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Types of solutions and multiplicity results for second order nonlinear boundary value problems

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  • We study the nonlinear BVP

    $x'' = f(t, x, x')$,     (i) $x(0)cos\alpha - x'(0)sin\alpha = 0$, $x(1)cos\Beta - x'(1)sin\Beta = 0$,    (ii)

    provided that $f$ : [0,1] $\times R^2 \rightarrow R$ is continuous together with the partial derivatives $f_(x'), 0 <= \alpha < \pi, 0 < \Beta <= \pi.$ If a quasi-linear ($F$ is bounded) equation

    $(L_2x)(t) := d/(dt) (e^(2mt)x') + e^(2mt)k^2x = F (t,x,x')$    (iii)

    can be constructed so that any solution of the problem (iii), (ii) solves also the BVP (i), (ii), then we say that the problem (i), (ii) allows for ($L_2x$)-quasilinearization. We show that if the problem (i), (ii) allows for quasilinearization with respect to essentially different linear parts then the problem (i), (ii) has multiple solutions.

    Mathematics Subject Classification: Primary: 34B15.


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