theorems from multivalued analysis.