Topics In Almost Automorphy

by N'Guerekata, Gaston M.

• ISBN13: 9780387228464
• ISBN10: 0387228462
• Format: Hardcover
• Copyright: 2004-12-30
• Publisher: Springer Verlag

Summary

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.

Author Biography

Gaston M. N'Guerekata is Professor and Chairman of the Mathematics Department at Morgan State University in Baltimore, Maryland and Editor-in-Chief of the International Journal of Evolution Equations.

Table of Contents

1 Introduction and Preliminaries	1	(40)
1.1 Measurable Functions	1	(4)
1.2 Sobolev Spaces	5	(2)
1.3 Semigroups of Linear Operators	7	(1)
1.4 Fractional Powers of Operators	8	(1)
1.5 Evolution Equations	9	(3)
1.6 Almost Automorphic Functions	12	(22)
1.6.1 Asymptotically Almost Automorphic Functions	23	(2)
1.6.2 Applications to Abstract Dynamical Systems	25	(9)
1.7 Almost Periodic Functions	34	(5)
1.8 Bibliographical Remarks and Open Problems	39	(2)
2 Almost Automorphic Evolution Equations	41	(54)
2.1 Linear Equations	41	(15)
2.1.1 The inhomogeneous equation χ¹ = Αχ + f	41	(5)
2.1.2 Method of Invariant Subspaces	46	(7)
2.1.3 Almost Automorphic Solutions to Some Second-Order Hyperbolic Equations	53	(3)
2.2 Nonlinear Equations	56	(17)
2.2.1 Existence of Almost Automorphic Mild Solutions-Case I	56	(6)
2.2.2 Existence of Almost Automorphic Mild Solutions-Case II	62	(11)
2.3 Optimal weak-almost periodic solutions	73	(9)
2.4 Existence of Weakly Almost Automorphic Solutions	82	(6)
2.5 A Correspondence Between Linear and Nonlinear Equations	88	(7)
3 Almost Periodicity in Fuzzy Setting	95	(28)
3.1 Fuzzy Sets	95	(3)
3.2 Almost Periodicity in Fuzzy Setting	98	(7)
3.3 Harmonics of Almost Periodic Functions in Fuzzy Setting	105	(11)
3.4 Applications to Fuzzy Differential Equations	116	(5)
3.5 Bibliographical Remarks and Open Problems	121	(2)
4 Almost Automorphy in Fuzzy Setting	123	(36)
4.1 Introduction	123	(1)
4.2 Preliminaries	124	(6)
4.3 Basic Definitions and Properties	130	(22)
4.4 Applications to Fuzzy Differential Equations	152	(5)
4.5 Bibliographical Remarks and Open Problems	157	(2)
References	159	(8)
Index	167