On a class of nonlinear singular Riemann-Liouville fractional differential equations
The main objective of this paper is to study existence and multiplicity of positive solutions for the fractional boundary value problem \[ D_{0^{+}}^{\alpha}u(t)+f(t,u(t))=0,\quad t\in(0,1),\tag{1} \] \[ u(0)=u^{'}(0)=\dots =u^{(n-2)}(0)=0,\quad D_{0^{+}}^{p}u(1)=\int_{0}^{1}D_{0^{+}}^{q}u(t)dH(t)\tag{2} \] in which \(\alpha\in(n-1,n],~n\in\mathbb{N}_{\geq3},~p\in[1,n-2]\) and \(q\in[0,p]\). \(D_{0^{+}}^{\beta}\) stands for the Riemann-Liouville fractional derivative of order \(\beta\). In the boundary conditions (2), the integral boundary condition is a Riemann-Stieltjes integral with \(H\) a function of bounded variation. Furthermore, the nonlinearity \(f\) potentially may change sign and can be singular at the end points and/or in the space variable \(u\). The author, using the Green function technique and the Guo-Krasnoselskii fixed point theorem together with some controls on the Green function, the function \(H\) and the nonlinearity \(f\), has been obtained some existence and multiplicity criteria for solutions of the fractional boundary value problem (1)--(2) on a positive cone in the Banach space \(C[0,1]\).
scientific article; zbMATH DE number 6965864
scientific article