Existence and uniqueness of solution for Sturm–Liouville fractional differential equation with multi-point boundary condition via Caputo derivative

Abstract

We investigate the existence and uniqueness of a solution for a Sturm-Liouville fractional differential equation with a multi-point boundary condition via the Caputo derivative; existence and uniqueness results for the given problem are obtained via the Banach fixed point theorem. Also we study its continuous dependence on coefficients of the nonlocal condition. We discuss our results for more general boundary conditions, we present the existence of solutions under nonlocal integral conditions and also extend our results to an ordinary Sturm-Liouville problem. Two examples illustrating the main results are also presented.

On a fractional hybrid version of the Sturm–Liouville equation

“…In this section we are going to investigate continuous dependence(on the coefficient ξ i and η j of the hybrid multi-point condition) of the solution of the fractional hybrid Sturm-Liouville differential equation (21) with the hybrid multi-point boundary condition (22). Note that the main theorem of this section is a hybrid version of Theorem 3.2 in [42].…”

“…Definition 4 (see [42]) The solution of the fractional hybrid Sturm-Liouville differential equation 21is continuously dependent on the data ξ i and η j if, for every > 0, there exist δ 1 ( ) and δ 2 ( ) such that, for any two solutions u(t) andũ(t) of (21) with the initial data (22) and…”

Fractional Coupled Hybrid Sturm–Liouville Differential Equation with Multi-Point Boundary Coupled Hybrid Condition

“…In 2019, El-Sayed et al [23] investigated the following fractional Sturm-Liouville differential equation: D α c (p(t)u (t)) + q(t)u(t) = h(t) f (u(t)), t ∈ I with multi-point boundary hybrid condition…”

Best approximations of the ϕ-Hadamard fractional Volterra integro-differential equation by matrix valued fuzzy control functions

“…Generally a functional equation is said to be stable provided that, for any function f satisfying the perturbed functional equation, there exists an exact solution f 0 of that equation which is not far from the given f . Based on this concept, the study of the stability of functional equations can be regarded as a branch of optimization theory [2].…”