Existence results for fractional order functional differential equations with infinite delay

Abstract

The Banach fixed point theorem and the nonlinear alternative of Leray-Schauder type are used to investigate the existence of solutions for fractional order functional and neutral functional differential equations with infinite delay.

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.…”

On fractional random differential equations with delay

“…Indeed, we have (ii) for all (t, ω) ∈ [0, a] × Ω, f ω (t, x t ) = tω 2 1 + ω 2 x(t − 1, ω) ≤ tω 2 1 + ω 2 x(t − 1, ω) ≤ t sup r∈ [−1,0] x(r, ω) ≤ a 2 (1 + ω 2 ).…”

“…See for example, M. Benchohra, et al [2], J. Deng, et al [5] and the references therein. In this paper, inspired and motivated by Lupulescu et al [22][23][24][25] and Dhage [6][7][8][9], under the condition of the Lipschitzean right-hand side we obtain the existence and uniqueness of the solutions of the fractional random differential equations with delay.…”

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.…”