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By applying the Leggett-Williams fixed point theorem in a suitably constructed cone, we obtain the existence of at least three unbounded positive solutions for a boundary value problem on the half line. Our result improves and complements some of the work in the literature.

We offer sufficient conditions for the oscillation of all solutions of the perturbed difference equation $$|{\Delta }^m{y\left(k\right)|}^{\alpha -1}{\Delta }^{m}y\left(k\right)+Q(k,y(k-{\sigma }_k),\Delta y(k-{\sigma }_k),\cdots ,{\Delta }^{m-2}y(k-{\sigma }_k))$$ $=P(k,y(k-{\sigma }_k),\Delta y(k-{\sigma }_k),\cdots ,{\Delta }^{m-1}y(k-{\sigma }_k)),k\ge k_0$ where $\alpha >0.$ Examples which dwell upon the importance of our results are also included.