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<p>This paper conducts a rigorous qualitative analysis of a class of implicit fractional Volterra integro-differential equations subject to anti-periodic boundary conditions, incorporating the recently developed <InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi mathvariant="normal">Ξ</mi><mo stretchy="false">)</mo></math></EquationSource><EquationSource Format="TEX">$(\sigma , \beta ,\Xi )$</EquationSource></InlineEquation>-tempered Caputo fractional derivative. This generalized operator provides a unified framework that encompasses several well-known fractional derivatives. By leveraging fixed-point theorems—namely, Banach’s, Schaefer’s, and Schauder’s—we establish sufficient criteria for the existence and uniqueness of solutions. Furthermore, we investigate various types of Ulam-Hyers stability (Ulam-Hyers, generalized Ulam-Hyers, and Ulam-Hyers-Rassias) for the proposed problem, deriving explicit stability constants. The theoretical findings are substantiated through two detailed numerical examples that illustrate the applicability of the main theorems and demonstrate the sensitivity of the results to the choice of the kernel function&#xa0;Ξ.</p>

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Analysis of anti-periodic boundary value problems for implicit fractional Volterra integro-differential equations with generalized tempered fractional derivatives

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  • 发表日期：

    2026年5月4日

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  • Abdulrahman A. Sharif,
  • Maha M. Hamood,
  • Kirtiwant P. Ghadle

摘要

This paper conducts a rigorous qualitative analysis of a class of implicit fractional Volterra integro-differential equations subject to anti-periodic boundary conditions, incorporating the recently developed (σ,β,Ξ)$(\sigma , \beta ,\Xi )$-tempered Caputo fractional derivative. This generalized operator provides a unified framework that encompasses several well-known fractional derivatives. By leveraging fixed-point theorems—namely, Banach’s, Schaefer’s, and Schauder’s—we establish sufficient criteria for the existence and uniqueness of solutions. Furthermore, we investigate various types of Ulam-Hyers stability (Ulam-Hyers, generalized Ulam-Hyers, and Ulam-Hyers-Rassias) for the proposed problem, deriving explicit stability constants. The theoretical findings are substantiated through two detailed numerical examples that illustrate the applicability of the main theorems and demonstrate the sensitivity of the results to the choice of the kernel function Ξ.