Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations

Consider the second-order linear delay differential equation ð ‘¥ î…ž î…ž ( ð ‘¡ ) + ð ‘ ( ð ‘¡ ) ð ‘¥ ( ð œ ( ð ‘¡ ) ) = 0 , ð ‘¡ ≥ ð ‘¡ 0 , where ð ‘ âˆˆ ð ¶ ( [ ð ‘¡ 0 , ∞ ) , â„ + ) , ð œ âˆˆ ð ¶ ( [ ð ‘¡ 0 , ∞ ) , â„ ) , ð œ ( ð ‘¡ ) is nondecreasing, ð œ ( ð ‘¡ ) ≤ ð ‘¡ for ð ‘¡ ≥ ð ‘¡ 0 and l i m ð ‘¡ → ∞ ð œ ( ð ‘¡ ) = ∞ , the (discrete analogue) second-order difference equation Δ 2 ð ‘¥ ( ð ‘› ) + ð ‘ ( ð ‘› ) ð ‘¥ ( ð œ ( ð ‘› ) ) = 0 , where Δ ð ‘¥ ( ð ‘› ) = ð ‘¥ ( ð ‘› + 1 ) − ð ‘¥ ( ð ‘› ) , Δ 2 = Δ ∘ Δ , ð ‘ âˆ¶ â„• → â„ + , ð œ âˆ¶ â„• → â„• , ð œ ( ð ‘› ) ≤ ð ‘› − 1 , and l i m ð ‘› → ∞ ð œ ( ð ‘› ) = + ∞ , and the second-order functional equation ð ‘¥ ( ð ‘” ( ð ‘¡ ) ) = 𠑃 ( ð ‘¡ ) ð ‘¥ ( ð ‘¡ ) + ð ‘„ ( ð ‘¡ ) ð ‘¥ ( ð ‘” 2 ( ð ‘¡ ) ) , ð ‘¡ ≥ ð ‘¡ 0 , where the functions 𠑃 , ð ‘„ ∈ ð ¶ ( [ ð ‘¡ 0 , ∞ ) , â„ + ) , ð ‘” ∈ ð ¶ ( [ ð ‘¡ 0 , ∞ ) , â„ ) , ð ‘” ( ð ‘¡ ) ≢ ð ‘¡ for ð ‘¡ ≥ ð ‘¡ 0 , l i m ð ‘¡ → ∞ ð ‘” ( ð ‘¡ ) = ∞ , and ð ‘” 2 denotes the 2th iterate of the function ð ‘” , that is, ð ‘” 0 ( ð ‘¡ ) = ð ‘¡ , ð ‘” 2 ( ð ‘¡ ) = ð ‘” ( ð ‘” ( ð ‘¡ ) ) , ð ‘¡ ≥ ð ‘¡ 0 . The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case where l i m i n f ð ‘¡ → ∞ ∫ ð ‘¡ ð œ ( ð ‘¡ ) ð œ ( ð ‘ ) ð ‘ ( ð ‘ ) ð ‘‘ ð ‘ â‰¤ 1 / ð ‘’ and l i m s u p ð ‘¡ → ∞ ∫ ð ‘¡ ð œ ( ð ‘¡ ) ð œ ( ð ‘ ) ð ‘ ( ð ‘ ) ð ‘‘ ð ‘ < 1 for the second-order linear delay differential equation, and 0 < l i m i n f ð ‘¡ → ∞ { ð ‘„ ( ð ‘¡ ) 𠑃 ( ð ‘” ( ð ‘¡ ) ) } ≤ 1 / 4 and l i m s u p ð ‘¡ → ∞ { ð ‘„ ( ð ‘¡ ) 𠑃 ( ð ‘” ( ð ‘¡ ) ) } < 1 , for the second-order functional equation, are presented.