A proof of Jones’ conjecture.

*(English)*Zbl 1416.34050In 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 \(\alpha\).

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 \(\alpha\).

Reviewer: Anatoli F. Ivanov (Lehman)

##### MSC:

34K13 | Periodic solutions to functional-differential equations |

##### Keywords:

Wright’s equation; slowly oscillating periodic solutions; Jones’ conjecture; computer assisted proof; Krawczyk method##### References:

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