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**Partial ordering methods in nonlinear problems.**
*(English)*
Zbl 1116.45007

Hauppauge, NY: Nova Science Publishers (ISBN 1-59454-018-7/hbk). viii, 346 p. (2004).

Publisher’s description: Special Interest Categories: Pure and applied mathematics, physics, optimization and control, mechanics and engineering, nonlinear programming, economics, finance, transportation and elasticity.

The usual method used in studying nonlinear problems such as the topological method, the variational method and others are generally only suited to nonlinear problems with continuity and compactness. However, a lot of problems appearing 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 be divided into three types as follows:

(I) Without using compactness conditions but only using some inequalities related to some ordering, the existence and uniqueness of a fixed point for increasing operators, decreasing operators and mixed monotone operators, and the convergence of iterative sequence are obtained. Also, these results have been used to nonlinear integral equations defined on unbounded regions.

(II) Without using continuity conditions but only using a very relaxed weakly compactness conditions, some new fixed point theorem of increasing operators are obtained. We have applied these results to nonlinear equations with discontinuous terms.

(III) They systematically use the partial ordering methods to nonlinear integro-differential equations (include impulsive type) in Banach space.

The usual method used in studying nonlinear problems such as the topological method, the variational method and others are generally only suited to nonlinear problems with continuity and compactness. However, a lot of problems appearing 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 be divided into three types as follows:

(I) Without using compactness conditions but only using some inequalities related to some ordering, the existence and uniqueness of a fixed point for increasing operators, decreasing operators and mixed monotone operators, and the convergence of iterative sequence are obtained. Also, these results have been used to nonlinear integral equations defined on unbounded regions.

(II) Without using continuity conditions but only using a very relaxed weakly compactness conditions, some new fixed point theorem of increasing operators are obtained. We have applied these results to nonlinear equations with discontinuous terms.

(III) They systematically use the partial ordering methods to nonlinear integro-differential equations (include impulsive type) in Banach space.

### MSC:

45N05 | Abstract integral equations, integral equations in abstract spaces |

45G10 | Other nonlinear integral equations |

45J05 | Integro-ordinary differential equations |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |