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Fixed point theorems for self and nonself Fcontractions in metric spaces endowed with a graph
Journal of the Egyptian Mathematical Society volume 28, Article number: 44 (2020)
Abstract
The main results obtained in this paper are fixed point theorems for self and nonself GFcontractions on metric spaces endowed with a graph. Our new results are generalization of recently fixed point theorems for self mappings on metric spaces and also fixed point theorems for nonself mappings in Banach spaces by using the concept of new type of contractive mappings namely Fcontractions.
Introduction
The wellknown Banach contraction theorem [1] has plenty of extensions in the literature (see, for example, [2, 3]). That theorem states that every selfmapping f defined on complete metric space (S,d) satisfying
where α∈(0,1) has a unique fixed point, i.e., there exists a unique r^{∗}∈S such that fr^{∗}=r^{∗}.
The extension of Banach contraction theorem for nonself multivalued mappings was first studied by Assad and Kirk [4] in 1972. After this initiation, lot of fixed point theorems for nonself mappings have been proved by various authors, see, for example, [5–7] and [8].
Firstly, the study of fixed point theorem for singlevalued monotone mappings in a metric space endowed with a partial ordering has been investigated by Ran and Reurings [9] and presented its applications to matrix equations. After this, many results in this direction were studied by different authors; see [10–12] and [13]. These theorems are actually hybrids of two fundamental theorems of fixed point theory: the KanasterTarski theorem [14] and the Banach Contraction Principle. Jachymski [15] established the fixed point theorems by using graphs which is the generalization of concept of partial ordering in metric spaces. Jachymski [15, Theorem 3.2] generalized the Banach contraction theorem for selfmappings on complete metric spaces endowed with the graph, where as Berinde [16, Theorem 3.1] for nonself mappings to Banach spaces endowed with a graph by using the inwardness condition defined in [17]. There are few other fixed point theorems for nonself mappings to Banach spaces endowed with a graph, see, for example, [18] and [19].
Recently, Wardowski [20] introduced a new type of contraction by using a particular function \(F:\mathbb {R}^{+}\rightarrow \mathbb {R}\) called Fcontraction and gave examples to show the validity of such extensions in complete metric spaces. The author proved a new fixed point theorem by using this concept of Fcontraction.
This paper has been organized in the following manner: In the “Preliminaries” section, we will give the brief introduction of a new type of contraction called F contraction. In the last section, we present a few preliminary notations and our main aim is to study the fixed point theorems for selfmappings as well as nonself mappings using Fcontractions for metric spaces endowed with a graph. These theorems are the generalization of fixed point theorems discussed by Berinde [16] on Banach spaces endowed with a graph and Wardowski [20] on complete metric spaces.
Preliminaries
In this section, we present some definitions, examples and results from [20], which will be used in this article. Throughout this paper, consider \(\mathbb {R}\) be the set of all real numbers, \(\mathbb {R}^{+}\) be the set of all positive real numbers and \( \mathbb {N} \) be the set of all positive integers.
Definition 1
Let the mapping \(F:\mathbb {R}^{+}\rightarrow \mathbb {R}\) satisfies the following conditions:

(f1)
F is strictly increasing;

(f2)
for each sequence \(\left \{ r_{n}\right \} \subset \mathbb {R}^{+} {\lim }_{n\rightarrow +\infty }r_{n}=0\) iff \({\lim }_{n\rightarrow +\infty }F\left (r_{n}\right) =\infty \);

(f3)
there exists k∈(0,1) provided that \({\lim }_{\lambda \rightarrow 0^{+}}\lambda ^{k}F\left (\lambda \right) =0.\)
The collection of all such mappings is denoted by Ω.
Definition 2
Let (S,d) be a metric space. A mapping Υ:S→S is said to be Fcontraction if there exist F∈Ω and τ>0 provided that
for all r,s∈S.
