Global existence and uniqueness for implicit differential equation of arbitrary order

Abstract

Abstract. The aim of this paper is to establish the existence result for implicit differential equation of fractional (arbitrary) order via topological transversality theorem known as Leray-Schauder alternative. Further we prove the uniqueness results. The Grownwall's lemma for singular kernels play an important role to prove our results. We verify our results by providing an example. Mathematics subject classification (2010): 26A33, 34A08.

Some existence and stability results of Hilfer-Hadamard fractional implicit differential fractional equation in a weighted space

“…Then, we use similar ideas to those used by several researchers in recent studies, involving the existence and uniqueness of solutions of implicit fractional differential equations,. Furthermore, for other examples of implicit fractional differential equations, we suggest [1,2,6,19,29]. Consider the following Hilfer-Hadamard FIDE of the form:…”

Implicit Fractional Differential Equations in Banach Spaces via Picard and Weakly Picard Operator Theory

“…Kucche et.al. in [30,31,32] have analyzed implicit FDEs for existence, uniqueness dependence of solution on various initial data and Ulam-Hyers and Ulam-Hyers-Rassias stabilities.…”

Existence and uniqueness results for nonlinear hybrid implicit Caputo-Hadamard fractional differential equations

“…Fractional differential equations with and without delay arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein.…”