Abstract
We consider the Sturm–Liouville differential equations with a power of the independent variable and sums of periodic functions as coefficients (including the case when the periodic coefficients do not have any common period). Using known results, one can show that the studied equations are conditionally oscillatory, i.e., there exists a threshold value which can be expressed by the coefficients and which separates oscillatory equations from non-oscillatory ones. It is very complicated to specify the behaviour of the treated equations in the borderline case. In this paper, applying the method of the modified Prüfer angle, we answer this question and we prove that the considered equations are non-oscillatory in the critical borderline case.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, 55. U.S. Government Printing. Office, Washington (1964)
Aghajani, A., Moradifam, A.: Oscillation of solutions of second-order nonlinear differential equations of Euler type. J. Math. Anal. Appl. 326(2), 1076–1089 (2007)
Aghajani, A., O’Regan, D., Roomi, V.: Oscillation of solutions to second-order nonlinear differential equations of generalized Euler type. Electron. J. Differ. Equ. 2013(185), 1–13 (2013)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)
Conde, S., Kalla, S.L.: On zeros of the hypergeometric function. Serdica 7(3), 243–249 (1981)
Deng, J.: Oscillation criteria for second-order linear differential equations. J. Math. Anal. Appl. 271(1), 283–287 (2002)
Došlý, O.: Half-linear Euler differential equation and its perturbations. Electron. J. Qual. Theory Differ. Equ., Proc. 10’th Coll. Qual. Theory Diff. Equ. 2016(10), 1–14 (2016)
Došlý, O., Bohner, M.: The discrete Prüfer transformation. Proc. Am. Math. Soc. 129(9), 2715–2726 (2013)
Došlý, O., Funková, H.: Euler type half-linear differential equation with periodic coefficients. Abstr. Appl. Anal. 2013, 1–6 (2013). Article ID 714263
Došlý, O., Hasil, P.: Critical oscillation constant for half-linear differential equations with periodic coefficients. Ann. Math. Pura Appl. 190(3), 395–408 (2011)
Došlý, O., Jaroš, J., Veselý, M.: Generalized Prüfer angle and oscillation of half-linear differential equations. Appl. Math. Lett. 64(2), 34–41 (2017)
Došlý, O., Řehák, P.: Half-Linear Differential Equations. Elsevier, Amsterdam (2005)
Došlý, O., Veselý, M.: Oscillation and non-oscillation of Euler type half-linear differential equations. J. Math. Anal. Appl. 429, 602–621 (2015)
Došlý, O., Yamaoka, N.: Oscillation constants for second-order ordinary differential equations related to elliptic equations with \(p\)-Laplacian. Nonlinear Anal. 113, 115–136 (2015)
Fišnarová, S., Mařík, R.: Local estimates for modified Riccati equation in theory of half-linear differential equation. Electron. J. Qual. Theory Differ. Equ. 2012(63), 1–15 (2012)
Fišnarová, S., Mařík, R.: On constants in nonoscillation criteria for half-linear differential equations. Abstr. Appl. Anal. 2011, 1–15 (2011). Article ID 638271
Gesztesy, F., Ünal, M.: Perturbative oscillation criteria and Hardy-type inequalities. Math. Nachr. 189(1), 121–144 (1998)
Gesztesy, F., Zinchenko, M.: Renormalized oscillation theory for Hamiltonian systems. Adv. Math. 311, 569–597 (2017)
Grigorian, G.A.: A new oscillatory criterion for the generalized Hill’s equation. Differ. Equ. Appl. 9(3), 369–377 (2017)
Grigorian, G.A.: On one oscillatory criterion for the second order linear ordinary differential equations. Opusc. Math. 36(5), 589–601 (2016)
Grigorian, G.A.: Two comparison criteria for scalar Riccati equations and some of their applications. Russ. Math. (Iz. VUZ) 56(11), 17–30 (2012)
Hasil, P.: Conditional oscillation of half-linear differential equations with periodic coefficients. Arch. Math. 44(2), 119–131 (2008)
Hasil, P., Mařík, R., Veselý, M.: Conditional oscillation of half-linear differential equations with coefficients having mean values. Abstr. Appl. Anal. 2014, 1–14 (2014). Article ID 258159
Hasil, P., Veselý, M.: Non-oscillation of half-linear differential equations with periodic coefficients. Electron. J. Qual. Theory Differ. Equ. 2015(1), 1–21 (2015)
Hasil, P., Veselý, M.: Non-oscillation of periodic half-linear equations in the critical case. Electron. J. Differ. Equ. 2016(120), 1–12 (2016)