Example 1
Let F∈Ωbe defined by F(α)= lnα. For any k∈(0,1), it is clear that every mapping Υ:S→S satisfying (2) is an Fcontraction such that
Example 2
Consider F∈Ωbe defined by \(F\left (\alpha \right) =\frac {1} {\sqrt {\alpha }}, \alpha >0.\) In this case, for any k∈(1/2,1), every Fcontraction Υ satisfies
Wardowski stated the Fcontraction theorem for self mappings in complete metric spaces as follows.
Theorem 1
Let a mapping T:S→S be an Fcontraction and (S,d) be a complete metric space. Then, T has a unique fixed point s^{∗}∈S and for every s∈S the sequence \(\left (T^{n}s\right)_{n\in \mathbb {N} }\) converges to s^{∗}.
Remark 1
From (f1) and (2), we can conclude that every Fcontraction Υ is a contractive mappping, i.e.,
Thus, every Fcontraction is continuous mapping.
Fixed point theorems in metric spaces endowed with a graph
By using the concept of Fcontractions, we establish fixed point theorems for self as well as nonself mappings in complete metric spaces endowed with a graph.
Some graph theory terminologies will be presented here. Let (S,d) be metric space and △ denote the diagonal of Cartesian product S×S. Let G=(V(G),E(G)) be a directed graph such that E(G), the set of its edges consists of all loops, that is, △⊂E(G) and V(G), the vertex set coincides with S. Let G has no parallel edges (arcs). For more details of these terminologies and notations see [21] and [22].
G^{−1} is the converse graph of G, i.e., the edge set of G^{−1} is obtained by reversing the edges of G, defined as:
If s,r are vertices in the graph G, then a path from s to r of length t is a sequence \(\left \{ s_{i}\right \}_{i=1}^{t}\) of t+1 vertices of G such that s_{0}=s,s_{t}=r and (s_{i−1},s_{i})∈E(G),i=1,2⋯t.
A graph G is called connected if there exist at least a path between two arbitrary vertices. If \(\tilde {G}=\left (S,E\left (\tilde {G}\right) \right) \) is the symmetric graph obtained by placing together the vertices of both G and G^{−1}, that is,
then G is said to be weakly connected whenever \(\tilde {G}\) is connected.
If G=(V(G),E(G)) is a graph and V(G)⊃H, then the graph (H,E(G)) with
is said to be the subgraph of G determined by H, denoted by G_{H}.
Self Fcontraction case
A mapping Υ:S→S is said to be defined on a metric space endowed with a graph G if it satisfies
A mapping Υ:S→S defined on metric space endowed with a graph G, is said to be a GFcontraction, if there is a constant τ>0 such that ∀r,s∈S with (r,s)∈E(G), we have
If Υr=r, then the element r∈S is said to be the fixed point of mapping Υ.
Theorem 2
Suppose (S,d,G) be a complete metric space endowed with a weakly connected and directed graph G such that the following property (T) holds, that is, for any sequence \(\left \{ r_{n}\right \} _{n=1}^{\infty }\subset S\) with r_{n}→r as n→∞ and (r_{n},r_{n+1})∈E(G) for all \(n\in \mathbb {N},\) there exists a subsequence \(\left \{ r_{s_{n}}\right \}_{n=1}^{\infty }\) satisfying
Let Υ:S→S be a GFcontraction. If the set
is nonempty, then the mapping Υ has a unique fixed point in S.
