Guo, Yingxin Nontrivial solutions for boundary-value problems of nonlinear fractional differential equations. (English) Zbl 1187.34008 Bull. Korean Math. Soc. 47, No. 1, 81-87 (2010). Summary: We consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP) \[ -D^\alpha_0+u(t)=\lambda[f(t,u(t))+q(t)],\quad 0<t<1,\quad u(0) = u(1) = 0, \]where \(\lambda> 0\) is a parameter, \(1 < \alpha \leq 2\), \(D^\alpha_{0+}\) is the standard Riemann-Liouville derivative, \(f : [0, 1]\times \mathbb R\to\mathbb R\) is continuous, and \(q : (0,1) \to [0,\infty]\) is Lebesgue integrable. We obtain sufficient conditions for the existence and uniqueness of a nontrivial solution of BVP when \(\lambda\) belongs to some interval. Our approach is based on Leray-Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity which was essential for the technique used in almost all existed literature on \(f\). Cited in 25 Documents MSC: 34A08 Fractional ordinary differential equations 47N20 Applications of operator theory to differential and integral equations 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:standard Riemann-Liouville differentiation; fractional differential equation; boundary-value problem; nontrivial solution; Leray-Schauder nonlinear alternative PDFBibTeX XMLCite \textit{Y. Guo}, Bull. Korean Math. Soc. 47, No. 1, 81--87 (2010; Zbl 1187.34008) Full Text: DOI