Abstract
Differential equations are often classified according to oscillatory/nonoscillatory properties of their solutions as equations having property A or property B. The aim of the paper is to state an equivalence theorem between property A and property B for third order differential equations. Some applications, to linear as well as to nonlinear equations, are given too. Particularly, we give integral criteria ensuring property A or B for nonlinear equations. Our only assumption on nonlinearity is its superlinearity in neighbourhood of infinity, hence our results apply also to Emden-Fowler type equations.
Article PDF
Similar content being viewed by others
References
M. Bartušek,On the structure of solutions of a system of three differential inequalities, Arch. Math.,30 (1994), pp. 117–130.
M. Cecchi -M. Marini -Gabriele Villari,On a cyclic disconjugate operator associated to linear differential equations, Annali Mat. Pura Appl., IV,CLXX (1996), pp. 297–309.
M.Cecchi - Z.Došlá - M.Marini,Some properties of third order differential operators, Czech. Math. J. (1996).
M. Cecchi -Z. Došlá -M. Marini -Gabriele Villari,On the qualitative behavior of solutions of third order differential equations, J. Math. Anal. Appl,197 (1996), pp. 749–766.
M. Cecchi -Z. Došlá -M. Marini,Comparison theorems for third order differential equations, Proceedings of Dynamic Systems and Appl.,2 (1996), pp. 99–106.
T. A. Chanturia,Some comparison theorems for ordinary differential equations of higher order (Russian), Bull. Acad. Polon. Sci. Ser. Sci. Math., Astr. Phys.,20 (1977), pp. 749–756.
T. A. Chanturia,On monotone and oscillatory solutions of ordinary differential equations, Annal. Polon. Math.,37 (1980), pp. 93–111.
T. A. Chanturia,On the oscillation of solutions of higher order linear differential equations (Russian), Rep. Sem. I. N. Vekua Inst. Appl. Math.,16 (1982), pp. 3–72.
T. A. Chanturia,On oscillatory properties of system of nonlinear ordinary differential equations, Proc. of I. N. Vekua Inst. of Appl. Math., Tbilisi,14 (1983), pp. 163–203.
J. Džurina,Comparison theorems for functional differential equations with advanced argument, Boll. U. M. I. (7),7-A (1993), pp. 461–470.
M. Gaudenzi,On the Sturm-Picone theorem for n-th-order differential equations, Siam J. Math. Anal.,21 (1990), pp. 980–994.
M. Greguš,Third Order Linear Differential Equation, D. Reidel Publ. Comp., Dordrecht, Boston, Lancaster, Tokyo, 1987.
M. Greguš -M. Greguš jr.,Asymptotic properties of solutions of a certain nonautonomous nonlinear differential equation of the third order, Boll. U. M. I. (7),7-A (1993), pp. 341–350.
M. Hanan,Oscillation criteria for third-order linear differential equation, Pacific J. Math.,11 (1961), pp. 919–944.
P. Hartman,Ordinary Differential Equations, 2nd ed., Birkhäuser, Boston, 1982.
I. T. Kiguradze -T. A. Chanturia,Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Academic Publishers, Dordrecht-Boston-London, 1993.
T. Kusano -M. Naito -K. Tanaka,Oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations, Proc. Royal Soc. Edinburgh,90 A (1981), pp. 25–40.
J. Ohriska,Oscillation of differential equations and ν-derivatives, Czech. Math. J.,39 (114) (1989), pp. 24–44.
J. Ohriska,Oscillatory and asymptotic properties of third and fourth order linear differential equations, Czech. Math. J.,39 (114) (1989), pp. 215–224.
J. Ohriska,Adjoint differential equations and oscillation, J. Math. Anal. Appl.,195 (1995), pp. 778–796.
C. A. Swanson,Comparison and Oscillation Theory of Linear Differential Equations, Acad. Press, New York, 1968.
M. Švec,Sur une propriété intégrale de l'équation y (n)+Q(x)y=0, n=3, 4, Czech. Math. J.,7 (1957), pp. 450–462.
M. Švec,Behaviour of nonoscillatory solutions of some nonlinear differential equations, Acta Math. Univ. Comenianae,34 (1980), pp. 115–130.
Gaetano Villari,Contributi allo studio asintotico dell'equazione x‴(t)+p(t)x(t)=0, Ann. Mat. Pura Appl., IV,LI (1960), pp. 301–328.
Author information
Authors and Affiliations
Additional information
The second author wishes to thank C.N.R. of Italy and Grant Agency of Czech Republic (grant 201/96/0410) which made this research possible.
Rights and permissions
About this article
Cite this article
Cecchi, M., Doslá, Z. & Marini, M. An equivalence theorem on properties A, B for third order differential equations. Annali di Matematica pura ed applicata 173, 373–389 (1997). https://doi.org/10.1007/BF01783478
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01783478