Solvability and bifurcations of nonlinear equations. (English) Zbl 0753.34002

Pitman Research Notes in Mathematics Series. 264. Harlow: Longman Scientific & Technical. New York: John Wiley & Sons, Inc.. xxi, 227 p. (1992).
This monograph reviews work by the author and others, mainly in the 1980s, on the functional analytic theory of problems which arise as boundary value problems for ordinary and partial differential equations. The most famous example of the general type of problem under consideration is the problem of Landesman-Lazer type. This is a nonlinear problem where it is possible to give necessary and sufficient conditions for its solvability. There is here an extensive account of developments along lines which say that the multiplicity of solutions of a certain equation for \(u\) of the form \(L(u)+N(u)=f\) can be intimately related to the behaviour of the nonlinearity \(N\) at infinity and zero relative to the spectrum of the linear operator \(L\). This covers large classes of semilinear boundary value problems and covers the first half of the book. The second part is directed towards results for quasilinear and nonlinear (strongly in a certain sense) problems and deals with questions of existence and local and global bifurcation.
The methodology is that of nonlinear functional analysis and is geometrical and topological in character including critical point theory and degree theory. The setting is infinite dimensional throughout.
Reviewer: J.F.Toland (Bath)


34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47J05 Equations involving nonlinear operators (general)
34G20 Nonlinear differential equations in abstract spaces
34B15 Nonlinear boundary value problems for ordinary differential equations
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces