Asymptotically periodic behavior of solutions to fractional non-instantaneous impulsive semilinear differential inclusions with sectorial operators

Zainab Alsheekhhussain,Ahmed Gamal Ibrahim,Jinrong Wang

doi:10.1186/s13662-021-03475-w

Zainab Alsheekhhussain, Ahmed Gamal Ibrahim + Show 1 more

Open Access

https://doi.org/10.1186/s13662-021-03475-w

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Abstract

In this paper, we prove two results concerning the existence of S-asymptotically ω-periodic solutions for non-instantaneous impulsive semilinear differential inclusions of order 1<alpha <2 and generated by sectorial operators. In the first result, we apply a fixed point theorem for contraction multivalued functions. In the second result, we use a compactness criterion in the space of bounded piecewise continuous functions defined on the unbounded interval J=[0,infty ). We adopt the fractional derivative in the sense of the Caputo derivative. We provide three examples illustrating how the results can be applied.

Highlights

Fractional calculus has become a well-established branch of mathematical analysis
Rezapour et al [3] showed the existence and uniqueness of solutions for a general multi-term fractional BVP involving the generalized ψ-RL operators. They suggested two numerical algorithms, namely, the Dafterdar-Gejji and Jafari method (DGJIM) and the Adomian decomposition method (ADM) in which a series of approximate solutions converge to the exact ones
To clarify the advantage of this study, we mention that two methods have been provided to demonstrate the existence of S-asymptotic ω-periodic solutions for semilinear fractional differential inclusions in the presence of non-instantaneous impulse effects, and in which the nonlinear part is a multivalued function, and the linear part is a sectorial operator

Summary

Introduction

Fractional calculus has become a well-established branch of mathematical analysis. It has many applications in engineering and science. To clarify the advantage of this study, we mention that two methods have been provided to demonstrate the existence of S-asymptotic ω-periodic solutions for semilinear fractional differential inclusions in the presence of non-instantaneous impulse effects, and in which the nonlinear part is a multivalued function, and the linear part is a sectorial operator. Lemma 3 ([49], Corollary 3.3.1) Let W be a closed convex subset of a Banach space X and N : E → Pck(W ) be a closed multifunction which is θ -condensing on every bounded subset of W , where θ is a non-singular measure of noncompactness defined on subsets of W , the set of fixed points for N is non-empty. According to Lemma 4, Steps 4, 5 and 6 imply the set T = (Dλ) is relatively compact in SAPωPC(J, E).

Examples

Conclusion