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Nontrivial solutions for boundary-value problems of nonlinear fractional differential equations. (English) Zbl 1187.34008

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 \(f\).

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
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