On the existence of mild solutions to the Cauchy problem for a class of fractional evolution equation

Abstract

We are concerned with the existence of mild solutions to the Cauchy problem for fractional evolution equations of neutral type with almost sectorial operatorswhere 0 <q < 1, the fractional derivative is understood in the Caputo sense, A is an almost sectorial operator on a complex Banach space, and f, h are given functions. With the help of the theory of measure of noncompactness and a fixed point theorem of Darbo type, we establish a new existence theorem of mild solutions for the Cauchy problem above. By the way, the global attractive property of the solutions is also obtained. Moreover, we give two examples to illustrate our abstract results.

Positive Solution for the Nonlinear Hadamard Type Fractional Differential Equation withp-Laplacian

“…Fractional differential equations have attracted more and more attention for their useful applications in various fields, such as economics, science, and engineering; see [1][2][3][4]. In the last few decades, much attention has been focused on the study of the existence and uniqueness of solutions for boundary value problems of Riemann-Liouville type or Caputo type fractional differential equations; see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. There are few papers devoted to the research of the -Laplacian fractional differential equations; see [20][21][22][23][24][25].…”

Erratum to: On the existence of mild solutions to the Cauchy problem for a class of fractional evolution equation

“…Author details1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China.…”