Some Fixed Point Theorems in 𝑏-Metric Space Endowed with Graph

Abstract

We define some notions of contraction mappings in 𝑏-metric space endowed with a graph 𝐺 and subsequently establish some fixed point results for such classes of contractions. According to the applications of our results, we obtain fixed point theorems for cyclic operators and an existence theorem for the solution of an integral equation.

1. Introduction and Preliminaries

The study of 𝑏-metric spaces was initiated in some works of Bakhtin, Heinonen, Bourbaki, and Czerwik [1–4]. Afterwards, several articles which deal with fixed point theorems for single-valued and multivalued functions in 𝑏-metric space appeared [1–8].

Definition 1 (see [1, 4]). Let 𝑋 be a set, and let 𝑠≥1 be a given real number. A function 𝑑∶𝑋×𝑋→ℝ⁺ is said to be a 𝑏-metric on 𝑋, and the pair (𝑋,𝑑) is called a 𝑏-metric space if, for all 𝑥,𝑦,𝑧∈𝑋,  (d1)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦,  (d2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥), (d3)𝑑(𝑥,𝑧)≤𝑠[𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧)].

Note that the class of 𝑏-metric spaces contains the class of metric spaces.

Example 2. Let 𝑋∶=𝑙_𝑝(ℝ) with 0<𝑝<1, where 𝑙_𝑝(ℝ)∶={𝑥={𝑥_𝑛}⊂ℝ∶∑^∞_𝑛=1|𝑥_𝑛|^𝑝<∞}. Then, 𝑑(𝑥,𝑦)=(∑^∞_𝑛=1|𝑥_𝑛−𝑦_𝑛|^𝑝)^1/𝑝 is a 𝑏-metric on 𝑋 with 𝑠=2^1/𝑝.

A sequence {𝑥_𝑛} in a 𝑏-metric space 𝑋 is said to be convergent if and only if there exists 𝑥∈𝑋 such that 𝑑(𝑥_𝑛,𝑥)→0 as 𝑛→∞. In this case, we write lim_𝑛→∞𝑥_𝑛=𝑥. A sequence {𝑥_𝑛} in a 𝑏-metric space 𝑋 is said to be Cauchy if and only if 𝑑(𝑥_𝑛,𝑥_𝑚)→0 as 𝑚,𝑛→∞. A 𝑏-metric space (𝑋,𝑑) is complete if every Cauchy sequence in 𝑋 converges. In general, a 𝑏-metric is not continuous.

The famous Banach contraction principle [9] infers that every contraction on a complete metric space has a unique fixed point. Recently, Jachymski [10] introduced the notion of Banach 𝐺-contraction to generalize Banach contraction principle as follows. Let (𝑀,𝛿) be a metric space, let Δ be the diagonal of the Cartesian product 𝑀×𝑀, and let 𝐺 be a directed graph such that the set 𝑉(𝐺) of its vertices coincides with 𝑀 and the set 𝐸(𝐺) of its edges contains all loops; that is, 𝐸(𝐺)⊇Δ. Assume that 𝐺 has no parallel edges. A mapping 𝑓∶𝑀→𝑀 is called a Banach 𝐺-contraction if (i)  for  all 𝑥,𝑦∈𝑋  ((𝑥,𝑦)∈𝐸(𝐺)⇒(𝑓𝑥,𝑓𝑦)∈𝐸(𝐺)),  (ii)  ∃𝛼, 0<𝛼<1 such that for all 𝑥,𝑦∈𝑋,(𝑥,𝑦)∈𝐸(𝐺)⇒𝛿(𝑓𝑥,𝑓𝑦)≤𝛼𝛿(𝑥,𝑦). A mapping 𝑓∶𝑀→𝑀 is known as Picard operator [11] if 𝑓 has a unique fixed point 𝑥^∗ and lim_𝑛→∞𝑓^𝑛𝑥=𝑥^∗ for all 𝑥∈𝑀.

Various generalizations of Banach's principle have been obtained by weakening contractive conditions. In this context, Matkowski [12] introduced class of 𝜑-contractions in metric fixed point theory, and subsequently further study was developed in this setting by different authors when underlying space was taken to be a partially ordered set (see, e.g., [13, 14]).

Let 𝜑∶ℝ⁺→ℝ⁺. Consider the following properties:(i)_𝜑𝑡₁≤𝑡₂⇒𝜑(𝑡₁)≤𝜑(𝑡₂), for all 𝑡₁,𝑡₂∈ℝ⁺,(ii)_𝜑𝜑(𝑡)<𝑡 for 𝑡>0,(iii)_𝜑𝜑(0)=0,(iv)_𝜑lim_𝑛→∞𝜑^𝑛(𝑡)=0 for all 𝑡≥0,(v)_𝜑∑^∞_𝑛=0𝜑^𝑛(𝑡) converges for all 𝑡>0. It is easily seen that (i)_𝜑 and (iv)_𝜑 imply (ii)_𝜑 and (i)_𝜑 and (ii)_𝜑 imply (iii)_𝜑.

We recall that a function 𝜑 satisfying (i)_𝜑 and (iv)_𝜑 is said to be a comparison function. A function 𝜑 satisfying (i)_𝜑 and (v)_𝜑 is known as (𝑐)-comparison function.

Any (𝑐)-comparison function is a comparison function, but converse may not be true. For example, 𝜑(𝑡)=𝑡/(1+𝑡); 𝑡∈ℝ⁺ is a comparison function but not a (𝑐)-comparison function. On the other hand, define 𝜑(𝑡)=𝑡/2; 0≤𝑡≤1 and 𝜑(𝑡)=𝑡−(1/2); 𝑡>1, and then 𝜑 is a (𝑐)-comparison function. For details on 𝜑 contractions, we refer the readers to [15, 16].

Berinde [17] took further step to investigate 𝜑 contractions when the framework was taken to be a 𝑏-metric space, and for some technical reasons, he had to introduce the notion of 𝑏-comparison function; in particular, he obtained some estimations for rate of convergence [17]. For other related results, see also [5, 7, 17–21].

Definition 3. Let 𝑠≥1 be a fixed real number. A function 𝜑∶ℝ⁺→ℝ⁺ is known as 𝑏-comparison function if it satisfies (i)_𝜑, and the following holds:(vi)_𝜑∑^∞_𝑛=0𝑠^𝑛𝜑^𝑛(𝑡) converges for all 𝑡∈ℝ⁺.

The concept of 𝑏-comparison function coincides with comparison function when 𝑠=1. Let (𝑋,𝑑) be a 𝑏-metric space with coefficient 𝑠≥1, and then 𝜑(𝑡)=𝑎𝑡; 𝑡∈ℝ⁺ with 0<𝑎<(1/𝑠) is a 𝑏-comparison function.

2. Main Results

Throughout this section, let (𝑋,𝑑) be a 𝑏-metric space with coefficient 𝑠≥1, and Δ is the diagonal of the Cartesian product 𝑋×𝑋. 𝐺 is a directed graph such that the set 𝑉(𝐺) of its vertices coincides with 𝑋, and the set 𝐸(𝐺) of its edges contains all loops; that is, 𝐸(𝐺)⊇Δ. Assume that 𝐺 has no parallel edges. We assign to each edge having vertices 𝑥 and 𝑦 a unique element 𝑑(𝑥,𝑦). Now, we introduce the following definition.

