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Two classes of asymptotically different positive solutions of the equation $\dot y (t)= -f(t,y_t)$. (English) Zbl 1169.34050
This paper is devoted to the problem of the existence of two classes of asymptotically different positive solutions of the delay equation $$\dot y (t)= -f(t,y_t)$$ as $t\rightarrow \infty,$ where $f:\Omega\to \mathbb{R}$ is a continuous quasi-bounded functional that satisfies a local Lipschitz condition with respect to the second argument and $\Omega$ is an open subset in $\mathbb{R}\times C([-r,0],\mathbb{R})$. Two approaches are used. One is the method of monotone sequences and the other is the retract method combined with Razumikhin’s technique. By means of linear estimates of the right-hand side of the equation considered, inequalities for both types of positive solutions are given as well. Finally, the authors give an illustrative example and formulate some open problems.

34K25Asymptotic theory of functional-differential equations
Full Text: DOI
[1] Berezansky, L.; Braverman, E.: On exponential stability of linear differential equations with several delays, J. math. Anal. appl. 324, 1336-1355 (2006) · Zbl 1112.34055 · doi:10.1016/j.jmaa.2006.01.022
[2] Berezansky, L.; Braverman, E.: On stability of some linear and nonlinear delay differential equations, J. math. Anal. appl. 314, 391-411 (2006) · Zbl 1101.34057 · doi:10.1016/j.jmaa.2005.03.103
[3] Burton, T. A.: Stability by fixed point theory for functional differential equations, (2006) · Zbl 1160.34001
[4] Čermák, J.: On a linear differential equation with a proportional delay, Math. nachr. 280, 495-504 (2007) · Zbl 1128.34051 · doi:10.1002/mana.200410498
[5] Diblík, J.: A criterion for existence of positive solutions of systems of retarded functional differential equations, Nonlinear anal. 38, 327-339 (1999) · Zbl 0935.34061 · doi:10.1016/S0362-546X(98)00199-0
[6] Diblík, J.; Koksch, N.: Positive solutions of the equation x\dot{}$(t)=-c(t)$x(t-${\tau}$) in the critical case, J. math. Anal. appl. 250, 635-659 (2000) · Zbl 0968.34057 · doi:10.1006/jmaa.2000.7008
[7] Diblík, J.; Kúdelčíková, M.: Nonoscillating solutions of the equation x\dot{}$(t)$=-(a+b/t)x(t-${\tau}$), Stud. univ. \V{z}ilina math. Ser. 15, 11-24 (2002) · Zbl 1064.34044
[8] Diblík, J.; Svoboda, Z.: Positive solutions of p-type retarded functional differential equations, Nonlinear anal. 64, 1831-1848 (2006) · Zbl 1109.34058 · doi:10.1016/j.na.2005.07.020
[9] Diblík, J.; R\ring užičková, M.: Asymptotic behavior of solutions and positive solutions of differential delayed equations, Funct. differ. Equ. 14, No. 1, 83-105 (2007) · Zbl 1129.34050
[10] Driver, R. D.: Ordinary and delay differential equations, (1977) · Zbl 0374.34001
[11] Hale, J. K.; Lunel, S. M. V.: Introduction to functional differential equations, (1993) · Zbl 0787.34002
[12] Kolmanovskii, V.; Myshkis, A.: Applied theory of functional differential equations, Mathematics and its application (Soviet series) 85 (1992) · Zbl 0785.34005
[13] Kozakiewicz, E.: Über das asymptotische verhalten der nichtschwingenden lösungen einer linearen differentialgleichung mit nacheilendem argument, Wiss. Z. Humboldt univ. Berlin, math. Nat. R. 13, No. 4, 577-589 (1964) · Zbl 0277.34085
[14] Kozakiewicz, E.: Zur abschätzung des abklingens der nichtschwingenden lösungen einer linearen differentialgleichung mit nacheilendem argument, Wiss. Z. Humboldt univ. Berlin, math. Nat. R. 15, No. 5, 675-676 (1966) · Zbl 0221.34025
[15] Kozakiewicz, E.: Über die nichtschwingenden lösungen einer linearen differentialgleichung mit nacheilendem argument, Math. nachr. 32, No. 1/2, 107-113 (1966) · Zbl 0182.15504 · doi:10.1002/mana.19660320112
[16] Pituk, M.: Asymptotic behavior of solutions of a differential equation with asymptotically constant delay, Nonlinear anal. 30, 1111-1118 (1997) · Zbl 0895.34058 · doi:10.1016/S0362-546X(97)00297-6
[17] Pituk, M.: Special solutions of functional differential equations, Stud. univ. \V{z}ilina math. Ser. 17, 115-122 (2003) · Zbl 1063.34072
[18] Ryabov, Yu.A.: Certain asymptotic properties of linear systems with small time lag, Trudy sem. Teor. diff. Uravnenii s otklon. Argumentom univ. Druzby narodov patrisa lumumby 3, 153-165 (1965) · Zbl 0196.38701
[19] Rybakowski, K. P.: Wa.zewski’s principle for retarded functional differential equations, J. differential equations 36, No. 1, 117-138 (1980) · Zbl 0407.34056 · doi:10.1016/0022-0396(80)90080-7
[20] Srzednicki, R.: A.cañadap.drábeka.fondawa.zewski method and Conley index, Handbook of differential equations, ordinary differential equations 1, 591-684 (2004) · Zbl 1091.37006
[21] Zeidler, E.: Nonlinear functional analysis and its applications, part I, fixed-point theorems, (1985) · Zbl 0583.47051