Existence and multiplicity of heteroclinic solutions for a non-autonomous boundary eigenvalue problem
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In this paper we investigate the boundary eigenvalue problem
x'' - β(c, t, x)x' + g(t, x) = 0
x(-∞) = 0, x(+∞) = 1
depending on the real parameter c. We take β continuous and positive and assume that g is bounded and becomes active and positive only when x exceeds a threshold value θ ∈]0, 1[. At the point θ we allow g(t, ·) to have a jump. Additional monotonicity properties are required, when needed. Our main discussion deals with the non-autonomous 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 β reduces to c + h(x) with h continuous, we also give a non-existence result, for any real c. Our methods combine comparison-type arguments, both for the the first and second order dynamics, with a shooting technique. Some applications of the obtained results are included.
CitationMalaguti, L., & Marcelli, C. (2003). Existence and multiplicity of heteroclinic solutions for a non-autonomous boundary eigenvalue problem. Electronic Journal of Differential Equations, 2003(118), pp. 1-21.
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