Definition 4. One says that a mapping 𝑓∶𝑋→𝑋 is a 𝑏-(𝜑,𝐺) contraction if for all 𝑥,𝑦∈𝑋: 𝑓𝑥,𝑓𝑦∈𝐸(𝐺)whenever𝑥,𝑦∈𝐸(𝐺);𝑑𝑓𝑥,𝑓𝑦≤𝜑𝑑𝑥,𝑦whenever𝑥,𝑦∈𝐸(𝐺), (1)(2) where 𝜑∶ℝ⁺→ℝ⁺ is a comparison function.

Remark 5. Note that a Banach 𝐺-contraction is a 𝑏-(𝜑,𝐺) contraction.

Example 6. Any constant mapping 𝑓∶𝑋→𝑋 is a 𝑏-(𝜑,𝐺) contraction for any graph 𝐺 with 𝑉(𝐺)=𝑋.

Example 7. Any self-mapping 𝑓 on 𝑋 is trivially a 𝑏-(𝜑,𝐺₁) contraction, where 𝐺₁=(𝑉(𝐺),𝐸(𝐺))=(𝑋,Δ).

Example 8. Let 𝑋=ℝ, and define 𝑑∶𝑋×𝑋→ℝ by 𝑑(𝑥,𝑦)=|𝑥−𝑦|². Then, 𝑑 is a 𝑏-metric on 𝑋 with 𝑠=2. Further, 𝑓𝑥=𝑥/2, for all 𝑥∈𝑋. Then, 𝑓 is a 𝑏-(𝜑,𝐺₀) contraction with 𝜑(𝑡)=𝑡/4 and 𝐺₀=(𝑋,𝑋×𝑋). Note that 𝑑 is not a metric on 𝑋.

Definition 9. Two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝑋 are said to be equivalent if lim_𝑛→∞𝑑(𝑥_𝑛,𝑦_𝑛)=0, and if each of them is a Cauchy sequence, then they are called Cauchy equivalent.

As a consequence of Definition 9, we get the following lemma.

Remark 10. Let {𝑥_𝑛} and {𝑦_𝑛} be equivalent sequences in 𝑋. (i) If {𝑥_𝑛} converges to 𝑥, then {𝑦_𝑛} also converges to 𝑥 and vice versa. (ii)  {𝑦_𝑛} is a Cauchy sequence whenever {𝑥_𝑛} is a Cauchy sequence and vice versa.

Now, we recollect some preliminaries from graph theory which we need for the sequel. Let 𝐺=(𝑉(𝐺), and let 𝐸(𝐺)) be a directed graph. By letter 𝐺, we denote the undirected graph obtained from 𝐺 by ignoring the direction of edges. If 𝑥 and 𝑦 are vertices in a graph 𝐺, then a path in 𝐺 from 𝑥 to 𝑦 of length 𝑙 is a sequence {𝑥_𝑖}^𝑙_𝑖=0 of 𝑙+1 vertices such that 𝑥₀=𝑥,𝑥_𝑙=𝑦, and (𝑥_𝑖−1,𝑥_𝑖)∈𝐸(𝐺) for 𝑖=1,…,𝑙. A graph 𝐺 is called connected if there is a path between any two vertices. 𝐺 is weakly connected if 𝐺 is connected. For a graph 𝐺 such that 𝐸(𝐺) is symmetric and 𝑥 is a vertex in 𝐺, the subgraph 𝐺_𝑥 consisting of all edges and vertices which are contained in some path beginning at 𝑥 is known as a component of 𝐺 containing 𝑥. So that 𝑉(𝐺_𝑥𝐺)=[𝑥], where 𝐺[𝑥] is the equivalence class of a relation 𝑅 defined on 𝑉(𝐺) by the rule: 𝑦𝑅𝑧 if there is a path in 𝐺 from 𝑦 to 𝑧. Clearly, 𝐺_𝑥 is connected. A graph 𝐺 is known as (𝐶)-graph in 𝑋 [22] if for any sequence {𝑥_𝑛} in 𝑋 with 𝑥_𝑛→𝑥 and (𝑥_𝑛,𝑥_𝑛+1)∈𝐸(𝐺) for 𝑛∈ℕ; then there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that (𝑥_𝑛𝑘,𝑥)∈𝐸(𝐺) for 𝑘∈ℕ.

Proposition 11. Let 𝑓∶𝑋→𝑋 be a 𝑏-(𝜑,𝐺) contraction, where 𝜑∶ℝ⁺→ℝ⁺ is a comparison function; then (i)𝑓 is a 𝑏-(𝜑,𝐺) contraction and a 𝑏-(𝜑,𝐺⁻¹) contraction as well,(ii)[𝑥₀]𝐺 is 𝑓-invariant, and 𝑓|_[𝑥0]𝐺 is a 𝑏-(𝜑,𝐺_𝑥) contraction provided that 𝑥₀∈𝑋 is such that 𝑓𝑥₀∈[𝑥₀]𝐺.

Proof. (i) By using (d2) (Definition 1) it can be easily proved.
(ii) Let 𝑥∈[𝑥₀]𝐺. Then, there is a path 𝑥=𝑧₀,𝑧₁,…,𝑧_𝑙=𝑥₀ between 𝑥 and 𝑥₀. Since 𝑓 is a 𝑏-(𝜑,𝐺) contraction, then (𝑓𝑧_𝑖−1,𝑓𝑧_𝑖)∈𝐸(𝐺) for all 𝑖=1,2,…,𝑙. Thus, 𝑓𝑥∈[𝑓𝑥₀]𝐺=[𝑥₀]𝐺.
Suppose (𝑥,𝑦)∈𝐸(𝐺_𝑥0). Then, (𝑓𝑥,𝑓𝑦)∈𝐸(𝐺), since 𝑓 is a 𝑏-(𝜑,𝐺) contraction. But [𝑥₀]𝐺 is 𝑓 invariant, so we conclude that (𝑓𝑥,𝑓𝑦)∈𝐸(𝐺_𝑥0). Condition (2) is satisfied automatically as 𝐺_𝑥0 is a subgraph of 𝐺.

From now on, we assume that coefficient of 𝑏-comparison function is at least as large as the coefficient of 𝑏-metric 𝑠.

Lemma 12. Let 𝑓∶𝑋→𝑋 be a 𝑏-(𝜑,𝐺) contraction, where 𝜑∶ℝ⁺→ℝ⁺ is a 𝑏-comparison function. Then, given any 𝑥∈𝑋 and 𝐺𝑦∈[𝑥], two sequences {𝑓^𝑛𝑥} and {𝑓^𝑛𝑦} are equivalent.

