Sidorov N.A.   Falaleev M.   Sidorov D.  

Some Equations with Fredholm Operator in Main Part: Existence Theorems, Bifurcation and Applications

Reporter: Sidorov N.A.

Creation and investigation of  parameter dependent functional equations in mathematical models with Fredholm operator in the main part has various applications in many areas of mathematical modeling. Contemporary branching (bifurcation) theory is one of the most important aspects in the state of the art applied mathematics. The goals of this theory are the qualitative theory of dynamical systems, analytical and numerical computation of their solutions under absence of conditions guaranteeing the uniqueness of the solution.Applications sphere of bifurcation theory and Lyapunov-Schmidt method is permanent extending. Besides their traditional applications in elasticity theory and hydrodynamics bifurcation theory methods turn out to be successful in the investigation of specific nonlinear problems of phase transitions and plasma physics, mathematical biology, filtration theory, non-Newtonian fluids movement theory. During the recent decades  the Lyapunov-Schmidt method has been applied in combination with representation and group analysis theories, finite-dimensional topological and variational methods, perturbation methods as well as the regularization theory (see Sidorov N., Loginov B., Sinitsin A. and Falaleev M.  Lyapunov-Schmidt Methods in  Nonlinear Analysis and Applications. Kluwer Ac. Publ. 2002; Nonlinear Analysis and Nonlinear Differential Equations.}  V.A.Trenogin, A.F.Fillipov (Eds.) Moscow,  Fizmatlit.  2003). Such a combined methods approaches have given the possibility to prove the most general existence theorems of bifurcating solutions, to make their algorithmic and qualitative  analysis, to develop asymptotical and iterative methods for differential-operator equations.This talk presents some our results in the above-mentioned areas. We also discuss our recent results of application of successive approximations method for construction of classic and generalized solutions for different classes of nonlinear equations in irregular cases.

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