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Global properties of decreasing solutions of the equation \(X'(T)=X(X(T))+X(T)\). (English) Zbl 0901.34064

The author attempts to describe the structure of maximal decreasing solutions \({\mathcal A}_-\) to the equation \[ x'(t)= x(x(t))+ x(t). \] It is shown that \({\mathcal A}_-= {\mathcal A}^1_-\cup{\mathcal A}^2_-\), where \({\mathcal A}^1_-\) and \({\mathcal A}^2_-\) are disjoint nonempty sets given by \[ {\mathcal A}^1_-= \Biggl\{x\in{\mathcal A}_-: \lim_{t\to \infty} x(t)< \infty\Biggr\} \] and \[ {\mathcal A}^2_-= \Biggl\{x\in{\mathcal A}_-: \lim_{t\to\infty} x(t)= \infty\Biggr\}. \] Many open problems are stated. The present work constitutes a supplement of the earlier work by the author [Dyn. Syst. Appl. 4, No. 2, 263-278 (1995; Zbl 0830.34064)].

MSC:

34K05 General theory of functional-differential equations
34K12 Growth, boundedness, comparison of solutions to functional-differential equations

Citations:

Zbl 0830.34064
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