content_image/325778.jpg content_image/325777.jpg content_image/325776.jpg content_image/325774.jpg
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. 

<img src='content_image/325778.jpg'>

<img src='content_image/325777.jpg'>

<img src='content_image/325776.jpg'>

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. 

Example 1. Let us consider the following differential equation 

<img src='content_image/325774.jpg'>

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. 

