Positive Solution for the Integral and Infinite Point Boundary Value Problem for Fractional-Order Differential Equation Involving a Generalized ϕ-Laplacian Operator

Abstract

In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving $\varphi$-Laplacian operator $D_{0^{+}}^{\alpha }\varphi (\text{x},D_{0^{+}}^{\beta }u(x))+f(x,u(x))=0,$ $x\in (0,1),D_{0^{+}}^{\sigma }u(0)=D_0^{\beta }+u(0)=0$, $u(1)={\displaystyle {\int }_0^{1}g(t)u(t)dt}+{\displaystyle {\sum }_{n=1}^{n=+\infty }{\alpha }_{n}u({\eta }_n)}$, where $\varphi :[0,1]\times R\to R$ is a continuous function and $D_{0^{+}}^p$ is the Riemann-Liouville derivative for $p\in \{a,\beta ,\text{ σ}\}.$ By using some properties of fixed point index, we obtain the existence results and give an example at last.