Boundedness Criteria for Solutions of Some Nonlinear Differential Equations of Second Order
Keywords:Boundedness, Nonlinear Differential Equations, Second Order, Gronwall's Inequality, Bonnet's Theorem
Mathematical modelling phenomena of most applied sciences is associated with second order nonlinear diﬀerential equations, which are not easily solvable. Therefore, the study of behavior of the solutions has attracted the attention of many mathematicians worldwide. In the present work, we discuss some clear assumptions for the boundedness of all solutions of some non-linear differential equations of second order. The main tools in the proofs of our results are Gronwall's inequality and Bonnet's Theorem. The results obtained here extend and/or improve some of well-known results in the literature. Further, some illustrative examples are provided to show the applicability of the new results
Ademola A, Arawomo P (2011) Stability, boundedness and asymptotic behaviour of solutions of certain nonlinear differential equation. Kragujevac J. of Mathematics, (35) 431-445.
Ahmed FN and Ali AD (2019) On the oscillation property for some nonlinear differential equations of second order. Journal of Pure & Applied Sciences, (18)342-345.
Amhalhil JJ (2021) Oscillations of solutions for nonlinear differential equations, Sirte University Scientific Journal, (11)1-16.
Athanassov ZS (1987) Boundedness criteria for solutions of certain second order nonlinear differential equations, J. Math. Anal. Appl., (123) 461-479.
Bartle ZS (1976) The elements of real analysis. (7th Edition). John Willey and Sons. Avenue, New York.
Bellman R (1953) Stability theory of differential equations, McGraw-Hill, New York.
Bihari I (1957) Researches on the boundedness and stability of the solutions of nonlinear differential equations. Acta Math. Sci. Hungar, (8) 261-278.
Burton TA (1970) On the equation Ann. Mat. Pura
. Appl., (85)277-285.
Burton TA and Townsend CG (1968) On the generalized linear equation with forcing term. J. Differential Equations, (4) 620-633.
Chang SH (1970) Boundedness theorems for certain second order nonlinear differential equations. J. Math. Anal. Appl., (31) 509-516.
Elabbasy EM and Elzeiny ShR (2011) Oscillation theorems concerning non-linear differential equations of the second order. Opuscula Mathematica J., (31) 373-391.
Graef JR and Spikes PW (1975) Asymptotic behavior of solutions of a second order nonlinear differential equation. J. Differential Equations, (17) 461-476.
Hartman P (1982) Ordinary differential equations, Birkhauser.
Kroopnick A (1995) General boundedness theorems to some second order nonlinear differential equations with integrable forcing term. Inter. J. Math. Sci., (18) 823-824.
Lalli BS (1969) On boundedness of solutions of certain second order nonlinear differential equations. J. Math. Anal. Appl., (25) 182-188.
Olehnik SN (1972) The boundedness and unboundedness of the solutions of a second order differential equation. Differentsial’nye Uravneniya, (8)1701-1704.
Olehnik SN (1973) The boundedness of the solutions of a certain second order differential equation. Differentsial’nye Uravneniya, (9) 1994-1999.
Saad MJ, Kumaresan N and Ratnavelu K (2013) Oscillation theorems for nonlinear second order equations with damping, Bull. Malays. Math. Sci. Soc., (36) 881-893.
Saker S (2006) B of solutions second-order forced nonlinear dynamics equation. Rocky Mountain Journal of Mathematics, (36) 2027-2039.
Salhin A. A (2019) Oscillation theorems for nonlinear second order forced differential equations, Sirte University Scientific Journal, (9) 1-19.
Tunc C (2010) Boundedness results for solutions of certain nonlinear differential equations of second order. J. Indones Math. Soc., (16) 115-126.
Waltman P (1963) Some properties of solutions of Monatsh. Math., (67) 50-54.
Wong JSW (1966) Some properties of solutions of , III, SIAM J., (14) 209-214.
Wong JSW (1967) Boundedness theorems for solutions of , IV, Enseign. Math., (13)157-168.
Wong JSW (1968) On second order nonlinear oscillation, Funkcial. Ekvac., (11) 207-234.
Wong JSW and Burton TA (1965) Some properties of solutions of , Monatsch. Math., (69) 364-374.