Let \((X,d)\) be a metric space, \(f\) a continuous mapping of \(X\) into \(X\), and \(T\) a mapping of \(X\) into \(2^X-\{\emptyset\}\). A subset \(K\) of \(X\) is called proximal if, for each \(x\in K\), there exists an element \(k\in K\) such that \(d(x,k)=d(x,K)\) where \(d(x,K)=\inf\{d(x,y):y\in K\}\). The family of all bounded proximal subsets of \(X\) is denoted by \(P(X)\). A map \(\phi:X\times X\to [0,\infty)\) is called compactly positive if \(\inf\{\phi(x,y): a\leq d(x,y)\leq b\}>0\) for each finite interval \([a,b]\subset(0,\infty)\). \(T:X\to F(X)\) is called \(f\)-contractive if \(H(Tx,Ty) < d(fx,fy)\) whenever \(fx\neq fy\), where \(H\) denotes the Hausdorff metric and \(F(X)\) denotes the family of all nonempty closed subsets of \(X\). An \(f\)-orbit of \(x\in X\) under \(T\) is a sequence \(\{x_n:x_n\in X\) and \(f(x_n)\in Tx_{n-1},x_0=x\}\). An \(f\)-orbit of \(x\) under \(T\) is called strongly regular if \(T:X\to P(X)\) and for each \(n\in\mathbb{N}\), \(d(fx_{n+1},fx_n)=d(fx_n,Tx_n)\). For each \(x\in X\), we denote the metric projection of \(X\) onto \(A\) by \(P_A(x)\), namely \(P_A(x)=\{a\in A:d(x,a)=d(x,A)\}\).
The following two theorems generalize results from [the authors, ibid. 8, 233-241 (1994; Zbl 0813.54013)] and from [J. Dugundji and A. Granas, Bull. Greek Math. Soc. 19, 141-151 (1978; Zbl 0417.54010)]:
Theorem 2.5. Let \((X,d)\) be a metric space and \(T:X\to P(X)\) a continuous \(f\)-contractive map with \(T(X)\subset f(X)\). Let \(f\) be continuous and one-to-one with \(f^{-1}\) continuous on \(f(X)\). If for some \(x\in X\), a strongly regular \(f\)-orbit of \(x\) under \(T\) has a cluster point \(x^*\), and if \(P_{Tx^*}(fx^*)\) is a compact set, then \(x^*\) is a coincidence point of \(f\) and \(T\).
Theorem 3.2. Let \((X,d)\) be a complete metric space and \(T:X\to P(X)\) be continuous. Suppose that \(T\) satisfies \(H(P_{Tx}(x), P_{Ty}(y))\leq d(x,y)-g(x,y)\), for all \(x,y\in X\), where \(g\) is compactly positive. Then \(T\) has a unique fixed point.