Common fixed point of generalized ($\psi$-$\varphi$)-weak contraction mappings

Document Type : Research Paper


Department of Mathematics, Faculty of Science, Arak University, Arak, 38156- 8-8349, Iran.


Let $(X, d)$ be a complete metric space and let $f,g : X \to X$ be two mappings which satisfy a ($\psi$-$\varphi$)-weak contraction condition or generalized ($\psi$-$\varphi$)-weak contraction condition. Then $f$ and $g$ have a unique common fixed point. Our results extend previous results given by Ciric (1971), Rhoades (2001), Branciari (2002), Rhoades (2003), Abbas and Ali Khan (2009), Zhang and Song (2009) and Moradi at. el. (2011).