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.


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


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