2007, 2007(Special): 1061-1069. doi: 10.3934/proc.2007.2007.1061

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

1. 

Daugavpils University, Parades str. 1, LV-5400 Daugavpils, Latvia, Latvia

Received  September 2006 Revised  June 2007 Published  September 2007

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.

Citation: Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061
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