So, from inequality (30), we have which contradicts (22). So, the proof is complete. Corollary 3. Assume that b ( t ) = 0, c ( t ) ≡ 0 and that Then every solution of Eq. (1)-(2) is oscillatory. Corollary 4. Assume that b ( t ) ≡ 0 , c ( t ) = 0 and that Then every solution of Eq. (1)-(2) is oscillatory. Now, we give some examples to illustrate our results. Note that previous results in the litarature can not be applied following differential equations to obtain existence of oscillatory solutions. where c ( t ) ≥ 0 is any continuous function. It can be shown that the hypotheses of Theorem 2 as well as Theorem 3 are satisfied. So, all solutions of Eq. (31) are oscillatory. 16     x $^{′}$( t ) + πx ( t − 1 2 ) + c ( t ) x ([ t − 1]) = 0 , t = n, n = 1 , 2 , ..., t > 0 , x ( n $^{+}$) − x ( n $^{−}$) = − x ( n $^{+}$) , n = 1 , 2 , ..., (31) Example 1. Let us consider the following differential equation lim n $_{→∞}$inf n +1 ∫ n   ∏ n − 1 <t$_{j}$ ≤ s (1 − b$_{j}$ $_{)}$  c ( s $_{) exp}$  s ∫ n − 1 a ( u ) $_{du}$  ds > 1 e . lim t $_{→∞}$inf t ∫ t − τ   ∏ s − τ<t$_{j}$ ≤ s (1 − b$_{j}$ $_{)}$  b ( s $_{) exp}$  s ∫ s − τ a ( u ) $_{du}$  ds > 1 e . lim n $_{→∞}$inf n +1 ∫ n +1 − τ   ∏ n − 1 <t$_{j}$ ≤ s (1 − b$_{j}$ $_{)}$  c ( s $_{) exp}$  s ∫ n − 1 a ( u ) $_{du}$  ds ≤ 1 e ,