Existence of the unique nontrivial solution for mixed fractional differential equations (Q2028659)

The authors investigate the existence of solutions for the fractional differential equation \[ ^CD_{1-}^{\alpha}D_{0+}^{\beta}x(t)+f(t,x(t))=b, \] where \(0<t<1\), and \(x(0)=x'(1)=D_{0+}^{\beta}x(1)=0\). Here \(^CD_{1-}^{\alpha}\) is the right-sided Caputo fractional derivative and \(D_{0+}^{\beta}\) is the leftsided Riemann-Liouville fractional derivative and \(\alpha \in (0,1]\), \( \beta \in (1,2]\) with \(\alpha +\beta >2\). The function \(f:[0,1]\times \mathbb{R} \to \mathbb{R}\) is continuous, \(b>0\) is a constant. The Greens function of the linear homogeneous version of the equation is deduced. Results on mixed monotone and on concave operators are given. Employing these results and making some monotonicity and growth assumptions on \(f\) allows the authors to deduce existence and uniqueness of solutions of the fractional differential equation above. Two numerical examples are given.