The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion

Piotr Kowalski

doi:10.3934/dcdsb.2014.19.2569

Article Contents

2014, Volume 19, Issue 8: 2569-2580. Doi: 10.3934/dcdsb.2014.19.2569

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The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion

Piotr Kowalski^1,

1.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warszawa

Received: November 30, 2013

Revised: February 28, 2014

Published: September 2014

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Abstract

We consider a differential inclusion which is somehow analogous to the classical Duffing's equation with Dirichlet boundary condition. We prove the existence of a solution using two-steps approach. Firstly we consider an auxiliary problem where we substitute $h := \frac{dx} {dt} \in L^2(0,1)$. Next, using the method of pseudomonotone and coercive operators, we prove the existence of a solution to the auxiliary problem. Finally we prove that under suitable assumptions, an iterative scheme converges to the solution of our inclusion, which appears to be unique.

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Mathematics Subject Classification: Primary: 49K15; Secondary: 34B15.

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