Nonlinear sequential Riemann–Liouville and Caputo fractional differential equations with generalized fractional integral conditions

Abstract

In this paper, we discuss the existence and uniqueness of solutions for two new classes of sequential fractional differential equations of Riemann-Liouville and Caputo types with generalized fractional integral boundary conditions, by using standard fixed point theorems. In addition, we also demonstrate the application of the obtained results with the aid of examples.

Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Equation with Iterated Fractional Integrals Conditions

“…For a systematic development on the topic we refer to the monographs as [3][4][5][6][7][8][9][10]. Fractional order boundary value problems attracted considerable attention and the literature on the topic was enriched with a huge number of articles, for instance, see [11][12][13][14][15][16][17][18][19][20][21][22][23] and references cited therein. In the literature there are several kinds of fractional derivatives, such as Riemann-Liouville, Caputo, Hadamard, Hilfer, Katugampola, and so on.…”

“…In many papers in the literature the authors studied existence and uniqueness results for boundary value problems and coupled systems of fractional differential equations by using mixed types of fractional derivatives. For example Riemann-Liouvile and Caputo fractional derivatives are used in the papers [14,19,21], Riemann-Liouville and Hadamard-Caputo fractional derivatives in the papers [15] and Caputo-Hadamard fractional derivatives in the papers [20,22]. Multiterm fractional differential equations also gained considerable importance in view of their occurrence in the mathematical models of certain real world problems, such as behavior of real materials [24], continuum and statistical mechanics [25], an inextensible pendulum with fractional damping terms [26], etc.…”

Existence Results for Sequential Riemann–Liouville and Caputo Fractional Differential Inclusions with Generalized Fractional Integral Conditions

“…In a recent work [22], the authors studied the existence of solutions for a nonlinear sequential Riemann-Liouville and Caputo fractional differential equation subject to generalized fractional integral conditions. The objective of the present paper is to investigate the multivalued analogue of the problem considered in [22]. Precisely, we consider the following inclusions problem: RL D q C D r x (t) ∈ F(t, x(t)), 0 < q ≤ 1, 0 < r ≤ 1, t ∈ (0, T),…”

“…. , n. For definitions of fractional derivatives and integrals involved in the problem (1) and (2), see [22]. Here we emphasize that the boundary conditions (2) correspond to different kinds of integral boundary conditions for appropriate choice of the parameters; for details, see Remark 2 in [22].…”

“…Lemma 1. [22] Let 0 < q ≤ 1, 0 < r ≤ 1, ρi , ρ j , q, r, ᾱi , α j > 0, ξ i , δ j ∈ (0, T), βi , ηi , κi , β j , η j , κ j ∈ R for i = 1, 2, . .…”