Bounded oscillation criteria for certain third order nonlinear difference equations with several delays and advances

This paper deals with the third order nonlinear difference equations with several delays and advances (*)@D(a"[email protected]^2x"n)+f(n,x"n,x"n"-"d"""1"""n,...,x"n"-"d"""k"""n)=0,n>=n"0 and (**)@D(a"[email protected]^2x"n)+b"ng(x"n,x"n"-"d"""1"""n,...,x"n"-"d"""k"""n)=0,n>=n"0. Sufficient conditions of the existence of uncountably many bounded positive, negative and nonoscillatory solutions, respectively, and necessary and sufficient conditions of the bounded oscillation of (*) and (**) with certain asymptotic behavior are given. In addition, a sufficient condition of the oscillatory behavior of all bounded solutions of (*) with a special of asymptotic property is also presented. Nine nontrivial examples which dwell upon the importance of the results obtained in this paper are also included.

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