Interval oscillation criteria for forced Emden-Fowler functional dynamic equations with oscillatory potential. (English) Zbl 1273.34095

Anderson, Douglas R.; Saker, Samir H.

Sci. China, Math. 56, No. 3, 561-576 (2013).

Summary: Interval oscillation criteria are established for a second-order functional dynamic equation of Emden-Fowler type with oscillatory potential by applying Riccati and generalized Riccati techniques. The results represent further improvements on those given even for differential and difference equations. Some examples are considered to illustrate the main results.

MSC:

34N05	Dynamic equations on time scales or measure chains
34K11	Oscillation theory of functional-differential equations

Keywords:

oscillation; forced dynamic equations; time scales; functional equations; Riccati technique

References:

[1]	Agarwal R P, Bohner M, Saker S H. Oscillation of second order delay dynamic equation. Can Appl Math Quart, 2005, 13: 1–17 · Zbl 1126.39003
[2]	Anderson D R. Oscillation of second-order forced functional dynamic equations with oscillatory potentials. J Differ Equ Appl, 2007, 13: 407–421 · Zbl 1123.34051
[3]	Anderson D R. Interval criteria for oscillation of nonlinear second-order dynamic equations on time scales. Nonlinear Anal Theor Meth Appl, 2008, 69: 4614–4623 · Zbl 1167.34008
[4]	Bohner M, Peterson A. Dynamic Equations on Time Scales: An Introduction with Applications. Boston: Birkhäuser, 2001 · Zbl 0978.39001
[5]	Bohner M, Peterson A. Advances in Dynamic Equations on Time Scales. Boston: Birkhäuser, 2003 · Zbl 1025.34001
[6]	Bohner M, Saker S H. Oscillation of second order nonlinear dynamic equations on time scales. Rocky Mountain J Math, 2004, 34: 1239–1254 · Zbl 1075.34028
[7]	Bohner M, Tisdell C. Oscillation and nonoscillation of forced second order dynamic equations. Pacific J Math, 2007, 230: 59–71 · Zbl 1160.34029
[8]	Erbe L, Hassan T S, Peterson A, et al. Oscillation criteria for half-linear delay dynamic equations on time scales. Nonlinear Dyn Syst Theor, 2009, 9: 51–68 · Zbl 1173.34037
[9]	Erbe L, Hassan T S, Peterson A, et al. Oscillation criteria for sublinear half-linear delay dynamic equations on time scales. Int J Differ Equ, 2008, 3: 227–245 · Zbl 1162.39005
[10]	Erbe L, Peterson A, Saker S H. Oscillation criteria for second-order nonlinear dynamic equations on time scales. J London Math Soc, 2003, 76: 701–714 · Zbl 1050.34042
[11]	Erbe L, Peterson A, Saker S H. Kamenev-type oscillation criteria for second-order linear delay dynamic equations. Dyn Syst Appl, 2006, 15: 65–78 · Zbl 1104.34026
[12]	Erbe L, Peterson A, Saker S H. Oscillation criteria for a forced second-order nonlinear dynamic equation. Dyn Equ J Differ Equ Appl, 2008, 14: 997–1009 · Zbl 1168.34025
[13]	Erbe L, Hassan T S, Peterson A, et al. Interval oscillation criteria for forced second-order nonlinear delay dynamic equations with oscillatory potential. Dyn Cont Discret Impuls Syst Ser A, 2010, 13: 533–542 · Zbl 1202.34162
[14]	Güvenilir A F, Zafer A. Second-order oscillation of forced functional differential equations with oscillatory potentials. Comput Math Appl, 2006, 51: 1395–1404 · Zbl 1138.34335
[15]	Hilger S. Analysis on measure chains – a unified approach to continuous and discrete calculus. Results Math, 1990, 18: 18–56 · Zbl 0722.39001
[16]	Huang M, Feng W. Oscillation for forced second-order nonlinear dynamic equations on time scales. Electron J Differ Equ, 2006, 2006: 1–8 · Zbl 1160.76444
[17]	Huang M, Feng W. Forced Oscillation of second-order nonlinear dynamic equations on time scales. Electron J Qual Theor Differ Equ, 2008, 36: 1–13 · Zbl 1183.34142
[18]	Kac V, Chueng P. Quantum Calculus. New York: Springer, 2002
[19]	Nasr A H. Necessary and sufficent conditions for the oscillation of forced nonlinear second order differential equations with delayed argument. J Math Anal Appl, 1997, 212: 51–59 · Zbl 0884.34075
[20]	Saker S H. Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. J Comp Appl Math, 2005, 177: 375–387 · Zbl 1082.34032
[21]	Saker S H. Oscillation of second-order forced nonlinear dynamic equations on time scales. Electron J Qual Theor Differ Equ, 2005, 23: 1–17 · Zbl 1097.34027
[22]	Saker S H. New oscillation criteria for second-order nonlinear dynamic equations on time scales. Nonlinear Funct Anal Appl, 2006, 11: 170–351 · Zbl 1126.34024
[23]	Saker S H. Oscillation of nonlinear dynamic equations on time scales. Appl Math Comp, 2004, 148: 81–91 · Zbl 1045.39012
[24]	Saker S H. Oscillation criteria of second-order half-linear dynamic equations on time scales. J Comp Appl Math, 2005, 177: 375–387 · Zbl 1082.34032
[25]	Saker S H. Oscillation Theorey of Dynamic Equations on Time Scales, Second and Third Orders. Berlin: Lambert Academic Publishing, 2010
[26]	Saker S H. Boundedness of solutions of second-order forced nonlinear dynamic equations. Rocky Mount J Math, 2006, 36: 2027–2039 · Zbl 1139.34030
[27]	Saker S H. Oscillation of second-order forced nonlinear dynamic equations on time scales. Electron J Qual Theor Differ Equ, 2005, 23: 1–17 · Zbl 1097.34027
[28]	Saker S H. Kamenev-type oscillation criteria for forced Emden-Fowler superlinear difference equations. Electron J Differ Equ, 2002, 2002: 1–9 · Zbl 1002.39013
[29]	Spedding V. Taming nature’s numbers. New Scientist, 2003, 179: 28–31
[30]	Sun Y G. A note on Nasr’s and Wong’s papers. J Math Anal Appl, 2003, 286: 363–367 · Zbl 1042.34096
[31]	Wang Q R. Interval criteria for oscillation of second-order nonlinear differential equations. J Comp Appl Math, 2007, 205: 231–238 · Zbl 1142.34331
[32]	Wong J S W. Oscillation criteria for a forced second-order linear differential equation. J Math Anal Appl, 1999, 231: 233–240 · Zbl 0922.34029