Fixed points for non-surjective expansion mappings
Let \((X,\tau)\) be a topological space and \(d:X \times X\to [0,\infty)\) such that \(d(x,y) =0\) if and only if \(x=y\). \(X\) is said to be \(d\)-complete [\textit{T. L. Hicks}, ibid. 15, No. 3, 435-439 (1992; Zbl 0773.54027)] if \(\sum^\infty_{n=1} d(x_n,x_{n+1}) <\infty\) implies that the sequence \(\{x_n\}\) is convergent. The mapping \(T:X\to X\) is \(w\)-continuous if \(x_n\to x\) implies \(Tx_n\to Tx\) as \(n\to \infty\). Let \(F,G\) be two self-mappings of a topological space \((X, \tau)\). These mappings are said to be semi-compatible [\textit{Y. J. Cho}, \textit{B. K. Sharma} and \textit{D. R. Sahu}, Math. Jap. 42, No. 1, 91-98 (1995; Zbl 0828.54032)] if the following conditions hold: \((D_1)\) \(p\in X\), \(Fp=Gp\) implies \(FGp=GFp\); \((D_2)\) the \(w\)-continuity of \(G\) at some point \(p\) in \(X\) implies \(\lim FGx_n=Gp\), whenever \(\{x_n\}\) is a sequence in \(X\) such that: \(\lim Fx_n= \lim Gx_n=p\), for some \(p\) in \(X\). Let \(\psi\) denote the family of all functions \(\psi: \mathbb{R}^3_+ \to\mathbb{R}_+\) satisfying the following properties: \(\psi\) is continuous; \(\varphi (1,1,1) =h>1\); \(a\geq \psi(b,b,a) =hb\), \(a\geq \psi (b,a,b) =hb\); \(a>0\) implies \(\psi (a,0, 0) >a\). If \(M\subset (X,\tau)\), where \((X,\tau)\) is a Hausdorff space, then the following holds: Theorem. Let \(A,B,S,T\) be self-maps of \(M\) such that: each of the pairs \((A,S)\) and \((B, T)\) are semi-compatible; \(A(M) \subset T(M)\), \(B(M) \subset S(M)\); \(S(M)\) is \(d\)-complete; for all \(x,y\in M,d(Sx,Ty) \geq\psi (d(Ax,By), d(Ax,Sx), d(By,Ty))\), where \(\psi \in\Psi\). Then \(A, B,S,T\) have a unique common fixed point in \(M\). This result improves and generalizes the main results from [\textit{M. A. Khan}, \textit{M. S. Khan} and \textit{S. Sessa}, Demonstr. Math. 19, 673-683 (1986; Zbl 0644.54023); \textit{V. Popa}, ibid. 23, No. 1, 213-218 (1990; Zbl 0731.54030); Fixed point theorems for expansion mappings, Prepr., ``Babeş-Bolyai'' Univ., Fac. Math., Res. Semin. 1987, No. 3, 25-30 (1987)] to \(d\)-complete topological spaces under semi-compatible conditions.
scientific article; zbMATH DE number 1140270