Extremal solutions of φ−Caputo fractional evolution equations involving integral kernels

Abstract

This paper deals with the existence and uniqueness of solution for the Cauchy problem of φ−Caputo fractional evolution equations involving Volterra and Fredholm integral kernels. We derive a mild solution in terms of semigroup and construct a monotone iterative sequence for extremal solutions under a noncompactness measure condition of the nonlinearity. These results can be reduced to previous works with the classical Caputo fractional derivative. Furthermore, we give an example of initial-boundary value problem for the time-fractional parabolic equation to illustrate the application of the results.

Keywords

• fractional evolution equation
• monotone iterative technique
• upper and lower solutions
• volterra operator
• fredholm operator