##
**Nonlinear integral equations in abstract spaces.**
*(English)*
Zbl 0866.45004

Mathematics and its Applications (Dordrecht). 373. Dordrecht: Kluwer Academic Publishers. viii, 341 p. (1996).

This book is dedicated to the theory of nonlinear integral equations in abstract spaces (usually Banach spaces), which makes it different of other books in the existing literature on this subject. It does contain, to a great extent, results obtained by the authors during the last 10-15 years. Their research associates also enjoy a good representation in the book.

The first chapter contains some preliminary material (mean value theorem, measure of noncompactness, comparison results for Volterra equations, linear integral equations in Banach spaces). The second chapter deals with nonlinear integral equations of Volterra type, Fredholm type and Hammerstein type. A model for the spread of infectious diseases is also studied. The third chapter is dedicated to nonlinear integro-differential equations in Banach spaces and treats first order and second order equations, covering also the case of unbounded domains. The fourth chapter contains results related to nonlinear integral equations of impulsive type. For an illustration of this type of equations we mention here \[ x(t)= x_0(t)+ \int^t_{t_0} H(t,s,x(s))ds+ \sum_{t_0<t_k<t} a_k(t) I_k(x(t_k)), \] where \(x_0(t)\), \(H(t,s,x)\), \(a_k(t)\) and \(I_k(x)\) are given. The connection between differential and integral equations in Banach spaces is stressed repeatedly.

Let us point out the fact that all equations contain a single independent variable, but they can either be finite-dimensional or infinite-dimensional (depending on the range of unknown functions). Each chapter is followed by references and comments. The list of references includes 110 items.

The book is a welcome addition to the existing literature of nonlinear integral equations and, at the same time, a good illustration for problems in nonlinear analysis. It is aimed, first of all, to the researchers in these fields of investigation. It is also useful for graduate teaching. Many methods used are commonly encountered in the literature of nonlinear analysis (like fixed point methods, comparison technique, method of upper and lower solutions). More applications, other than those dealing with differential equations, would have been welcome.

The first chapter contains some preliminary material (mean value theorem, measure of noncompactness, comparison results for Volterra equations, linear integral equations in Banach spaces). The second chapter deals with nonlinear integral equations of Volterra type, Fredholm type and Hammerstein type. A model for the spread of infectious diseases is also studied. The third chapter is dedicated to nonlinear integro-differential equations in Banach spaces and treats first order and second order equations, covering also the case of unbounded domains. The fourth chapter contains results related to nonlinear integral equations of impulsive type. For an illustration of this type of equations we mention here \[ x(t)= x_0(t)+ \int^t_{t_0} H(t,s,x(s))ds+ \sum_{t_0<t_k<t} a_k(t) I_k(x(t_k)), \] where \(x_0(t)\), \(H(t,s,x)\), \(a_k(t)\) and \(I_k(x)\) are given. The connection between differential and integral equations in Banach spaces is stressed repeatedly.

Let us point out the fact that all equations contain a single independent variable, but they can either be finite-dimensional or infinite-dimensional (depending on the range of unknown functions). Each chapter is followed by references and comments. The list of references includes 110 items.

The book is a welcome addition to the existing literature of nonlinear integral equations and, at the same time, a good illustration for problems in nonlinear analysis. It is aimed, first of all, to the researchers in these fields of investigation. It is also useful for graduate teaching. Many methods used are commonly encountered in the literature of nonlinear analysis (like fixed point methods, comparison technique, method of upper and lower solutions). More applications, other than those dealing with differential equations, would have been welcome.

Reviewer: C.Corduneanu (Arlington)

### MSC:

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

45J05 | Integro-ordinary differential equations |

45Gxx | Nonlinear integral equations |

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