SOLVABILITY OF BOUNDARY-VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations D(alpha)u(t) + lambda[f(t, u(t)) + q(t)] = 0, 0 < t < 1, u(0) = 0, u(1) = beta u(eta), where lambda > 0 is a parameter, 1 < alpha <= 2, eta epsilon (0, 1), beta epsilon R = (-infinity, +infinity), beta eta(alpha-1) not equal 1, D(alpha) is a Riemann-Liouville differential operator of order alpha, f:(0, 1) x R -> R is continuous, f may be singular for t = 0 and/or t = 1, and q(t):[0, 1] -> [0, +infinity] is continuous. We give some sufficient conditions for the existence of nontrivial solutions to the formulated boundary-value problems. Our approach is based on the Leray-Schauder nonlinear alternative. In particular, we do not use the assumption of nonnegativity and monotonicity of f essential for the technique used in almost all available literature.