具有p -Laplacian算子的delta-nabla分数阶 差分边值问题正解的存在性

考虑具有p -Laplacian算子的delta -nabla分数阶差分方程边值问题: {Δ^β_{α -2}(φ_p(_b▽^αx(t)))+λ f(t-α+β+1,x(t-α+β+1),_b▽^εx(t)]_{t -α +β + ε +1})=0, t∈T; x(b)=0, _{b -1}▽^{α -1}x(α-2)=_{b +α -2}▽^-ωg(t,x(t))]_{t =α -ω -1}; _b▽^αx(t)]_{α -2}=0, _b▽^αx(t)]_{α + b -2}=0. 其中b∈Z⁺, T=α-β-1,b+α-β-1]_{Ν<sup>α -β -1}, 1≤α, β≤2, 3<α+β≤4, 0<ω<1, λ∈(0,+∞), Δ^β_{α -2}和 _b▽^α分别是左右分数阶差分算子,并且φ_p(s)=|s|^{p -2}s, p>1.利用上下解方法和Schauder不动点定理,得到了上述边值问题正解的存在性.

Existence of positive solutions for p-Laplacian fractional difference involving the discrete delta-nabla fractional boundary value problem

Consider the boundary value problem of delta-nabla fractional difference equations with p -Laplacian operator: {Δ^β_{α -2}(φ_p(_b▽^αx(t)))+λ f(t-α+β+1,x(t-α+β+1),_b▽^εx(t)]_{t -α +β + ε +1})=0, t∈T; x(b)=0, _{b -1}▽^{α -1}x(α-2)=_{b +α -2}▽^-ωg(t,x(t))]_{t =α -ω -1}; _b▽^αx(t)]_{α -2}=0, _b▽^αx(t)]_{α +b -2}=0. Where b∈Z⁺, T=α-β-1,b+α-β-1]_{Ν<sup>α -β -1}, 1≤α,β≤2, 3<α+β≤4, 0<ω<1, λ∈(0,+∞), Δ^β_{α -2}, _b▽^α are left and right fractional difference operator, and φ_p(s)=|s|^{p -2}s, p>1. By using the upper and lower solution method and the Schauder fixed point theorem, the existence of the positive solution of the above boundary value problem is obtained.