Generalized existence results for solutions of nonlinear fractional differential equations with nonlocal boundary conditions

Abstract

This research delves into investigating the presence of solutions to fractional differential equations with an order $\sigma \in (2,3]$. These equations include the Caputo derivative and introduce innovative nonlocal antiperiodic boundary conditions. These boundary conditions, defined at a nonlocal intermediary point $0\le \delta <c$ and the fixed endpoint c of the interval $[0,c]$, where $\psi \left(\delta \right)=-\psi \left(c\right)$, ${\psi }^{\prime }\left(\delta \right)=-{\psi }^{\prime }\left(c\right)$, and ${\psi }^{″}\left(\delta \right)=-{\psi }^{″}\left(c\right)$. They are specifically designed to enhance measurement accuracy in applied mathematics and physics. The research demonstrates the existence and uniqueness of solutions by employing Krasnoselskii's fixed-point theorem and the contraction mapping principle. A thorough analysis of the fractional differential equations supports this mathematical framework. This work verifies the viability of such equations and emphasizes their practical importance in representing intricate physical phenomena. Finally, examples are provided to illustrate the results.