On modeling and complete solutions to general fixpoint problems in multi-scale systems with applications

Ning Ruan,David Yang Gao

doi:10.1186/s13663-018-0648-x

Abstract

This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

Highlights

1 Introduction The fixed point problem is a well-established subject in the area of nonlinear analysis [3, 4, 7], which is usually formulated in the following form: (P0) : x = F(x), (1)
5 Conclusions Based on the canonical duality theory, a unified model is proposed such that the general fixed point problems can be reformulated as a global optimization problem
This model is directly related to many other challenging problems in variational inequality, d.c. programming, chaotic dynamics, nonconvex analysis/PDEs, post-buckling of large deformed structures, phase transitions in solids, computer science, etc

Summary

Introduction

The fixed point problem is a well-established subject in the area of nonlinear analysis [3, 4, 7], which is usually formulated in the following form: (P0) : x = F(x), (1). It can be used to “model” a large class of mathematical problems, one must pay a price: it is impossible to develop a unified mathematical theory with powerful real-world applications. This dilemma is due to a gap between mathematics and physics. Arnold [1]:“In the middle

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Conclusion