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Global solution branches of two point boundary value problems. (English) Zbl 0780.34010
Lecture Notes in Mathematics. 1458. Berlin etc.: Springer-Verlag. xvii, 141 p. (1990).
One of the basic problems in this monograph is the study (in Chapter 1) of the global solution branches of two point BVP of the form: (1) $$u''(t)+f(u)=0$$, $$u(0)=u(\lambda)=0$$, where the nonlinearity $$f$$ is such that $$f(0)=0$$ and $$f'(0)>0$$ up to its first zeros $$>0$$ or $$<0$$. This is done by using an integral formula for the so called time map $$T$$ of $$f$$. Conditions on $$f$$ are given which result in bifurcating branches that have at most one turning point at the bifurcation point itself (Chafee- Infante, Opial), or at a point away from it. In the latter case, one allows a change of stability along branches, which can happen at most twice. A wide class of the so called $$A-B$$-functions $$f$$ has been introduced for which these conditions can be easily verified. Chapter 2 is devoted to the study of the Neumann problem: (2) $$u''(t)+f(u)=0$$, $$u'(0)=u'(\lambda)=0$$ and of the period map of the Hamiltonian system: (3) $$u'=f_ 2(v)$$, $$v'=-f_ 1(u)$$. By using the corresponding time map $$T$$, it has been shown that some bifurcating branches of (2) have a single turn at the bifurcation point when $$f$$ is an $$A-B$$-function. By a similar method, monotonicity results for the period map of (3) have been obtained. After looking at some examples, the author has indicated how these results can be used to prove bifurcation of subharmonic solutions if (3) is perturbed by a small nonautonomous $$t$$-periodic term and the existence and uniqueness of periodic-4-solutions to the time delay equations $$u'(t)+f(u(t-1))=0$$ with $$f$$ antisymmetric.
Positive solution branches of (1) with $$f\geq0$$ have been studied in Chapter 3 using again a similar time map formula. Multiple zeros of $$f$$ at 0 are allowed and it is indicated that problems with singularities at $$u=0$$ could be treated too. For $$A-B$$-functions $$f$$, as well as for more general ones, it has been shown that the branch curve is also $$U$$-shaped. Nonbifurcating branches and the case $$f(0)<0$$ have also been studied. If $$f$$ is positive on some interval $$(0,a)$$, the inverse problem consisting in deciding whether or not a given curve $$p\to T(p)$$ is a time map of some $$f$$ for (1), and if $$f$$ can be calculated from $$T$$, has been studied in Chapter 4. The set of all time maps of such $$f's$$ has been characterized by an integral condition and the operator $$f\to T$$ has been shown to be invertible. Using the inverse problem results, it has been shown that the set of $$f\in F_ a=\{f\in L^ 1_{loc}([0,a))\mid f>0$$ a.e. in $$(0,a)$$, $$1/f\in L^ \infty_{loc}((0,a))\}$$, which have a $$C^ \infty$$-time map for (1) with only nondegenerate critical points, is a dense subset of $$F_ a$$. Applications to estimates for $$\lambda$$- regions of existence and nonexistence for (1) have been also given. As illustrations of possible applications of the results of this work, the author has discussed various specific examples. Finally, we note that the exposition in the work is clear and easy to follow. Due to the elementary nature of the analytical tools used in the exposition, this monograph is also a good text for undergraduate and graduate students.

##### MSC:
 34B05 Linear boundary value problems for ordinary differential equations 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations