Stability and convergence of the Dirichlet difference problem

Estimation of a solution of the Dirichlet difference problem. We make use of a priori estimates obtained in Section 2 for a grid equation of common structure for constructing a uniform estimate of a solution of the Dirichlet difference problem (24)-(26) arising in Section 1  

Other ideas are connected with the operator A, which coincides with A at the near-boundary nodes and with A at the remaining inner nodes, leading to an alternative form of writing 

We are going to evaluate separately each of the functions v(x) and w(x). In order to estimate v(x), it is necessary to construct a majorant Y(x). Assuming that the origin is inside the domain G, we try to determine a majorant of the type 

IMIc 1 licit is easily seen from the expression for Y that Y c KR2. So, for a solution of problem (7) the estimate 

Our next step is the estimation of the function w x). First, we are going to show that for problem (8) 

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