Zhang, Bo Fixed points and stability in differential equations with variable delays. (English) Zbl 1159.34348 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 63, No. 5-7, e233-e242 (2005). Summary: We consider a linear scalar differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically stable by means of fixed point theory. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic stability theorem with a necessary and sufficient condition is proved. Cited in 1 ReviewCited in 39 Documents MSC: 34K20 Stability theory of functional-differential equations 34K40 Neutral functional-differential equations Keywords:fixed points; stability; delay equations; variable delays PDFBibTeX XMLCite \textit{B. Zhang}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 63, No. 5--7, e233--e242 (2005; Zbl 1159.34348) Full Text: DOI References: [1] Burton, T. A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations (1985), Academic Press: Academic Press New York · Zbl 0635.34001 [2] Burton, T. A., Stability by fixed point theory or Liapunov’s theory: a comparison, Fixed Point Theory, 4, 15-32 (2003) · Zbl 1061.47065 [3] T.A. Burton, Stability and fixed points: addition of terms, Dynamic Systems Appl. 13 (2004) 459-478.; T.A. Burton, Stability and fixed points: addition of terms, Dynamic Systems Appl. 13 (2004) 459-478. · Zbl 1133.34364 [4] Burton, T. A.; Haddock, J. R., On the delay differential equations \(x^\prime + a(t) f(x(t - r(t)) = 0\) and \(x^{\prime \prime}(t) + af(x(t - r(t)) = 0\), J. Math. Anal. Appl., 54, 37-48 (1976) · Zbl 0344.34065 [5] Graef, J. R.; Qian, C.; Zhang, B., Asymptotic behavior of solutions of differential equations with variable delays, Proc. London Math. Soc., 81, 72-92 (2000) · Zbl 1030.34075 [6] Kolmanovskii, V. B.; Torelli, L.; Vermiglio, R., Stability of some test equations with delay, SIAM J. Math. Anal., 25, 948-961 (1994) · Zbl 0808.34086 [7] Krisztin, T., On the stability properties for one dimensional functional differential equations, Funkcial Ekvac., 34, 241-256 (1991) · Zbl 0746.34045 [8] Smart, D. R., Fixed Point Theory (1980), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0427.47036 [9] Yoneyama, T., On the \(\frac{3}{2}\) stability theorem for one-dimensional delay-differential equations, J. Math. Anal. Appl., 125, 161-173 (1987) · Zbl 0655.34062 [10] Yorke, J. A., Asymptotic stability for one-dimensional functional differential-delay equations, J. Differential Equations, 7, 189-202 (1970) · Zbl 0184.12401 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.