Abstract
The iterative methods to solve the system of the difference equations derived from the nonlinear elliptic equation with integral condition are considered. The convergence of these methods is proved using the properties of M-matrices, in particular, the regular splitting of an M-matrix. To our knowledge, the theory of M-matrices has not ever been applied to convergence of iterative methods for system of nonlinear difference equations. The main results for the convergence of the iterative methods are obtained by considering the structure of the spectrum of the two-dimensional difference operators with integral condition. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014).
Highlights
Over the last few decades, in both the theory of differential equations and numerical analysis, much attention is paid to various types of differential equations with nonlocal conditions
In one of the first papers written on this topic [10], the finite difference method for a linear second order elliptic equation with Bitsadze–Samarskii nonlocal condition was considered
Convergence of iterative method to some systems of linear difference equations using the M-matrix properties was analysed in [33]. We develop further this idea and, according to the M-matrix methodology, we investigate the convergence of iterative method for a system of nonlinear difference equations with a nonlocal condition
Summary
Over the last few decades, in both the theory of differential equations and numerical analysis, much attention is paid to various types of differential equations with nonlocal conditions. In one of the first papers written on this topic [10], the finite difference method for a linear second order elliptic equation with Bitsadze–Samarskii nonlocal condition was considered. An iterative method for solving a system of difference equations with a nonlocal condition has been considered possibly for the first time. Many papers were written in order to justify the finite difference method for elliptic equations with various types of nonlocal conditions [1, 2, 4,5,6, 13, 18]. Even in the first articles [24, 28] of the investigation of the spectrum for such operators, it has been noted that the spectrum structure of both differential and difference operators with rather simple nonlocal conditions can be quite complex. This structure very sensitively depends on the parameters and functions under nonlocal conditions
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