Existence of solutions for a higher order Riemann–Liouville fractional differential equation by Mawhin's coincidence degree theory

In this paper, we investigate the existence of at least one solution to following higher order Riemann–Liouville fractional differential equation with Riemann–Stieltjes integral boundary condition resonance: − ( D 0 + α x ) t = f , 1 n < ≤ ∈ [ 0,1 ] ′ … 2 k ∫ d A $$ {\displaystyle \begin{array}{cc}\hfill -\left({D}_{0+}^{\alpha }x\right)(t)=& f\left(t,x(t),{D}_{0+}^{\alpha -1}x(t)\right),n-1<\alpha \le n,t\in \left[0,1\right],\hfill \\ {}\hfill x(0)=& {x}^{\prime }(0)=\dots ={x}^{\left(n-2\right)}(0)=0,{x}^{(k)}(1)=\int_0^1{x}^{(k)}(t) dA(t),\hfill \end{array}} by using Mawhin's coincidence degree theory. Here, {D}_{0+}^{\alpha } is standard derivative : × ℝ → \alpha, f:\left[0,1\right]\times {\mathbb{R}}^2\to \mathbb{R} and {\int}_0^1{x}^{(k)}(t) dA(t) {x}^{(k)} respect . Our choice in can be any integer between n-1 which supplements many conditions assumed literature. Several examples are given strengthen our result.