Abstract
We establish two results regarding the existence of fixed points for strict contractions on complete metric spaces with graphs.
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1 Introduction
For more than 60 years now, there has been a lot of research activity regarding the fixed point theory of nonexpansive (that is, 1-Lipschitz) and contractive mappings. See, for instance, [2, 4, 7, 8, 11,12,13,14,15,16,17,18,, 19, 21, 24, 25] and references cited therein. This activity stems from Banach’s classical theorem [1] concerning the existence of a unique fixed point for a strict contraction on a complete metric space. It also concerns the convergence of (inexact) iterates of a nonexpansive mapping to one of its fixed points. Since that seminal result, many developments have taken place in this field including, in particular, studies of feasibility, common fixed point problems, nonlinear operator theory and variational inequalities, which find important applications in engineering, medical and the natural sciences [3, 5, 6, 22,23,24,, 25]. In particular, the study of nonexpansive and contractive mappings on complete metric spaces with graphs has recently become a rapidly growing area of research. See, for example, [9, 10, 20]. In the present paper we establish two results regarding the existence of fixed points for strict contractions on complete metric spaces with graphs.
2 Preliminaries and the first main result
Let \((X,\rho )\) be a complete metric space and let G be a (directed) graph on X. Let V(G) be the set of its vertices and let E(G) be the set of its edges. We identify the graph G with the pair (V(G), E(G)).
Denote by \({{\mathcal {M}}}\) the set of all mappings \(T: X \rightarrow X\) such that for each \(x,y \in X\) satisfying \((x,y) \in E(G)\), we have
A mapping \(T \in {{\mathcal {M}}}\) is called G-nonexpansive. If \(T \in {{\mathcal {M}}}\), \(\alpha \in (0,1)\) and for each \(x,y \in X\) satisfying \((x,y) \in E(G)\), we have
then T is said to be a G-strict contraction.
Fix \(\theta \in X\). For each \(x \in X\) and each \(r>0\), set
In the sequel we assume that any sum over the empty set is zero and that the infimum of the empty set is \(\infty \). We also assume that \(a+\infty =\infty \) for each \(a \in R^1\). Finally, we assume that if \(x,y \in X\) satisfy \((x,y) \in E(G)\), then
For each \(x,y\in X\), define
It is not difficult to see that for each \(x,y,z \in X\), we have
and if \(\rho _1(x,y)=0\), then \(x=y\). Note that it is may happen that \(\rho _1(x,y)=\infty \) for some \(x,y \in X\). The pseudometric \(\rho _1\) plays an important role in this paper because it turns out that if a mapping T is a G-strict contraction, then it is a strict contraction with respect to the pseudometric \(\rho _1\).
Let \(T \in {{\mathcal {M}}}\) be a G-strict contraction. It is known [10] that under certain mild assumptions, the mapping T has a unique fixed point which attracts all the iterates of T. In [20] we provided a very simple proof of this fact by using the pseudometric \(\rho _1\) under the assumption that it is always finite-valued. Then we established uniform convergence of the iterates of T on bounded subsets of X and showed that this convergence is stable in the presence of small perturbations of these iterates. We also showed there that under certain assumptions, a typical G-nonexpansive mapping has a unique fixed point which attracts all its iterates, uniformly on bounded subsets of X. In the present paper we prove the existence result of [20] under a weakened assumption. We do not assume that the pseudometric \(\rho _1\) is always finite-valued. Instead, we assume that there exists a point \(x \in X\) such that \(\rho _1(x,T(x))\) is finite. This is our first main result. It is proved in this section. In our second result we show the existence of a fixed point for a mapping defined on a closed ball in X which takes values in X.
Given a mapping \(S: X \rightarrow X\), we define \(S^0=I\), the identity self-mapping on X, \(S^1=S\), and \(S^{i+1}=S\circ S^i\) for all integers \(i \ge 0\).
