The focus of our investigation was on determining the existence of solutions for fractional differential equations (FDEs) of order $1<\gamma \le 2$ involving the boundary conditions ${\kappa }_0\varphi (0)+{\eta }_0\varphi (v)={\mu }_0$, and ${\kappa }_1{\varphi }^{{}^{\prime }}(0)+{\eta }_1{\varphi }^{{}^{\prime }}(v)={\mu }_1$, for ${\kappa }_i,{\eta }_i,{\mu }_i\in {\mathbb{R}}^{+}$. The existence results were based on the Schauder fixed point theorem and the nonlinear alternative of the Leray-Schauder type. Examples were provided to illustrate the results.

This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point $0\le c<d$ and fixed points at the end of the interval $[0,d]$. For $i=0,1$, the boundary conditions are as follows: $a_i,b_i>0$, $a_{0}p(c)=-b_{0}p(d),a_1{p}^{{}^{\prime }}(c)=-b_1{p}^{{}^{\prime }}(d)$. Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixed-point theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved.

In this work, we introduce a analytical framework for proving the existence and uniqueness of solutions to nonlinear fractional differential equations (NFDEs) of order $\nu \in (1,2]$, subject to irregular boundary conditions, in a Banach space. Several fundamental definitions, theorems, and auxiliary lemmas are presented for the analysis of the proposed problem. We establish sufficient conditions for existence and uniqueness for the proposed system by applying Krasnosel'skii's fixed-point theorem and the Banach contraction principle. Finally, an illustrative example is constructed to validate the main theoretical results and demonstrate the effectiveness and applicability of the proposed approach, showing how the generalized framework can be systematically applied to analyze fractional boundary value problems.

In this paper, we study a class of nonlinear fractional boundary value problems involving the Caputo fractional derivative of order $\tau \in (1,2]$. We establish sufficient conditions for the existence and uniqueness of solutions. Our analysis is based on Krasnoselskii's fixed-point theorem and the Banach contraction principle in appropriate Banach spaces. The obtained results not only guarantee the solvability of the proposed equation but also generalize and improve several related results available in the literature. An example is provided in several orders to validate the results.