Gaston M. N'Guérékata – författare

This volume introduces a unified and homogeneous exposition of the theory of almost automorphic functions and its applications to the fast-growing field of differential equations in abstract spaces (Banach and Hilbert spaces). It is based essentially on the work of M. Zaki, S. Zaidman, and the author during the last three decades of the 20th century. It is also the first one to present the theory of almost periodic functions in general topological vector spaces.

Since the publication of our first book [80], there has been a real resiu-gence of interest in the study of almost automorphic functions and their applications ([16, 17, 28, 29, 30, 31, 32, 40, 41, 42, 46, 51, 58, 74, 75, 77, 78, 79]). New methods (method of invariant s- spaces, uniform spectrum), and new concepts (almost periodicity and almost automorphy in fuzzy settings) have been introduced in the literature. The range of applications include at present linear and nonlinear evolution equations, integro-differential and functional-differential equations, dynamical systems, etc...It has become imperative to take a bearing of the main steps of the the­ ory. That is the main purpose of this monograph. It is intended to inform the reader and pave the road to more research in the field. It is not a self contained book. In fact, [80] remains the basic reference and fimdamental source of information on these topics. Chapter 1 is an introductory one. However, it contains also some recent contributions to the theory of almost automorphic functions in abstract spaces. VIII Preface Chapter 2 is devoted to the existence of almost automorphic solutions to some Unear and nonUnear evolution equations. It con­ tains many new results. Chapter 3 introduces to almost periodicity in fuzzy settings with applications to differential equations in fuzzy settings. It is based on a work by B. Bede and S. G. Gal [40].

Since the publication of our first book [80], there has been a real resiu-gence of interest in the study of almost automorphic functions and their applications ([16, 17, 28, 29, 30, 31, 32, 40, 41, 42, 46, 51, 58, 74, 75, 77, 78, 79]). New methods (method of invariant s- spaces, uniform spectrum), and new concepts (almost periodicity and almost automorphy in fuzzy settings) have been introduced in the literature. The range of applications include at present linear and nonlinear evolution equations, integro-differential and functional-differential equations, dynamical systems, etc...It has become imperative to take a bearing of the main steps of the the­ ory. That is the main purpose of this monograph. It is intended to inform the reader and pave the road to more research in the field. It is not a self contained book. In fact, [80] remains the basic reference and fimdamental source of information on these topics. Chapter 1 is an introductory one. However, it contains also some recent contributions to the theory of almost automorphic functions in abstract spaces. VIII Preface Chapter 2 is devoted to the existence of almost automorphic solutions to some Unear and nonUnear evolution equations. It con­ tains many new results. Chapter 3 introduces to almost periodicity in fuzzy settings with applications to differential equations in fuzzy settings. It is based on a work by B. Bede and S. G. Gal [40].

Almost Automorphic and Almost Periodic Functions in Abstract Spaces introduces and develops the theory of almost automorphic vector-valued functions in Bochner's sense and the study of almost periodic functions in a locally convex space in a homogenous and unified manner. It also applies the results obtained to study almost automorphic solutions of abstract differential equations, expanding the core topics with a plethora of groundbreaking new results and applications. For the sake of clarity, and to spare the reader unnecessary technical hurdles, the concepts are studied using classical methods of functional analysis.

This monograph presents recent contributions to the topics of almost periodicity and almost automorphy. Several new methods, including the methods of invariant subspaces and uniform spectrum, as well as various classical methods, such as fixed point theorems, are used to obtain almost periodic and almost automorphic solutions to some linear and non-linear evolution equations and dynamical systems. Almost periodicity and almost automorphy are also intensively developed on the more general structures called fuzzy-number type spaces. They have further potential applications to the study of differential equations, which model the real-world problems governed by imprecision due to uncertainty or vagueness, rather than randomness. In conclusion, the author indicates several open problems and directions for future research. This monograph is a great source of information and inspiration for researchers and graduate students from many mathematical fields.

​​​ Topics in Fractional Differential Equations is devoted to the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative. ​​Fractional calculus generalizes the integrals and derivatives to non-integer orders. During the last decade, fractional calculus was found to play a fundamental role in the modeling of a considerable number of phenomena; in particular the modeling of memory-dependent and complex media such as porous media. It has emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations. Some equations present delays which may be finite, infinite, or state-dependent. Others are subject to an impulsive effect. The above problems are studied using the fixed point approach, the method of upper and lower solution, and the Kuratowski measure of noncompactness. This book is addressed to a wide audience of specialists such as mathematicians, engineers, biologists, and physicists. ​

​​​ Topics in Fractional Differential Equations is devoted to the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative. ​​Fractional calculus generalizes the integrals and derivatives to non-integer orders. During the last decade, fractional calculus was found to play a fundamental role in the modeling of a considerable number of phenomena; in particular the modeling of memory-dependent and complex media such as porous media. It has emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations. Some equations present delays which may be finite, infinite, or state-dependent. Others are subject to an impulsive effect. The above problems are studied using the fixed point approach, the method of upper and lower solution, and the Kuratowski measure of noncompactness. This book is addressed to a wide audience of specialists such as mathematicians, engineers, biologists, and physicists. ​

