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Fixed point theorems for self and non-self F-contractions 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 non-self GF-contractions 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 non-self mappings in Banach spaces by using the concept of new type of contractive mappings namely F-contractions.
Introduction
The well-known Banach contraction theorem [1] has plenty of extensions in the literature (see, for example, [2, 3]). That theorem states that every self-mapping 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 non-self multi-valued mappings was first studied by Assad and Kirk [4] in 1972. After this initiation, lot of fixed point theorems for non-self mappings have been proved by various authors, see, for example, [5–7] and [8].
Firstly, the study of fixed point theorem for single-valued 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 Kanaster-Tarski 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 self-mappings on complete metric spaces endowed with the graph, where as Berinde [16, Theorem 3.1] for non-self mappings to Banach spaces endowed with a graph by using the inwardness condition defined in [17]. There are few other fixed point theorems for non-self 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 F-contraction 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 F-contraction.
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 self-mappings as well as non-self mappings using F-contractions 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 F-contraction 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 F-contraction 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 F-contraction Υ satisfies
Wardowski stated the F-contraction theorem for self mappings in complete metric spaces as follows.
Theorem 1
Let a mapping T:S→S be an F-contraction 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 F-contraction Υ is a contractive mappping, i.e.,
Thus, every F-contraction is continuous mapping.
Fixed point theorems in metric spaces endowed with a graph
By using the concept of F-contractions, we establish fixed point theorems for self as well as non-self 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 s0=s,st=r and (si−1,si)∈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 GH.
Self F-contraction 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 GF-contraction, 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 rn→r as n→∞ and (rn,rn+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 GF-contraction. If the set
is nonempty, then the mapping Υ has a unique fixed point in S.
Proof
Let r0∈SΥ. It follows from (7) that (r0,Υr0)∈E(G) and by using (4), we obtain
Denote rn:=Υnr0 for all \(n\in \mathbb {N}.\) Then, by the fact that Υ is a GF-contraction and in view of (4), we get
for all \(n\in \mathbb {N}.\) Denote αn=d(rn,rn+1),n=0,1,…
Let rn+1≠rn, for every \(n\in \mathbb {N} \cup \left \{ 0\right \}.\) Then, αn>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 αn→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 {rn} 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 GF-contraction 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 GF-contraction (5) is actually a F-contraction (2) which reduces to Banach contraction, i.e.,
for any k∈(0,1) and τ>0.
Non-self F-contraction case
Let S be a Banach space, A be a nonempty, closed subset of S and Υ:A→S be a non-self 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 non-self mappings. The inward condition is more general because it does not need s in (18) to belong to ∂A.
A non-self 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 {rn}⊂S along with rn→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 GAF-contraction, that is, there exists a constant τ>0 such that
where GA 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 self-map of the closed set A and the conclusion follows by Theorem 2. Now, we consider the case that Υ(A)⊄A. Let r0∈AΥ. It follows that (r0,Υr0)∈E(G) and in view of equation (4), we have
Let we denote rn:=Υnr0, for all \(n\in \mathbb {N}.\) By virtue of (22) Υr0∈A.
Consider r1≡s1=Υr0. Let Υr1∈A, set r2≡s2=Υr1. If Υr1∉A, then we can select an element r2∈∂A on the segment [r1,Υr1], that is,
By following the same method, we obtain two sequences {rn} and {sn} whose terms satisfy one of the succeeding properties:
-
(i)
rn≡sn=Υrn−1, if Υrn−1∈A;
-
(ii)
rn=(1−μ)rn−1+μΥrn−1∈∂A,μ∈(0,1),Υrn−1∉A.
For the simplicity of arguments in the proof, let us denote
and
Note that {rn}⊂A for all \(n\in \mathbb {N}.\) Moreover, if ra∈Z, then both ra−1 and ra+1 belong to set U. The sequence {rn} can have consecutive terms ra and ra+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 ra−1≠Υra−2 then ra−1∈∂A. Since Υ(∂A)⊂A then Υra−1∈A. Hence, ra=Υra−1 which is a contradiction.
Here, we have three different cases to show that {rn} is Cauchy sequence which are following: □
Case 1. rn,rn+1∈U.
Since both elements belong to set U, therefore, we have rn=sn=Υrn−1 and rn+1=sn+1=Υrn. Hence,
where (sn,sn−1)∈E(G) by virtue of (23). Therefore, we have
Consequently, we get the following inequality
by using (21).
Case 2. rn∈U,rn+1∈Z.
In this case, we have rn=sn=Υrn−1, but rn+1≠sn+1=Υrn; therefore, we have
The above equality implies d(rn+1,Υrn)≠0 and hence
since rn∈U. By using (25), we obtain
We can obtain again inequality (24) by using the similar arguments to that in case 1.
Case 3. rn∈Z,rn+1∈U.
In this case, we have rn+1=Υrn, and rn≠sn=Υrn−1. Since rn∈Z, so we have
Hence, by triangle inequality
By virtue of (23) (sn−1,sn)∈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), (rn−2,rn−1)=(sn−2,sn−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(rn,rn+1)} satisfies the inequality
for all n≥2. Denote αn=d(rn,rn+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 αn→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{α0,α1}. 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 {rn} is a Cauchy sequence, hence convergent in (S,d,G). Since {rn}⊂A and A is closed, {rn} 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 F-contraction 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 GAF -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 GA 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 non-self GAF-contraction on A with \(\tau =\dfrac {1}{\sqrt {d\left (r,s\right) }},\) since
(for the rest of edges of E(GA), the F-contraction condition (21) is obvious, since the quantity in its left hand side is always zero). Property (T) holds with constant sequences {rn=r} satisfying the property (rn,rn+1)∈E(GA), 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 AΥ={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 non-self G, F-contractions on metric spaces endowed with a graph. These theorems immediately imply the extension of recently fixed point theorems for self-mappings on metric spaces and fixed point theorems for non-self 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 non-self F-contractions in metric spaces endowed with a graph. J Egypt Math Soc 28, 44 (2020). https://doi.org/10.1186/s42787-020-00100-9
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DOI: https://doi.org/10.1186/s42787-020-00100-9
Keywords
- Fixed point
- F-contraction
- Non-self G F-contraction
- Directed graph
- G F-contraction
- Non-self mapping
- Rothe’s boundary condition