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

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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

Josef Diblík ^1, , Klara Janglajew ^2, and Mária Kúdelčíková ^3,

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

Abstract
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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.

Keywords: positive solution, implicit criterion, explicit criterion, Advanced linear differential equation, asymptotic expansion..

Mathematics Subject Classification: 34K15, 34K2.

Citation: Josef Diblík, Klara Janglajew, Mária Kúdelčíková. An explicit coefficient criterion for the existence of positive solutions to the linear advanced equation. Discrete & 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, (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. 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. 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. doi: 10.1006/jmaa.2000.7008.
[6]	J. Diblík and M. Kúdelčíková, Positive solutions of advanced differential systems,, The Scientific World Journal, 2013 (2013), 1. 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. 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. 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). 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. 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, (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. 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.

show all references

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, (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. 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. 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. doi: 10.1006/jmaa.2000.7008.
[6]	J. Diblík and M. Kúdelčíková, Positive solutions of advanced differential systems,, The Scientific World Journal, 2013 (2013), 1. 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. 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. 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). 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. 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, (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. 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.

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