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Oscillations of certain functional differential equations

Said R. Grace — 1999

Czechoslovak Mathematical Journal

Sufficient conditions are presented for all bounded solutions of the linear system of delay differential equations ( - 1 ) m + 1 d m y i ( t ) d t m + ∑ j = 1 n q i j y j ( t - h j j ) = 0 , m ≥ 1 , i = 1 , 2 , ... , n , to be oscillatory, where q i j ε ( - ∞ , ∞ ) , h j j ∈ ( 0 , ∞ ) , i , j = 1 , 2 , ... , n . Also, we study the oscillatory behavior of all bounded solutions of the linear system of neutral differential equations ( - 1 ) m + 1 d m d t m ( y i ( t ) + c y i ( t - g ) ) + ∑ j = 1 n q i j y j ( t - h ) = 0 , where c , g and h are real constants and i = 1 , 2 , ... , n .

Oscillation of certain difference equations

Said R. Grace — 2000

Czechoslovak Mathematical Journal

Some new criteria for the oscillation of difference equations of the form Δ 2 x n - p n Δ x n - h + q n | x g n | c s g n x g n = 0 and Δ i x n + p n Δ i - 1 x n - h + q n | x g n | c s g n x g n = 0 , i = 2 , 3 , are established.

On the oscillation of certain difference equations

Said R. Grace;  H. A. El-Morshedy — 2000

Mathematica Bohemica

In this paper we study the oscillation of the difference equations of the form 2xn+pnxn+f(n, xn-g, xn-h)=0, in comparison with certain difference equations of order one whose oscillatory character is known. The results can be applied to the difference equation 2xn+pnxn+qn|xn-g||xn-h|sgnxn-g=0, where λ and μ are real constants, λ > 0 and μ ≥ 0 .

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