Coercive nonlocal elements in fractional differential equations

Abstract

We consider the fractional boundary problem $$\begin{aligned} -\left[ D_{0^+}^{\nu }y\right] (t)= & {} \lambda f\big (t,y(t)\big )\text {, }0<t<1\\ y^{(i)}(0)= & {} 0\text {, }0\le i\le n-2\\ \left[ D_{0^+}^{\alpha }y\right] (1)= & {} H\big (\varphi (y)\big ),\nonumber \end{aligned}$$ where \(n\in \mathbb {N}_4\), \(n-1 0\) is a parameter. Here the element \(\varphi \) is a linear functional that represents a nonlocal boundary condition. We show that by introducing a new order cone, we can ensure that this functional is coercive, which is of importance in proving existence results for the above boundary value problem under minimal assumptions on the functions f and H. We also develop a new open set attendant to the cone. By means of examples we investigate both the usefulness of the new set as well as the strength of the coercivity condition and its dependence on the order, \(\nu \), of the fractional derivative. Finally, the methods we develop are applicable to a range of fractional-order boundary value problems.