Fixed points and stability in neutral nonlinear differential equations with variable delays

Abstract

By means of Krasnoselskii's fixed point theorem we obtain boundedness and stability results of a neutral nonlinear differential equation with variable delays. A stability theorem with a necessary and sufficient condition is given. The results obtained here extend and improve the work of C. H. Jin and J. W. Luo [Nonlinear Anal. 68 (2008), 3307-3315], and also those of T. A. Burton [Fixed Point Theory 4 (2003), 15-32; Dynam. Systems Appl. 11 (2002), 499-519] and B. Zhang [Nonlinear Anal. 63 (2005), e233-e242]. In the end we provide an example to illustrate our claim.

Keywords

• fixed points
• stability
• nonlinear neutral differential equation
• integral equation
• variable delays