THE EXISTENCE AND ULAM-HYERS STABILITY RESULTS FOR MULTI-POINT GENERALIZED CAPUTO FRACTIONAL BOUNDARY VALUE PROBLEMS

This paper addresses a class of multi-point boundary value problems involvinggeneralized fractional derivatives with integral conditions. Specifically, we consider the following problem <p class="disp_formula">$\begin{aligned}&amp; { }^c D_{a^{+}}^{\alpha, h} u(t)+f(t, u(t))=0, \quad t \in[a, b], \\&amp; u^{(i)}(a)=0, i=0,1,2, \ldots, n-2, \\&amp; u(b)=\sum\limits_{j=1}^{m-2} \beta_j \int_a^{\eta_j} g(s) u(s) d s+\sum\limits_{j=1}^{m-2} \lambda_j u\left(\eta_j\right) .\end{aligned}$ We establish necessary and sufficient conditions for the existence and uniqueness of solutionsto this problem. Our approach relies on the Banach fixed point theorem and Schaefer's fixedpoint theorem to prove the existence of solutions. Additionally, we introduce the conceptof Ulam-Hyers stability for this class of boundary value problems and provide a stabilityanalysis. To illustrate the applicability of our theoretical results, we present two concreteexamples.