Proof. Let 𝑥∈𝑋, and let 𝐺𝑦∈[𝑥]; then there exists a path {𝑥_𝑖}^𝑙_𝑖=0 in 𝐺 from 𝑥 to 𝑦 with 𝑥₀=𝑥, 𝑥_𝑙=𝑦, and (𝑥_𝑖−1,𝑥_𝑖)∈𝐸(𝐺). From Proposition 11, 𝑓 is a 𝑏-(𝜑,𝐺) contraction. So, 𝑓^𝑛𝑥_𝑖−1,𝑓^𝑛𝑥_𝑖∈𝐸𝐺implies𝑑𝑓^𝑛𝑥_𝑖−1,𝑓^𝑛𝑥_𝑖≤𝜑𝑑𝑓^𝑛−1𝑥_𝑖−1,𝑓^𝑛−1𝑥_𝑖 (3) for all 𝑛∈ℕ and 𝑖=0,1,2,…,𝑙. Hence, 𝑑𝑓^𝑛𝑥_𝑖−1,𝑓^𝑛𝑥_𝑖≤𝜑^𝑛𝑑𝑥_𝑖−1,𝑥_𝑖∀𝑛∈ℕ,𝑖=0,1,2,…,𝑙. (4) We observe that {𝑓^𝑛𝑥_𝑖}^𝑙_𝑖=0 is a path in 𝐺 from 𝑓^𝑛𝑥 to 𝑓^𝑛𝑦. Using (d3) Definition 1 and (4), we have 𝑑𝑓^𝑛𝑥,𝑓^𝑛𝑦≤^𝑙_𝑖=1𝑠^𝑖𝑑𝑓^𝑛𝑥_𝑖−1,𝑓^𝑛𝑥_𝑖≤^𝑙_𝑖=1𝑠^𝑖𝜑^𝑛𝑑𝑥_𝑖−1,𝑥_𝑖. (5) Letting 𝑛→∞, we obtain 𝑑(𝑓^𝑛𝑥,𝑓^𝑛𝑦)→0.

Proposition 13. Let 𝑓 be a 𝑏-(𝜑,𝐺) contraction, where 𝜑∶ℝ⁺→ℝ⁺ is a 𝑏-comparison function. Suppose that there is 𝑧₀ in 𝑋 such that 𝑓𝑧₀∈[𝑧₀]𝐺. Then, {𝑓^𝑛𝑧₀} is a Cauchy sequence in 𝑋.

Proof. Since 𝑓𝑧₀∈[𝑧₀]𝐺, let {𝑦_𝑖}^𝑟_𝑖=0 be a path from 𝑧₀ to 𝑓𝑧₀. Then using the same arguments as in Lemma 12, we arrive at 𝑑𝑓^𝑛𝑧₀,𝑓^𝑛+1𝑧₀≤^𝑟_𝑖=1𝑠^𝑖𝜑^𝑛𝑑𝑦_𝑖−1,𝑦_𝑖,∀𝑛∈ℕ. (6) Let 𝑚>𝑛≥1, and then from above inequality; it follows for 𝑝≥1𝑑𝑓^𝑛𝑧₀,𝑓^𝑛+𝑝𝑧₀≤𝑠𝑑𝑓^𝑛𝑧₀,𝑓^𝑛+1𝑧₀+𝑠²𝑑𝑓^𝑛+1𝑧₀,𝑓^𝑛+2𝑧₀+⋯+𝑠^𝑝𝑑𝑓^{𝑛+𝑝−1}𝑧₀,𝑓^𝑛+𝑝𝑧₀≤1𝑠^{𝑛−1𝑛+𝑝−1}_𝑗=𝑛𝑠^𝑗𝑑𝑓^𝑗𝑧₀,𝑓^𝑗+1𝑧₀≤1𝑠^𝑛−1𝑟_𝑖=1𝑠^{𝑖𝑛+𝑝−1}_𝑗=𝑛𝑠^𝑗𝜑^𝑗𝑑𝑦_𝑖−1,𝑦_𝑖. (7) Denoting for each 𝑖=1,2,…,𝑟𝑆^𝑖_𝑛=^𝑛_𝑘=0𝑠^𝑘𝜑^𝑘𝑑𝑦_𝑖−1,𝑦_𝑖,𝑛≥1, (8) relation (7) becomes 𝑑𝑓^𝑛𝑧₀,𝑓^𝑛+𝑝𝑧₀≤1𝑠^𝑛−1𝑟_𝑖=1𝑠^𝑖𝑆^𝑖_{𝑛+𝑝−1}−𝑆^𝑖_𝑛−1, (9) since 𝜑 is a 𝑏-comparison function, so that for each 𝑖=1,2,…,𝑟, ^∞_𝑘=0𝑠^𝑘𝜑^𝑘𝑑𝑦_𝑖−1,𝑦_𝑖<∞. (10) Then, corresponding to each 𝑖, there is a real number 𝑆^𝑖 such that lim_𝑛→∞𝑆^𝑖_𝑛=𝑆^𝑖. (11) In view of (11), relation (9) gives 𝑑(𝑓^𝑛𝑧₀,𝑓^𝑛+𝑝𝑧₀)→0 as 𝑛→∞. Which shows that {𝑓^𝑛𝑧₀} is a Cauchy sequence in 𝑋.

Definition 14. Let 𝑓∶𝑋→𝑋, and let 𝑦∈𝑋, and the sequence {𝑓^𝑛𝑦} in 𝑋 is such that 𝑓^𝑛𝑦→𝑥^∗ with (𝑓^𝑛𝑦,𝑓^𝑛+1𝑦)∈𝐸(𝐺) for 𝑛∈ℕ. One says that a graph 𝐺 is (𝐶_𝑓)-graph if there exists a subsequence {𝑓^𝑛𝑘𝑦} and a natural number 𝑝 such that (𝑓^𝑛𝑘𝑦,𝑥^∗)∈𝐸(𝐺) for all 𝑘≥𝑝 [23]. One says that a graph 𝐺 is (𝐻_𝑓)-graph if 𝑓^𝑛𝑦∈[𝑥^∗]𝐺 for 𝑛≥1; then 𝑟(𝑓^𝑛𝑦,𝑥^∗)→0(as𝑛→∞), where 𝑟(𝑓^𝑛𝑦,𝑥^∗)=∑^𝑀_𝑖=1𝑠^𝑖𝑑(𝑧_𝑖−1,𝑧_𝑖); {𝑧_𝑖}^𝑀_𝑖=0 is a path from 𝑓^𝑛𝑦 to 𝑥^∗ in 𝐺.

Obviously every (𝐶)-graph is a (𝐶_𝑓)-graph for any self-mapping 𝑓 on 𝑋, but converse may not hold as shown in the following.

Example 15. Let 𝑋=[0,1] with respect to 𝑏-metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|². Consider a graph 𝐺 consisting of 𝑉(𝐺)=𝑋 and 𝐸(𝐺)={(𝑛/(𝑛+1), (𝑛+1)/(𝑛+2))∶𝑛∈ℕ}∪{(𝑥/2^𝑛,𝑥/2^𝑛+1)∶𝑛∈ℕ, 𝑥∈[0,1]}∪{((𝑥/2^2𝑛), 0)∶𝑛∈ℕ, 𝑥∈[0,1]}. Note that 𝐺 is not a (𝐶)-graph as 𝑛/(𝑛+1)→1. Define 𝑓∶𝑋→𝑋 as 𝑓𝑥=𝑥/2. Then, 𝐺 is a (𝐶_𝑓)-graph, since 𝑓^𝑛𝑥=𝑥/2^𝑛→0.

