Existence of Solutions for Fractional Differential Equations Involving Two Riemann-Liouville Fractional Orders

Abstract

In this work, we study existence and uniqueness of solutions for multi-point boundary value problem of nonlinear fractional differential equations with two fractional derivatives. By using the variety of fixed point theorems, such as Banach's fixed point theorem, Leray-Schauder's nonlinear alternative and Leray-Schauder's degree theory, the existence of solutions is obtained. At the end, some illustrative examples are discussed.

A Generic FPGA Implementation of the Fractional-Order Derivative and Its Application

“…The equilibrium points of formula (4) can be calculated by setting the right side of the equation to 0 as listed in Tab. And the Jacobian matrix of formula ( 4) can be obtained by solving the partial derivatives which is shown in formula (8).…”

“…The values of the integer-order system's equilibrium points are substituted into formula (8) For a 3-D integer-order chaotic system, all the equilibrium points mentioned above are the type of index 2, which means one of the eigenvalues is stable and the other two are unstable. In contrast, the points of index 1 indicate one of the eigenvalues is unstable and other two are stable.…”

“…And fractional calculus has gained more attention in inverter [5], oscillator [6], neural networks [6] and gene regulatory networks [7]. The fractional-order derivative has three frequently-used definitions -Riemann-Liouville definition [8], Caputo definition [9] and Grunwald-Letnikov definition [10]. Equilibrium point, time-domain waveform, phase diagram, maximum Lyapunov exponent and bifurcation diagram are the basic methods for analyzing fractional-order chaotic systems [9].…”

SOLVABILITY FOR A CLASS OF NONLINEAR CAPUTO-HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN OPERATOR IN BANACH SPACES

“…Although there are many methods (such as Banach contraction principle, Schauder's fixed point theorem, and Krasnoselskii's fixed point theorem, etc.) to analyze, under suitable conditions, the existence and uniqueness of solution of various problems with initial conditions, boundary conditions, integral boundary conditions, nonlinear boundary conditions, and periodic boundary conditions for fractional differential equations, for more details see for instance [5,13,18,19,20,21,26,29,43]. We focus here on the so-called measures of noncompactness, see for instance [1,2,4,12,14,39].…”

Existence and Stability Results for Fractional Differential Equations With Two Caputo Fractional Derivatives

“…For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. For more details, we refer the reader to [6,25]. Recently, by applying different fixed point theorems such as the Banach fixed point theorem, Schaefer's fixed point theorem, Krasnoselskii's fixed point theorem, the Leray-Schauder nonlinear alternative and the fixed point theorem of O'Regan, many researchers have obtained some interesting results of the existence and uniqueness of solutions to boundary value problems for fractional differential equations with nonlocal boundary value problems [1,2,7,8,9,14,15,18,23,24] and the references therein.…”