Ding, Liming; Li, Xiang; Li, Zhixiang Fixed points and stability in nonlinear equations with variable delays. (English) Zbl 1214.34061 Fixed Point Theory Appl. 2010, Article ID 195916, 14 p. (2010). The following nonlinear delay differential equation is considered\[ x'(t)=-a(t) f(x(t-r_1(t)))+b(t) g(x(t-r_2(t))) \](with special attention to \(f(y)=y\)), where \(r_1, r_2:[0,\infty)\to [0,\infty)\), \(a,b:[0,\infty)\to\mathbb R\), \(f,g:\mathbb R\to\mathbb R\) are continuous functions. The main assumptions are that \(r_1\) is differentiable and the functions \(t\to t-r_i(t)\), \(i=1,2\), are strictly increasing and tending to \(\infty\) as \(t\to\infty\). On the basis of the contraction mapping principle, new results are obtained for the boundedness and the stability of the solutions, which improve some recently obtained results of the same type. It is pointed out that, investigating stability, the fixed point method is some alternative of the Lyapunov direct method. The main difference is that while the Lyapunov method usually requires pointwise conditions, the fixed point method, needs average conditions. Reviewer: Ivan Ginchev (Varna) Cited in 1 Document MSC: 34K20 Stability theory of functional-differential equations 47N20 Applications of operator theory to differential and integral equations Keywords:retarded differential equations; nonlinear equations; fixed point theory PDFBibTeX XMLCite \textit{L. Ding} et al., Fixed Point Theory Appl. 2010, Article ID 195916, 14 p.. (2010; Zbl 1214.34061) Full Text: DOI EuDML References: [1] Burton TA: Stability by fixed point theory or Liapunov theory: a comparison.Fixed Point Theory 2003,4(1):15-32. · Zbl 1061.47065 [2] Zhang B: Fixed points and stability in differential equations with variable delays.Nonlinear Analysis: Theory, Methods and Applications 2005,63(5-7):e233-e242. · Zbl 1159.34348 [3] Becker LC, Burton TA: Stability, fixed points and inverses of delays.Proceedings of the Royal Society of Edinburgh. Section A 2006,136(2):245-275. 10.1017/S0308210500004546 · Zbl 1112.34054 [4] Jin C, Luo J: Stability in functional differential equations established using fixed point theory.Nonlinear Analysis: Theory, Methods & Applications 2008,68(11):3307-3315. 10.1016/j.na.2007.03.017 · Zbl 1165.34042 [5] Burton TA: Stability and fixed points: addition of terms.Dynamic Systems and Applications 2004,13(3-4):459-477. · Zbl 1133.34364 [6] Zhang, B., Contraction mapping and stability in a delay-differential equation, No. 4, 183-190 (2004) · Zbl 1079.34543 [7] Burton TA: Stability by fixed point methods for highly nonlinear delay equations.Fixed Point Theory 2004,5(1):3-20. · Zbl 1079.34057 [8] Hale JK, Verduyn Lunel SM: Introduction to Functional Differential Equations. Springer, New York, NY, USA; 1993. · Zbl 0787.34002 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.