Proof
Let r_{0}∈S_{Υ}. It follows from (7) that (r_{0},Υr_{0})∈E(G) and by using (4), we obtain
Denote r_{n}:=Υ^{n}r_{0} for all \(n\in \mathbb {N}.\) Then, by the fact that Υ is a GFcontraction and in view of (4), we get
for all \(n\in \mathbb {N}.\) Denote α_{n}=d(r_{n},r_{n+1}),n=0,1,…
Let r_{n+1}≠r_{n}, for every \(n\in \mathbb {N} \cup \left \{ 0\right \}.\) Then, α_{n}>0 for all \(n\in \mathbb {N} \cup \left \{ 0\right \} \) and by using (2), we get
Hence, \({\lim }_{n\rightarrow \infty }F\left (\alpha _{n}\right) =\infty.\) By the property (f2), we obtain that α_{n}→0 as n→∞. From (f3), there exists k∈(0,1) such that \({\lim }_{n\rightarrow \infty }\alpha _{n}^{k}F\left (\alpha _{n}\right) =0.\) By (10), the following holds for all \(n\in \mathbb {N} \)
Letting n→∞ in (11), we deduce \({\lim }_{n\rightarrow \infty }n\alpha _{n}^{k}=0\). From (11), we observe that there exists \(n^{\prime }\in \mathbb {N} \) such that \(n\alpha _{n}^{k}\leq 1\) for all n≥n^{′}. Consequently, we have
Choose \(m,n\in \mathbb {N} \) such that m≥n≥n^{′} and from (12), we have
The convergence of the series \({\sum \nolimits }_{j=n}^{\infty }\frac {1}{j^{1/k}}\) implies that {r_{n}} is a Cauchy sequence, hence convergent in (S,d,G). The limit of this sequence is denoted as:
By using property (T) of (S,d,G), there exists a subsequence \(\left \{ r_{s_{n}}\right \} \) satisfying
Hence, by inequality (5) and in view of (4), we get
which implies
Therefore, by triangle inequality, we have
By using (15), inequality (16) yields
for all n≥1. In Eq. (17), assuming n→∞ and using (13), we have d(r^{∗},Υr^{∗})=0, which implies r^{∗}=Υr^{∗}, i.e., r^{∗} a fixed point of mapping Υ.
Note that the uniqueness of r^{∗} follows by the GFcontraction condition (5).
Remark 2If we use the mapping F∈Ω defined by the formula F(α)= lnα in Theorem 2, then for all k∈(0,1), we obtain the extension of [16], Theorem 2.1. □
Example 3Let (S,d) be the complete metric space and G be the complete graph on the set S, that is, E(G)=S×S. Let the mapping F∈Ω be defined as: F(α)= lnα, then the GFcontraction (5) is actually a Fcontraction (2) which reduces to Banach contraction, i.e.,
for any k∈(0,1) and τ>0.
Nonself Fcontraction case
Let S be a Banach space, A be a nonempty, closed subset of S and Υ:A→S be a nonself mapping. We choose r∈A such that Υr∉A, then there is an element s∈∂A such that
which represents the fact that
where d(r,s)=∥r−s∥.
Caristi [17] used a condition related to (18), called inward condition, to get the generalization of Banach contraction theorem for nonself mappings. The inward condition is more general because it does not need s in (18) to belong to ∂A.
A nonself mapping Υ:A→S is said to be defined on the Banach space S endowed with a graph G, if it satisfies the property that
for the subgraph of G induced by A.
Theorem 3
Suppose (S,d,G) be a Banach space endowed with a weakly connected and directed graph G provided that following property (T) holds, that is, for any sequence {r_{n}}⊂S along with r_{n}→r as n→∞ and
there exists a subsequence \(\left \{ r_{s_{n}}\right \} \) satisfying
Let A be a nonempty, closed subset of S and Υ:A→S be a G_{A}Fcontraction, that is, there exists a constant τ>0 such that
where G_{A} is the subgraph of G determined by A. If the set
is nonempty and Υ satisfies Rothe ^{’}s boundary condition
then the mapping Υ has a unique fixed point.
Proof
If Υ(A)⊂A, then Υ is a selfmap of the closed set A and the conclusion follows by Theorem 2. Now, we consider the case that Υ(A)⊄A. Let r_{0}∈A_{Υ}. It follows that (r_{0},Υr_{0})∈E(G) and in view of equation (4), we have
Let we denote r_{n}:=Υ^{n}r_{0}, for all \(n\in \mathbb {N}.\) By virtue of (22) Υr_{0}∈A.
