Oscillation and nonoscillation criteria for second-order nonlinear difference equations of Euler type

Yamaoka, Naoto

doi:10.1090/proc/13888

Oscillation and nonoscillation criteria for second-order nonlinear difference equations of Euler type
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Proc. Amer. Math. Soc. 146 (2018), 2069-2081 Request permission

Abstract:

This paper deals with the oscillatory behavior of solutions of difference equations corresponding to second-order nonlinear differential equations of Euler type. The obtained results are represented as a pair of oscillation and nonoscillation criteria, and are best possible in a certain sense. Linear difference equations corresponding to the Riemann–Weber version of the Euler differential equation and its extended equations play an important role in proving our results. The proofs of our results are based on the Riccati technique and the phase plane analysis of a system.

References

A. Aghajani and A. Moradifam, Oscillation of solutions of second-order nonlinear differential equations of Euler type, J. Math. Anal. Appl. 326 (2007), no. 2, 1076–1089. MR 2280964, DOI 10.1016/j.jmaa.2006.03.065
Z. Došlá and N. Partsvania, Oscillatory properties of second order nonlinear differential equations, Rocky Mountain J. Math. 40 (2010), no. 2, 445–470. MR 2646451, DOI 10.1216/RMJ-2010-40-2-445
L. Erbe, A. Peterson, and S. H. Saker, Hille-Kneser-type criteria for second-order dynamic equations on time scales, Adv. Difference Equ. (2006), Art. ID 51401, 18. MR 2255165
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
Einar Hille, Non-oscillation theorems, Trans. Amer. Math. Soc. 64 (1948), 234–252. MR 27925, DOI 10.1090/S0002-9947-1948-0027925-7
Akane Hongyo and Naoto Yamaoka, General solutions of second-order linear difference equations of Euler type, Opuscula Math. 37 (2017), no. 3, 389–402. MR 3605618, DOI 10.7494/OpMath.2017.37.3.389
Başak Karpuz, Nonoscillation and oscillation of second-order linear dynamic equations via the sequence of functions technique, J. Fixed Point Theory Appl. 18 (2016), no. 4, 889–903. MR 3571312, DOI 10.1007/s11784-016-0334-8
Jitsuro Sugie and Tadayuki Hara, Nonlinear oscillations of second order differential equations of Euler type, Proc. Amer. Math. Soc. 124 (1996), no. 10, 3173–3181. MR 1350965, DOI 10.1090/S0002-9939-96-03601-5
Jitsuro Sugie and Kazuhisa Kita, Oscillation criteria for second order nonlinear differential equations of Euler type, J. Math. Anal. Appl. 253 (2001), no. 2, 414–439. MR 1808146, DOI 10.1006/jmaa.2000.7149
Jitsuro Sugie and Naoto Yamaoka, An infinite sequence of nonoscillation theorems for second-order nonlinear differential equations of Euler type, Nonlinear Anal. 50 (2002), no. 3, Ser. A: Theory Methods, 373–388. MR 1906468, DOI 10.1016/S0362-546X(01)00768-4
C. A. Swanson, Comparison and oscillation theory of linear differential equations, Mathematics in Science and Engineering, Vol. 48, Academic Press, New York-London, 1968. MR 0463570
James S. W. Wong, Oscillation theorems for second-order nonlinear differential equations of Euler type, Methods Appl. Anal. 3 (1996), no. 4, 476–485. MR 1437791, DOI 10.4310/MAA.1996.v3.n4.a5
Naoto Yamaoka, Oscillation criteria for second-order nonlinear difference equations of Euler type, Adv. Difference Equ. , posted on (2012), 2012:218, 14. MR 3017364, DOI 10.1186/1687-1847-2012-218
N. Yamaoka and J. Sugie, Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations, Ukraïn. Mat. Zh. 58 (2006), no. 12, 1704–1714 (English, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 58 (2006), no. 12, 1935–1949. MR 2347855, DOI 10.1007/s11253-006-0178-2