Example 16. Let 𝑋={1/𝑛∶𝑛∈ℕ}∪{0}∪ℕ with respect to 𝑏-metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|², and let 𝐼 be identity map on 𝑋. Consider a graph 𝐺₂ consisting of 𝑉(𝐺₂)=𝑋 and 𝐸𝐺₂=1𝑛,1,1𝑛+1,(),1𝑛+1,𝑛𝑛,0,5𝑛,0;𝑛∈ℕ (12) since 𝑥_𝑛=1/𝑛→0 as 𝑛→∞. We note that 𝐺₂ is a (𝐶_𝐼)-graph, but 𝑟(𝑥_𝑛,0)=2|(1/𝑛)−(1/(𝑛+1))|²+2²|(1/(𝑛+1))−𝑛|²+2²𝑛²↛0 as 𝑛→∞. Thus, 𝐺₂ is not an (𝐻_𝐼)-graph.

Example 17. Let 𝑋={1/𝑛∶𝑛∈ℕ}∪{√5/𝑛∶𝑛∈ℕ}∪{0} with respect to 𝑏-metric 𝑑(𝑥,𝑦)=|𝑥−𝑦|², and let 𝐼 be identity map on 𝑋. Consider a graph 𝐺₃ consisting of 𝑉(𝐺₃)=𝑋 and 𝐸𝐺₃=1𝑛,1,1𝑛+1,√𝑛+15𝑛,√5𝑛,0;𝑛∈ℕ, (13) since 𝑥_𝑛=1/𝑛→0 as 𝑛→∞. Clearly 𝐺₃ is not a (𝐶_𝐼)-graph, but it is easy to verify that 𝐺₃ is an (𝐻_𝐼)-graph.

Above examples show that for a given 𝑓 notions, (𝐶_𝑓)-graph and (𝐻_𝑓)-graph are independent even if 𝑓 is identity map.

Theorem 18. Let (𝑋,𝑑) be a complete 𝑏-metric space, and let 𝑓 be a 𝑏-(𝜑,𝐺) contraction, where 𝜑 is 𝑏-comparison function. Assume that 𝑑 is continuous and there is 𝑧₀ in 𝑋 for which (𝑧₀,𝑓𝑧₀) is an edge in 𝐺. Then, the following assertions hold. (1)If 𝐺 is a (𝐶_𝑓)-graph, then 𝑓 has a unique fixed point 𝑡∈[𝑧₀]𝐺, and for any 𝑦∈[𝑧₀]𝐺, 𝑓^𝑛𝑦→𝑡. Further, if 𝐺 is weakly connected, then 𝑓 is Picard operator. (2)If 𝐺 is weakly connected (𝐻_𝑓)-graph, then 𝑓 has a unique fixed point 𝑡∈𝑋 and for any 𝑦∈𝑋, 𝑓^𝑛𝑦→𝑡.

Proof. (1) It follows from Proposition 13 that {𝑓^𝑛𝑧₀} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑡∈𝑋 such that 𝑓^𝑛𝑧₀→𝑡. Since (𝑓^𝑛𝑧₀,𝑓^𝑛+1𝑧₀)∈𝐸(𝐺), for all 𝑛∈ℕ, and 𝐺 is a (𝐶_𝑓) graphs, there exists a subsequence {𝑓^𝑛𝑘𝑧₀} of {𝑓^𝑛𝑧₀} and 𝑝∈ℕ such that (𝑓^𝑛𝑘𝑧₀,𝑡)∈𝐸(𝐺) for all 𝑘≥𝑝. Observe that (𝑧₀,𝑓𝑧₀,𝑓²𝑧₀,…,𝑓^𝑛1𝑧₀,…,𝑓^𝑛𝑝𝑧₀,𝑡) is a path in 𝐺. Therefore, 𝑡∈[𝑧₀]𝐺. From (2), we get 𝑑𝑓^𝑛𝑘+1𝑧₀,𝑓𝑡≤𝜑𝑑𝑓^𝑛𝑘𝑧₀,𝑡<𝑑𝑓^𝑛𝑘𝑧₀,𝑡∀𝑘≥𝑛₀. (14) Letting 𝑘→∞, we obtain lim_𝑘→∞𝑓^𝑛𝑘+1𝑧₀=𝑓𝑡, as 𝑑 is continuous. Since {𝑓^𝑛𝑘𝑧₀} is a subsequence of {𝑓^𝑛𝑧₀}, we conclude that 𝑓𝑡=𝑡. Finally, if 𝑦∈[𝑧₀]𝐺, it follows from Lemma 12 that 𝑓^𝑛𝑦→𝑡.
(2) Let 𝐺 be weakly connected (𝐻_𝑓)-graph. From Proposition 13, 𝑓^𝑛𝑧₀→𝑡∈𝑋, and then 𝑟(𝑓^𝑛𝑧₀,𝑡)→0 as 𝑛→∞. Now, for each 𝑛∈ℕ, let {𝑦^𝑛_𝑖};𝑖=0,1,…,𝑀_𝑛 be a path from 𝑓^𝑛𝑧₀ to 𝑡 with 𝑦₀=𝑡 and 𝑦^𝑛_𝑀𝑛=𝑓^𝑛𝑧₀ in 𝐺; then 𝑑𝑡,𝑓𝑡≤𝑠𝑑𝑡,𝑓^𝑛+1𝑧₀+𝑑𝑓^𝑛+1𝑧₀𝑑,𝑓𝑡≤𝑠𝑡,𝑓^𝑛+1𝑧₀+^𝑀𝑛_𝑖=1𝑠^𝑖𝑑𝑓𝑦^𝑛_𝑖−1,𝑓𝑦^𝑛_𝑖≤𝑠𝑑𝑡,𝑓^𝑛+1𝑧₀+^𝑀𝑛_𝑖=1𝑠^𝑖𝜑𝑑𝑦^𝑛_𝑖−1,𝑦^𝑛_𝑖<𝑠𝑑𝑡,𝑓^𝑛+1𝑧₀+^𝑀𝑛_𝑖=1𝑠^𝑖𝑑𝑦^𝑛_𝑖−1,𝑦^𝑛_𝑖=𝑠𝑑𝑡,𝑓^𝑛+1𝑧₀+𝑟𝑓^𝑛𝑧₀,𝑡. (15) Letting 𝑛→∞, the above inequality yields 𝑓𝑡=𝑡. Let 𝑦∈[𝑧₀]𝐺∶=𝑋 be arbitrary; then from Lemma 12 and Remark 10, it is easily seen that 𝑓^𝑛𝑦→𝑡.

The following example shows that the condition of (𝐶_𝑓) or (𝐻_𝑓)-graph in the hypothesis of Theorem 18 can not be dropped.

