Fixed point theorems for self and non-self F-contractions in metric spaces endowed with a graph

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

$$ d\left(fr,fs\right) \leq\alpha d\left(r,s\right) \text{ }\forall\text{ }r,s\in S, $$

(1)

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

$$ d\left(\Upsilon s,\Upsilon r\right) >0\Longrightarrow\tau+F\left(d\left(\Upsilon s,\Upsilon r\right) \right) \leq F\left(d\left(s,r\right) \right), $$

(2)

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

$$d\left(\Upsilon s,\Upsilon r\right) \leq e^{-\tau}d\left(r,s\right) \text{ }\forall r,s\in S,\text{ }\Upsilon r\neq\Upsilon s. $$

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

$$d\left(\Upsilon r,\Upsilon s\right) \leq\frac{1}{\left(1+\tau \sqrt{d\left(r,s\right) }\right)^{2}}d\left(r,s\right) \text{ } \forall\text{ }r,s\in S,\text{ }\Upsilon r\neq\Upsilon s. $$

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.,

$$ d\left(\Upsilon r,\Upsilon s\right) \leq d\left(r,s\right) { for\, all }\, r,s\in S\text{ and }\Upsilon r\neq\Upsilon s. $$

(3)

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:

$$E\left(G^{-1}\right) =\left\{ \left(r,s\right) \in S\times S:\left(s,r\right) \in E\left(G\right) \right\}. $$

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,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,

$$E\left(\tilde{G}\right) =E\left(G\right) \cup E\left(G^{-1}\right), $$

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

$$E\left(H\right) =E\left(G\right) \cap\left(H\times H\right) $$

is said to be the subgraph of G determined by H, denoted by G_H.

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

$$ \forall\text{ }r,s\in S,\text{ }\left(r,s\right) \in E\left(G\right) \text{ implies }\left(\Upsilon r,\Upsilon s\right) \in E\left(G\right). $$

(4)

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

$$ \left[ d\left(\Upsilon r,\Upsilon s\right) >0\Longrightarrow\tau+F\left(d\left(\Upsilon r,\Upsilon s\right) \right) \leq F\left(d\left(r,s\right) \right) \right]. $$

(5)

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

$$ \left(r_{s_{n}},r\right) \in E\left(G\right),\text{ }\forall\text{ }n\in\mathbb{N}. $$

(6)

Let Υ:S→S be a GF-contraction. If the set

$$ S_{\Upsilon}=\left\{ r\in S:\left(r,\Upsilon r\right) \in E\left(G\right) \right\} $$

(7)

is nonempty, then the mapping Υ has a unique fixed point in S.

Proof

Let r₀∈S_Υ. It follows from (7) that (r₀,Υr₀)∈E(G) and by using (4), we obtain

$$ \left(\Upsilon^{n}r_{0},\Upsilon^{n+1}r_{0}\right) \in E\left(G\right),\text{ }\forall\text{ }n\in \mathbb{N}. $$

(8)

Denote r_n:=Υⁿr₀ for all $n\in \mathbb {N}.$ Then, by the fact that Υ is a GF-contraction and in view of (4), we get

$$ F\left(d\left(r_{n},r_{n+1}\right) \right) \leq F\left(d\left(r_{n-1},r_{n}\right) \right) -\tau, $$

(9)

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

$$ F\left(\alpha_{n}\right) \leq F\left(\alpha_{n-1}\right) -\tau\leq F\left(\alpha_{n-2}\right) -2\tau\leq\cdots\leq F\left(\alpha_{0}\right) -n\tau. $$

(10)

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 $n\in \mathbb {N}$

$$ \alpha_{n}^{k}F\left(\alpha_{n}\right) -\alpha_{n}^{k}F\left(\alpha_{0}\right) \leq\alpha_{n}^{k}\left(F\left(\alpha_{0}\right) -n\tau\right) -\alpha_{n}^{k}F\left(\alpha_{0}\right) =-\alpha_{n} ^{k}n\tau. $$

(11)

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

$$ \alpha_{n}\leq\frac{1}{n^{1/k}}\text{ for all }n\geq n^{\prime}. $$

(12)

Choose $m,n\in \mathbb {N}$ such that m≥n≥n^′ and from (12), we have

$$d\left(r_{m},r_{n}\right) \leq\alpha_{m-1}+\cdots+\alpha_{n}<\sum\limits_{j=n}^{\infty}\alpha_{n}\leq\sum\limits_{j=n}^{\infty}\frac{1}{j^{1/k}}. $$

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:

$$ \underset{n\rightarrow\infty}{\lim}r_{n}=r^{^{\ast}}. $$

(13)

By using property (T) of (S,d,G), there exists a subsequence $\left \{ r_{s_{n}}\right \}$ satisfying

$$\left(r_{s_{n}},r^{\ast}\right)\in E\left(G\right),\text{ }\forall\text{ }n\in \mathbb{N}. $$

Hence, by inequality (5) and in view of (4), we get

$$ F\left(d\left(\Upsilon r_{s_{n}},\Upsilon r^{\ast}\right) \right) \leq Fd\left(r_{s_{n}},r^{\ast}\right) -\tau<F\left(d\left(r_{s_{n}},r^{\ast}\right) \right), $$

(14)

which implies

$$ d\left(\Upsilon r_{s_{n}},\Upsilon r^{\ast}\right) \leq d\left(r_{s_{n} },r^{\ast}\right). $$

(15)

Therefore, by triangle inequality, we have

$$ \begin{aligned} d\left(r^{\ast},\Upsilon r^{\ast}\right) & \leq d\left(r^{\ast},r_{s_{n}+1}\right) +d\left(r_{s_{n}+1},\Upsilon r^{\ast}\right) \\ & =d\left(r^{\ast},r_{s_{n}+1}\right) +d\left(\Upsilon r_{s_{n}},\Upsilon r^{\ast}\right).\\ \end{aligned} $$

(16)

By using (15), inequality (16) yields

$$ d\left(r^{\ast},\Upsilon r^{\ast}\right) \leq d\left(r^{\ast},r_{s_{n} +1}\right) +d\left(r_{s_{n}},r^{\ast}\right), $$

(17)

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.,

$$d\left(\Upsilon r,\Upsilon s\right) \leq e^{-\tau}d\left(\Upsilon r,\Upsilon s\right),{ for\, all }\, r,s\in S,\text{ }\Upsilon r\neq\Upsilon s, $$

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

$$s=\left(1-\mu\right) r+\mu\Upsilon r\text{ where }\mu\in\left(0,1\right), $$

which represents the fact that

$$ d\left(r,\Upsilon r\right) =d\left(r,s\right) +d\left(s,\Upsilon r\right),\text{ }s\in\partial A $$

(18)

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

$$ \begin{aligned} &{for\, all }\, r,s\in A\text{ }\left(r,s\right) \in E\left(G\right) \\ &with\ \Upsilon r,\Upsilon s\in A,\text{ implies }\left(\Upsilon r,\Upsilon s\right) \in E\left(G\right) \cap\left(A\times A\right), \end{aligned} $$

(19)

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

$$\left(r_{n},r_{n+1}\right) \in E\left(G\right),\text{ }\forall\text{ }n\in \mathbb{N}, $$

there exists a subsequence $\left \{ r_{s_{n}}\right \}$ satisfying

$$ \left(r_{s_{n}},r\right) \in E\left(G\right),\text{ }\forall\text{ }n\in \mathbb{N}. $$

(20)

Let A be a nonempty, closed subset of S and Υ:A→S be a G_AF-contraction, that is, there exists a constant τ>0 such that

$$ \tau+F\left(d\left(\Upsilon r,\Upsilon s\right) \right) \leq F\left(d\left(r,s\right) \right) \text{ for all }(r,s)\in E\left(G_{A}\right), $$

(21)

where G_A is the subgraph of G determined by A. If the set

$$A_{\Upsilon}:=\left\{ r\in\partial A:\left(r,\Upsilon r\right) \in E\left(G\right) \right\} $$

is nonempty and Υ satisfies Rothe ^’s boundary condition

$$ \Upsilon\left(\partial A\right) \subset A, $$

(22)

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∈A_Υ. It follows that (r₀,Υr₀)∈E(G) and in view of equation (4), we have

$$ \left(\Upsilon^{n}r_{0},\Upsilon^{n+1}r_{0}\right)\in E\left(G\right),\text{ for all }n\in \mathbb{N}. $$

(23)

Let we denote r_n:=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,

$$r_{2}=\left(1-\mu\right) r_{1}+\mu\Upsilon r_{1},\text{ where }\mu \in\left(0,1\right). $$

