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A proof of Jones’ conjecture. (English) Zbl 1416.34050
In this paper the author completes a proof of a long standing conjecture that the well-known Wright’s differential delay equation \[ y^{\prime}(t)=-\alpha y(t-1)\left[1+y(t)\right]\tag{1} \] has a unique slowly oscillating periodic solution for every \(\alpha>\frac\pi2\). A solution of equation (1) is defined as slowly oscillating if the distance between its consecutive zeros is greater than the delay \(\tau=1\). The existence of slowly oscillating periodic solutions to equation (1) for all \(\alpha>\frac\pi2\) is a well-known fact that was established in mid fifties – early sixties through the works of E. M. Wright and G. S. Jones. G. S. Jones had also conjectured that such periodic solution is unique for every \(\alpha\) [J. Math. Anal. Appl. 5, 435–450 (1962; Zbl 0106.29504)].
The earlier work by X. Xie has in particular established the uniqueness of the slowly oscillating periodic solution for all \(\alpha\in[5.67,\infty)\) [J. Differ. Equations 103, No. 2, 350–374 (1993; Zbl 0780.34054)]. By using similar ideas and a computer assisted proof J. Jaquette et al. have shown the uniqueness of the slowly oscillating periodic solution for every \(\alpha\in[1.9,6.0]\) [J. Differ. Equations 263, No. 11, 7263–7286 (2017; Zbl 1382.34069)]. The present paper completes the proof of the conjecture by showing the uniqueness of the periodic solution for the remaining range of the parameter values \(\alpha\in(\frac\pi2, 1.9]\). The proof is computer assisted through a particular method of interval arithmetic developed for the case of Wright’s equation by the author and others in a series of earlier publications. The paper also shows that Wright’s equation has no isolas of slowly oscillating periodic solutions: every slowly oscillating periodic solution belongs to the primary branch that bifurcates off at \(\alpha=\frac\pi2\) and is a continuous curve for all \(\alpha>\frac\pi2\) in the \((\alpha, z)\)-plane, where \(z>0\) is the amplitude of the slowly oscillating periodic solution that exists at the parameter value \(\alpha\).

MSC:
34K13 Periodic solutions to functional-differential equations
Software:
CkAnalytic; INTLAB
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References:
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