Multiple Solutions of Singular Perturbation Problems

Abstract

Under certain conditions on g(x, u) we establish the existence and asymptotic behavior for small ε > 0 of multiple asymptotic solutions of the nonlinear boundary value problem εu" + u' - g(x,u) = 0, 0 < x < 1, u'(0) - au(0)= A ≥ 0, a > 0, u'(1) + bu(1) = B > 0, b > 0. Formal techniques of singular perturbation theory clearly reveal the mechanism which controls the appearance of multiple solutions. Their existence is then established rigorously by iteration schemes and the so-called "shooting method" for ordinary differential equations.

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