Hasil, P., Veselý, M.: Non-oscillation of perturbed half-linear differential equations with sums of periodic coefficients. Adv. Differ. Equ. 2015(190), 1–17 (2015)
Hasil, P., Veselý, M.: Oscillation and non-oscillation criteria for linear and half-linear difference equations. J. Math. Anal. Appl. 452(1), 401–428 (2017)
Hasil, P., Veselý, M.: Oscillation and non-oscillation criterion for Riemann-Weber type half-linear differential equations. Electron. J. Qual. Theory Differ. Equ. 2016(59), 1–22 (2016)
Hasil, P., Veselý, M.: Oscillation constant for modified Euler type half-linear equations. Electron. J. Differ. Equ. 2015(220), 1–14 (2015)
Hasil, P., Vítovec, J.: Conditional oscillation of half-linear Euler-type dynamic equations on time scales. Electron. J. Qual. Theory Differ. Equ. 2015(6), 1–24 (2015)
Jaroš, J., Veselý, M.: Conditional oscillation of Euler type half-linear differential equations with unbounded coefficients. Studia Sci. Math. Hungar. 53(1), 22–41 (2016)
Kelley, W.G., Peterson, A.C.: Difference Equations: An Introduction with Applications. Academic Press, San Diego (2001)
Kneser, A.: Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen. Math. Ann. 42(3), 409–435 (1893)
Krüger, H.: On perturbations of quasiperiodic Schrödinger operators. J. Differ. Equ. 249(6), 1305–1321 (2010)
Krüger, H., Teschl, G.: Effective Prüfer angles and relative oscillation criteria. J. Differ. Equ. 245(12), 3823–3848 (2008)
Krüger, H., Teschl, G.: Relative oscillation theory for Sturm-Liouville operators extended. J. Funct. Anal. 254(6), 1702–1720 (2008)
Misir, A., Mermerkaya, B.: Critical oscillation constant for half linear differential equations which have different periodic coefficients. Gazi Univ. J. Sci. 29(1), 79–86 (2016)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)
Řehák, P.: New results on critical oscillation constants depending on a graininess. Dyn. Syst. Appl. 19, 271–288 (2010)
Schmidt, K.M.: Critical coupling constant and eigenvalue asymptotics of perturbed periodic Sturm-Liouville operators. Commun. Math. Phys. 211, 465–485 (2000)
Schmidt, K.M.: Oscillation of perturbed Hill equation and lower spectrum of radially periodic Schrödinger operators in the plane. Proc. Am. Math. Soc. 127, 2367–2374 (1999)
Sugie, J.: Nonoscillation criteria for second-order nonlinear differential equations with decaying coefficients. Math. Nachr. 281(11), 1624–1637 (2008)
Sugie, J., Hara, T.: Nonlinear oscillations of second order differential equations of Euler type. Proc. Am. Math. Soc. 124(10), 3173–3181 (1996)
Sugie, J., Kita, K.: Oscillation criteria for second order nonlinear differential equations of Euler type. J. Math. Anal. Appl. 253(2), 414–439 (2001)
Sugie, J., Matsumura, K.: A nonoscillation theorem for half-linear differential equations with periodic coefficients. Appl. Math. Comput. 199(2), 447–455 (2008)
Sugie, J., Onitsuka, M.: A non-oscillation theorem for nonlinear differential equations with \(p\)-Laplacian. Proc. R. Soc. Edinburgh Sect. A 136(3), 633–647 (2006)
Sugie, J., Yamaoka, N.: Comparison theorems for oscillation of second-order half-linear differential equations. Acta Math. Hungar. 111(1–2), 165–179 (2006)
Veselý, M., Hasil, P.: Conditional oscillation of Riemann-Weber half-linear differential equations with asymptotically almost periodic coefficients. Studia Sci. Math. Hungar. 51(3), 303–321 (2014)
Vítovec, J.: Critical oscillation constant for Euler-type dynamic equations on time scales. Appl. Math. Comput. 243, 838–848 (2014)
Wong, J.S.W.: Oscillation theorems for second-order nonlinear differential equations of Euler type. Methods Appl. Anal. 3(4), 476–485 (1996)
Zarzo, A., Dehesa, J.S., Yáñez, R.J.: Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2(1–4), 457–472 (1995)
Zettl, A.: Sturm–Liouville Theory. Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence (2005)
Acknowledgements
The both authors are supported by Grant GA17-03224S of the Czech Science Foundation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Teschl.
Rights and permissions
About this article
Cite this article
Hasil, P., Veselý, M. Prüfer angle and non-oscillation of linear equations with quasiperiodic data. Monatsh Math 189, 101–124 (2019). https://doi.org/10.1007/s00605-018-1232-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-018-1232-5