# American Institute of Mathematical Sciences

October  2014, 19(8): 2461-2467. doi: 10.3934/dcdsb.2014.19.2461

## An explicit coefficient criterion for the existence of positive solutions to the linear advanced equation

 1 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering and Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno 2 Department of Difference Equations and Discrete Systems, Institute of Mathematics, University of Białystok, Białystok, Poland 3 Department of Mathematics, University of Žilina, Žilina, Slovak Republic

Received  March 2014 Revised  April 2014 Published  August 2014

The paper is devoted to the investigation of a linear differential equation with advanced argument $\dot y(t)=c(t)y(t+\tau),$ where $\tau>0$, and the function $c\colon [t_0,\infty)\to (0,\infty)$, $t_0\in \mathbb{R}$ is bounded and locally Lipschitz continuous. New explicit coefficient criterion for the existence of a positive solution in terms of $c$ and $\tau$ is derived.
Citation: Josef Diblík, Klara Janglajew, Mária Kúdelčíková. An explicit coefficient criterion for the existence of positive solutions to the linear advanced equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2461-2467. doi: 10.3934/dcdsb.2014.19.2461
##### References:
 [1] R. P. Agarwal, L. Berezansky, E. Braverman and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, 2012. doi: 10.1007/978-1-4614-3455-9. [2] R. P. Agarwal, M. Bohner and W.-T. Li, Nonoscillation and Oscillation: Theory for Functional Differential Equations, Marcel Dekker, Inc., 2004. doi: 10.1201/9780203025741. [3] H. Bereketoǧlu, F. Karakoç and G. Seyhan, Oscillatory and periodic solutions of impulsive differential equations with piecewise constant argument, Acta Appl. Math., 110 (2010), 499-510. doi: 10.1007/s10440-009-9458-9. [4] J. Diblík, A note on explicit criteria for the existence of positive solutions to the linear advanced equation $\dot x(t) = c(t)x(t + \tau)$, Appl. Math. Lett., 35 (2014), 72-76. doi: 10.1016/j.aml.2013.11.010. [5] J. Diblík and N. Koksch, Positive solutions of the equation $\dotx(t)=-c(t)x(t-\tau )$ in the critical case, J. Math. Anal. Appl., 250 (2000), 635-659. doi: 10.1006/jmaa.2000.7008. [6] J. Diblík and M. Kúdelčíková, Positive solutions of advanced differential systems, The Scientific World Journal, 2013 (2013), Article ID 613832, 1-7. doi: 10.1155/2013/613832. [7] A. Domoshnitsky and M. Drakhlin, Nonoscillation of first order differential equations with delay, J. Math. Anal. Appl., 206 (1997), 254-269. doi: 10.1006/jmaa.1997.5231. [8] Y. Domshlak and I. P. Stavroulakis, Oscillation of first-order delay differential equations in a critical case, Appl. Anal., 61 (1996), 359-371. doi: 10.1080/00036819608840464. [9] B. Dorociaková, M. Kubjatková and R. Olach, Existence of positive solutions of neutral differential equations, Abstr. Appl. Anal., 2012 (2012), Art. ID 307968, 14 pp. doi: 10.1155/2012/307968. [10] Á. Elbert and I. P. Stavroulakis, Oscillation and non-oscillation criteria for delay differential equations, Proc. Amer. Math. Soc., 123 (1995), 1503-1510. doi: 10.1090/S0002-9939-1995-1242082-1. [11] L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, Basel, Hong Kong, 1994. [12] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, 1991. [13] V. E. Sljusarchuk, The necessary and sufficient conditions for oscillation of solutions of nonlinear differential equations with pulse influence in the Banach space, Ukrain. Mat. Zh., 51 (1999), 98-109. doi: 10.1007/BF02591918. [14] B. G. Zhang, Oscillation of the solutions of the first-order advanced type differential equations, (Chinese summary), Sci. Exploration, 2 (1982), 79-82.

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##### References:
 [1] R. P. Agarwal, L. Berezansky, E. Braverman and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, 2012. doi: 10.1007/978-1-4614-3455-9. [2] R. P. Agarwal, M. Bohner and W.-T. Li, Nonoscillation and Oscillation: Theory for Functional Differential Equations, Marcel Dekker, Inc., 2004. doi: 10.1201/9780203025741. [3] H. Bereketoǧlu, F. Karakoç and G. Seyhan, Oscillatory and periodic solutions of impulsive differential equations with piecewise constant argument, Acta Appl. Math., 110 (2010), 499-510. doi: 10.1007/s10440-009-9458-9. [4] J. Diblík, A note on explicit criteria for the existence of positive solutions to the linear advanced equation $\dot x(t) = c(t)x(t + \tau)$, Appl. Math. Lett., 35 (2014), 72-76. doi: 10.1016/j.aml.2013.11.010. [5] J. Diblík and N. Koksch, Positive solutions of the equation $\dotx(t)=-c(t)x(t-\tau )$ in the critical case, J. Math. Anal. Appl., 250 (2000), 635-659. doi: 10.1006/jmaa.2000.7008. [6] J. Diblík and M. Kúdelčíková, Positive solutions of advanced differential systems, The Scientific World Journal, 2013 (2013), Article ID 613832, 1-7. doi: 10.1155/2013/613832. [7] A. Domoshnitsky and M. Drakhlin, Nonoscillation of first order differential equations with delay, J. Math. Anal. Appl., 206 (1997), 254-269. doi: 10.1006/jmaa.1997.5231. [8] Y. Domshlak and I. P. Stavroulakis, Oscillation of first-order delay differential equations in a critical case, Appl. Anal., 61 (1996), 359-371. doi: 10.1080/00036819608840464. [9] B. Dorociaková, M. Kubjatková and R. Olach, Existence of positive solutions of neutral differential equations, Abstr. Appl. Anal., 2012 (2012), Art. ID 307968, 14 pp. doi: 10.1155/2012/307968. [10] Á. Elbert and I. P. Stavroulakis, Oscillation and non-oscillation criteria for delay differential equations, Proc. Amer. Math. Soc., 123 (1995), 1503-1510. doi: 10.1090/S0002-9939-1995-1242082-1. [11] L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, Basel, Hong Kong, 1994. [12] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, 1991. [13] V. E. Sljusarchuk, The necessary and sufficient conditions for oscillation of solutions of nonlinear differential equations with pulse influence in the Banach space, Ukrain. Mat. Zh., 51 (1999), 98-109. doi: 10.1007/BF02591918. [14] B. G. Zhang, Oscillation of the solutions of the first-order advanced type differential equations, (Chinese summary), Sci. Exploration, 2 (1982), 79-82.
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