Complex valued approach to the system of non-linear second order Boundary value problem and multivalued mapping via fixed point method

The main aim of this manuscript is to work on the application part of CVMS. In this work we have demonstrated some common fixed results and then we deal primarily with two parts of applications, part(I) Complex valued version of existence and common solution for second order nonlinear boundary value problem using greens function,
\begin{equation} \nonumber \begin{cases} \mu''(x)= Im (x, \mu(x), \mu'(x)),&\text{when } x\in [ 0, \intercal], \intercal >0\\ \mu(x_1)= \mu_1, \\ \mu(x_2)=\mu_2 ,&\text{when } x_1, x_2 \in [ 0, \intercal]. \end{cases} \end{equation}
part(II) Application of fixed point results for multivalued mapping in setting of CVMS without using notion of continuity. Eventually several equivalent results and examples are presented to sustain our Main result.