Resonance Problems for Some Non-autonomous Ordinary Differential Equations
Recent years have seen a lot of activity in the study of quasilinear non-autonomous ordinary differential equations or systems of the form (ϕ(y′))′ = f(t, y, y′), where \(\phi : A \subset {\mathbb{R}}^{n} \rightarrow B \subset {\mathbb{R}}^{n}\) is some h
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Abstract Recent years have seen a lot of activity in the study of quasilinear non-autonomous ordinary differential equations or systems of the form ..y 0 //0 D f .t; y; y 0 /, where W A Rn ! B Rn is some homeomorphism such that .0/ D 0 between the open sets A and B. The situation generalizes the classical case where A D B D Rn and is the identity, and the well-studied case of the p-Laplacian .p > 1/ where .s/ D kskp2 s. Contemporary researches concern less standard situations where W B.a/ ! Rn (singular homeomorphism) and W Rn ! B.a/ (bounded homeomorphism), where B.a/ is the open ball of centre 0 and radius a. For n D 1, a model for the first case, namely .s/ D ps , corresponds to acceleration in special relativity, and a model for the second 1s 2 situation, namely .s/ D p s 2 , corresponds to problem with curvature satisfying 1Cs various conditions. In those case, both topological and variational methods, and sometimes combination of them give new complementary existence and multiplicity results. We will describe some of them. Some attention will be given to the generalized forced pendulum equation ..y 0 //0 C A sin y D h.t/ when is singular or bounded.
1 Introduction, Notations and Preliminary Results 1.1 Introduction Recent years have seen a lot of activity in the study of quasilinear non-autonomous ordinary differential equations of the form J. Mawhin () Institut de Recherche en Math´ematique et Physique, Universit´e Catholique de Louvain, chemin du cyclotron, 2, 1348 Louvain-la-Neuve, Belgium e-mail: [email protected] A. Capietto et al., Stability and Bifurcation Theory for Non-Autonomous Differential Equations, Lecture Notes in Mathematics 2065, DOI 10.1007/978-3-642-32906-7 3, © Springer-Verlag Berlin Heidelberg 2013
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..u0 //0 D f .t; u; u0 / where W .a; a/ ! .b; b/ is an increasing homeomorphism such that .0/ D 0 between the open intervals .a; a/ and .b; b/, with 0 < a; b C1. The situation generalizes the classical case where a D b D C1 and is the identity, and the well-studied case of the p-Laplacian .p > 1/ where a D b D C1 and .s/ D jsjp2 s. In this last case, the Fredholm alternative for the solvability of .ju0 jp2 u0 /0 jujp2 u D h.t/ with classical Dirichlet, Neumann or periodic boundary conditions on Œ0; T u.0/ D 0 D u.T /; u0 .0/ D 0 D u0 .T /; u.0/ D u.T /; u0 .0/ D u0 .T /
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is far to be fully understood, despite of recent interesting partial results [6, 27, 29, 40, 43, 44, 70, 128, 129]. Contemporary researches concern less standard situations where W .a; a/ ! R (singular homeomorphism) and W R ! .a; a/ (bounded homeomorphism). A model for the first case, namely .s/ D p s 2 , corresponds to acceleration 1s in special relativity. A model for the second situation, namely .s/ D p s 2 , 1Cs corresponds to curvature or capillarity problem. In those cases, both topological and variational methods give new complementary existence and multiplicity results. The topological approach essentially makes use of Brouwer and
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