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

$$U=\left\{ r_{a}\in\left\{ r_{n}\right\} :r_{a}=s_{a}=\Upsilon r_{a-1}\right\} $$

and

$$Z=\left\{ r_{a}\in\left\{ r_{n}\right\} :r_{a}\neq\Upsilon r_{a-1}\right\}. $$

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

$$r_{a}\neq\Upsilon r_{a-1}\,\, { implies}\,\, r_{a-1}=\Upsilon r_{a-2}. $$

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,

$$d\left(r_{n+1},r_{n}\right) =d\left(s_{n+1},s_{n}\right) =d\left(\Upsilon s_{n},\Upsilon s_{n-1}\right), $$

where (s_n,s_n−1)∈E(G) by virtue of (23). Therefore, we have

$$d\left(\Upsilon s_{n},\Upsilon s_{n-1}\right) =d\left(\Upsilon r_{n},\Upsilon r_{n-1}\right) >0. $$

Consequently, we get the following inequality

$$ \tau+F\left(d\left(\Upsilon s_{n},\Upsilon s_{n-1}\right) \right) \leq F\left(d\left(s_{n},s_{n-1}\right) \right), $$

(24)

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

$$d\left(r_{n},\Upsilon r_{n}\right) =d\left(r_{n},r_{n+1}\right) +d\left(r_{n+1},\Upsilon r_{n}\right). $$

The above equality implies d(r_n+1,Υr_n)≠0 and hence

$$ d\left(r_{n},r_{n+1}\right) =d\left(r_{n},\Upsilon r_{n}\right) -d\left(r_{n+1},\Upsilon r_{n}\right) < d\left(r_{n},\Upsilon r_{n}\right) =d\left(\Upsilon r_{n-1},\Upsilon r_{n}\right), $$

(25)

since r_n∈U. By using (25), we obtain

$$d\left(r_{n},r_{n+1}\right) < d\left(\Upsilon r_{n-1},\Upsilon r_{n}\right) =d\left(\Upsilon s_{n-1},\Upsilon s_{n}\right) >0. $$

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

$$ d\left(r_{n-1},\Upsilon r_{n-1}\right) =d\left(r_{n-1},r_{n}\right) +d\left(r_{n},\Upsilon r_{n-1}\right). $$

(26)

Hence, by triangle inequality

$$ \begin{aligned} d\left(r_{n},r_{n+1}\right) & \leq d\left(r_{n},\Upsilon r_{n-1}\right) +d(\Upsilon r_{n-1},r_{n+1})\\ & =d\left(r_{n},\Upsilon r_{n-1}\right) +d(\Upsilon r_{n-1},\Upsilon r_{n})\\ & =d\left(r_{n},\Upsilon r_{n-1}\right) +d(\Upsilon s_{n-1},\Upsilon s_{n}). \end{aligned} $$

(27)

By virtue of (23) (s_n−1,s_n)∈E(G), and the following inequality is obtained by the contraction condition (21)

$$ F\left(d\left(\Upsilon s_{n-1},\Upsilon s_{n}\right)\right) \leq F\left(d\left(s_{n-1},s_{n}\right) \right) -\tau<F\left(d\left(s_{n-1},s_{n}\right) \right), $$

(28)

which implies

$$ d(\Upsilon s_{n-1},\Upsilon s_{n})\leq d\left(s_{n-1},s_{n}\right) =d\left(r_{n-1},r_{n}\right). $$

(29)

Thus, by using (26) and (29) in inequality (27), we have

$$\begin{aligned} d\left(r_{n},r_{n+1}\right) &\leq d\left(r_{n},\Upsilon r_{n-1}\right) +d(\Upsilon s_{n-1},\Upsilon s_{n})\\ & < d\left(r_{n},\Upsilon r_{n-1}\right) +d(r_{n-1},r_{n})\\ & =d\left(r_{n-1},\Upsilon r_{n-1}\right).\\ \end{aligned} $$

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

$$ d\left(r_{n},r_{n+1}\right) < d\left(r_{n-1},\Upsilon r_{n-1}\right) =d\left(\Upsilon r_{n-2},\Upsilon r_{n-1}\right) \leq d\left(r_{n-2},r_{n-1}\right). $$

(30)

