Abstract
A nonlinear boundary value problem related to an equation of Kirchhoff type is considered. The existence of multiple positive solutions is proved through Avery-Peterson Fixed Point Theorem. A numerical method based on Levenberg-Marquadt algorithm combined with a heuristic process is present in order to align numerical and theoretical aspects.
Highlights
In this paper we present a study on second order equation Kirchhoff problem, given byM( u 22)u + q(t) f (t, u, u ) = 0 (1.1)u(0) = 0, u(1) = 0 where M : R → R, f : [0, 1] × R × R → R and q : R+ → R are continuous maps.Variations of (1.1) can be related to stationary state of Kirchhoff equation [8]
Numerical studies related to second order equations, normally, are presented as illustration of the existence Banach’s Fixed Point Theorem applied to equation but no strategy is given to ilustraste more general results
4 FINAL REMARKS Considering the numerical aspects of this work, we present a new algorithm (Algorithm 1) and a new heuristic that allows obtain multiple solutions
Summary
In this paper we present a study on second order equation Kirchhoff problem, given by. Variations of (1.1) can be related to stationary state of Kirchhoff equation [8]. 560 MULTIPLE SOLUTIONS FOR AN EQUATION OF KIRCHHOFF TYPE by [12] that studies with theoretical aspects were developed using Banach’s Fixed Point Theorem or Leray-Schauder Alternative combined with Krasnoselskii’s Theorem. It is natural to ask about existence results by using Avery-Peterson Theorem [4] Numerical studies related to second order equations, normally, are presented as illustration of the existence Banach’s Fixed Point Theorem applied to equation but no strategy is given to ilustraste more general results (like the existence results provided by Avery-Peterson Theorem).
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