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Dirichlet boundary value problem for the second order asymptotically linear system. (English) Zbl 1356.34031

In this paper the authors study the existence and multiplicity of solutions of the second-order autonomous ordinary differential systems \[ {\mathbf x}'' = {\mathbf f}({\mathbf x}), \] subject to the Dirichlet boundary condition \[ {\mathbf x}(0)={\mathbf 0} = {\mathbf x}(1). \] 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 \[ {\mathbf x}'' = {\mathbf A}{\mathbf x}, \] the authors give some existence and multiplicity results for nontrivial solutions of the boundary value problems.
In some sense, the results and the methods used are standard in literature.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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