Existence of Solution for a Katugampola Fractional Differential Equation Using Coincidence Degree Theory

In this paper, the authors study the existence of positive solutions to the fractional boundary value problem at resonance \(\begin{aligned} -(D^{\alpha ,\rho }_{a+}x)(t)= & {} f(t,x(t),D^{\alpha -1, \rho }_{a+}x(t)), \ \ t\in (a,b), \\ x(a)= & {} 0, \ \ x(b)=\int _{a}^{b} x(t){\text {d}}A(t), \end{aligned}\) $$\begin{array}{cc}\hfill -(D_{a+}^{\alpha ,\rho }x)(t)=& f(t,x(t),D_{a+}^{\alpha -1,\rho }x(t)),t\in (a,b),\hfill \\ \hfill x(a)=& 0,x(b)={\int }_a^{b}x(t)\text{d}A(t),\hfill \end{array}$$ where \(1<\alpha \le 2\) $1<\alpha \le 2$ , and \(D^{\alpha ,\rho }_{a+}\) $D_{a+}^{\alpha ,\rho }$ is a Katugampola fractional derivative, which generalizes the Riemann–Liouville and Hadamard fractional derivatives, and \(\int _{a}^{b} x(t){\text {d}}A(t)\) ${\int }_a^{b}x(t)\text{d}A(t)$ denotes a Riemann–Stieltjes integral of x with respect to A, where A is a function of bounded variation. Coincidence degree theory is applied to obtain existence results. This appears to be the first work in the literature to deal with a resonant fractional differential equation with a Katugampola fractional derivative. Examples are given to illustrate the application of their results.