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Existence and iterative algorithms of positive solutions for a higher order nonlinear neutral delay differential equation. (English) Zbl 1266.65114

Summary: This paper is concerned with the higher order nonlinear neutral delay differential equation \([a(t)(x(t) + b(t)x(t - \tau))^{(m)}]^{(n-m)} + [h(t, x(h_1(t)), \dots, x(h_l(t)))]^{(i)} + f(t, x(f_1(t)), \dots, x(f_l(t))) = g(t)\) for all \(t \geq t_0\). Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

MSC:

65L03 Numerical methods for functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34K40 Neutral functional-differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
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[1] Candan, T., The existence of nonoscillatory solutions of higher order nonlinear neutral equations, Applied Mathematics Letters, 25, 3, 412-416 (2012) · Zbl 1242.34124 · doi:10.1016/j.aml.2011.09.025
[2] Kulenović, M. R. S.; Hadžiomerspahić, S., Existence of nonoscillatory solution of second order linear neutral delay equation, Journal of Mathematical Analysis and Applications, 228, 2, 436-448 (1998) · Zbl 0919.34067 · doi:10.1006/jmaa.1997.6156
[3] Liu, Z.; Chen, L.; Kang, S. M.; Cho, S. Y., Existence of nonoscillatory solutions for a third-order nonlinear neutral delay differential equation, Abstract and Applied Analysis, 2011 (2011) · Zbl 1230.34056 · doi:10.1155/2011/693890
[4] Liu, Z.; Gao, H.; Kang, S. M.; Shim, S. H., Existence and Mann iterative approximations of nonoscillatory solutions of \(n\) th-order neutral delay differential equations, Journal of Mathematical Analysis and Applications, 329, 1, 515-529 (2007) · Zbl 1116.34051 · doi:10.1016/j.jmaa.2006.06.079
[5] Liu, Z.; Kang, S. M., Infinitely many nonoscillatory solutions for second order nonlinear neutral delay differential equations, Nonlinear Analysis: Theory, Methods & Applications, 70, 12, 4274-4293 (2009) · Zbl 1172.34042 · doi:10.1016/j.na.2008.09.013
[6] Liu, Z.; Kang, S. M.; Ume, J. S., Existence and iterative approximations of nonoscillatory solutions of higher order nonlinear neutral delay differential equations, Applied Mathematics and Computation, 193, 1, 73-88 (2007) · Zbl 1193.34131 · doi:10.1016/j.amc.2007.03.040
[7] Liu, Z.; Kang, S. M.; Ume, J. S., Existence of bounded nonoscillatory solutions of first-order nonlinear neutral delay differential equations, Computers & Mathematics with Applications, 59, 11, 3535-3547 (2010) · Zbl 1206.34088 · doi:10.1016/j.camwa.2010.03.047
[8] Liu, Z.; Wang, L.; Kang, S. M.; Ume, J. S., Solvability and iterative algorithms for a higher order nonlinear neutral delay differential equation, Applied Mathematics and Computation, 215, 7, 2534-2543 (2009) · Zbl 1208.65098 · doi:10.1016/j.amc.2009.08.054
[9] Zhang, W.; Feng, W.; Yan, J.; Song, J., Existence of nonoscillatory solutions of first-order linear neutral delay differential equations, Computers & Mathematics with Applications, 49, 7-8, 1021-1027 (2005) · Zbl 1087.34539 · doi:10.1016/j.camwa.2004.12.006
[10] Zhou, Y., Existence for nonoscillatory solutions of second-order nonlinear differential equations, Journal of Mathematical Analysis and Applications, 331, 1, 91-96 (2007) · Zbl 1111.34049 · doi:10.1016/j.jmaa.2006.08.048
[11] Zhou, Y.; Zhang, B. G., Existence of nonoscillatory solutions of higher-order neutral differential equations with positive and negative coefficients, Applied Mathematics Letters, 15, 7, 867-874 (2002) · Zbl 1025.34065 · doi:10.1016/S0893-9659(02)00055-1
[12] Zhou, Y.; Zhang, B. G.; Huang, Y. Q., Existence for nonoscillatory solutions of higher order nonlinear neutral differential equations, Czechoslovak Mathematical Journal, 55, 1, 237-253 (2005) · Zbl 1081.34068 · doi:10.1007/s10587-005-0018-9
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