φ-Hilfer分数阶共振边值问题解的存在性

Existence of Solutions ofφ-Hilfer Fractional Order Boundary Value Problems at Resonance

用Mawhin的重合度理论研究共振情形下φ-Hilfer分数阶Riemman-Stieltjes积分边值问题■解的存在性,其中n-1<α≤n,0≤β≤1,γ=α+nβ-αβ,n=1,2,…,φ∈C^(n)[0,1]且φ′(t)>0于[0,1],A(t)是一个有界变差函数.结果表明,在合适的Banach空间中,φ-Hilfer分数阶微分方程在Riemman-Stieltjes积分边界条件下的解存在.

By using Mawhin’s coincidence degree theory,we study the existence of solutions for theφ-Hilfer fractional order Riemman-Stieltjes integral boundary value problem ■ in resonance case,where n-1<α≤n,0≤β≤1,γ=α+nβ-αβ,n=1,2,…,φ∈C^(n)[0,1]and for all t∈[0,1]we haveφ′(t)>0,A(t)is a bounded variation function.The results show that the solutions ofφ-Hilfer fractional differential equation exist under the Riemann-Stieltjes integral boundary value conditions in suitable Banach spaces.