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: $$\begin{aligned} &x(l)= \int _{0}^{1}K_{1}\bigl(l,v,x(v) \bigr)\,dv+g(l), \\ &y(l)= \int _{0}^{1}K_{2}\bigl(l,v,y(v) \bigr)\,dv+g(l), \end{aligned}$$ x(l)=∫01K1(l,v,x(v))dv+g(l),y(l)=∫01K2(l,v,y(v))dv+g(l), where $l\in [0,1]$l∈[0,1] and $x,y,g\in \mathbf{E}$x,y,g∈E, where E is a real Banach space and $K_{1},K_{2}:[0,1]\times [0,1]\times \mathbb{R}\to \mathbb{R}$K1,K2:[0,1]×[0,1]×R→R.

In this paper, we introduce the concept of coupled type and cyclic coupled type fuzzy cone contraction mappings in fuzzy cone metric spaces. We establish some coupled fixed point results without the mixed monotone property, and also present some coupled fixed results using the partial order metric in the said space. We present some strong coupled fixed point theorems using cyclic coupled type fuzzy cone contraction mappings in fuzzy cone metric spaces. Moreover, we present an application of nonlinear integral equations for the existence of a unique solution to support our work.

In this paper, we present some common fixed point theorems for a pair of self-mappings in fuzzy cone metric spaces under the generalized fuzzy cone contraction conditions. We extend and improve some recent results given in the literature.

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<abstract><p>In this paper, we present some generalized multi-valued contraction results on cone metric spaces. We use some maximum and sum types of contractions for a pair of multi-valued mappings to prove some common fixed point theorems on cone metric spaces without the condition of normality. We present an illustrative example for multi-valued contraction mappings to support our work. Moreover, we present a supportive application of nonlinear integral equations to validate our work. This new theory, can be modified in different directions for multi-valued mappings to prove fixed point, common fixed point and coincidence point results in the context of different types of metric spaces with the application of different types of integral equations.</p></abstract>