A new version of the Ćirić quasi-contraction principle in the modular space

Abdolrahman Razani, Simin Homaei Pour, Elmira Nabizadeh, Maryam Beyg Mohamadi

Let $T$ be a mapping of a modular space $(M_{\rho},\rho)$ into itself which is $(q,c)$-quasi-contraction, i.e. if there exist numbers $0\leq q<1<c$ such that $T$ satisfies $\rho(c(Tx-Ty))\leq q\max\{\rho(x-y),\rho(x-Tx),\rho(y-Ty),\rho(x-Ty),\rho(y-Tx)\}$ for all $x, y\in M_\rho$. In this article, the existence of a unique fixed point of $T$ is proved. Moreover, the same result holds for the multi-valued $(q,c)$-quasi-contraction map.