Oscillation via Difference Inequalities

Here, we shall use difference inequalities to study the oscillatory behavior of solutions of the difference equation 12.1 $$\Delta \left( {a\left( k \right){{\left( {\Delta y\left( k \right)} \right)}^\sigma }} \right) + q\left( {k + 1} \right)f\left( {y\left( {k + 1} \right)} \right) = r\left( k \right),k \in N$$ where σ is a positive quotient of odd integers (odd/odd), or even over odd integers (even/odd). The technique employed here is an extension of the methods in the work of Graef and Spikes [119], Kwong and Wong [170] for differential equations, and Thandapani et. al. [288,295,297] for difference equations.