Convexity argument for monotone potential operators and its application

Abstract

Modelling of many applied problems leads to the nonlinear equations elliptic type with monotone potential operators [6, 11]. Some methods have been developed for di erent classes of nonlinear operators to study such qualitative questions as existence, uniqueness, linearization of the nonlinear operator and convergence of an approximate solution [2, 7, 11]. In some cases it is possible to construct an abstract iteration technique for linearization of the nonlinear operator and to prove the convergence of an approximate solution. In this paper we prove an abstract iterative scheme for monotone potential operators by using the convexity argument. Note that in the literature di erent convexity arguments (or inequalities) have been applied to problems of mathematical physics [14]. In particular, in [9, 10] the logarithmic convexity argument was used to establish continuous dependence on the initial data for the Cauchy problem. The convexity argument introduced here is used as a su cient condition for the convergence of the abstract iteration scheme for monotone potential operators. Then using the result the convergence of the iteration scheme is proved for concrete nonlinear problems. The paper is organized as follows. In Section 2, we introduce the functional inequality which is called the convexity argument. Then we consider an abstract operator equation and by using the argument we prove the strong convergence of the iteration