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

Abstract

AbstractIn this paper, the authors study the existence of positive solutions to the fractional boundary value problem at resonance $$\begin{aligned} -(D^{\alpha ,\rho }_{a+}x)(t)= & {} f(t,x(t),D^{\alpha -1, \rho }_{a+}x(t)), \ \ t\in (a,b), \\ x(a)= & {} 0, \ \ x(b)=\int _{a}^{b} x(t){\text {d}}A(t), \end{aligned}$$ - ( D a + α , ρ x ) ( t ) = f ( t , x ( t ) , D a + α - 1 , ρ x ( t ) ) , t ∈ ( a , b ) , x ( a ) = 0 , x ( b ) = ∫ a b x ( t ) d A ( t ) , where $$1<\alpha \le 2$$ 1 < α ≤ 2 , and $$D^{\alpha ,\rho }_{a+}$$ D a + α , ρ is a Katugampola fractional derivative, which generalizes the Riemann–Liouville and Hadamard fractional derivatives, and $$\int _{a}^{b} x(t){\text {d}}A(t)$$ ∫ a b x ( t ) 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.