| Summary: | Abstract In this paper, we consider a discrete delta-nabla boundary value problem for the fractional difference equation with p-Laplacian △ v − 2 β ( φ p ( b ∇ ν x ( t ) ) ) + λ f ( t − ν + β + 1 , x ( t − ν + β + 1 ) , [ b ∇ ε x ( t ) ] t − ν + β + ε + 1 ) = 0 , x ( b ) = 0 , [ b ∇ ν x ( t ) ] ν − 2 = 0 , x ( − 1 ) = ∑ t = 0 b − 1 x ( t ) A ( t ) , $$\begin{aligned}& {\triangle_{v-2}^{\beta}} \bigl({\varphi_{p} \bigl({_{b}\nabla^{\nu }}x(t) \bigr)} \bigr)+{\lambda} {f \bigl(t-\nu+\beta+1,x(t-\nu+\beta +1), \bigl[_{b}\nabla^{\varepsilon}x(t) \bigr]_{t-\nu+\beta+\varepsilon+1} \bigr)}=0, \\& x(b)=0,\quad\quad \bigl[_{b}\nabla^{\nu}x(t) \bigr]_{\nu-2}=0,\quad\quad x(-1)=\sum_{t=0}^{b-1}{x(t)A(t)}, \end{aligned}$$ where t ∈ T = [ ν − β − 1 , b + ν − β − 1 ] N ν − β − 1 $t\in\mathbb{T}=[\nu-\beta-1,b+\nu-\beta-1]_{\mathbb{N}_{\nu-\beta-1}}$ . △ ν − 2 β ${\triangle_{\nu-2}^{\beta}}$ , ∇ ν b ${_{b}\nabla^{\nu}}$ are left and right fractional difference operators, respectively, and φ p ( s ) = | s | p − 2 s $\varphi_{p}(s)=|s|^{p-2}s$ , p > 1 $p>1$ . By using the method of upper and lower solution and the Schauder fixed point theorem, we obtain the existence of positive solutions for the above boundary value problem; and applying a monotone iterative technique, we establish iterative schemes for approximating the solution.
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