Example 19. Let 𝑋=[0,1], let 𝑑(𝑥,𝑦)=|𝑥−𝑦|², and let 𝑓𝑥=𝑥/2 for all 𝑥∈(0,1] and 𝑓0=1/2. Then, (𝑋,𝑑) is a complete 𝑏-metric space with 𝑠=2. Further, 𝑑 is continuous, and 𝑓 is a 𝑏-(𝜑,𝐺₁) contraction (with 𝜑(𝑡)=𝑡/4), where 𝑉(𝐺₁)=𝑋 and 𝐸(𝐺₁)={(𝑥,𝑦)∈(0,1]×(0,1]; 𝑥≥𝑦}∪{(0,0),(0,1)}. Note that 𝐺₁ is weakly connected, but 𝑓 has no fixed point in [𝑧₀]𝐺₁=𝑋. Observe that 𝐺₁ is not a (𝐶_𝑓)-graph because the sequence 𝑓^𝑛𝑥=𝑥/2^𝑛→0 for 𝑥∈(0,1] and (𝑓^𝑛𝑥,𝑓^𝑛+1𝑥)∈𝐸(𝐺₁);𝑛∈ℕ, but it does not contain any subsequence such that (𝑥_𝑛𝑘,0)∈𝐸(𝐺₁). Also, we note that for any fixed 𝑥∈(0,1], 𝑟(𝑓^𝑛𝑥,0)=2[|𝑥/2^𝑛−1|²+|1−0|²]↛0 as 𝑛→∞.

Definition 20. Let (𝑋,𝑑) be a 𝑏-metric space. A mapping 𝑓∶𝑋→𝑋 is called orbitally continuous if for all 𝑥,𝑦∈𝑋 and any sequence {𝑘_𝑛}_𝑛∈ℕ of positive integers, 𝑓^𝑘𝑛𝑥→𝑦 implies 𝑓(𝑓^𝑘𝑛𝑥)→𝑓𝑦 as 𝑛→∞. A mapping 𝑓∶𝑋→𝑋 is called orbitally 𝐺-continuous if for all 𝑥,𝑦∈𝑋 and any sequence {𝑘_𝑛}_𝑛∈ℕ of positive integers, 𝑓^𝑘𝑛𝑥→𝑦 and (𝑓^𝑘𝑛𝑥,𝑓^𝑘𝑛+1𝑥)∈𝐸(𝐺) for all 𝑛∈ℕ imply 𝑓(𝑓^𝑘𝑛𝑥)→𝑓𝑦.

Theorem 21. Let (𝑋,𝑑) be a complete 𝑏-metric space, and let 𝑓 be a 𝑏-(𝜑,𝐺) contraction, where 𝜑 is a 𝑏-comparison function. Assume that 𝑑 is continuous, 𝑓 is orbitally 𝐺-continuous, and there is 𝑧₀ in 𝑋 for which (𝑧₀,𝑓𝑧₀) is an edge in 𝐺. Then, 𝑓 has a fixed point 𝑡∈𝑋. Moreover, for any 𝑦∈[𝑧₀]𝐺, 𝑓^𝑛𝑦→𝑡.

Proof. It follows from Proposition 13 that {𝑓^𝑛𝑧₀} is a Cauchy sequence in (𝑋,𝑑). Since 𝑋 is complete, there exists 𝑡∈𝑋 such that lim_𝑛→∞𝑓^𝑛𝑧₀=𝑡. Since (𝑓^𝑛𝑧₀,𝑓^𝑛+1𝑧₀)∈𝐸(𝐺) for all 𝑛∈ℕ, 𝑓 is orbitally 𝐺-continuous. Therefore, continuity of 𝑑 implies that 𝑓𝑡=𝑡. Let 𝑦∈[𝑧₀]𝐺 be arbitrary; then it follows from Lemma 12 that lim_𝑛→∞𝑓^𝑛𝑦=𝑡.

Slightly strengthening the continuity condition on 𝑓, our next theorem deals with the graph 𝐺 which may fail to have the property that there is 𝑧₀ in 𝑋 for which (𝑧₀,𝑓𝑧₀) is an edge in 𝐺.

Theorem 22. Let (𝑋,𝑑) be a complete 𝑏-metric space, and let 𝑓 be a 𝑏-(𝜑,𝐺) contraction, where 𝜑 is a 𝑏-comparison function. Assume that 𝑑 is continuous, 𝑓 is orbitally continuous, and there is 𝑧₀ in 𝑋 for which 𝑓𝑧₀∈[𝑧₀]𝐺. Then, for any 𝑦∈[𝑧₀]𝐺, 𝑓^𝑛𝑦→𝑡∈𝑋, where 𝑡 is a fixed point of 𝑓.

Proof. It follows from Proposition 13 that {𝑓^𝑛𝑧₀} is a Cauchy sequence in 𝑋. Since 𝑋 is complete, there exists 𝑡∈𝑋 such that 𝑓^𝑛𝑧₀→𝑡. Since 𝑓 is orbitally continuous, then lim_𝑛→∞𝑓𝑓^𝑛𝑧₀=𝑓𝑡 which yields 𝑓𝑡=𝑡. Let 𝑦∈[𝑧₀]𝐺 be arbitrary; then from Lemma 12, lim_𝑛→∞𝑓^𝑛𝑦=𝑡.

Remark 23. In addition to the hypothesis of Theorems 21 and 22, if we assume that 𝐺 is weakly connected, then 𝑓 will become Picard operator [11] on 𝑋.

Remark 24. Theorem 18 generalizes/extends claims 4⁰ and 5⁰ of [10, Theorem 3.2] and [7, Theorem 4(1)]. Theorem 21 generalizes claims 2⁰ and 3⁰ of [10, Theorem 3.3]. Theorem 22 generalizes claims 2⁰ and 3⁰ of [10, Theorem 3.4], and thus generalizes extends results of Nieto and Rodríguez-López [24, Theorems 2.1 and 2.3], Petrusel and Rus [11, Theorem 4.3], and Ran and Reurings [25, Theorem 2.1]. We mention here that Theorem 18 can not be improved using comparison function instead of 𝑏-comparison function (see, Gwóźdź-Łukawska and Jachymski [26, Example 2]).

We observe that Theorem 22 can be used to extend famous fixed point theorem of Edelstein to the case of 𝑏-metric space. We need to define notion of 𝜖-chainable property for 𝑏-metric space.

Definition 25. A 𝑏-metric space (𝑋,𝑑) is said to be 𝜖-chainable, for some 𝜖>0, if for 𝑥,𝑦∈𝑋 there exist 𝑥_𝑖∈𝑋;𝑖=0,1,2,…,𝑙 with 𝑥₀=𝑥,𝑥_𝑙=𝑦 such that 𝑑(𝑥_𝑖−1,𝑥_𝑖)<𝜖 for 𝑖=1,2,…,𝑙.

