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Asymptotically Almost Periodicity for a Class of Weyl–Liouville fractional Evolution Equations

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Abstract

This paper is devoted to study a class of abstract fractional evolution equation in a Banach space X:

$$\begin{aligned} \hbox {D}_{+}^{\alpha }x(t)+Ax(t)=F(t,x(t)), \quad t\in \mathbb {R}, \end{aligned}$$
(1)

where \(0<\alpha <1\), \(-A\) is the infinitesimal generator of a \(C_{0}\)-semigroup on X, and F(t, x) is an appropriate function defined on phase space; the fractional derivative is understood in the Weyl–Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we obtain some new sufficient conditions to ensure the existence of asymptotically almost periodic mild solutions for (1). Our result generalizes and improves some previous results, since the Lipschitz continuity on the nonlinearity F(t, x) with respect to x is not required. An example is also presented as an application to illustrate the feasibility of the abstract result.

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Authors and Affiliations

  1. Department of Mathematics, Guangdong University of Education, Guangzhou, 510303, People’s Republic of China

    Junfei Cao

  2. Department of Mathematics, Guelma University, 24000, Guelma, Algeria

    Amar Debbouche

  3. Faculty of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, People’s Republic of China

    Yong Zhou

  4. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Yong Zhou

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  1. Junfei Cao
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  2. Amar Debbouche
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  3. Yong Zhou
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Correspondence to Amar Debbouche.

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Cao, J., Debbouche, A. & Zhou, Y. Asymptotically Almost Periodicity for a Class of Weyl–Liouville fractional Evolution Equations. Mediterr. J. Math. 15, 155 (2018). https://doi.org/10.1007/s00009-018-1208-7

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  • DOI: https://doi.org/10.1007/s00009-018-1208-7

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