Partial ordering methods in nonlinear problems

Special Interest Categories: Pure and applied mathematics, physics, optimisation and control, mechanics and engineering, nonlinear programming, economics, finance, transportation and elasticity. The usual method used in studying nonlinear problems such as topological method, variational method and others are generally only suited to the nonlinear problems with continuity and compactness. However, a lots of the problems appeared in theory and applications have no continuity and compactness, For example, differential equations and integral equations in infinite dimensional spaces, various equations defined on unbounded region are generally having no compactness. The problems can been divided into three types as follows: (1) Without using compact conditions but only using some inequalities related to some ordering, the existence and uniqueness of the fixed point for increasing operators, decreasing operators and mixed monotone operators, and the convergence of the iterative sequence are obtained. Also, these results have been used to nonlinear integral equations defined on unbounded regions. (2) Without using continuity conditions but only using a very relaxed weakly compact conditions, some new fixed point theorem of increasing operators are obtained. We have applied these results to nonlinear equations with discontinuous terms. (3) They systemly use the partial ordering methods to nonlinear integro-differential equations (include impulsive type) in Banach space.