Functional differential equations. Advances and applications (Q2808519)

In this monograph, the authors present their results in the field of functional differential equations (FDEs) obtained during the past two decades. The book is organized in five chapters, an appendix, an extensive bibliography and an index. Chapter 1 contains a short history on classical FDEs, new types of these equations, and the main directions in the study of FDEs. There are also presented the tools used in the study of various classes of FDEs, namely: metric spaces and related concepts, function spaces, fixed point theorems (the contraction mapping principle, Schauder theorem, Schauder-Tychonoff theorem), the Leray-Schauder principle, the method based on monotone operators, and the step method for FDEs with finite delay, and further types of functional equations (the discrete equations, the fractional-order differential equations and the difference equations). Chapter 2 deals with various results on the local existence for functional equations in the classes of continuous or measurable functions. Global existence on a preassigned interval, and the existence of solutions in spaces of measurable functions like Lebesgue spaces \(L^p\), \(1\leq p\leq \infty\), are also discussed. The authors investigate some ordinary differential equations, integral equations in a single variable, integro-differential equations, finite or infinite delay equations, and equations involving operators acting on function spaces, such as causal operators. Some properties of the solutions, namely: positiveness, constant sign, boundedness on unbounded intervals, asymptotic behavior, uniqueness and dependence on data, and properties of the solution sets are studied. An application to the optimal control theory for a first-order FDE and a characterization of the flow invariance of a set are finally presented. Chapter 3 is concerned with problems of stability for FDEs (particularly for ODEs) and of some refinements of this concept, such as uniform stability, asymptotic stability, uniform asymptotic stability and partial stability. The preservation of a given kind of stability is also investigated in this chapter. The basic method used in the theory of stability is the comparison method, which uses various differential inequalities and one or more Liapunov functions/functionals subject to such inequalities, and it has a wide range of applications. The stability of a set, not necessarily formed by solutions of a given equation/system for finite delay equations, and the Ulam-Hyers-Rassias stability for a FDE are also studied here. Chapter 4 is focused on the oscillatory properties of solutions, especially of almost periodic type (which includes periodic type), for which the series approach can be applied. Several examples of functional differential equations, integral differential equations and dynamical systems with regard to the existence of \(AP_r\)-almost periodic solutions (\(r\in [1,2]\)), solutions in Besicovitch spaces \(B_2\) of almost periodic functions, and also in the classical case (Bohr), are given in this chapter. Chapter 5 deals with neutral functional differential equations with continuous and discrete argument, for which existence results, boundedness and stability properties of the solutions (especially asymptotic stability) are presented. We remark that each chapter is finalized by a section entitled``Bibliographical Notes'' which contains some comments, several papers and authors whose results complete the study related to the subjects of the chapter. In Appendix, the first author presents generalized Fourier series of the form \(\sum_{k=1}^{\infty}a_ke^{i\lambda_k t}\) with \(a_k\in \mathbb{C}\) and \(\lambda_k\) some real-valued function on \(\mathbb{R}\) that represents the third stage of the development of the Fourier analysis, after periodicity and almost periodicity. A survey of problems which occur in the construction of new spaces of oscillatory functions is addressed here. The extensive Bibliography has 562 titles of papers and books which are connected with the topics considered in this monograph. The book is well-written, and the presentation is very clear and rigorous. This monograph is a great source for graduate students in mathematics, science and engineering, and for all researchers interested in functional differential equations and related applied fields.