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Existence and multiplicity of heteroclinic solutions for a non-autonomous boundary eigenvalue problem. (English) Zbl 1054.34046
Authors’ summary: We investigate the boundary eigenvalue problem \[ x''-\beta(c,t,x)x'+g(t,x)=0,\quad x(-\infty)=0,\quad x(+\infty)=1, \] depending on the real parameter \(c.\) We take \(\beta\) continuous and positive and assume that \(g\) is bounded and becomes active and positive only when \(x\) exceeds a threshold value \(\theta\in ]0,1[.\) At the point \(\theta\) we allow \(g(t,\cdot)\) to have a jump. Additional monotonicity properties are required, when needed. Our main discussion deals with the nonautonomous case. In this context, we prove the existence of a continuum of values \(c\) for which this problem is solvable and we estimate the interval of such admissible values. In the autonomous case, we show its solvability for at most one \(c^*.\) In the special case when \(\beta\) reduces to \(c+h(x)\) with \(h\) continuous, we also give a nonexistence result, for any real \(c.\) Our methods combine comparison-type arguments, both for first and second-order dynamics, with a shooting technique. Some applications of the obtained results are included.

34B40 Boundary value problems on infinite intervals for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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