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In this paper, we present some weakly compatible and quasi-contraction results for self-mappings in fuzzy cone metric spaces and prove some coincidence point and common fixed point theorems in the said space. Moreover, we use two Urysohn type integral equations to get the existence theorem for common solution to support our results. The two Urysohn type integral equations are as follows: $$\begin{aligned} &x(l)= \int _{0}^{1}K_{1}\bigl(l,v,x(v) \bigr)\,dv+g(l), \\ &y(l)= \int _{0}^{1}K_{2}\bigl(l,v,y(v) \bigr)\,dv+g(l), \end{aligned}$$  where $l\in [0,1]$ and $x,y,g\in \mathbf{E}$, where E is a real Banach space and $K_{1},K_{2}:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}$.

Weakly compatible and quasi-contraction results in fuzzy cone metric spaces with application to the Urysohn type integral equations