- Open Access
Remarks on Caristi’s fixed point theorem in metric spaces with a graph
© Alfuraidan; licensee Springer. 2014
- Received: 17 September 2014
- Accepted: 19 November 2014
- Published: 8 December 2014
We discuss some extensions of Caristi’s fixed point theorem for mappings defined on a metric space endowed with a graph. This work should be seen as a generalization of the classical Caristi’s fixed point theorem where the assumptions in Caristi’s theorem can, a priori, be weakened. It extends some recent works on Caristi’s fixed point theorem for mappings defined on metric spaces with a graph.
MSC:47H09, 46B20, 47H10, 47E10.
- acyclic digraph
- Banach contraction principle
- directed graph
- fixed point
- partially ordered metric spaces
This work was motivated by some recent work on Caristi’s fixed point theorem for mappings defined on metric spaces with a graph . It seems that the terminology of graph theory instead of partial ordering gives clearer pictures and yields generalized fixed point theorems. This paper is kind of a revisit of  in graph theory terminology.
for every . Recall that is called a fixed point of T if . This general fixed point theorem has found many applications in nonlinear analysis. It is shown, for example, that this theorem yields essentially all the known inwardness results  of geometric fixed point theory in Banach spaces. Recall that inwardness conditions are the ones which assert that, in some sense, points from the domain are mapped toward the domain. Possibly the weakest of the inwardness conditions, the Leray-Schauder boundary condition, is the assumption that a map points x of ∂M anywhere except to the outward part of the ray originating at some interior point of M and passing through x.
The proofs given to Caristi’s result vary and use different techniques (see [3, 6–8]). It is worth to mention that because of Caristi’s theorem close connection to Ekeland’s  variational principle, many authors refer to it as the Caristi-Ekeland fixed point theorem. For more on Ekeland’s variational principle and the equivalence between Caristi-Ekeland fixed point result and the completeness of metric spaces, the reader is advised to read .
In this work we present a characterization to the existence of minimal elements in partially ordered sets in terms of fixed point of multivalued maps, see . In fact, we state that the assumptions on the graph G needed to insure the existence of the minimal element among its vertices. Then we show how Caristi’s theorem in metric spaces with a graph may be generalized.
Let A be an abstract set partially ordered by ≺. We will say that is a minimal element of A if and only if implies . The concept of minimal element is crucial in the proofs given for Caristi’s fixed point theorem.
Theorem 2.1 
A contains a minimal element.
Any multivalued map T defined on A, such that for any there exists with , has a fixed point, i.e., there exists a in A such that .
Remark 2.1 Recall that Taskovic  showed that Zorn’s lemma is equivalent to the following:
for all and all . If each chain in A has an upper bound (resp. lower bound), then the family ℱ has a common fixed point.
Therefore, Theorem 2.1 is different from the remark as theorem  considers the existence of minimal elements, which in general does not imply that any chain has a lower bound.
Throughout this section we assume that is a metric space, and G is a directed graph (digraph) with a set of vertices and a set of edges contains all the loops, i.e., for any . We also assume that G has no parallel edges (arcs) and so we can identify G with the pair . Our graph theory notations and terminology are standard and can be found in all graph theory books, like [12, 13]. A digraph G is called an oriented graph if whenever , then .
Let be a partially ordered set. We define the oriented graph in X as follows: the vertices of are the elements of X, and two vertices are connected by a directed edge (arc) if . Therefore, has no parallel arcs as and .
A closed directed path of length from x to y, i.e., , is called a directed cycle. An acyclic digraph is a digraph that has no directed cycle.
Given an acyclic digraph G, we can always define a partial order on the set of vertices of G by whenever there is a directed path from x to y.
Using the definition of , one can easily show that is a Cauchy sequence and therefore converges to . Finally, it is straightforward to see that for all , which means that x is a lower bound for . In order to see that x is also a lower bound for , let be such that for all . Then we have for all , which implies . Since , we get , which implies . Therefore, for any , there exists such that , which implies , i.e., x is a lower bound of . Zorn’s lemma will therefore imply that has minimal elements.
Corollary 2.1 
(i.e., ) fixes a, i.e., .
One can now give the graph theory version of the above corollary as follows.