​​​ Topics in Fractional Differential Equations is devoted to the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative. ​​Fractional calculus generalizes the integrals and derivatives to non-integer orders. During the last decade, fractional calculus was found to play a fundamental role in the modeling of a considerable number of phenomena; in particular the modeling of memory-dependent and complex media such as porous media. It has emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations. Some equations present delays which may be finite, infinite, or state-dependent. Others are subject to an impulsive effect. The above problems are studied using the fixed point approach, the method of upper and lower solution, and the Kuratowski measure of noncompactness. This book is addressed to a wide audience of specialists such as mathematicians, engineers, biologists, and physicists. ​

This monograph is aimed at presenting smooth and unified generalized fractional programming (or a program with a finite number of constraints). Under the current interdisciplinary computer-oriented research environment, these programs are among the most rapidly expanding research areas in terms of its multi-facet applications and empowerment for real world problems that can be handled by transforming them into generalized fractional programming problems. Problems of this type have been applied for the modeling and analysis of a wide range of theoretical as well as concrete, real world, practical problems. More specifically, generalized fractional programming concepts and techniques have found relevance and worldwide applications in approximation theory, statistics, game theory, engineering design (earthquake-resistant design of structures, design of control systems, digital filters, electronic circuits, etc.), boundary value problems, defect minimization for operator equations, geometry, random graphs, graphs related to Newton flows, wavelet analysis, reliability testing, environmental protection planning, decision making under uncertainty, geometric programming, disjunctive programming, optimal control problems, robotics, and continuum mechanics, among others. It is highly probable that among all industries, especially for the automobile industry, robots are about to revolutionize the assembly plants forever. That would change the face of other industries toward rapid technical innovation as well.The main focus of this monograph is to empower graduate students, faculty and other research enthusiasts for more accelerated research advances with significant applications in the interdisciplinary sense without borders. The generalized fractional programming problems have a wide range of real-world problems, which can be transformed in some sort of a generalized fractional programming problem. Consider fractional programs that arise from management decision science; by analyzing system efficiency in an economical sense, it is equivalent to maximizing system efficiency leading to fractional programs with occurring objectives:Maximizing productivityMaximizing return on investmentMaximizing return/ riskMinimizing cost/timeMinimizing output/inputThe authors envision that this monograph will uniquely present the interdisciplinary research for the global scientific community (including graduate students, faculty, and general readers). Furthermore, some of the new concepts can be applied to duality theorems based on the use of a new class of multi-time, multi-objective, variational problems as well.

This volume of Advances in Evolution Equations is dedicated to the memory of Professor Vasilii Vasilievich Zhikov, an outstanding Russian mathematician. Zhikov's scientific interest ranged from almost periodic differential equations and topological dynamics to spectral theory of elliptic operators, qualitative theory of parabolic equations, calculus of variations, homogenization, and hydrodynamics, to name a few. Many of his results are now classical.

This book presents the foundation of the theory of almost automorphic functions in abstract spaces and the theory of almost periodic functions in locally and non-locally convex spaces and their applications in differential equations.

This book presents the foundation of the theory of almost automorphic functions in abstract spaces and the theory of almost periodic functions in locally and non-locally convex spaces and their applications in differential equations. Since the publication of Almost automorphic and almost periodic functions in abstract spaces (Kluwer Academic/Plenum, 2001), there has been a surge of interest in the theory of almost automorphic functions and applications to evolution equations. Several generalizations have since been introduced in the literature, including the study of almost automorphic sequences, and the interplay between almost periodicity and almost automorphic has been exposed for the first time in light of operator theory, complex variable functions and harmonic analysis methods. As such, the time has come for a second edition to this work, which was one of the most cited books of the year 2001.

This new edition clarifies and improves upon earlier materials, includes many relevant contributions and references in new and generalized concepts and methods, and answers the longtime open problem, "What is the number of almost automorphic functions that are not almost periodic in the sense of Bohr?" Open problems in non-locally convex valued almost periodic and almost automorphic functions are also indicated.As in the first edition, materials are presented in a simplified and rigorous way. Each chapter is concluded with bibliographical notes showing the original sources of the results and further reading.

This book presents the foundation of the theory of almost automorphic functions in abstract spaces and the theory of almost periodic functions in locally and non-locally convex spaces and their applications in differential equations.

This volume features recent development and techniques in evolution equations by renown experts in the field.

This volume features recent development and techniques in evolution equations by renown experts in the field. Each contribution emphasizes the relevance and depth of this important area of mathematics and its expanding reach into the physical, biological, social, and computational sciences as well as into engineering and technology.The reader will find an accessible summary of a wide range of active research topics, along with exciting new results. Topics include: Impulsive implicit Caputo fractional q-difference equations in finite and infinite dimensional Banach spaces; optimal control of averaged state of a population dynamic model; structural stability of nonlinear elliptic p(u)-Laplacian problem with Robin-type boundary condition; exponential dichotomy and partial neutral functional differential equations, stable and center-stable manifolds of admissible class; global attractor in Alpha-norm for some partial functional differential equations of neutral and retarded type; and more.Researchers in mathematical sciences, biosciences, computational sciences and related fields, will benefit from the rich and useful resources provided. Upper undergraduate and graduate students may be inspired to contribute to this active and stimulating field.