Corollary 26. Let (𝑋,𝑑) be a complete 𝜖-chainable 𝑏-metric space. Assume that 𝑑 is continuous, and there exists a 𝑏-comparison function 𝜑∶ℝ⁺→ℝ⁺ such that 𝑓∶𝑋→𝑋 satisfying, 𝑑𝑥,𝑦<𝜖𝑖𝑚𝑝𝑙𝑖𝑒𝑠𝑑𝑓𝑥,𝑓𝑦≤𝜑𝑑𝑥,𝑦, (16) for all 𝑥,𝑦∈𝑋. Then, 𝑓 is a Picard operator.

Proof. Consider a graph 𝐺 consisting of 𝑉(𝐺)∶=𝑋 and (𝑥,𝑦)∈𝐸(𝐺) if and only if 𝑑(𝑥,𝑦)<𝜖. Since 𝑋 is 𝜖-chainable, 𝐺 is weakly connected. Let (𝑥,𝑦)∈𝐸(𝐺), and from (16), we have 𝑑𝑓𝑥,𝑓𝑦≤𝜑𝑑𝑥,𝑦<𝑑𝑥,𝑦<𝜖. (17) Then, 𝑑(𝑓𝑥,𝑓𝑦)∈𝐸(𝐺). Therefore, in view of (16), 𝑓 is 𝑏-(𝜑,𝐺) contraction. Further, (16) implies that 𝑓 is continuous. Now, the conclusion follows by using Theorem 22.

Now, we establish a fixed point theorem using a general contractive condition.

Theorem 27. Let (𝑋,𝑑) be a complete 𝑏-metric space, and let 𝐺 be a (𝐶_𝑓)-graph in 𝑋×𝑋 such that 𝑉(𝐺)=𝑋 and 𝑓∶𝑋→𝑋 is an edge-preserving mapping. Assume that 𝑑 is continuous and there exist 𝛼,𝛽,𝛾≥0 with 𝑠𝛼+(𝑠+1)𝛽+𝑠(𝑠+1)𝛾<1 and for all (𝑥,𝑦)∈𝐸(𝐺)𝑑𝑑𝑑.𝑓𝑥,𝑓𝑦≤𝛼𝑑𝑥,𝑦+𝛽𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦+𝛾𝑥,𝑓𝑦+𝑑𝑦,𝑓𝑥 (18) If there is 𝑧₀ in 𝑋 for which (𝑧₀,𝑓𝑧₀) is an edge in 𝐺, then 𝑓 has a fixed point in [𝑧₀]𝐺.

Proof. Since 𝑓 is edge-preserving, then (𝑓^𝑛𝑧₀,𝑓^𝑛+1𝑧₀)∈𝐸(𝐺) for all 𝑛∈ℕ. From (18) and using (d3), it follows that 𝑑𝑓^𝑛𝑧₀,𝑓^𝑛+1𝑧₀≤𝛼𝑑𝑓^𝑛−1𝑧₀,𝑓^𝑛𝑧₀+𝛽𝑑𝑓^𝑛−1𝑧₀,𝑓^𝑛𝑧₀+𝑑𝑓^𝑛𝑧₀,𝑓^𝑛+1z₀+𝛾𝑠𝑑𝑓^𝑛−1𝑧₀,𝑓^𝑛𝑧₀+𝑑𝑓^𝑛𝑧₀,𝑓^𝑛+1𝑧₀. (19) On rearranging, 𝑑𝑓^𝑛𝑧₀,𝑓^𝑛+1𝑧₀≤𝛼+𝛽+𝛾𝑠𝑑𝑓1−𝛽−𝛾𝑠^𝑛−1𝑧₀,𝑓^𝑛𝑧₀. (20) Repeating iteratively, we have 𝑑𝑓^𝑛𝑧₀,𝑓^𝑛+1𝑧₀≤𝛼+𝛽+𝛾𝑠1−𝛽−𝛾𝑠^𝑛𝑑𝑧₀,𝑓𝑧₀. (21) For 𝑚>𝑛≥1 and using (d3) Definition 1, we have 𝑑𝑓^𝑛𝑧₀,𝑓^𝑚𝑧₀≤𝑠𝑑𝑓^𝑛𝑧₀,𝑓^𝑛+1𝑧₀+𝑠²𝑑𝑓^𝑛+1𝑧₀,𝑓^𝑛+2𝑧₀+⋯+𝑠^𝑚−𝑛𝑑𝑓^𝑚−1𝑧₀,𝑓^𝑚𝑧₀≤1𝑠^𝑛−1𝑑𝑧₀,𝑓𝑧₀^𝑚−1_𝑗=𝑛𝑠^𝑗𝛼+𝛽+𝛾𝑠1−𝛽−𝛾𝑠^𝑗using()<121𝑠^𝑛−1𝑑𝑧₀,𝑓𝑧₀^∞_𝑗=𝑛𝑠^𝑗𝛼+𝛽+𝛾𝑠1−𝛽−𝛾𝑠^𝑗. (22) Since 𝑠[(𝛼+𝛽+𝛾𝑠)/(1−𝛽−𝛾𝑠)]<1, then {𝑓^𝑛𝑧₀} is a Cauchy sequence in 𝑋. By completeness of 𝑋, the sequence {𝑓^𝑛𝑧₀} converges to 𝑡∈𝑋. Since 𝐺 is a (𝐶_𝑓)-graph, there exists a subsequence {𝑓^𝑛𝑘𝑧₀} and a natural number 𝑝 such that (𝑓^𝑛𝑘𝑧₀,𝑡)∈𝐸(𝐺) for all 𝑘≥𝑝. Using (18) for all 𝑘≥𝑝, we have 𝑑𝑓^𝑛𝑘+1𝑧₀𝑓,𝑓𝑡≤𝛼𝑑^𝑛𝑘𝑧₀,𝑡+𝛽𝑑𝑓^𝑛𝑘𝑧₀,𝑓^𝑛𝑘+1𝑧₀𝑑𝑓+𝑑𝑡,𝑓𝑡+𝛾^𝑛𝑘𝑧₀,𝑓𝑡+𝑑𝑡,𝑓^𝑛𝑘+1𝑧₀. (23) Since the 𝑏-metric 𝑑 is continuous and 𝛽+𝛾<1, letting 𝑘→∞ inequality (23) yields 𝑓𝑡=𝑡. Also note that (𝑧₀,𝑓𝑧₀,𝑓²𝑧₀,…,𝑓^𝑛1𝑧₀,…,𝑓^𝑛𝑝,𝑡) is a path in 𝐺 and hence in 𝐺, therefore 𝑡∈[𝑧₀]𝐺.

We note that Theorem 27 does not guarantee the uniqueness of fixed point, but this can be accomplished under some assumptions as in the following theorem.

Theorem 28. In addition to the hypothesis of Theorem 27, one further assumes that if 𝛼+2𝑠𝛽+2𝛾<1 for the same set of 𝛼,𝛽,𝛾≥0 and for any two fixed points 𝑡₁,𝑡₂∃𝑧∈𝑋 such that (𝑡₁,𝑧) and (𝑡₂,𝑧)∈𝐸(𝐺). Then, 𝑓 has a unique fixed point.

