American Institute of Mathematical Sciences

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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

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Diblík and M. Kúdelčíková, Positive solutions of advanced differential systems,, The Scientific World Journal, 2013 (2013), 1.  doi: 10.1155/2013/613832.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations,, Marcel Dekker, (1994).   Google Scholar

[12]

I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations,, Clarendon Press, (1991).   Google Scholar

[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.  Google Scholar

[14]

B. G. Zhang, Oscillation of the solutions of the first-order advanced type differential equations, (Chinese summary),, Sci. Exploration, 2 (1982), 79.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Diblík and M. Kúdelčíková, Positive solutions of advanced differential systems,, The Scientific World Journal, 2013 (2013), 1.  doi: 10.1155/2013/613832.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations,, Marcel Dekker, (1994).   Google Scholar

[12]

I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations,, Clarendon Press, (1991).   Google Scholar

[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.  Google Scholar

[14]

B. G. Zhang, Oscillation of the solutions of the first-order advanced type differential equations, (Chinese summary),, Sci. Exploration, 2 (1982), 79.   Google Scholar

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