Existence Results for Hilfer Fractional Differential Equations with Variable Coefficient

Abstract

The aim of this paper is to establish the existence and uniqueness results for differential equations of Hilfer-type fractional order with variable coefficient. Firstly, we establish the equivalent Volterra integral equation to an initial value problem for a class of nonlinear fractional differential equations involving Hilfer fractional derivative. Secondly, we obtain the existence and uniqueness results for a class of Hilfer fractional differential equations with variable coefficient. We verify our results by providing two examples.

Well-posedness analysis and pseudo-Galerkin approximations using Tau Legendre algorithm for fractional systems of delay differential models regarding Hilfer (α,β)-framework set

“…and H D a;b 0 W ϐ relate to the FHD [27][28][29][30][31] of ϑ ϐ of fraction α and type β. Our primary concern is to obtain a quantitative estimate of (1) through examination of the elegances of the SGLA, which depends on orthogonal spline basis functions [32][33][34][35][36].…”

“…Lemma 2. [28] Let ρ2(0,1), u 2 ðÀ 1; 0�, and ρ+υ>0, then at 0 < p < rþu 2 and W 2 L 1 p , one has ðI r 0 þ s u WðsÞÞðƻÞ 2 ACðJ; RÞ. Lemma 3.…”

A Result Regarding Finite-Time Stability for Hilfer Fractional Stochastic Differential Equations with Delay

“…By using Hilfer fractional calculus, the fixed-point method, and the Mittag-Leffer function, Gao and Yu [31] studied the uniqueness and existence of the nonlocal values of solutions for a kind of semi-linear system with Hilfer fractional derivatives. Li et al [32] first established the equivalent Volterra integral equation, and then existence and uniqueness for a kind of fractional system of the Hilfer type with variable coefficients were discussed.…”

Passivity Analysis of Fractional-Order Neutral-Type Fuzzy Cellular BAM Neural Networks with Time-Varying Delays

“…e study of fractional calculus can be dated back to 1695, and the fractional operator concept was put forward by Leibnitz, which did not acquire sufficient attention for a long period since it is complicated. Many actual systems can be described by fractional-order differential equations, making the slowly developed fractional calculus be a renewal of interest [1][2][3][4][5]. Generally speaking, fractional calculus is a generalization of classical calculus and is more accurate to describe reality models compared to the corresponding integer-order calculus in different research communities, such as particle physics, wave mechanics, electrical systems, and computational methods for mathematical physics, and many references cited therein [6][7][8][9].…”