Proof. Let 𝑡₁ and 𝑡₂ be two fixed points of 𝑓, and then there exists 𝑧∈𝑋 such that (𝑡₁,𝑧),(𝑡₂,𝑧)∈𝐸(𝐺). By induction, we have (𝑡₁,𝑓^𝑛𝑧),(𝑡₂,𝑓^𝑛𝑧)∈𝐸(𝐺) for all 𝑛=0,1,…. From (18), we have 𝑑𝑡₁,𝑓^𝑛𝑧≤𝛼𝑑𝑡₁,𝑓^𝑛−1𝑧+𝛽𝑑𝑓^𝑛−1𝑧,𝑓^𝑛𝑧+𝛾𝑑𝑡₁,𝑓^𝑛𝑧+𝑑𝑓^𝑛−1𝑧,𝑡₁≤𝛼𝑑𝑡₁,𝑓^𝑛−1𝑧+𝛽𝑠𝑑𝑓^𝑛−1𝑧,𝑡₁+𝑑𝑡₁,𝑓^𝑛𝑧+𝛾𝑑𝑡₁,𝑓^𝑛𝑧+𝑑𝑓^𝑛−1𝑧,𝑡₁. (24) On rearranging, 𝑑𝑡₁,𝑓^𝑛𝑧≤𝛼+𝑠𝛽+𝛾𝑑𝑡1−𝑠𝛽−𝛾¹,𝑓^𝑛−1𝑧,∀𝑛=1,2,…. (25) Continuing recursively, (25) gives 𝑑𝑡₁,𝑓^𝑛𝑧≤𝛼+𝑠𝛽+𝛾1−𝑠𝛽−𝛾^𝑛𝑑𝑡₁,𝑧. (26)
Since [(𝛼+𝑠𝛽+𝛾)/(1−𝑠𝛽−𝛾)]<1, then lim_𝑛→∞𝑑(𝑡₁,𝑓^𝑛𝑧)=0. Similarly, one can show that lim_𝑛→∞𝑑(𝑡₂,𝑓^𝑛𝑧)=0, which by using (d3) Definition 1 infers that 𝑑(𝑡₁,𝑡₂)=0.

Suppose that (𝑋,⪯) is a partially ordered set. Consider graph 𝐺₂ consisting of 𝐸(𝐺₂)={(𝑥,𝑦)∈𝑋×𝑋∶𝑥⪯𝑦or𝑦⪯𝑥}, and 𝑉(𝐺₂) coincides with 𝑋. We note that if a self-mapping 𝑓 is monotone with respect to the order, then, for graph 𝐺₂, it is obvious that 𝑓 is edge-preserving, or equivalently we can say that 𝑓 maps comparable elements onto comparable elements.

Following corollaries are the direct consequences of Theorem 28.

Corollary 29. Let (𝑋,𝑑) be a complete metric space, where 𝑋 is partially ordered set with respect to ⪯. Let 𝑓∶𝑋→𝑋 be nondecreasing (or nonincreasing) with respect to ⪯. Assume that there exists 𝛼,𝛽,𝛾≥0 with 𝛼+2𝛽+2𝛾<1 such that, 𝑑𝑑𝑑,𝑓𝑥,𝑓𝑦≤𝛼𝑑𝑥,𝑦+𝛽𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦+𝛾𝑥,𝑓𝑦+𝑑𝑦,𝑓𝑥 (27) for all comparable 𝑥,𝑦∈𝑋. If the following conditions hold: (i)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑓𝑥₀, (ii)for nondecreasing (or nonincreasing) sequence {𝑥_𝑛}→𝑥∈𝑋, there exists a subsequence {𝑥_𝑛𝑘} such that 𝑥_𝑛𝑘⪯𝑥, for  all  𝑘.Then, 𝑓 has a fixed point. Moreover, if for  all 𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, then the fixed point is unique.

Corollary 30. Let (𝑋,𝑑) be a complete metric space, where 𝑋 is partially ordered set with respect to ⪯. Let 𝑓∶𝑋→𝑋 be nondecreasing (or nonincreasing) with respect to ⪯. Assume that there exists a constant 0<𝑐<1/2 such that 𝑑𝑓𝑥,𝑓𝑦≤𝑐𝑑𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦, (28) for  all comparable 𝑥,𝑦∈𝑋. If the following conditions hold: (i)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑓𝑥₀, (ii)for nondecreasing (or nonincreasing) sequence {𝑥_𝑛}→𝑥∈𝑋, there exists a subsequence {𝑥_𝑛𝑘} such that 𝑥_𝑛𝑘⪯𝑥, for  all  𝑘.Then, 𝑓 has a fixed point. Moreover, if for all 𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, then the fixed point is unique.

Corollary 31. Let (𝑋,𝑑) be a complete metric space, where 𝑋 is partially ordered set with respect to ⪯. Let 𝑓∶𝑋→𝑋 be nondecreasing (or nonincreasing) with respect to ⪯. Assume that there exists a constant 0<𝑐<1/2 such that 𝑑𝑓𝑥,𝑓𝑦≤𝑐𝑑𝑥,𝑓𝑦+𝑑𝑦,𝑓𝑥, (29) for all comparable 𝑥,𝑦∈𝑋. If the following conditions hold: (i)there exists 𝑥₀∈𝑋 such that 𝑥₀⪯𝑓𝑥₀, (ii)for nondecreasing (or nonincreasing) sequence {𝑥_𝑛}→𝑥∈𝑋, there exists a subsequence {𝑥_𝑛𝑘} such that 𝑥_𝑛𝑘⪯𝑥, for  all 𝑘.Then, 𝑓 has a fixed point. Moreover, if for all 𝑥,𝑦∈𝑋 there exists 𝑧∈𝑋 such that 𝑥⪯𝑧 and 𝑦⪯𝑧, then the fixed point is unique.

Remark 32. We note that in Theorem 27, the condition “there is 𝑧₀ in 𝑋 for which (𝑧₀,𝑓𝑧₀) is an edge in 𝐺” yields 𝑓^𝑛𝑧₀→𝑡, where 𝑡∈𝑋 is a fixed point of 𝑓. Consider a graph 𝐺∶=(𝑋,𝑋×𝑋). For such graph under the assumptions of Theorems 27 and 28, it infers that 𝑓 is a Picard operator. Thus, many standard fixed point theorems can be easily deduced from Theorem 28 as follows.

Corollary 33 (Hardy and Rogers [27]). Let (𝑋,𝑑) be a complete metric space, and let 𝑓∶𝑋→𝑋. Suppose that there exists constants 𝛼,𝛽,𝛾≥0 such that 𝑑𝑑𝑑,𝑓𝑥,𝑓𝑦≤𝛼𝑑𝑥,𝑦+𝛽𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦+𝛾𝑥,𝑓𝑦+𝑑𝑦,𝑓𝑥 (30) for all 𝑥,𝑦∈𝑋, where 𝛼+2𝛽+2𝛾<1; then 𝑓 has a unique fixed point in 𝑋.

Corollary 34 (Kannan [28]). Let (𝑋,𝑑) be a complete metric space, and let 𝑓∶𝑋→𝑋. Suppose that there exists a constants 𝑐 such that 𝑑𝑓𝑥,𝑓𝑦≤𝑐𝑑𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦, (31) for all 𝑥,𝑦∈𝑋, where 0<𝑐<1/2; then 𝑓 is Picard operator.