Consider r_{1}≡s_{1}=Υr_{0}. Let Υr_{1}∈A, set r_{2}≡s_{2}=Υr_{1}. If Υr_{1}∉A, then we can select an element r_{2}∈∂A on the segment [r_{1},Υr_{1}], that is,
By following the same method, we obtain two sequences {r_{n}} and {s_{n}} whose terms satisfy one of the succeeding properties:

(i)
r_{n}≡s_{n}=Υr_{n−1}, if Υr_{n−1}∈A;

(ii)
r_{n}=(1−μ)r_{n−1}+μΥr_{n−1}∈∂A,μ∈(0,1),Υr_{n−1}∉A.
For the simplicity of arguments in the proof, let us denote
and
Note that {r_{n}}⊂A for all \(n\in \mathbb {N}.\) Moreover, if r_{a}∈Z, then both r_{a−1} and r_{a+1} belong to set U. The sequence {r_{n}} can have consecutive terms r_{a} and r_{a+1} in set U, but this assertion is not true for the set Z. First of all we have to prove that
Suppose contrary that r_{a−1}≠Υr_{a−2} then r_{a−1}∈∂A. Since Υ(∂A)⊂A then Υr_{a−1}∈A. Hence, r_{a}=Υr_{a−1} which is a contradiction.
Here, we have three different cases to show that {r_{n}} is Cauchy sequence which are following: □
Case 1. r_{n},r_{n+1}∈U.
Since both elements belong to set U, therefore, we have r_{n}=s_{n}=Υr_{n−1} and r_{n+1}=s_{n+1}=Υr_{n}. Hence,
where (s_{n},s_{n−1})∈E(G) by virtue of (23). Therefore, we have
Consequently, we get the following inequality
by using (21).
Case 2. r_{n}∈U,r_{n+1}∈Z.
In this case, we have r_{n}=s_{n}=Υr_{n−1}, but r_{n+1}≠s_{n+1}=Υr_{n}; therefore, we have
The above equality implies d(r_{n+1},Υr_{n})≠0 and hence
since r_{n}∈U. By using (25), we obtain
We can obtain again inequality (24) by using the similar arguments to that in case 1.
Case 3. r_{n}∈Z,r_{n+1}∈U.
In this case, we have r_{n+1}=Υr_{n}, and r_{n}≠s_{n}=Υr_{n−1}. Since r_{n}∈Z, so we have
Hence, by triangle inequality
By virtue of (23) (s_{n−1},s_{n})∈E(G), and the following inequality is obtained by the contraction condition (21)
which implies
Thus, by using (26) and (29) in inequality (27), we have
By using (23), (r_{n−2},r_{n−1})=(s_{n−2},s_{n−1})∈E(G) and by virtue of contraction condition (21), we get
Now, we summarize all the above mentioned three cases. By virtue of (24) and (30), it follows that the sequence {d(r_{n},r_{n+1})} satisfies the inequality
for all n≥2. Denote α_{n}=d(r_{n},r_{n+1}) for n=2,3,⋯.