(i.e., ) fixes a, i.e., .
This corollary can be seen as a generalization of Caristi’s result endowed with a graph, see [, Theorem 3]. Indeed, the regular assumptions made in Caristi’s theorem imply that any chain (for ) has a lower bound, which is stronger than having a minimal element. Corollary 2.2 in fact contains implicitly a conclusion of the existence of a common fixed point graph. See [2, 15] for a similar conclusion in the partial order version.
for some positive function η, has a fixed point. In fact Kirk’s original question was stated when for some . In , the author gave a good example which answers Kirk’s problem in the negative where the order is used implicitly. We present a similar example but in terms of a graph.
where , for all . An easy computation shows that ϕ is lower semi-continuous on . Furthermore, one can show that T is nonexpansive, i.e., for all . And it is obvious that T has no fixed point.
Though the above example gives a negative answer to Kirk’s problem, some positive partial answers may also be found. Note that the order approach to Caristi’s traditional result is no longer possible. Indeed, if we define on the metric space the relation whenever , then ≺ is reflexive and anti-symmetric. But it is not in general transitive. Of course, if η is subadditive, i.e., for any , then ≺ is transitive. So one may wonder how to approach this general case when ≺ is not transitive and therefore is not a partial order set.
Definition 3.1 We say that a map is a well-behaved map if it is nondecreasing, continuous, and such that there exist and such that for any we have .
Notice that as η is continuous, there exists such that .
where and are functions.
Theorem 3.1 Let G be an acyclic oriented graph with containing all loops. Suppose that there exists a metric d in such that is a complete metric space. Let η be a well-behaved map and be a lower semi-continuous map, then G has a root .
which implies . Since is a minimal element in , we get . This completes the proof of Theorem 3.1. □
The next result is a positive partial answer to Kirk’s problem via a graph.
Theorem 3.2 Let G be an acyclic oriented graph with containing all loops. Suppose that there exists a distance d in such that is a complete metric space. Let be a G-edge preserving and a Caristi-Kirk G-mapping. Then T has a fixed point if and only if there exists with .
Proof Define the relation ≺ as in Theorem 3.1 on the vertex set . G has all the loops. In particular, if is a fixed point of T, i.e., , then we have . Assume that there exists such that . Clearly, we have . Theorem 3.1 implies the existence of a root of G and hence we must have . □
This is an amazing result because the relation ≺ is not a partial order. In particular, it is not . Also the minimal point is fixed by any map T that is G-edge preserving and a Caristi-Kirk G-mapping. So the fixed point is independent of the map T and only depends on the functions ϕ and η.
The field of mathematics plays a vital role in various fields. One of the important areas in mathematics is graph theory which is used in structural models. These structural arrangements of various objects or technologies lead to new inventions and modifications in the existing environment for enhancement in those fields. This section gives an idea of the implementation of our main results in computer science applications that uses graph theoretical concepts.
A graph model for fault tolerant computing systems
Graphical representation of algorithm
An algorithm will be defined in the form of a facility graph whose vertices represent the facilities required to execute the algorithm and whose edges represent the links required among these facilities. An algorithm A is executable by a computing system S if A is isomorphic to a subgraph of S. This means that there is a 1-1 mapping from the vertices of A into the vertices of S that preserves vertex labels and adjacencies between vertices. This implies that S contains all the facilities and connections between facilities required by A. So, A can be embedded in S.
A system S is fault tolerant with respect to algorithm A and fault F if A is executable by SF.
S is fault tolerant with respect to a set of algorithms and a set of faults if is executable by for all i and j, where .
If S is k-fault tolerant with respect to A, then S is j-fault tolerant with respect to A for all j, where .
Now let G be an acyclic oriented facility graph with containing all loops. Suppose that there exists a metric d in such that is a complete metric space. Let η be a well-behaved map and be a lower semi-continuous map, then by Theorem 3.1, G has a root , i.e., all system faults will be accessed from the root vertex (mother keyboard of the computer) and .
The author is grateful to King Fahd University of Petroleum and Minerals for supporting this research. The author would also like to thank Professor MA Khamsi who read carefully the earlier versions of this paper and suggested some improvements.
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