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Applying the convexification of nonconvex function and the theorem in Chapter 2, we establish the perturbed optimization in the bounded subsets of the dual of locally convex spaces

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Based on the principle of local thermal dynamic equilibrium, a general mathematical model and its solving process were presented.

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The model in this paper takes the form as belowThe main contents and results in this paper are as follows:In section 1, we state our main analytical results on equilibria of the above system. First, several sufficient conditions for coexistence of the steady-state are given by the standard fixed-point index theory in cone. Second, the global structure of the coexistence solutions and their local stability are established by using bifurcation theory. Third, the multiplicity, uniqueness and stability of positive steady-state solutions to this system are derived by means of perturbation theory of eigenvalues, standard regularity theory, Sobolev embedding theorem and fixed-point index theory.

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Aiming at the dual affects to classification results by each input data, this method has punish item be fuzzy, compensates weight to classification, reconstructs the optimization problem and its restrictions, reconstructs Langrage formula, and presents the theories deduction. This method is applied to the credit evaluating system of personal loan.

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Moreover, this paper makes an effort in incorporating theCosserat theory in the VCFEM for the elastic-plastic analysis of heterogeneous materialsbased on the PVP. The chapters are divided as follows: In Chapter 1, the author summarizes the study on the equivalent mechanics properties ofheterogeneous materials (including the development of the universal relationship which isirrelevant to the micro structure, the discussion of the limitations of the effective moduli, thecomputation of the effective properties of heterogeneous materials with the micromechanicsmethod and the FEM, and the equivalence of the heterogeneous micropolar materials) and thepresentation, the evolution, the corresponding numerical studies of the Cosserat model and itsapplications in engineering.

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