Existence of positive solutions to nonlinear fractional boundary value problem with changing sign nonlinearity and advanced arguments (Q1722218)

Summary: We discuss the existence of positive solutions to a class of fractional boundary value problem with changing sign nonlinearity and advanced arguments \(D^\alpha x(t) + \mu h(t) f(x(a(t))) = 0\), \(t \in(0,1)\), \(2 < \alpha \leq 3\), \(\mu > 0\), \(x(0) = x'(0) = 0\), \(x(1) = \beta x(\eta) + \lambda [x]\), \(\beta > 0\), and \(\eta \in(0,1)\), where \(D^\alpha\) is the standard Riemann-Liouville derivative, \(f : [0, \infty) \rightarrow [0, \infty)\) is continuous, \(f(0) > 0\), \(h : [0,1] \rightarrow(- \infty, + \infty)\), and \(a(t)\) is the advanced argument. Our analysis relies on a nonlinear alternative of Leray-Schauder type. An example is given to illustrate our results.