Dirichlet boundary value problem for the second order asymptotically linear system (Q341227)

In this paper the authors study the existence and multiplicity of solutions of the second-order autonomous ordinary differential systems NEWLINE\[NEWLINE {\mathbf x}'' = {\mathbf f}({\mathbf x}), NEWLINE\]NEWLINE subject to the Dirichlet boundary condition NEWLINE\[NEWLINE {\mathbf x}(0)={\mathbf 0} = {\mathbf x}(1). NEWLINE\]NEWLINE Here the nonlinearity \({\mathbf f}({\mathbf x})\) is asymptotically linear at \(\infty\), i.e. \({\mathbf f}'(\infty)={\mathbf A}\) exists. After finding the explicit formulas for the (topological) index of the system NEWLINE\[NEWLINE {\mathbf x}'' = {\mathbf A}{\mathbf x}, NEWLINE\]NEWLINE the authors give some existence and multiplicity results for nontrivial solutions of the boundary value problems.NEWLINENEWLINEIn some sense, the results and the methods used are standard in literature.