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An Existence Result for Some Fractional Evolution Equation with Nonlocal Conditions and Compact Resolvent Operator

  • G. M. N’GuérékataEmail author
Conference paper
  • 43 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 302)

Abstract

We are concerned with the existence of mild solutions to the fractional differential equation \(D^{\alpha }_{t}u(t)=Au(t)+J_{t}^{1-\alpha }f(t,u(t)),\;\;0<t\le T,\) with nonlocal conditions \(u_0=u(0)+g(u)\), where \(0<\alpha <1\), \(D^{\alpha }_{t}\) is the Caputo derivative, \(J_{t}^{1-\alpha }h(t):=\frac{1}{\Gamma (1-\alpha )}\int _{0}^{t}\frac{h(s)}{(t-s)^{\alpha }}ds\) is the fractional integral of order \(\alpha \) of the function h, and \(A:D(A)\subset X\rightarrow X\) is a linear operator which generates a compact analytic resolvent family \((R_{\alpha }(t))_{t\ge 0}\), X being a Banach space. We obtain our results using the Krasnoleskii’s fixed point theorem.

Keywords

S-asymptotically \(\omega \)-periodic functions Bloc-periodic functionc Fractional differential equations 

Notes

Acknowledgements

We would like to thank the referee for his/her careful reading and valuable suggestions.

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Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of MathematicsMorgan State UniversityBaltimoreUSA

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