We obtain the following inequality by simple induction for n≥2, and using (31)
where \(\left [ \frac {n}{2}\right ] \) denotes the greatest integer not exceeding \(\frac {n}{2}.\)
Hence, \({\lim }_{n\rightarrow \infty }F\left (\alpha _{n}\right) =\infty.\) By the property (f2), we obtain that α_{n}→0 as n→∞. From (f3), there exists k∈(0,1) such that \({\lim }_{n\rightarrow \infty }\alpha _{n}^{k}F\left (\alpha _{n}\right) =0.\) Denote γ= max{α_{0},α_{1}}. By (32), the following holds for all n≥2:
Assuming n→∞ in (33), we deduce \({\lim }_{n\rightarrow \infty }\left [ \frac {n}{2}\right ] \alpha _{n}^{k}=0\). From (33), we observe that there exists \(n^{\prime }\in \mathbb {N} \) such that \(\left [ \frac {n}{2}\right ] \alpha _{n}^{k}< n\alpha _{n}^{k}\leq 1 \) for all n≥n^{′}. Consequently, we have
Choose \(m,n\in \mathbb {N} \) such that m≥n≥n^{′} and from (34), we have
The convergence of the series \({\sum \nolimits }_{j=n}^{\infty }\frac {1}{j^{1/k}}\) implies that {r_{n}} is a Cauchy sequence, hence convergent in (S,d,G). Since {r_{n}}⊂A and A is closed, {r_{n}} converges to some point \(r^{^{\prime }}\in A,\) i.e., \({\lim }_{n\rightarrow \infty }r_{n}=r^{^{\prime }}.\)
By property (T), there exists a subsequence \(\left \{ r_{s_{n}}\right \} \) satisfying
Hence, by the Fcontraction condition (21), we get
Therefore, by triangle inequality, we have
By using (35), the above inequality yields
for all n≥1. Taking limit n→∞ and using (36), we obtain \(d\left (r^{^{\prime }},\Upsilon r^{^{\prime }}\right) =0\) and get \(r^{^{\prime }}=\Upsilon r^{^{\prime }}\), which shows that \(r^{^{\prime }}\) is a fixed point of Υ.
The uniqueness of r^{∗} immediately follows by the G_{A}F contraction condition (21).
Remark 3
If we use the mapping F∈Ω defined by the formula F(α)= lnα in Theorem 3, then for all k∈(0,1), we obtain the extension of [16, Theorem 3.1].
Example 4
Let \(S=\mathbb {R}\) be a Banach space with the usual norm and A=(−∞,0] is a closed subset of S. Let the mapping Υ:A→S be defined as:
Let the mapping F∈Ω be given by the formula \(F\left (\alpha \right) =\frac {1}{\sqrt {\alpha }}\), and the edge set of graph G and the subgraph G_{A} determined by A is defined as:
and
respectively. It is easy to check that (19) holds, that is,
In view of (19), for t,u∈(−∞,−1) and r,s∈[−1,0], the edges (t,u),(t,r) has to be removed and for the rest of edges we have
Moreover, G is a weakly connected and for any k∈(0.5,1),Υ is a nonself G_{A}Fcontraction on A with \(\tau =\dfrac {1}{\sqrt {d\left (r,s\right) }},\) since
(for the rest of edges of E(G_{A}), the Fcontraction condition (21) is obvious, since the quantity in its left hand side is always zero). Property (T) holds with constant sequences {r_{n}=r} satisfying the property (r_{n},r_{n+1})∈E(G_{A}), for all \(n\in \mathbb {N}.\) Rothe’s boundary condition is also satisfied, as ∂A={0} and so Υ(∂A)⊂A. Finally, since we have A_{Υ}={0}≠∅, all assumptions in Theorem 3 are satisfied and \(r^{^{\prime }}=0\) is the fixed point of Υ.
Conclusion
In this paper, we have presented the fixed point theorems for self and nonself G, Fcontractions on metric spaces endowed with a graph. These theorems immediately imply the extension of recently fixed point theorems for selfmappings on metric spaces and fixed point theorems for nonself mappings in Banach spaces.
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The authors thank the anonymous reviewers for their careful reading of this paper and their many insightful comments and suggestions.
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Younus, A., Azam, M.U. & Asif, M. Fixed point theorems for self and nonself Fcontractions in metric spaces endowed with a graph. J Egypt Math Soc 28, 44 (2020). https://doi.org/10.1186/s42787020001009
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DOI: https://doi.org/10.1186/s42787020001009
Keywords
 Fixed point
 Fcontraction
 Nonself G Fcontraction
 Directed graph
 G Fcontraction
 Nonself mapping
 Rothe’s boundary condition
AMS Subject Classifications
 47H10
 54H25
 47J26
 54E50
 05C40