Distribution of zeros of sublinear dynamic equations with a damping term on time scales

Yıl 2016, Cilt: 45 Sayı: 2, 455 - 471, 01.04.2016

Samir. H. Saker Ramy. R. Mahmoud

Öz

In this paper, for a second order sublinear dynamic equation with a
damping term we will study the lower bounds of the distance between
zeros of a solution and/or its derivatives and then establish some new
criteria for disconjugacy and disfocality. Our results present a slight
improvement to some results proved in the litrature. As a special case
when T = R, for a second order linear differential equation, we get some
results proved by Brown and Harris as a consequence of our results. The
results will be proved by employing the time scales Hölder inequality,
the time scales chain rule and some new dynamic Opial-type inequalities
designed and proved for this purpose. 

Anahtar Kelimeler

Disconjugacy, Disfocality, Lyapunov inequality, Opial’s inequality, Sublinear dynamic equations, Time Scales

Kaynakça

• Agarwal, R.P. and Pang, P.Y.H. Opial inequalities with Applications in Differential and Difference Equations, Kluwer, Dordrechet (1995).
• Bohner, M. and Peterson, A. Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA (2001).
• Bohner, M. and Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston (2003).
• Bohner, M. and Kaymakçalan, B. Opial Inequalities on time scales, Annal. Polon. Math. 77 (2001), 11-20.
• Bohner, M., Clark, S. and Ridenhour, J. Lyapunov inequalities for time scales, J. Ineq. Appl. 7 (2002), 61-77.
• Brown, R.C. and Hinton, D.B. Lyapunov inequalities and their applications, in: T. M. Rassias (Ed.), Survey on Classical Inequalities, Kluwer Academic Publishers, Dordrecht, The Netherlands (2000), 1-25.
• Brown, R.C. and Hinton, D.B. Opial’s inequality and oscillation of 2 nd order equations, Proc. Amer. Math. Soc. 125 (1997), 1123-1129.
• Cheng, S. Lyapunov inequalities for differential and difference equations, Fasc. Math. 23 (1991) 25–41.
• Cohn, J.H.E. Consecutive zeros of solutions of ordinary second order differential equations, J. Lond. Math. Soc. 2 (5) (1972), 465-468.
• Harris, B. J. and Kong, Q. On the oscillation of differential equations with an oscillatory coefficient, Transc. Amer. Math. Soc. 347 (5) (1995), 1831-1839.
• Hartman, P. Ordinary Differential Equations, Wiley, New York, (1964) and Birkhäuser, Boston (1982).
• Hartman, P. and Wintner, A. On an oscillation criterion of de la Vallée Poussin, Quart. Appl. Math. 13 (1955), 330-332.
• Karpuz, B., Kaymakçalan, B. and Öclan, Ö. A generalization of Opial’s inequality and applications to second order dynamic equations, Diff. Eqns. Dyn. Sys. 18 (2010), 11-18.
• M. K. Kwong, On Lyapunov’s inequality for disfocality, J. Math. Anal. Appl. 83 (1981), 486-494.
• Lasota, A. A discrete boundary value problem, Annal. Polon. Math. 20 (1968), 183-190.
• Lettenmeyer, F. Ueber die von einem punktausgehenden Integralkurven einer Differentialgleichung 2. Ordnung, Deutsche Math. 7 (1944), 56-74.
• Lyapunov, A.M. Probleme général de la stabilité du mouvement, Ann. Math. Stud. 17 (1947), 203-474.
• Mitrinović, D.S., Pečarić, J.E. and Fink, A.M. Classical and New Inequalities in Analysis, vol. 61, Kluwer Academic Publishers, Dordrecht, The Netherlands, (1993).
• Nehari, Z. On the zeros of solutions of second order linear differential equations, Amer. J. Math. 76 (1954), 689-697.
• Opial, Z. Sur une inegalite, Annal. Polon. Math. 8 (1960), 29-32.
• Opial, Z. Sur une inégalité de C. de La Vallée Poussin dans la théorie de l’équation différentielle linéaire du second ordre, Annal. Polon. Math. 6 (1959), 87-91.
• Poussin, C. de la Vallée Sur l’équation différentielle linéaire du second ordre, J. Math. Pures Appl. 8 (1929), 125-144.
• Saker, S.H. Applications of Opial inequalities on time scales on dynamic equations with damping terms, Math. Comp. Model. 58 (2013), 1777-1790. 471
• Saker, S.H. Opial’s type inequalities on time scales and some applications, Annal. Polon. Math. 104 (2012), 243-260.
• Saker, S.H. New inequalities of Opial’s type on time scales and some of their applications, Disc. Dynam. Nat. Soc. 2012 (2012), 1-23.
• Saker, S.H. Some New Inequalities of Opial’s Type on Time Scales, Abs. Appl. Anal. 2012 (2012), 1-14.
• Saker, S.H. Lyapunov inequalities for half-linear dynamic equations on time scales and disconjugacy, Dyn. Contin. Discrete Impuls. Syst. Ser. B, Appl. Algorithms 18 (2011), 149- 161.
• Saker, S.H. Oscillation Theory of Dynamic Equations on Time Scales: Second and Third Orders, Lambert Academic Publishing, Germany (2010).
• Saker, S.H. Some new disconjugacy criteria for second order differential equations with a middle term, Bull. Math. Soc. Sci. Math. Roum. 57 (1) (2014), 109-120.
• Szmanda, B. The distance between the zeros of certain solutions of nth order linear differential equations (Polish), Fasciculi Math. Nr. 4 (1969), 65-70.
• Tiryaki, A. Recent development of Lyapunov-type inequalities, Adv. Dyn. Syst. Appl. 5 (2) (2010) 231-248.
• Willett, D. Generalized de la Vallée Poussin disconjugacy tests for linear differential equations, Canad. Math. Bull. 14 (1971), 419-428.