Existence and uniqueness of periodic solutions of mixed monotone functional differential equations. (English) Zbl 1187.34090

Kang, Shugui; Cheng, Sui Sun

Abstr. Appl. Anal. 2009, Article ID 162891, 13 p. (2009).

The authors investigate the existence and uniqueness of periodic solutions for
\[ y'(t)=-a(t)y(t)+f_1(t,y(t-\tau(t)))+f_2(t,y(t-\tau(t))), \tag{1} \]
where \(a\) and \(\tau\) are continuous and \(T\)-periodic functions, \(f_1,f_2\in C(\mathbb{R}^2,\mathbb{R})\) are \(T\)-periodic with respect to their first variable \(t\), and \(a(t)>0\) for all \(t\in\mathbb{R}\). It is assumed also that \(f_1\) is increasing with respect to its second variable, while \(f_2\) is decreasing with respect to its second variable. Another basic assumption is the existence of an ordered pair of lower and upper “quasisolutions”.
The main tool for proving the results is a fixed point theorem in ordered Banach spaces for mixed monotone operators. We remind that an operator \(A:E\times E\to E\), where \(E\) is an ordered Banach space, is called mixed monotone whenever \(A(x_1,y_1)\leq A(x_2,y_2)\) for any \(x_2,x_2,y_1,y_2\in E\) that satisfy \(x_1\leq x_2\) and \(y_2\leq y_1\). Also, \(x^*\in E\) is called a fixed point of \(A\) if \(A(x^*,x^*)=x^*\).

Reviewer: Adriana Buică (Cluj-Napoca)

MSC:

34K13	Periodic solutions to functional-differential equations
47H10	Fixed-point theorems

Keywords:

first order periodic functional-differential equation; periodic solution; existence and uniqueness; fixed point theorem; mixed monotone operators; ordered Banach spaces

References:

[1]	S. S. Cheng and G. Zhang, “Existence of positive periodic solutions for non-autonomous functional differential equations,” Electronic Journal of Differential Equations, vol. 2001, no. 59, pp. 1-8, 2001. · Zbl 1003.34059
[2]	G. Zhang and S. S. Cheng, “Positive periodic solutions of nonautonomous functional differential equations depending on a parameter,” Abstract and Applied Analysis, vol. 7, no. 5, pp. 279-286, 2002. · Zbl 1007.34066
[3]	D. Q. Jiang and J. J. Wei, “Existence of positive periodic solutions for nonautonomous delay differential equations,” Chinese Annals of Mathematics. Series A, vol. 20, no. 6, pp. 715-720, 1999 (Chinese). · Zbl 0948.34046
[4]	S. Kang and G. Zhang, “Existence of nontrivial periodic solutions for first order functional differential equations,” Applied Mathematics Letters, vol. 18, pp. 101-107, 2005. · Zbl 1075.34064
[5]	S. Kang, G. Zhang, and S. S. Cheng, “Periodic solutions of a class of integral equations,” Topological Methods in Nonlinear Analysis, vol. 22, pp. 245-252, 2003. · Zbl 1053.45006
[6]	D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1988. · Zbl 0661.47045
[7]	S. Y. Xu, C. Y. Zeng, and C. X. Zhu, “Existence and uniqueness of fixed points for \varphi concave-( - \psi ) convex mixed monotone operators and its applications,” Acta Mathematica Sinica. Chinese Series, vol. 48, no. 6, pp. 1055-1064, 2005. · Zbl 1125.47314