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Stochastic controllability for a non-autonomous fractional neutral differential equation with infinite delay in abstract space

Areefa Khatoon, Abdur Raheem, Asma Afreen

Abstract

This paper deals with the controllability for a class of non-autonomous neutral differential equations of fractional order with infinite delay in an abstract space. The semi-group theory of bounded linear operators, fractional calculus, and stochastic analysis techniques have been implemented to achieve the main result. We prove the existence of mild solution and controllability of the system by using the theory of measure of non-compactness, fixed point theorems, and $k$-set contractive mapping. An example is given to demonstrate the effectiveness of the abstract result.

Stochastic controllability for a non-autonomous fractional neutral differential equation with infinite delay in abstract space

Abstract

This paper deals with the controllability for a class of non-autonomous neutral differential equations of fractional order with infinite delay in an abstract space. The semi-group theory of bounded linear operators, fractional calculus, and stochastic analysis techniques have been implemented to achieve the main result. We prove the existence of mild solution and controllability of the system by using the theory of measure of non-compactness, fixed point theorems, and -set contractive mapping. An example is given to demonstrate the effectiveness of the abstract result.
Paper Structure (5 sections, 13 theorems, 76 equations)

This paper contains 5 sections, 13 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Assumptions
  3. Main Result
  4. Application
  5. Conclusion

Key Result

Lemma 2.2

M2001 For any $\vartheta_\ell\in L^p(\Upsilon,X)$ and $p\geq2,$ there exists a function $\kappa \in L_{\Upsilon}^p([0,\ell],L_2^0)$ such that

Theorems & Definitions (23)

  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 13 more