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He, Tieshan; Xu, Yuantong; Zeng, Yanshan

Higher order functional differential systems with parameter dependence. (English) Zbl 1252.34078

Appl. Math. Comput. 218, No. 15, 7783-7791 (2012).
The authors study a class of higher-order functional differential systems with a real bifurcation parameter \(p\). Criteria for the existence, nonexistence and multiplicity of positive periodic solutions are established by using fixed point index theory. If certain assumptions are satisfied, they show that there exists a positive number \(q\) such that at least two positive periodic solutions exist for \(p\) in one of the two intervals \((0,q)\) and \((q, \infty)\), one positive periodic solution for \(p = q\) and no positive periodic solution for \(p\) in the other interval.
Reviewer: Kwok-wai Chung (Hong Kong)

MSC:

34K13 Periodic solutions to functional-differential equations
47N20 Applications of operator theory to differential and integral equations
34K18 Bifurcation theory of functional-differential equations

Keywords:

high-order functional differential systems; fixed point index; positive periodic solution; existence and nonexistence
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References:

[1] Bai, D.; Xu, Y., Periodic solutions of first order functional differential equations with periodic deviations, Comput. Math. Appl., 53, 1361-1366 (2007) · Zbl 1123.34328
[2] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0559.47040
[3] Graef, J. R.; Kong, L., Periodic solutions for functional differential equations with sign-changing nonlinearities, Proc. Edinburgh Math. Soc., 140, 597-616 (2010) · Zbl 1200.34077
[4] Guo, D.; Lakskmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045
[5] Islam, M. N.; Raffoul, Y. N., Periodic solutions of neutral nonlinear system of differential equations with functional delay, J. Math. Anal. Appl., 331, 1175-1186 (2007) · Zbl 1118.34057
[6] Jiang, W., The existence of multiple positive periodic solutions for functional differential equations, Appl. Math. Comput., 208, 165-171 (2009) · Zbl 1167.34371
[7] Kang, S.; Shi, B.; Wang, G., Existence of maximal and minimal periodic solutions for first-order functional differential equations, Appl. Math. Lett., 23, 22-25 (2010) · Zbl 1186.34092
[8] Liu, Y., Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations, J. Math. Anal. Appl., 327, 435-452 (2007) · Zbl 1119.34062
[9] Luo, Y.; Wang, W.; Shen, J., Existence of positive periodic solutions for two kinds of neutral functional differential equations, Appl. Math. Lett., 21, 581-587 (2008) · Zbl 1149.34040
[10] Liu, P.; Li, Y., Positive periodic solutions of infinite delay functional differential equations depending on a parameter, Appl. Math. Comput., 150, 159-168 (2004) · Zbl 1054.34113
[11] Liu, X.; Li, W., Existence and uniqueness of positive periodic solutions of functional differential equations, J. Math. Anal. Appl., 293, 28-39 (2004) · Zbl 1057.34094
[12] Padhi, S.; Srivastava, S.; Pati, S., Three periodic solutions for a nonlinear first order functional differential equation, Appl. Math. Comput., 216, 2450-2456 (2010) · Zbl 1198.34140
[13] Weng, A.; Sun, J., Positive periodic solutions of first-order functional differential equations with parameter, J. Comput. Appl. Math., 229, 327-332 (2009) · Zbl 1171.34342
[14] Wang, H., Positive periodic solutions of functional differential equations, J. Differential Eqs., 202, 354-366 (2004) · Zbl 1064.34052
[15] Wan, A.; Jiang, D., A new existence theory for positive periodic solutions to functional differential equations, Comput. Math. Appl., 47, 1257-1262 (2004) · Zbl 1073.34082
[16] Zeng, Z.; Bi, L.; Fan, M., Existence of multiple positive periodic solutions for functional differential equations, J. Math. Anal. Appl., 325, 1378-1389 (2007) · Zbl 1110.34043
[17] Zhang, G.; Cheng, S., Positive periodic solutions of nonautonomuos functional differential equations depending on a parameter, Abstr. Appl. Anal., 7, 279-286 (2002) · Zbl 1007.34066
[18] Zhang, N.; Dai, B.; Chen, Y., Positive periodic solutions of nonautonomous functional differential systems, J. Math. Anal. Appl., 333, 667-678 (2007) · Zbl 1125.34052
[19] Zhang, W.; Zhu, D.; Bi, P., Existence of periodic solutions of a scalar functional differential equations via a fixed point theorem, Math. Comput. Model., 46, 718-729 (2007) · Zbl 1145.34041
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