A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions

Abstract

In this survey paper, we shall establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fractional differential equations and inclusions involving the Caputo fractional derivative. The both cases of convex and nonconvex valued right hand side are considered. The topological structure of the set of solutions is also considered.

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

“…Many physical processes appear to exhibit fractional order behavior that may vary with time or space. In recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; we only enumerate here the monographs of Kilbas et al [26,27], Diethelm [28], Hilfer [29], Podlubny [30], Miller [31], and Zhou [32] and the papers of Agarwal et al [33,34], Benchohra et al [35,36], El-Borai [37], Lakshmikantham et al [38][39][40][41], Mophou et al [42][43][44][45], N'Guérékata [46], and Zhou et al [47][48][49][50] and the reference therein.…”

Remarks for one-dimensional fractional equations

“…For several results on fractional differential equations, one can see, for example, the monographs of Miller and Ross [15], Samko et al [18], Podlubny [16], Hilfer [11], Kilbas et al [13] and the papers [1,[3][4][5][6][7] We cite a recent monograph by Kristály, Rădulescu and Varga [14] as a general reference on variational methods adopted here.…”

Existence of Positive Solutions to a Coupled System of Fractional Hybrid Differential Equations

“…[1][2][3][4][5]. Moreover, most of the authors also considered the fractional differential equations as an object of mathematical investigations, we refer the readers and the references therein for recent development of the theory [5][6][7][8][9][10][11][12][13][14][15][16]. Caputo's fractional derivatives play vital role in applied problems as it provides known physical interpretation for initial and boundary conditions.…”