Let $(X,d)$ and $(Y,e)$ be two complete metric spaces. It is proved that if $T$ is a mapping of $X$ into $Y$ and $S$ is a mapping of $Y$ into $X$ satisfying the inequalites \begin{gather*} e^2(Tx,TSy)eq c_1\max\{d(x,Sy)e(y,Tx),d(x,Sy)e(y,TSy),e(y,Tx)e(y,TSy)\}. d^2(Sy,STx)eq c^2\max\{e(,Tx)d(x,Sy),e(y,Tx)d(x,STx),d(x,Sy)d(x,STx)\}. \end{gather*} for all $x$ in $X$ and $y$ in $Y$, where $0\leq c_1\cdot c_2<1$ or the inequalities \begin{align*} e(Tx,TSy)\cdot&\max\{e(,Tx),e(TSy,y)\} &\quadeq c_1d(x,Sy)\cdot\max\{d(x,Sy),e(y,TSy)\}d(Sy,STx) &\qquad\cdot\max\{d(x,Sy),d(x,STx)\} &\quadeq c_2e(y,Tx)\cdot\max\{e(y,Tx),d(x,STx)\} \end{align*} for all $x$ in $X$ and $y$ in $Y$, where $0\leq c_1,c_2<1$, then $ST$ has a unique fixed point $z$ in $X$ and $TS$ has a unique fixed point $w$ in $Y$. Further, $Tz=w$ and $Sw=z$.