Condensing multivalued maps and semilinear differential inclusions in Banach spaces.

*(English)*Zbl 0988.34001
de Gruyter Series in Nonlinear Analysis and Applications. 7. Berlin: de Gruyter. xi, 231 p. (2001).

The book is devoted to some aspects of multivalued analysis. It is organized into six chapters. Chapter 1 recalls general definitions and properties of multivalued maps and focuses special attention to measurable multimaps and to the superposition multioperator. Chapter 2 describes the main results of the theory of measures of noncompactness in Banach spaces and specifies the notion of condensing multimap relative to a measure of noncompactness. Chapter 3 investigates the topological degree for different types of condensing multifields in Banach spaces. A study of the existence of solutions to a system of inclusions with condensing multioperators is developed and applied to obtain optimal control for systems governed by a neutral functional-differential equation. Chapter 4 summarizes some results dealing with strongly continuous semigroups that are necessary to the study of semilinear inclusions. This is done in the last two chapters. Chapter 5 presents results concerning the existence of local and global solutions, the topological structure of the solution set and the dependence of the solutions on parameters and initial data. In Chapter 6, the authors develop methods for justifying the averaging principle in periodic problems and for proving the existence of a periodic solution and the existence of a global attractor of semilinear differential inclusions satisfying a dissipativity condition.

The presentation is self-contained, and the subject is addressed to graduate students as well as to researchers in applied functional analysis.

The presentation is self-contained, and the subject is addressed to graduate students as well as to researchers in applied functional analysis.

Reviewer: Patrick Saint-Pierre (Paris)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34G25 | Evolution inclusions |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47H04 | Set-valued operators |

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

47H10 | Fixed-point theorems |

47H11 | Degree theory for nonlinear operators |

34A60 | Ordinary differential inclusions |

34C25 | Periodic solutions to ordinary differential equations |

34C29 | Averaging method for ordinary differential equations |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49J24 | Optimal control problems with differential inclusions (existence) (MSC2000) |

49J27 | Existence theories for problems in abstract spaces |

49J53 | Set-valued and variational analysis |

54C60 | Set-valued maps in general topology |

54C65 | Selections in general topology |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

55M20 | Fixed points and coincidences in algebraic topology |

55M25 | Degree, winding number |

34H05 | Control problems involving ordinary differential equations |

34D45 | Attractors of solutions to ordinary differential equations |