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Multi-valued nonlinear perturbations of time fractional evolution equations in Banach spaces

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Abstract

The paper is concerned with the fractional evolution inclusion \(^\mathrm{c}D_t^q u(t)\in Au(t)+F(t,u(t))\) in Banach spaces, where \(^\mathrm{c}D_t^q\), \(0<q<1\), is the regularized Caputo fractional derivative of order q, A generates a compact semigroup, and \(F\) is a multi-valued function with convex, closed values. Constructing a suitable directionally \(L^p\)-integrable selection from \(F\), we study the compactness and \(R_\delta \)-structure of the set of trajectories on a closed domain. Moreover, we discuss the \(R_\delta \)-structure of the set of trajectories to the control problem corresponding to the inclusion above. Finally, we apply our abstract theory to boundary value problems of fractional diffusion inclusions.

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Acknowledgments

Research supported by the NNSF of China (Nos. 11101202, 61104138) and the University Scientific and Technological Innovation Project of Guangdong Province (No. 2013KJCX0068). The authors would like to thank the referees very much for their valuable suggestions and comments.

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Authors and Affiliations

  1. Department of Applied Mathematics, Guangdong University of Foreign Studies, Guangzhou, 510420, People’s Republic of China

    Rong-Nian Wang & Qing-Hua Ma

  2. Department of Mathematics, Nanchang University, Nanchang, 330031, Jiangxi, People’s Republic of China

    Peng-Xian Zhu

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  1. Rong-Nian Wang
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Correspondence to Qing-Hua Ma.

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Wang, RN., Zhu, PX. & Ma, QH. Multi-valued nonlinear perturbations of time fractional evolution equations in Banach spaces. Nonlinear Dyn 80, 1745–1759 (2015). https://doi.org/10.1007/s11071-014-1453-7

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