Existence and Uniqueness of Solutions for a Boundary Value Problem of Fractional Type with Nonlocal Integral Boundary Conditions in Hölder Spaces

In this paper, we prove the existence and uniqueness of solutions for the following fractional boundary value problem cD0+αu(t)=λf(t,u(t)),t∈[0,1],u(0)=γI0+ρu(η)=γ∫0η(η-s)ρ-1Γ(ρ)u(s)ds,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{aligned}&^cD_{0+}^\alpha u(t)=\lambda f(t,u(t)),\quad t\in [0,1],\\&u(0)=\gamma I_{0+}^\rho u(\eta )=\gamma \int _0^\eta \frac{(\eta -s)^{\rho -1}}{\Gamma (\rho )}u(s)\mathrm {d}s, \end{aligned} \right. \end{aligned}$$\end{document}where 0<α≤1,0<η<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha \le 1, 0<\eta <1$$\end{document} and λ,γ,ρ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ,\gamma ,\rho \in \mathbb {R}$$\end{document}. Our solutions are placed in the space of functions satisfying the Hölder condition. Our analysis relies on a fixed point theorem in complete metric spaces. Moreover, we present some examples illustrating our results.