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Hao, Zhaocai; Huang, Yubo

Existence of positive solutions to nonlinear fractional boundary value problem with changing sign nonlinearity and advanced arguments. (English) Zbl 1470.34069

Abstr. Appl. Anal. 2014, Article ID 158436, 7 p. (2014).
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.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34A08 Fractional ordinary differential equations
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\textit{Z. Hao} and \textit{Y. Huang}, Abstr. Appl. Anal. 2014, Article ID 158436, 7 p. (2014; Zbl 1470.34069)
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References:

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