On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions

Palaniyappan Kalamani,Mani Mallika Arjunan,Dumitru Baleanu,Siva Selvarasu

doi:10.1186/s13662-016-0885-4

Palaniyappan Kalamani, Mani Mallika Arjunan + Show 2 more

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https://doi.org/10.1186/s13662-016-0885-4

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Abstract

This manuscript deals with a new set of sufficient conditions for the existence of solutions for a class of impulsive fractional neutral stochastic integro-differential systems (IFNSIDS) with nonlocal conditions (NLCs) and state-dependent delay (SDD) in Hilbert spaces. An example is provided to illustrate the obtained theory.

Highlights

In this paper, we establish the existence of mild solutions for impulsive fractional neutral stochastic integro-differential systems (IFNSIDS) with state-dependent delay (SDD) in Hilbert spaces through the utilization of the fixed point theorem of Krasnoselskii [ ], Lemma
1 Introduction In this paper, we establish the existence of mild solutions for IFNSIDS with SDD in Hilbert spaces through the utilization of the fixed point theorem of Krasnoselskii [ ], Lemma
3 Existence results we show the existence of solutions for model ( . )-( . ) under the Krasnoselskii fixed point theorem [ ] using the operator semigroup theory and fractional calculus

Summary

Introduction

We establish the existence of mild solutions for IFNSIDS with SDD in Hilbert spaces through the utilization of the fixed point theorem of Krasnoselskii [ ], Lemma. N, u( ) + h(u) = φ ∈ B, where CDαt is the Caputo fractional derivative of order α ∈ ( , ), t ∈ I = [ , T] is an operational interval, the state variable u takes values in a Hilbert space H, u(tk) = u(tk+) – u(tk–), k = , , . N, are jumps of the solution at impulsive points tk ( < t < t < · · · < tn < T ), A : D(A ) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators {T(t) : t ≥ }, that is, T(t) ≤ M for some constant M ≥ and Kalamani et al Advances in Difference Equations (2016) 2016:163 all t ≥.

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