Partial Integral Operators and Integro-Differential Equations: Pure and Applied Mathematics

by Appell; Jurgen

• ISBN13: 9780824703967
• ISBN10: 0824703960
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-02-29
• Publisher: CRC Press

Summary

Provides the first self-contained account of integro-differential equations of the Barbashin type and partial integral operators--including existence, uniqueness, stability, and perturbation results.

Author Biography

Jurgen M. Appell is Professor of Mathematical Analysis, University of Wurzburg, Germany. Dr. Appell is an editorial board member of several journals and a member of the German National Mathematical Society (DMV), the Italian Mathematical Union (UMI), and the European Mathematical Society. He received the Dr. rer. nat. degree (1978) from the Free University of Berlin, Germany, and the Dr. rer. nat. habil. degree (1985) from the University of Augsburg, Germany. Anatolij S. Kalitvin is Head of the Chair of Mathematical Analysis, Algebra and Geometry at the Pedagogical Institute of Lipetsk, Russia. Dr. Kalitvin received the Dr. of Mathematics degree (1986) from the Voronezh State University, USSR (now Russia). Petr P. Zabrejko is Head of the Chair of Mathematical Methods in Control Theory, Belgos University, and Senior Researcher, Mathematical Institute of the National Academy of Sciences, Minsk, Belorussia. He is a member of the American Mathematical Society and the Belorussian Mathematical Society. Dr. Zabrejko received the Dr. of Philosophy (1964) and the Dr. of Mathematics (1968) degrees from the Voronezh State University, USSR (now Russia).

Table of Contents

Preface	iii
Introduction	1	(16)
Equations of Barbashin Type	17	(94)
Differential equations in Banach spaces	18	(16)
Integro-differential equations of Barbashin type	18	(4)
The evolution operator	22	(2)
Special cases and examples	24	(2)
Some auxiliary results	26	(3)
Degenerate kernels	29	(2)
Symmetric kernels	31	(3)
Barbashin equations in the space C	34	(22)
Linear operators in the space C	34	(3)
The spaces £b (C) and £m (C) Å £i (C)	37	(6)
Compact and weakly compact operators	43	(2)
Strongly continuous operator functions	45	(2)
Norm-continuous operator functions	47	(2)
Representation of the evolution operator	49	(2)
Stamping and infra-stamping operators	51	(5)
Barbashin equations in Lebesgue spaces	56	(29)
Linear operators in the space Lp	56	(6)
The space £br (Lp)	62	(5)
Sufficient conditions for regularity	67	(7)
Strongly continuous operator functions	74	(2)
Norm-continuous operator functions	76	(6)
Representation of the evolution operator	82	(3)
Barbashin equations in ideal spaces	85	(26)
Ideal spaces	85	(6)
Functions of two variables and vector functions	91	(4)
Barbashin operators in ideal spaces	95	(3)
The space £br (X)	98	(1)
Sufficient conditions for regularity	99	(1)
Representation of the evolution operator	100	(1)
Some spectral theory for Barbashin operators	101	(10)
Theory of Linear Barbashin Equations	111	(102)
Stability of solutions	112	(14)
Ljapunov and Ljapunov-Bohl exponents	112	(2)
Perturbation of the Ljapunov-Bohl exponent	114	(2)
Application to Barbashin operators	116	(4)
Ljapunov functions	120	(6)
Continuous dependence on parameters	126	(14)
Continuous dependence of the evolution operator	126	(2)
Application to Barbashin operators	128	(2)
The first Bogoljubov theorem	130	(7)
Smooth dependence	137	(3)
Bounded and periodic solutions	140	(27)
A fixed point principle	140	(3)
Application to Barbashin operators	143	(5)
The use of majorant functions	148	(5)
The Green function	153	(2)
Comparison of Green functions	155	(3)
The second Bogoljubov theorem	158	(3)
Application of Darbo's fixed point principle	161	(4)
Application of the Fredholm alternative	165	(2)
Degenerate kernels	167	(12)
A general approach	167	(5)
Explicit solution for a particular class	172	(3)
An abstract degeneration result	175	(4)
Stationary boundary value problems	179	(22)
The abstract problem	179	(1)
Equivalent operator equations	180	(4)
Reduction to a two-dimensional integral equation	184	(8)
Application of K-normed spaces	192	(5)
Unbounded multipliers	197	(4)
Non-stationary boundary value problems	201	(12)
Equations with variable operators	201	(2)
Reduction to a two-dimensional integral equation	203	(6)
Application of K-normed spaces	209	(4)
Partial Integral Operators	213	(140)
General properties	214	(15)
Continuity properties	215	(2)
Regularity properties	217	(4)
The associate operator	221	(3)
Algebras of partial integral operators	224	(5)
Operators in spaces with mixed norm	229	(29)
Ideal spaces with mixed norm	229	(8)
Lebesgue spaces with mixed norm	237	(5)
Orlicz spaces with mixed norm	242	(9)
Operator functions with values in £p (L∞) and £p (L1)	251	(4)
Operator functions with values in £p (Lp)	255	(3)
Partial integral operators in the space C	258	(29)
Weakly continuous functions	259	(3)
Acting and boundedness conditions	262	(6)
Algebras of partial integral operators	268	(8)
Operator functions with values in £p (C)	276	(11)
Spectral properties	287	(41)
Essential spectra of bounded linear operators	287	(2)
Application to partial integral operators in L2	289	(9)
Application to other partial integral operators	298	(15)
An index formula for partial integral operators	313	(9)
The case of positive kernels	322	(6)
Linear partial integral equations	328	(25)
Fredholm equations	328	(6)
Volterra equations	334	(2)
Bounded and continuous solutions	336	(1)
Using tensor products	337	(6)
Using eigenfunction expansions	343	(10)
Generalizations and Applications	353	(136)
Generalized equations of Barbashin type	354	(12)
Reduction to partial integral equations	354	(3)
Volterra operators and Barbashin equations	357	(3)
Generalized Barbashin equations	360	(2)
Differential equations in Banach spaces	362	(1)
Representation of the evolution operator	363	(3)
Nonlinear equations and operators	366	(28)
Barbashin equations with Uryson operators	366	(5)
Generalized Barbashin equations	371	(7)
Equations with Hammerstein operators	378	(7)
Surjectivity results for monotone operators	385	(1)
Equations with monotone operators	385	(4)
Variational methods	389	(5)
The Newton-Kantorovich method	394	(27)
The abstract Newton-Kantorovich method	394	(2)
Lipschitz conditions for derivatives	396	(6)
Lipschitz conditions for partial Uryson operators	402	(5)
The case X = C(T x S)	407	(5)
The case X = L∞ (T x S)	412	(5)
The case X = Lp (T x S) (1 ≤p <∞)	417	(4)
Applications of Barbashin equations	421	(29)
Applications to probability theory	421	(6)
Applications to evolution equations	427	(4)
Systems with substantially distributed parameters	431	(2)
The Kimura continuum-of-alleles model	433	(6)
An application to a radiation problem	439	(5)
Applications to astrophysics	444	(3)
Plane-parallel transport problems	447	(3)
Applications of partial integral equations	450	(39)
Applications to elasticity theory	450	(9)
Applications to mechanics of continuous media	459	(4)
Mixed problems of evolutionary type	463	(4)
Axially symmetric contact problems	467	(5)
Creeping of non-uniformly aging bodies	472	(4)
A unified approach to some equations of mechanics	476	(4)
Other applications	480	(9)
References	489	(50)
List of symbols	539	(8)
Index	547