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

$$ \tau+F\left(\max\left\{ d\left(r_{n-2},r_{n-1}\right),d\left(r_{n-1},r_{n}\right) \right\} \right) \leq F\left(d\left(r_{n},r_{n+1}\right) \right), $$

(31)

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)

$$ F\left(\alpha_{n}\right) \leq F\left(\max\left\{ \alpha_{0},\alpha_{1}\right\} \right) -\left[ \frac{n}{2}\right] \tau, $$

(32)

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{α₀,α₁}. By (32), the following holds for all n≥2:

$$ \alpha_{n}^{k}F\left(\alpha_{n}\right) -\alpha_{n}^{k}F\left(\gamma\right) \leq\alpha_{n}^{k}\left(F\left(\gamma\right) -\left[ \frac{n}{2}\right] \tau\right) -\alpha_{n}^{k}F\left(\gamma\right) =-\alpha_{n}^{k}\left[ \frac{n}{2}\right] \tau. $$

(33)

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

$$ \alpha_{n}\leq\frac{1}{n^{1/k}}\text{ for all }n\geq n^{\prime}. $$

(34)

Choose $m,n\in \mathbb {N}$ such that m≥n≥n^′ and from (34), we have

$$d\left(r_{m},r_{n}\right) \leq\alpha_{m-1}+\cdots+\alpha_{n}<\sum\limits_{j=n}^{\infty}\alpha_{n}\leq\sum\limits_{j=n}^{\infty}\frac{1}{j^{1/k}}. $$

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

$$\left(r_{s_{n}},r^{^{\prime}}\right)\in E\left(G\right),\text{ for all }n\in \mathbb{N}. $$

Hence, by the F-contraction condition (21), we get

$$ d\left(\Upsilon r_{s_{n}},\Upsilon r^{^{\prime}}\right) \leq d\left(r_{s_{n}},r^{^{\prime}}\right). $$

(35)

Therefore, by triangle inequality, we have

$$\begin{aligned} d\left(r^{^{\prime}},\Upsilon r^{^{\prime}}\right) & \leq d\left(r^{^{\prime}},r_{s_{n}+1}\right) +d\left(r_{s_{n}+1},\Upsilon r^{^{\prime} }\right) \\ & =d\left(r^{^{\prime}},r_{s_{n}+1}\right) +d\left(\Upsilon r_{s_{n} },\Upsilon r^{^{\prime}}\right). \end{aligned} $$

By using (35), the above inequality yields

$$ d\left(r^{^{\prime}},\Upsilon r^{^{\prime}}\right) \leq d\left(r^{^{\prime}},r_{s_{n}+1}\right) +d\left(r_{s_{n}},r^{^{\prime}}\right), $$

(36)

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_AF -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:

$$\Upsilon r=\left\{\begin{array}{lc} 0\text{ \ \ \ \ \ if }r\in\left[ -1,0\right] \\ 0.5\text{ \ \ \ if }r\in\left(-\infty,-1\right).\\ \end{array}\right. $$

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:

$$E\left(G\right) =\left\{ \left(r,s\right) \in S\times S:r\leq s\right\} $$

and

$$E\left(G_{A}\right) =\left\{ \left(r,s\right) \in A\times A:r\leq s\right\}, $$

respectively. It is easy to check that (19) holds, that is,

$$\begin{aligned} &{for\, all }\, r,s\in A\text{ }\left(r,s\right) \in E\left(G\right) \\ &{with }\,\, \Upsilon r,\Upsilon s\in A,\text{ implies }\left(\Upsilon r,\Upsilon s\right) \in E\left(G\right) \cap\left(A\times A\right). \end{aligned} $$

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

$$\left(\Upsilon r,\Upsilon s\right) =\left(0,0\right) \in E\left(G_{A}\right). $$

Moreover, G is a weakly connected and for any k∈(0.5,1),Υ is a non-self G_AF-contraction on A with $\tau =\dfrac {1}{\sqrt {d\left (r,s\right) }},$ since

$$d\left(\Upsilon r,\Upsilon s\right) =\frac{1}{2}<\frac{1}{4}\times d\left(r,s\right) \text{ for }r\in\left(-\infty,-1\right) \text{ and }s\in\left[ -1,0\right] \text{.} $$

(for the rest of edges of E(G_A), the F-contraction 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 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.