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Sign-preserving solutions for a class of asymptotically linear systems of second-order ordinary differential equations. (English) Zbl 1510.34042

In this paper, the existence of multiple solutions is proved for the Dirichlet problem associated with a system of ordinary differential equations of the form \[ u'' + A(t,u)u = 0, \quad u \in \mathbb{R}^2 \] where \(A\) is a symmetric \(2 \times 2\) matrix.
More precisely, the matrix \(A\) is required to be asymptotically linear at zero and at infinity, that is \[ \lim_{u \to 0} A(t,u) = A_0(t) \] and \[ \lim_{\vert u \vert \to \infty} A(t,u) = A_\infty(t). \] Moreover, it is assumed that the entry \(a_{12}\) of the matrix \(A\) is sign-definite.
Under these hypotheses, the existence of two solutions is proved whenever the Morse index of \(A_0\) is zero while the Morse index of \(A_\infty\) is non-zero (or viceversa). Moreover, it is shown that such solutions are sign-preserving component-wise, meaning that each component of the two vector solutions \(u_i = (x_i,y_i)\), with \(i=1,2\), has definite sign (depending on the sign of \(a_{12}\)).
The proof relies on a higher-dimensional shooting technique, a crucial role being played by a careful analysis of the behavior of the phase angles (in connection with the index theory for linear Hamiltonian systems).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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