Fractional Retarded Differential Equations Involving Mixed Nonlocal Plus Local Initial Conditions

Abstract

The aim of this paper is to discuss the existence of mild solutions and positive mild solutions for a general class of semilinear fractional retarded evolution equations subjected to mixed nonlocal plus local initial conditions on infinite dimensional Banach spaces. Under the situation that the nonlinear term and nonlocal function satisfy some appropriate growth conditions and a noncompactness measure condition, we obtained the existence of mild solutions and positive mild solutions by utilizing a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness. In addition, the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example about the retarded parabolic partial differential equation involving a general mixed nonlocal plus local initial conditions is also given to illustrate the feasibility of our abstract results.

Keywords:

• Convex-condensing operator
• equicontinuous semigroup
• fractional delay evolution equations
• Kuratowski measure of noncompactness
• nonlocal delay initial condition

MATHEMATICS SUBJECT CLASSIFICATION:

• 26A33
• 34K30
• 47D06
• 47G20

Acknowledgments

The authors would like to express sincere thanks to the anonymous referee for his/her carefully reading the manuscript and valuable comments and suggestions.

Additional information

Funding

This research was supported by NNSF of China (11661071), NNSF of China (11501455) and Doctoral Research Fund of Northwest Normal University (no. 6014/0002020209).