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Nonlinear singular perturbation phenomena: theory and applications. (English) Zbl 0559.34013
Applied Mathematical Sciences, 56. New York etc.: Springer-Verlag. viii, 180 p. DM 58.00 (1984).
This monograph presents a study of the existence and asymptotic behavior of solutions of certain singularly perturbed nonlinear boundary value problems for second order differential equations. There are eight chapters. The introductory chapter 1 sets the basic theme of the monograph; that is, a study of quasilinear and nonlinear problems to which the use of differential inequality techniques appears to be particularly suited, as opposed to other well-known methods such as matched asymptotic expansions and two-variable expansions. Chapter 2 reviews basic existence theorems for the scalar and vector boundary value problem based on differential inequality techniques, such as those developed by M. Nagumo [Proc. Phys. Math. Soc. Japan 19, 861–866 (1937; Zbl 0017.30801)] and L. K. Jackson [Adv. Math. 2, 307–363 (1968; Zbl 0197.06401)]. Chapters 3, 4, and 5 consider boundary and interior layer phenomena for the Dirichlet and Robin problems for semilinear, quasilinear, and quadratic scalar singular perturbation problems. (These are, respectively, the equations \(\varepsilon y''=h(t,y)\), \(a<t<b\), \(\varepsilon y''=f(t,y)y'+g(t,y),\) \(a<t<b\), and \(\varepsilon y''=p(t,y)y'{}^ 2+g(t,y),\) \(a<t<b.)\) Chapter 6 is devoted to the superquadratic problem \(\varepsilon y''=f(t,y,y')\), \(a<t<b\), where \(f(t,y,z)=O(| z|^ n)\) as \(| z| \to \infty\) for \(n>2\). Chapter 7 deals with the vector situation where the results are much less complete and the last chapter is concerned with examples and applications. At the conclusion of each chapter is a section on Notes and Remarks giving an indication of the origin and possible extensions of the results. It appears that the authors have succeeded admirably in their stated two-fold purpose in writing this book: “to collect in one place many of the recent results on the existence and asymptotic behavior of solutions of certain classes of singularly perturbed nonlinear boundary value problems” and “to raise along the way a number of questions for further study”.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34E15 Singular perturbations for ordinary differential equations