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September 10, 2025Open Access

A Unified Approach to Implicit Fractional Differential Equations with Anti-Periodic Boundary Conditions

Key Points

The study establishes the existence and uniqueness of solutions for implicit fractional differential equations.
It employs Banach’s and Schaefer’s fixed-point theorems under Lipschitz conditions to prove results.
Stability is analyzed through Ulam–Hyers and generalized Ulam–Hyers stability concepts involving fractional derivatives.
Examples demonstrate the practical applicability of the developed analytical framework in solving complex equations.

Abstract

This paper develops a unified analytical framework for implicit fractional differential equations subject to anti-periodic boundary conditions. The study considers two main cases: fractional derivatives of order α∈(0,1) and α∈(1,2), both defined with respect to a general kernel function. The existence and uniqueness of solutions are established using Banach’s and Schaefer’s fixed-point theorems under suitable Lipschitz conditions. Furthermore, Ulam–Hyers stability and generalized Ulam–Hyers stability are investigated for each problem. Examples are provided to illustrate the applicability of the main results.

A Unified Approach to Implicit Fractional Differential Equations with Anti-Periodic Boundary Conditions

Key Points

Abstract

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