A Singular Boundary Value Problem

Here, we shall offer sufficient conditions for the existence of solutions for the nth order difference equation 28.1 $${\Delta ^n}y\left( k \right) + f\left( {k,y\left( k \right),\Delta y\left( k \right), \cdots {\Delta ^{n - 2}}y\left( k \right)} \right) = 0,n2,k \in N\left( {0,J - 1} \right)$$ satisfying the boundary conditions 28.2 $${\Delta ^i}y\left( 0 \right) = 0,0in - 3$$ 28.3 $$\alpha {\Delta ^{n - 2}}y\left( 0 \right) - \beta {\Delta ^{n - 1}}y\left( 0 \right) = 0$$ 28.4 $$\gamma {\Delta ^{n - 2}}y\left( J \right) + \delta {\Delta ^{n - 1}}y\left( J \right) = 0$$ where α, β, γ and δ are constants such that 28.5 $$\rho = \alpha \gamma J + \alpha \delta + \beta \gamma succ0$$ and 28.6 $$\alpha \succ 0,\gamma \succ 0,\beta 0,\delta \gamma $$