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A minimum condition and some related fixed-point theorems. (English) Zbl 0931.47042

Let \((X,d)\) be a metric space. Suppose that \({\mathcal F}\) is a set of self-maps of \(X\). Instead of requiring that a single operator \(T\) satisfies a restriction for certain pairs belonging to \(X\times X\), the authors require that, for each pair in a given collection of pairs, there exists an operator \(T\in{\mathcal F}\) (dependent on the pairs) satisfying that restriction. The authors consider the following conjectured generalization of the Banach contraction principle.
Generalized Banach contraction conjecture (GBCC): Let \((X,d)\) be a complete metric space, \(0<M<1\), and \(T\) a self-map of \(X\). Let \(J\) be a finite set of positive integers. Assume that \(T\) satisfies the following condition: \(\min\{d(T^k x, T^ky):k\in J\}\leq Md(x,y)\). Then \(T\) has a fixed point.
This conjecture has been proved when \(J= \{1\}\) (by Banach), and for the cases \(J= \{1,2\}\), \(J= \{1,3\}\) and \(J= \{2,3\}\) [J. R. Jachymski, B. Schröder and J. D. Stein jun., “A connection between fixed-point theorems and tiling problems”, to appear in J. Comb. Theory Ser. A (1999)]. The main results of this paper are the following theorems:
Theorem 1. Let \((X,d)\) be a complete metric space, \(0< M<1\), and let \(p\) be an integer. Let \(T\) be a self-map of \(X\). Assume that, given \(p\) pairs \((x_1,y_1),\dots,(x_p, y_p)\) of points from \(X\), for some \(k\), \(1\leq k\leq 2p\), we have \(d(T^k x_j, T^ky_j)\leq Md(x_j, y_j)\), for \(1\leq j\leq p\). Then \(T\) has a fixed point.
Theorem 2. Let \((X,d)\) be a complete metric space, and assume that \(T: X\to Y\) is uniformly continuous. Let \(u\) be an upper semicontinuous function defined on the nonnegative reals such that \(u(t)< t\) for all \(t>0\) and \(\liminf_{t\to\infty}(t-u(t))> 0\). Let \(N\) be a positive integer. Assume that for all \(x,y\in X\), \[ \min\{d(Tx, Ty), d(T^2x, T^2y),\dots, d(T^Nx, T^Ny)\}\leq u(d(x,y)).\tag{1} \] Then \(T\) has a unique fixed point.
Theorem 4. Let \((X,d)\) be complete, and assume that \(T: X\to X\). Let \(u\) satisfy conditions of Theorem 2 and assume that \(T\) satisfies condition (1). Then \(T\) has a contractive fixed point (a fixed point to which every sequence of iterates \(T^nx\) converges) if and only if \(T\) is asymptotically regular.
Theorem 5. Suppose that \(\alpha: X\to \mathbb{R}_+\) is such that, for each \(x\in X\) and \(k= 1,2\); either \[ d(x,T^kx)\leq \alpha(x)- \alpha(T^kx),\tag{2} \] or \[ d(Tx, T^{k+ 1}x)\leq \alpha(Tx)- \alpha(T^{k+ 1}x).\tag{3} \] Then \(T\) has a fixed point.
The observation that there is a relation between GBCC and certain tiling problems in combinatorics is made in the Jachymski-Schröder-Stein (paper cited above). This relation is exploited in proving GBCC in the cases \(J= \{1,2\}\), \(J= \{1,3\}\) and \(J= \{2,3\}\) replacing the tedious and highly computational process of constructing bounds for iterates with a set of rules for manipulation of titles.
In the third section of this paper, the authors show that other combinatorial arguments can also be useful in the fixed point theory.
Definition. Let \({\mathcal F}= \{T_1,\dots, T_n\}\) be a collection of self-maps of \(X\). We say that \({\mathcal F}\) has property \(\Sigma_p\) if, for every set of \(p\) pairs of points \((x_1,y_1),\dots, (x_p, y_p)\), there is a \(Q\in{\mathcal F}\) such that \[ \sum^\infty_{n=1} d(Q^n x_k, Q^ny_k)< \infty\quad\text{for }1\leq k\leq p. \] Theorem 6. Let \(q\) be a positive integer. Then there is a positive integer \(N_0(q)\) with the following property: if \(N> N_0(q)\) and \({\mathcal F}= \{T_1,\dots, T_N\}\) consists of continuous maps and has property \(\Sigma_{N-q}\), then some \(Q\in{\mathcal F}\) has a fixed point.
Theorem 7. Let \({\mathcal F}= \{T_1,\dots, T_N\}\) consist of continuous maps having property \(\Sigma_{N-1}\). Then some \(Q\in{\mathcal F}\) has a fixed point.
Reviewer: V.Popa (Bacau)

MSC:

47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
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