Asymptotically convergent solutions of a system of nonlinear functional differential equations of neutral type with iterated deviating arguments. (English) Zbl 1278.34085
Summary: Some conditions that guarantee the existence of a family of solutions converging to zero of a system of nonlinear functional differential equations with iterated deviating arguments are given.
MSC:
| 34K25 | Asymptotic theory of functional-differential equations |
| 34K40 | Neutral functional-differential equations |
Keywords:
nonlinear system; functional differential equation; convergence to zero; iterated deviating argumentReferences:
| [1] | Bellman, R.; Cooke, K. L., Differential-Difference Equations (1963), Academic Press: Academic Press New York · Zbl 0105.06402 |
| [2] | Cruz, M. A.; Hale, J. K., Existence, uniqueness, and continuous dependence for hereditary system, Ann. Mat. Pura Appl., 85, 4, 63-82 (1970) · Zbl 0194.41002 |
| [3] | Demidovich, B. P., Lectures on the Mathematical Theory of Stability (1970), Nauka: Nauka Moscow, (in Russian) |
| [4] | Diblik, J., Asymptotic behaviour of solutions of a differential equation partly solved with respect to its derivative, Siberian Math. J., 23, 5, 80-91 (1982) · Zbl 0521.34003 |
| [5] | Diblik, J., On conditional stability of the solutions of the systems of differential equations, not solved with respect to the derivatives, Sbornik VUT, 1-4, 5-9 (1985) · Zbl 0672.34044 |
| [6] | Diblik, J., Functional Differential Equations (2008), University of Žilina · Zbl 0749.34040 |
| [7] | Driver, R. D., Existence and continuous dependence of solutions of a neutral functional-differential equation, Arch. Ration. Mech. Anal., 19, 2, 149-166 (1965) · Zbl 0148.05703 |
| [8] | Grimm, L. J., Existence and continuous dependence for a class of nonlinear neutral-differential equation, Proc. Am. Math. Soc., 29, 3, 467-473 (1971) · Zbl 0222.34061 |
| [9] | Hale, J., Theory of Functional Differential Equations (1977), Springer: Springer Berlin · Zbl 0352.34001 |
| [10] | Kwapisz, M., On the existence and uniqueness of solutions of certain integral-functional equation, Ann. Pol. Math., 31, 1, 23-41 (1975) · Zbl 0261.45005 |
| [11] | Oliinychenko, O. P., Asymptotic properties of solutions of linear functional differential equations, Nelin. Kolyv., 3, 4, 489-496 (2000) · Zbl 1007.34060 |
| [12] | Pelyukh, G. P., On global solutions of systems of nonlinear functional differential equations with deviating argument dependent on unknown functions, Ukrainian Math. J., 54, 3, 496-503 (2001) |
| [13] | Pelyukh, H. P.; Oliinychenko, O. P., Asymptotic properties of global solutions of systems of functional differential equations of neutral type with nonlinear deviations of an argument, Nonlinear Oscill., 5, 4, 489-503 (2002) · Zbl 1098.34576 |
| [14] | Samoilenko, A. M.; Pelyukh, G. P., Solutions of systems of nonlinear functional differential equations bounded on the entire real axis and their properties, Ukr. Mat. Zh., 46, 6, 737-747 (1994) · Zbl 0843.34073 |
| [15] | Shlapak, Yu. D., On periodic solutions of nonlinear second order differential equations, not solved with respect to the highest derivative, Ukrainian Math. J., 26, 6, 850-854 (1974), (in Russian) · Zbl 0314.34049 |
| [16] | Stević, S., On stability results for a new approximating fixed points iteration process, Demonstratio Math., 34, 4, 873-880 (2001) · Zbl 1011.47050 |
| [17] | Stević, S., Stability of a new iteration method for strongly pseudocontractive mappings, Demonstratio Math., 36, 2, 417-424 (2003) |
| [18] | Stević, S., Stability results for \(\phi \)-strongly pseudocontractive mappings, Yokohama Math. J., 50, 71-85 (2003) · Zbl 1068.47076 |
| [19] | Stević, S., Approximating fixed points of strongly pseudocontractive mappings by a new iteration method, Appl. Anal., 84, 1, 89-102 (2005) · Zbl 1068.47084 |
| [20] | Stević, S., Approximating fixed points of nonexpansive mappings by a new iteration method, Bull. Inst. Math. Acad. Sin., 1, 3, 437-450 (2006) · Zbl 1117.47056 |
| [21] | Stević, S., On the recursive sequence \(x_{n + 1} = \max \{c, x_n^p / x_{n - 1}^p \}\), Appl. Math. Lett., 21, 8, 791-796 (2008) · Zbl 1152.39012 |
| [22] | Stević, S., On a nonlinear generalized max-type difference equation, J. Math. Anal. Appl., 376, 317-328 (2011) · Zbl 1208.39014 |
| [23] | Stević, S., On a system of difference equations, Appl. Math. Comput., 218, 3372-3378 (2011) · Zbl 1242.39017 |
| [24] | Stević, S., Periodicity of a class of nonautonomous max-type difference equations, Appl. Math. Comput., 217, 9562-9566 (2011) · Zbl 1225.39018 |
| [25] | Stević, S., On some solvable systems of difference equations, Appl. Math. Comput., 218, 5010-5018 (2012) · Zbl 1253.39011 |
| [26] | Stević, S., Bounded solutions of some systems of nonlinear functional differential equations with iterated deviating argument, Appl. Math. Comput., 218, 10429-10434 (2012) · Zbl 1261.34048 |
| [27] | Stević, S., Existence of bounded solutions of some systems of nonlinear functional differential equations with complicated deviating argument, Appl. Math. Comput., 218, 9974-9979 (2012) · Zbl 1272.34096 |
| [28] | Stević, S., Globally bounded solutions of a system of nonlinear functional differential equations with iterated deviating argument, Appl. Math. Comput., 219, 2180-2185 (2012) · Zbl 1298.34128 |
| [29] | Stević, S., Local existence of Lipschitz-continuous solutions of systems of nonlinear functional equations with iterated deviations, Appl. Math. Comput., 218, 10542-10547 (2012) · Zbl 1251.39013 |
| [30] | Stević, S., On continuous solutions of a class of systems of nonlinear functional difference equations with deviating argument, Appl. Math. Comput., 218, 10188-10193 (2012) · Zbl 1246.39002 |
| [31] | Stević, S., On some periodic systems of max-type difference equations, Appl. Math. Comput., 218, 11483-11487 (2012) · Zbl 1280.39012 |
| [32] | Stević, S., Solutions converging to zero of some systems of nonlinear functional differential equations with iterated deviating argument, Appl. Math. Comput., 219, 4031-4035 (2012) · Zbl 1311.34158 |
| [33] | Stević, S., Unique existence of bounded continuous solutions on the real line of a class of nonlinear functional equations with complicated deviations, Appl. Math. Comput., 218, 7813-7817 (2012) · Zbl 1243.39018 |
| [35] | Zhivotovskii, L. A., On the existence and uniqueness of solutions of differential equations with delay dependent on a solution and its derivative, Differents. Uravn., 5, 5, 880-889 (1969) · Zbl 0183.37303 |
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