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1 Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
1 ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

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On the construction of two-sided approximations to positive solutions of some elliptic problem

Kolosova S., Lukhanin V.

ECONTECHMOD : An International Quarterly Journal on Economics of Technology and Modelling Processes

2016

Vol. 5, No 4

11--19

In this paper we have investigated the existence, uniqueness and possibility of constructing of two-sided approximations to the positive solution of a heat conduction problem with two sources. The investigation is based on methods in operator equations theory in half-ordered spaces. In this case we have considered a nonlinear operator equation that corresponds to the initial boundary value problem in a cone of non-negative continous functions. The properties of the corresponding operator define conditions which provide the existence and uniqueness of the solution. The conditions link the parameters of the problem implicitly meaning that they don’t provide the range of allowed values but need to be verified for each specific parameters value set separately. During the investigation we have provided the scheme of a two-sided iteration process which must satisfy the conditions in order to converge to the positive solution from both sides. The computational experiment have been conducted in two domains – unit disk and unit half disk. We have applied both two-sided approximations method and Green’s quasifunction method for the problem solving. The obtained results are presented as a surface and level lines plots and also as a table. The results in corresponding domains obtained by different methods have been compared with each other.

The Generalized saddle-node bifurcation of degenerate solution

Liu P., Wang Y-W.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2005

Vol. 45, [Z] 2

145--150

In this paper we discuss the bifurcation problem for the abstract operator equation of the form F(u,λ)=θ with a parameter lambda, where F:X x R -> Y is a C^1 mapping, X, Y are Banach spaces. By the bounded linear generalized inverse A^+ of A=F_u(u_0 λ_0), an abstract bifurcation theorem for the case dim N(F_u(u_0 λ_0)) >= codim R(F_u(u_0,λ_0))=1 has been obtained