Proposition 2.1
Let \(T \in {{\mathcal {M}}}\), \(\alpha \in [0,1)\) and assume that for each \(x,y \in X\) satisfying \((x,y) \in E(G)\), the inequality
holds. Then for each \(x,y \in X\), we have
Proof
Let \(x,y \in X\). We may assume without any loss of generality that \(\rho _1(x,y)<\infty \). Let \(\epsilon >0\) be given. By (2.3), there exist an integer \(q\ge 1\) and points \(x_i \in X, \; i=0,\dots ,q\), such that
By (2.1) and (2.5)–(2.7), we have
and
When combined with (2.3) and (2.8), these relations imply that
Since \(\epsilon \) is any positive number, we conclude that
as asserted. This completes the proof of Proposition 2.1. \(\square \)
Theorem 2.2
Let \(T \in {{\mathcal {M}}}\), \(\alpha \in [0,1)\) and assume that for each \(z,y \in X\) satisfying \((z,y) \in E(G)\),
\(x \in X\) and \(\rho _1(x,T(x))<\infty \). Then for each integer \(k \ge 0\),
there exists
in \((X,\rho )\) and
Moreover, if T is continuous at \(x_*\) as a self-mapping of \((X,\rho )\), then \(T(x_*)=x_*\).
Proof
Proposition 2.1 and (2.1), (2.9) imply that for each integer \(i \ge 0\),
and
This implies that for each integer \(k \ge 0\),
In view of (2.10), \(\{T^i(x)\}_{i=0}^{\infty }\) is a Cauchy sequence in \((X,\rho )\). Therefore there exists
in \((X,\rho )\) and for each integer \(k \ge 0\),
as asserted. This completes the proof of Theorem 2.2. \(\square \)
The following proposition easily implies the well-posedness of our fixed point problem.
Proposition 2.3
Let \(T \in {{\mathcal {M}}}\), \(\alpha \in [0,1)\) and assume that (2.9) holds for each \((z,y) \in E(G)\). Let \(x,y \in X\),
Then
Proof
Proposition 2.1 and (2.1) imply that
and
This completes the proof of Proposition 2.3. \(\square \)
3 The second main result
Assume that \(x \in X\), \(r >0\), \(\alpha \in (0,1)\) and that for each \((z,y) \in E(G) \cap (B(x,r) \times B(x,r))\), \(T: B(x,r) \rightarrow X\) satisfies
Theorem 3.1
Assume that
Then
for each integer \(k \ge 1\),
and there exists
in \((X,\rho )\). Moreover, if T is continuous at \(x_*\) with respect to \(\rho \), then \(T(x_*)=x_*\).
Proof
By (2.4), we have
Assume that \(k \ge 1\) is an integer,
and that for each \(i \in \{0,\dots ,k-1\}\),
(In view of (3.3), our assumption does hold for \(k=1\).) By (3.5), we have
Let
It follows from (2.3) that there exist an integer \(q\ge 1\) and points \(x_i \in X, \; i=0,\dots ,q\), such that
There are two cases: \(k=1\) and \(k>1\). Assume first that \(k=1\). In view of (3.7), (3.8) and (3.10),
and for each \(i=0,\dots ,q\),
and
By (3.8) and (3.11), for each \(i=0,\dots ,q-1\), we have
and
It follows from (3.8), (3.10) and the above relations that
Since \(\epsilon \) is any positive number, we conclude that
by (3.2). Thus
and our assumption holds for \(k+1=2\).
Assume now that \(k>1\). Proposition 2.1 and (3.5) imply that
By (3.8), (3.10) and (3.12), for \(i=1,\dots ,q\), we have
and
Thus
By (3.1), (3.9) and (3.13), for \(i=0,\dots ,q-1\), we have
and
When combined with (3.8) and (3.10), the above relations imply that
Since \(\epsilon \) is any positive number, using (3.6), we conclude that
and
Thus our assumption holds for \(k+1\) too and by mathematical induction it is now seen to hold for each integer \(k \ge 1\). This implies that
and so \(\{T^i(x)\}_{i=0}^{\infty }\) is a Cauchy sequence in \((X,\rho )\). Therefore there exists
in \((X,\rho )\).
Let \(k \ge 1\) be an integer. Then
as \(k \rightarrow \infty \). Clearly, if T is continuous at \(x_*\), then \(x_*=T(x_*)\). This completes the proof of Theorem 3.1. \(\square \)
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Acknowledgements
Simeon Reich was partially supported by the Israel Science Foundation (Grant No. 820/17), by the Fund for the Promotion of Research at the Technion (Grant No. 2001893) and by the Technion General Research Fund (Grant No. 2016723). Both authors are grateful to the Editor-in-Chief and to two anonymous referees for their helpful comments and suggestions.
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Reich, S., Zaslavski, A.J. Two existence results regarding strict contractions on metric spaces with graphs. J Anal (2024). https://doi.org/10.1007/s41478-024-00828-y
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DOI: https://doi.org/10.1007/s41478-024-00828-y