Boundary value problems for systems of second-order functional differential equations

Abstract

Systems of second-order functional differential equations $(x'(t)+L(x)(t))'=F(x)(t)$ together with nonlinear functional boundary conditions are considered. Here $L:C^1([0,T];\mathbb{R}^n) \rightarrow C^0([0,T];\mathbb{R}^n)$ and $F:C^1([0,T];\mathbb{R}^n) \rightarrow L_1([0,T];\mathbb{R}^n)$ are continuous operators. Existence results are proved by the Leray-Schauder degree and the Borsuk antipodal theorem for $\alpha$-condensing operators. Examples demonstrate the optimality of conditions.