Xiangling Li, S. Ur Rehman, S. Ullah Khan, H. Aydi, J. Ahmad, and N. Hussain. Strong Coupled Fixed Point Results and Applications to Urysohn Integral Equations. Dynamic Systems and Applications 30 (2021) No.2, 197-218

https://doi.org/10.46719/dsa20213023

ABSTRACT. The aim of this paper is to  establish some strong coupled fixed point theorems via a new concept of cyclic contractive type mappings in the context of fuzzy metric spaces. Moreover, we ensure the existence of a common solution of the two Urysohn type integral equations:% for our result to get the existence theorem for common solution. The two Urysohn type integral equations are
\begin{align*}
&\xi(l)=\int_{a}^{b}K_1(l,s,\xi(s))ds+h_1(l),\\
&\xi(l)=\int_{a}^{b}K_2(l,s,\xi(s))ds+h_2(l),
\end{align*}
where $l\in[a,b]\subset\mathbb{R}$, $\xi,h_1,h_2\in C([a,b],\mathbb{R})$ and $K_1,K_2:[a,b]^2\times \mathbb{R}\to\mathbb{R}$.

 AMS (MOS) Subject Classification. 46T99; 47H10; 54H25.