Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

"Matrix Mathematics" is a reference work for users of matrices in all branches of engineering, science, and applied mathematics. This book brings together a vast body of results on matrix theory for easy reference and immediate application. Each chapter begins with the development of relevant background theory followed by a large collection of specialized results. Hundreds of identities, inequalities, and matrix facts are stated rigorously and clearly with cross references, citations to the literature, and illuminating remarks. Twelve chapters cover all of the major topics in matrix theory: preliminaries; basic matrix properties; matrix classes and transformations; matrix polynomials and rational transfer functions; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; norms; functions of matrices and their derivatives; the matrix exponential and stability theory; and linear systems and control theory. A detailed list of symbols, a summary of notation and conventions, an extensive bibliography with author index, and an extensive index are provided for ease of use. The book will be useful for students at both the undergraduate and graduate levels, as well as for researchers and practitioners in all branches of engineering, science, and applied mathematics.

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Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory

Some Gronwall type inequalities for kernels of L-type and application in qualitative theory of Volterra integral equations and for systems of differential equations are presented.

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Some Gronwall Type Inequalities and Applications

By critical point theory, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equation is a positive integer. This is probably the first time critical point theory has been applied to deal with the existence of periodic solutions of difference systems.This project is supported by TRATOYT of China and by the State Education Commission Trans-Century Training Program Foundation for the Talents.

The Existence of Periodic and Subharmonic Solutions of Subquadratic Second Order Difference Equations

Generalization of a retarded Gronwall-like inequality and its applications

Lipschitz stability of nonlinear systems of differential equations

Integral Inequalities of Gronwall-Bellman-Bihari Type and Asymptotic Behavior of Certain Second Order Nonlinear Differential Equations

Non-standard Discretization Methods for Some Biological Models

We give an overview of results on the existence of periodic solutions of difference equations that have been obtained in the last two decades. The survey covers both ordinary and Volterra difference systems. Some extensions and generalizations of known result are also presented.

Periodic solutions of difference equations

We present a complete theory for the stability of non-hyperbolic fixed points of one-dimensional continuous maps. As well as we give simple criteria for the global stability of general maps without using derivatives.

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Fozi M. Dannan

Papers

Lipschitz stability of nonlinear systems of differential equations

Integral Inequalities of Gronwall-Bellman-Bihari Type and Asymptotic Behavior of Certain Second Order Nonlinear Differential Equations

Non-standard Discretization Methods for Some Biological Models

Periodic solutions of difference equations

Stability of Hyperbolic and Nonhyperbolic Fixed Points of One-dimensional Maps