## 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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### References:

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