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Nonlocal initial and boundary value problems a survey. (English) Zbl 1098.34011
Cañada, A.(ed.) et al., Ordinary differential equations. Vol. II. Amsterdam: Elsevier/North Holland (ISBN 0-444-52027-9/hbk). Handbook of Differential Equations, 461-557 (2005).
In this survey paper, the author considers the nonlocal initial value problem
\[ x'=f(t,\,x),\;x(t_0)+g(x(\cdot))=x_0,\quad x\in \mathbb R^n, \]
and other boundary value problems for first-order, second-order and third-order ordinary differential equations. Some background of these problems is introduced, the existence and uniqueness of solutions of nonlocal initial value problems, the solvability of nonlocal nonresonance and resonance problems, and the existence of positive solutions of multipoint boundary value problems are studied by using the Leray-Schauder continuation theorem, Mawhin’s coincidence degree theorem and Krasnoselskii’s fixed-point theorem in cones.
For the entire collection see [Zbl 1074.34003].
Reviewer: Ruyun Ma (Lanzhou)

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators
47H10 Fixed-point theorems
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations