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Global attractivity of the zero solution for Wright’s equation. (English) Zbl 1301.34094

Summary: In 1955 E. M. Wright proved that all solutions of the delay differential equation \[ \dot x(t) = -\alpha (e^{x(t-1)}-1) \] converge to zero as \(t\to\infty\) for \(\alpha\in(0,3/2]\) and conjectured that this is even true for \(\alpha\in(0,\pi/2)\). The present paper proves the conjecture for \(\alpha\in[1.5,1.5706]\) (compare with \(\pi/2=1.570796\dots\)). The first part of the proof verifies that it is sufficient to guarantee the nonexistence of slowly oscillating periodic solutions, and it shows that slowly oscillating periodic solutions with small amplitudes cannot exist. In the second part a computer-assisted proof is given to exclude slowly oscillating periodic solutions with large amplitudes.

MSC:

34K25 Asymptotic theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
65G20 Algorithms with automatic result verification
65L03 Numerical methods for functional-differential equations

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References:

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