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Existence of Coupled Quasi-solutions of Systems of Nonlinear Elliptic Boundart Value Problems

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Existence of Coupled Quasi-solutions of Systems of Nonlinear Elliptic Boundart Value Problems

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Title: Existence of Coupled Quasi-solutions of Systems of Nonlinear Elliptic Boundart Value Problems
Author: Lakshmikantham, V.; Ladde, G. S.; Vatsala, A. S.
Abstract: Systems of nonlinear elliptic boundary value problems arise in many applications such as multiple chemical reactions that take place in an isothermal or nonisothermal catalyst pellet and simple models of tubular chemical reactors [7,8,9]. Moreover, such problems also occur in a natural way as auxiliary problems in stability analysis of steady states of dynamic systems governed by reaction-diffusion systems. Constructive methods of proving existence results of such boundary value problems, which can also provide numerical procedures for the computation of solutions, are of greater value than theoretical existence results. The method of upper and lower solutions coupled with monotone iterative technique has been employed successfully to prove existence of multiple solutions of nonlinear elliptic boundary value problems, in special cases, by various authors [3,4,5,7,8,9,12,17,18,19,20]. In [6], the general nonlinear elliptic boundary value problem in the scalar case has been considered. Recently, in [21] weakly coupled systems of elliptic boundary value problems, when the nonlinear terms are independent of the gradient terms, are discussed and some special type of results were obtained. We, in this paper, investigate general systems of nonlinear elliptic boundary value problems when the nonlinear terms possess a mixed quasimonotone property. We discuss a very general situation and obtain coupled extremal quasisolutions in [10,16], which in special cases, reduce to min-max solutions and minimal and maximal solutions [15]. We shall also indicate, following [21], how one step cyclic monotone iterative schemes can be generated which yield accelerated rate of convergence of iterates.
URI: http://hdl.handle.net/10106/2288
Date: 1983-06

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