Nonlinear integral operators and applications.

*(English)*Zbl 1030.47003
de Gruyter Series in Nonlinear Analysis and Applications. 9. Berlin: Walter de Gruyter. xii, 201 p. (2003).

The present book is devoted to the description of some new theoretical results and methods for the investigation of nonlinear integral operators and equations. This investigation is given in a general setting based on the so-called modular linear spaces instead of the standard setting of normed linear spaces. This allows to extend a range of applications to nonlinear integral equations. In particular, an application to nonlinear summability and to nonlinear sampling theory and signals is given.

Chapters 1-2 are preliminary ones. There the authors introduce a notion of modular spaces, modular convergence and modulus of continuity of functions in this spaces. The kernel functions (kernel functionals) generating the integral operators are also described. Chapters 3-5 are devoted to convolution integral operators. In these chapters, the authors present applications of nonlinear integral equations to approximation theory including error estimates, embedding theorems, the error of modular approximations and convergence theorems. In Chapter 3, the same questions are considered for the usual convolution operator \[ (Tf)(s)=\int_\Omega K(t, f(t+s)) d\mu (t),\quad s\in\Omega, \] and in Chapter 4 for the Urysohn type integral operator \[ (Tf)(s)=\int_\Omega K(s,t, f(t)) d\mu (t), \quad s\in\Omega \] with homogeneous kernel functions. The kernel \(K(t,u)\) (or \(K(s,t,u)\)) is in general a Cathéodory kernel function satisfying the Lipshitz \((L,\psi)\) condition in \(u.\) Namely, \(|K(t,u)-K(t,v)|\leq L(t)\psi (|u-v|)\) with a summable function \(L(t)\) (or \(|K(s,t,u)-K(s,t,v)|\leq L(s,t)\psi (t, |u-v|)\)) and a function \(\psi\). Finally, in Chapter 5 the conservative nonlinear summability methods are considered. In Chapter 6 it is assumed that the modular is a function of generalized bounded variation while before it was supposed that a modular is monotone. Convergence in generalized variations in Chapter 6 is considered for nonlinear Mellin-type convolution operators. Chapters 7-9 contain some applications to nonlinear integral equations and embedding theorems, existence and uniqueness results, applications to signal analysis and to modular convergence for sampling type operators.

The bibliography consists of 212 papers and books and all chapters also contain short bibliographical notes.

The book may be useful for specialists in the theory of linear and nonlinear integral equations and operators and for postgraduate students.

Contents: Chapter 1. Kernel functionals and modular spaces. Chapter 2. Absolutely continuous modulars and moduli of continuity. Chapter 3. Approximation by convolution type operators. Chapter 4. Urysohn integral operators with homogeneous kernel functions. Application to nonlinear Mellin-type convolution operators. Chapter 5. Summability methods by convolution-type operators. Chapter 6. Nonlinear integral operators in the space \(BV_\varphi\). Chapter 7. Application to nonlinear integral equations. Chapter 8. Uniform approximation by sampling operators. Application in signal analysis. Chapter 9. Modular approximation by sampling type operators. References. Index.

Chapters 1-2 are preliminary ones. There the authors introduce a notion of modular spaces, modular convergence and modulus of continuity of functions in this spaces. The kernel functions (kernel functionals) generating the integral operators are also described. Chapters 3-5 are devoted to convolution integral operators. In these chapters, the authors present applications of nonlinear integral equations to approximation theory including error estimates, embedding theorems, the error of modular approximations and convergence theorems. In Chapter 3, the same questions are considered for the usual convolution operator \[ (Tf)(s)=\int_\Omega K(t, f(t+s)) d\mu (t),\quad s\in\Omega, \] and in Chapter 4 for the Urysohn type integral operator \[ (Tf)(s)=\int_\Omega K(s,t, f(t)) d\mu (t), \quad s\in\Omega \] with homogeneous kernel functions. The kernel \(K(t,u)\) (or \(K(s,t,u)\)) is in general a Cathéodory kernel function satisfying the Lipshitz \((L,\psi)\) condition in \(u.\) Namely, \(|K(t,u)-K(t,v)|\leq L(t)\psi (|u-v|)\) with a summable function \(L(t)\) (or \(|K(s,t,u)-K(s,t,v)|\leq L(s,t)\psi (t, |u-v|)\)) and a function \(\psi\). Finally, in Chapter 5 the conservative nonlinear summability methods are considered. In Chapter 6 it is assumed that the modular is a function of generalized bounded variation while before it was supposed that a modular is monotone. Convergence in generalized variations in Chapter 6 is considered for nonlinear Mellin-type convolution operators. Chapters 7-9 contain some applications to nonlinear integral equations and embedding theorems, existence and uniqueness results, applications to signal analysis and to modular convergence for sampling type operators.

The bibliography consists of 212 papers and books and all chapters also contain short bibliographical notes.

The book may be useful for specialists in the theory of linear and nonlinear integral equations and operators and for postgraduate students.

Contents: Chapter 1. Kernel functionals and modular spaces. Chapter 2. Absolutely continuous modulars and moduli of continuity. Chapter 3. Approximation by convolution type operators. Chapter 4. Urysohn integral operators with homogeneous kernel functions. Application to nonlinear Mellin-type convolution operators. Chapter 5. Summability methods by convolution-type operators. Chapter 6. Nonlinear integral operators in the space \(BV_\varphi\). Chapter 7. Application to nonlinear integral equations. Chapter 8. Uniform approximation by sampling operators. Application in signal analysis. Chapter 9. Modular approximation by sampling type operators. References. Index.

Reviewer: Nikolai K.Karapetyants (Rostov-na-Donu)

##### MSC:

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

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |

45P05 | Integral operators |

45Gxx | Nonlinear integral equations |

47N20 | Applications of operator theory to differential and integral equations |

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