Extensions of Banach's and Kannan's results in fuzzy metric spaces (Q2883956)

Two main results of the paper are stated below:NEWLINENEWLINE Theorem 1: Let \((X,M,*)\) be a fuzzy metric space such that \(M(x,y,t)\to 1\) as \(t\to\infty\,\forall x,y\in X\), where \(*\) is a Hadzic type \(t\)-norm and let \(A,B,g: X\to X\) be three mappings such thatNEWLINENEWLINE (1) \(gX\) is closed,NEWLINENEWLINE (2) \(AX\subseteq gX\) and \(BX\subseteq gX\),NEWLINENEWLINE (3) \(M(Ax, By,kt)+ q(1-\max\{M(gx,By,kt)\), \(M(gy, Ax,kt)\})\geq M(gx,gy,t)\), where \(x,y\in X\), \(x\neq y\), \(t> 0\) and \(0< k< 1\).NEWLINENEWLINE Then the mapping \(A\), \(B\) and \(g\) have a coincidence point.NEWLINENEWLINE Theorem 2: Let \((X,M,*)\) be a complete fuzzy metric space such that \(M(x,y,t)\) is strictly increasing in the variable \(t\) and \(M(x,y,t)\to 1\) as \(t\to\infty\,\forall x,y\in x\), where \(*\) is a Hadzic type \(t\)-norm. Let \(A,b: X\to X\) be two self mappings on \(X\) such thatNEWLINENEWLINE (1) \(gX\) is closed,NEWLINENEWLINE (2) \(Ax\subseteq gx\),NEWLINENEWLINE (3) \(M(Ax,Ay,kt)+ q(1-\max\{M(gx, Ay,kt)\), \(M(gy,Ax,kt)\}\geq \psi(M(gx,Ax,t)\), \(M(gy, Ay,t))\,\forall x,\) \(y\in X\),NEWLINENEWLINE where \(q= q(x,y,t)\geq 0\), \(t> 0\), \(0< k< 1\) and \(\psi\) is a \(\Psi\)-function. Then \(A\) and \(g\) have a coincidence point. Also, if \((A,g)\) is a weakly compatible pair, then \(A\) and \(g\) have a unique common fixed point.