Half-linear differential equations. (English) Zbl 1090.34027

Došlý, O.

Cañada, A. (ed.) et al., Ordinary differential equations. Vol. I. Amsterdam: Elsevier/North Holland (ISBN 0-444-51128-8/hbk). Handbook of Differential Equations, 161-357 (2004).

This text, which forms one chapter in the Handbook of Differential Equations, deals primarily with the theory of the half-linear second-order differential equation \[ (r(t)\Phi(x'))'+c(t)\Phi(x)=0,\quad \Phi(x):=| x| ^{p-1}\text{ sgn\,} x,\;p>1 \text{ (HLDE)}. \] The author has successfully tried to gather many important existing facts about this equation. In particular, the qualitative theory of (HLDE) is processed, which has attracted the attention of many mathematicians in the last 30 years. The text is very well organized: it is divided into four chapters, each of them consisting of sections and subsections. The first chapter is devoted to basic results, in particular, to the ones concerning the existence and uniqueness of the initial value problem involving (HLDE), transformations, Sturmian theory, and some elementary half-linear differential equations. Actually, in this part, there is presented the most of sophisticated tools that are used to obtain many results in the next chapters. Within the whole text, on one hand there are stressed similarities with the theory of half-linear equations (when it makes sense), and on the other hand, the difficulties are shown which are connected with the fact that not all of the results which play important roles in the linear theory can be extended to the half-linear case. This lack causes big difficulties in the proofs, and one has to use different approaches very often. The main subject of the second chapter is the nonoscillatory equation (HLDE). Firstly, there are presented characterizations of nonoscillation in terms of the definiteness of the corresponding \(p\)-degree energy functional, and of the solvability of the associated generalized Riccati inequality. These handy results then are used to prove many refined nonoscillation, comparison and (dis)conjugacy criteria for (HLDE), as well as an extension of the well-known Hartman-Wintner theorem. In this chapter, one can also find the classification of nonoscillatory solutions with respect to the asymptotic behavior, and an extension of the concept of the principal solution, where its different constructions, characterizations and basic properties are given. The third chapter may be partially understood as a complement of the previous chapter – there are many nice extensions of linear oscillation criteria for (HLDE) based on the variational principle and the Riccati technique. In addition to these ones, there can be found results on strong and conditional oscillation, oscillation of forced and retarded half-linear differential equations, and half-linear Sturm-Liouville problem. The last chapter deals with problems concerning (HLDE) which are somehow of different types comparing with the previous ones, and with another objects which are related to (HLDE) in a certain way. In the part devoted to boundary value problems involving (HLDE), more precisely in the Fredholm alternative for the nonlinear case, one can meet the phenomena which wouldn’t be expected when motivating ourselves by the linear case. Half-linear differential equations can be viewed also as special cases of quasilinear equations, of partial differential equations with \(p\)-Laplacian, and PDEs involving pseudo-Laplacian – some of the results on these equations can also be found in the fourth chapter. This chapter ends with mentioning that the basic parts of oscillation theory of (HLDE) extend to the discrete case, namely to the half-linear difference equations. As already written, one of the main features of the theory of half-linear equations is its similarity, in some aspects, with the theory of linear equations, even though the additivity of the solution space is lost. The author utilizes his rich experiences from the work in the field of linear ordinary differential equations. This, together with the facts that he is one of the leading experts in the area of half-linear equations (many important results for (HLDE) are of his own) and he very carefully puts together existing results of other authors that are the best conditions for doing a good job. The resultant text is very readable, comprehensible, and it will surely serve as a good source and survey of information on half-linear differential equations (not only) for experts. It should be said that it is actually the first kind of cover-all description of this topic, even though many mathematicians have worked on it in the last years. Finally, note that this chapter of Handbook of Differential Equations became a basis for the very recent monograph [O. Došlý, P. Řehák, Half-linear differential equations. North-Holland Mathematics Studies 202. Amsterdam: Elsevier (2005, Zbl 1090.34001)].
For the entire collection see [Zbl 1067.34001].

Reviewer: Pavel Řehák (Brno)

MSC:

34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34C11	Growth and boundedness of solutions to ordinary differential equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations

Keywords:

half-linear differential equation; oscillation theory; \(p\)-Laplacian

Citations:

Zbl 1090.34001