Monotone iterative technique for semilinear elliptic systems
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We develop monotone iterative technique for a system of semilinear elliptic boundary value problems when the forcing function is the sum of Caratheodory functions which are nondecreasing and nonincreasing, respectively. The splitting of the forcing function leads to four different types of coupled weak upper and lower solutions. In this paper, relative to two of these coupled upper and lower solutions, we develop monotone iterative technique. We prove that the monotone sequences converge to coupled weak minimal and maximal solutions of the nonlinear elliptic systems. One can develop results for the other two types on the same lines. We further prove that the linear iterates of the monotone iterative technique converge monotonically to the unique solution of the nonlinear BVP under suitable conditions. 1. Introduction Semilinear systems of elliptic equations arise in a variety of physical contexts, specially in the study of steady-state solutions of time-dependent problems. See [1, 4, 5], for example. Existence and uniqueness of classical solutions of such systems by monotone method has been established in [2, 4]. Using generalized monotone method, the existence and uniqueness of coupled weak minimal and maximal solutions for the scalar semilinear elliptic equation has been established in [3]. They have utilized the existence and uniqueness result of weak solution of the linear equation from [1]. In [3], the authors have considered coupled upper and lower solutions and have obtained natural sequences as well as alternate sequences which converge to coupled weak minimal and maximal solutions of the scalar semilinear elliptic equation. In this paper, we develop generalized monotone method combined with the method of upper and lower solutions for the system of semilinear elliptic equations. For this purpose, we have developed a comparison result for the system of semilinear elliptic equations which yield the result of the scalar comparison theorem of [3] as a special case. One can derive analog results for the other two types of coupled weak upper and lower solutions on the same lines. We develop two main results related to two different types of coupled weak upper and lower solutions of the nonlinear semilinear elliptic systems. Copyright © 2005 Hindawi Publishing Corporation Boundary Value Problems 2005:2 (2005) 93–106 DOI: 10.1155/BVP.2005.93
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Semilinear elliptic systems
We obtain natural as well as intertwined monotone sequences which converge uniformly to coupled weak minimal and maximal solutions of the semilinear elliptic system. Further using the comparison theorem for the system, we establish the uniqueness of the weak solutions for the nonlinear semilinear elliptic systems. The existence of the solution of the linear system has been obtained as a byproduct of our main results. 2. Preliminaries In this section, we present some known comparison results, existence and uniqueness results related to scalar semilinear elliptic BVP without proofs. See [1, 3] for details. Consider the semilinear elliptic BVP ᏸu =
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