Methods in nonlinear integral equations.

*(English)*Zbl 1060.65136
Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0844-9/hbk). xiv, 218 p. (2002).

This book deals with several methods of nonlinear analysis for the investigation of nonlinear integral equations such as Fredholm, Volterra and Hammerstein equations, and integral equations with delay. Necessary abstract results of nonlinear analysis such as compactness criteria, fixed point theorems, critical point results and principles of iterative approximation are provided. As applications of these results, existence, uniqueness, continuous dependence and approximation results are discussed relative to nonlinear integral equations of different kinds.

The selected topics reflect the interset of the author. Although most of the methods are classical, new points of view, extensions and applications are presented. For example, the selected topics include a unified existence theory of continuous and integrable equations, Schechler’s bounded mountain pass lemma, new fixed point results for increasing and decreasing operators in ordered Banach spaces, monotone iterative techniques, and quasilinearization methods. The presentation is self-contained and therefore should be useful to find the current ideas and methods.

The selected topics reflect the interset of the author. Although most of the methods are classical, new points of view, extensions and applications are presented. For example, the selected topics include a unified existence theory of continuous and integrable equations, Schechler’s bounded mountain pass lemma, new fixed point results for increasing and decreasing operators in ordered Banach spaces, monotone iterative techniques, and quasilinearization methods. The presentation is self-contained and therefore should be useful to find the current ideas and methods.

Reviewer: V. Lakshmikantham (Melbourne/Florida)

##### MSC:

65R20 | Numerical methods for integral equations |

45L05 | Theoretical approximation of solutions to integral equations |

45G10 | Other nonlinear integral equations |

47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |

47H10 | Fixed-point theorems |

47J25 | Iterative procedures involving nonlinear operators |

45-02 | Research exposition (monographs, survey articles) pertaining to integral equations |