Journal of Mathematical Sciences, Vol. 155, No. 2, 2008 NONCLASSICAL BOUNDARY-VALUE PROBLEMS. I A. L. Skubachevskii UDC 517.9 Devoted to my wife Olesia CONTENTS Notation ................................................. 199 Introduction .............................................. 200 Chapter 1. Nonlocal Problems in the One-Dimensional Case ................... 209 1.1. Second-Order Ordinary Differential Equations with Nonlocal Boundary Conditions .... 209 1.2. Ordinary Differential Equations with Integral Conditions .................. 215 1.3. Difference Operators in the One-Dimension case ....................... 224 1.4. Boundary-Value Problems for Second-Order Differential-Difference Equations ....... 234 1.5. Nonlocal Problems for Systems of Ordinary Differential Equations with a Parameter . . . 243 Remarks to Chapter 1 ...................................... 251 Chapter 2. Elliptic Problems with Nonlocal Conditions Inside Domain ............. 252 2.1. Nonlocal Elliptic Problems with a Parameter ......................... 252 2.2. Solvability and Index of Nonlocal Elliptic Problems ...................... 259 2.3. Second-Order Elliptic Equations with Nonlocal Boundary Conditions in a Cylinder .... 266 Remarks to Chapter 2 ...................................... 275 Chapter 3. Weighted Spaces ..................................... 277 3.1. Weighted Spaces in Angles and R n ............................... 277 3.2. Functional Spaces with Nonhomogeneous Weight ....................... 288 3.3. Weighted Spaces in Bounded Domains ............................. 295 Remarks to Chapter 3 ...................................... 302 Appendix ................................................ 302 A. Linear Operators ......................................... 302 B. Functional Spaces ........................................ 309 C. Generalized Solutions of Elliptic Problems ........................... 318 References ............................................. 327 NOTATION R — the real numbers; C — the complex numbers; Z — the integers; N — the positive integers; R n — the n-dimensional real space; C n — the n-dimensional complex space; B R (x 0 ) — the ball {x ∈ B : ‖x - x 0 ‖ <R} in a Banach space B; B R = B R (0); [a, b] — the closed interval {x ∈ R : a ≤ x ≤ b}; Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 26, Nonclassical boundary-value problems. I, 2008. 1072–3374/08/1552–0199 c  2008 Springer Science+Business Media, Inc. 199