We establish existence, multiplicity, and nonexistence of periodic solutions for a class of first-order neutral difference equations modelling physiological processes and conditions. Our approach is based on a fixed point theorem in cones as well as some analysis techniques.
R. P. Agarwal and W. N. Zhang, “Periodic solutions of difference equations with general periodicity,” Computers & Mathematics with Applications, vol. 42, no. 3–5, pp. 719–727, 2001.
View at: Google Scholar | Zentralblatt MATH | MathSciNetS. N. Chow, “Existence of periodic solutions of autonomous functional differential equations,” Journal of Differential Equations, vol. 15, no. 2, pp. 350–378, 1974.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetK. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
View at: Zentralblatt MATH | MathSciNetM. Fan and K. Wang, “Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system,” Mathematical and Computer Modelling, vol. 35, no. 9-10, pp. 951–961, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetD. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Massachusetts, 1988.
View at: Zentralblatt MATH | MathSciNetD. Jiang, J. Chua, and M. Zhang, “Multiplicity of positive periodic solutions to superlinear repulsive singular equations,” Journal of Differential Equations, vol. 211, no. 2, pp. 282–302, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetM. A. Krasnosel'skiĭ, Positive Solutions of Operators Equations, Noordhoff, Groningen, 1964.
View at: Zentralblatt MATH | MathSciNetB. S. Lalli and B. G. Zhang, “On existence of positive solutions and bounded oscillations for neutral difference equations,” Journal of Mathematical Analysis and Applications, vol. 166, no. 1, pp. 272–287, 1992.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetY. K. Li, L. F. Zhu, and P. Liu, “Positive periodic solutions of nonlinear functional difference equations depending on a parameter,” Computers & Mathematics with Applications, vol. 48, no. 10-11, pp. 1453–1459, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetS. Lu and W. Ge, “Periodic solutions of neutral differential equation with multiple deviating arguments,” Applied Mathematics and Computation, vol. 156, no. 3, pp. 705–717, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetM. C. Mackey and L. Glass, “Oscillations and chaos in physiological control systems,” Science, vol. 197, pp. 287–289, 1997.
View at: Google ScholarH. Péics, “Positive solutions of neutral delay difference equation,” Novi Sad Journal of Mathematics, vol. 32, no. 2, pp. 111–122, 2005.
View at: Google ScholarH. L. Smith and Y. Kuang, “Periodic solutions of differential delay equations with threshold-type delays,” in Oscillation and Dynamics in Delay Equations (San Francisco, CA, 1991), vol. 129 of Contemporary Mathematics, pp. 153–176, American Mathematical Society, Rhode Island, 1992.
View at: Google Scholar | Zentralblatt MATH | MathSciNetH. Wang, “Positive periodic solutions of functional differential equations,” Journal of Differential Equations, vol. 202, no. 2, pp. 354–366, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetM. Ważewska-Czyżewska and A. Lasota, “Mathematical problems of the dynamics of a system of red blood cells,” Roczniki Polskiego Towarzystwa Matematycznego. Seria III. Matematyka Stosowana, vol. 6, pp. 23–40, 1976.
View at: Google Scholar | Zentralblatt MATH | MathSciNetX. Y. Zeng, B. Shi, and M. J. Gai, “A discrete periodic Lotka-Volterra system with delays,” Computers & Mathematics with Applications, vol. 47, no. 4-5, pp. 491–500, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetM. R. Zhang, “Periodic solutions of linear and quasilinear neutral functional-differential equations,” Journal of Mathematical Analysis and Applications, vol. 189, no. 2, pp. 378–392, 1995.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetG. Zhang and S. S. Cheng, “Positive periodic solutions of coupled delay differential systems depending on two parameters,” Taiwanese Journal of Mathematics, vol. 8, no. 4, pp. 639–652, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNet