λ-FIXED POINT THEOREM WITH KINDS OF FUNCTIONS OF MIXED MONOTONE OPERATOR

Our work is related to the existence and uniqueness of positive solution to the fractional boundary value problem(BVP) with Riemann-Liouville fractional derivative. We employ the fixed point theorem of mixed monotone operator and the attributes of the Green function to consider the following:

$ \begin{aligned}& -D_{0+}^\nu \mathfrak{u}(t)=\lambda^{-1}(\mathfrak{f}(t, \mathfrak{u}(t), \mathbf{v}(t))+\mathfrak{g}(t, \mathfrak{u}(t))+\mathbf{k}(t, \mathbf{v}(t))), 0<t<1,3 \leq \nu \leq 4, \\& \mathfrak{u}(0)=\mathfrak{u}^{\prime}(0)=\mathfrak{u}^{\prime \prime}(0)=0, \\& {\left[D_{0^{+}}^\rho \mathfrak{u}(t)\right]_{t=1}=0, \quad 1 \leq \rho \leq 2 .}\end{aligned} $

$ \lambda $ is a positive number. $ D_{0^{+}}^{\nu} $ and $ D_{0^{+}}^{\rho} $ are the standard Riemann-Liouville fractional derivatives of degree $ \nu $ and $ \rho $, respectively. In the end, we provide an exemplar to illustrate the outcome. It should also be noted that in this paper we have assumed the variable $ \mathbf{v} $ as follows:

$ \mathbf{v}(t)=1-\frac{\Gamma(2-\rho)}{t^{1-\rho}} D_{0^{+}}^\rho \mathfrak{u}(t). $