Corollary 35 (Chatterjea [29]). Let (𝑋,𝑑) be a complete metric space, and let 𝑓∶𝑋→𝑋. Suppose that there exists a constants 𝑐 such that 𝑑𝑓𝑥,𝑓𝑦≤𝑐𝑑𝑥,𝑓𝑦+𝑑𝑦,𝑓𝑥, (32) for all 𝑥,𝑦∈𝑋, where 0<𝑐<1/2; then 𝑓 is Picard operator.

3. Applications

Let 𝑋 be a nonempty set, let 𝑚 be a positive integer, {𝑋_𝑖}^𝑚_𝑖=1 be nonempty closed subsets of 𝑋, and let 𝑓∶⋃^𝑚_𝑖=1𝑋_𝑖→⋃^𝑚_𝑖=1𝑋_𝑖 be an operator. Then, 𝑋∶=⋃^𝑚_𝑖=1𝑋_𝑖 is known as cyclic representation of 𝑋 w.r.t. 𝑓 if 𝑓𝑋₁⊂𝑋₂,…,𝑓𝑋_𝑚−1⊂𝑋_𝑚,𝑓𝑋_𝑚⊂𝑋₁, (33) and operator 𝑓 is known as cyclic operator [30].

Theorem 36. Let (𝑋,𝑑) be complete 𝑏-metric space such that 𝑑 is continuous functional on 𝑋×𝑋. Let 𝑚 be positive integer, let {𝑋_𝑖}^𝑚_𝑖=1 be nonempty closed subsets of 𝑋, let 𝑌∶=⋃^𝑚_𝑖=1𝑋_𝑖, 𝜑∶ℝ⁺→ℝ⁺ be a 𝑏-comparison function, and let 𝑓∶𝑌→𝑌. Further, suppose that (i)⋃^𝑚_𝑖=1𝑋_𝑖 is cyclic representation of  𝑌 w.r.t. 𝑓,(ii)𝑑(𝑓𝑥,𝑓𝑦)≤𝜑(𝑑(𝑥,𝑦)) whenever 𝑥∈𝑋_𝑖,𝑦∈𝑋_𝑖+1, where 𝑋_𝑚+1=𝑋₁.Then, 𝑓 has a unique fixed point 𝑡∈⋂^𝑚_𝑖=1𝑋_𝑖 and 𝑓^𝑛𝑦→𝑡 for any 𝑦∈⋃^𝑚_𝑖=1𝑋_𝑖.

Proof. We note that (𝑌,𝑑) is complete 𝑏-metric space. Let us consider a graph 𝐺 consisting of 𝑉(𝐺)∶=𝑌 and 𝐸(𝐺)∶=Δ∪{(𝑥,𝑦)∈𝑌×𝑌∶𝑥∈𝑋_𝑖,𝑦∈𝑋_𝑖+1;𝑖=1,…,𝑚}. By (i), it follows that 𝑓 preserves edges. Now, let 𝑓^𝑛𝑥→𝑥^∗ in 𝑌 such that (𝑓^𝑛𝑥,𝑓^𝑛+1𝑥)∈𝐸(𝐺) for all 𝑛≥1; then in view of (33), sequence {𝑓^𝑛𝑥} has infinitely many terms in each 𝑋_𝑖 so that one can easily extract a subsequence of {𝑓^𝑛𝑥} converging to 𝑥^∗ in each 𝑋_𝑖. Since 𝑋_𝑖's are closed, then 𝑥^∗∈⋂^𝑚_𝑖=1𝑋_𝑖. Now, it is easy to form a subsequence {𝑓^𝑛𝑘𝑥} in some 𝑋_𝑗, 𝑗∈{1,…,𝑚} such that (𝑓^𝑛𝑘𝑥,𝑥^∗)∈𝐸(𝐺) for 𝑘≥1, and it indicates that 𝐺 is weakly connected (𝐶_𝑓)-graph, and thus conclusion follows from Theorem 18.

Remark 37 (see [31, Theorem  2.1(1)]). It can be deduced from Theorem 36 if (𝑋,𝑑) is a metric space.

On the same lines as in proof of Theorem 36, we obtain the following consequence of Theorem 28.

Theorem 38. Let (𝑋,𝑑) be a complete 𝑏-metric space such that 𝑑 is continuous functional on 𝑋×𝑋. Let 𝑚 be positive integer, let {𝑋_𝑖}^𝑚_𝑖=1 be nonempty closed subsets of 𝑋, let 𝑌∶=⋃^𝑚_𝑖=1𝑋_𝑖, and let 𝑓∶𝑌→𝑌. Further, suppose that (i)⋃^𝑚_𝑖=1𝑋_𝑖 is cyclic representation of 𝑌 w.r.t. 𝑓,(ii)there exist 𝛼,𝛽,𝛾≥0 with 𝑠𝛼+𝑠(𝑠+1)𝛽+𝑠(𝑠+1)𝛾<1 such that 𝑑𝑑𝑑,𝑓𝑥,𝑓𝑦≤𝛼𝑑𝑥,𝑦+𝛽𝑥,𝑓𝑥+𝑑𝑦,𝑓𝑦+𝛾𝑥,𝑓𝑦+𝑑𝑦,𝑓𝑥 (34)
whenever 𝑥∈𝑋_𝑖,𝑦∈𝑋_𝑖+1, where 𝑋_𝑚+1=𝑋₁.
Then, 𝑓 has a fixed point 𝑡∈⋂^𝑚_𝑖=1𝑋_𝑖.

Remark 39. [32, Theorem 7] and [33, Theorem 3.1] can be deduced from Theorem 38, but it does not ensure 𝑓 to be a Picard operator.

Remark 40. We note that in proof [32, Theorem 7], the author's argument to prove that is 𝐺 (as assumed in proof of Theorem 36), a (𝐶)-graph is valid only if the terms of sequence {𝑥_𝑛} are Picard iterations; otherwise it is void. For example, let 𝑌=⋃³_𝑖=1𝑋_𝑖 where 𝑋₁={1/𝑛∶𝑛isodd}∪{0}, 𝑋₂={1/𝑛∶𝑛iseven}∪{0}, 𝑋₃={0}, and define 𝑓∶𝑌→𝑌 as 𝑓𝑥=𝑥/2;𝑥∈𝑌⧵𝑋₂ and 𝑓𝑥=0;𝑥∈𝑋₂. We see that (33) is satisfied and 1/𝑛→0, but 𝑋₃ does not contain infinitely many terms of {𝑥_𝑛}. In the following, we give the corrected argument to prove that 𝐺 is a (𝐶)-graph.

Let 𝑥_𝑛→𝑥 in 𝑋 such that (𝑥_𝑛,𝑥_𝑛+1)∈𝐸(𝐺) for all 𝑛≥1. Keeping in mind construction of 𝐺, there exists at least one pair of closed sets {𝑋_𝑗,𝑋_𝑗+1} for some 𝑗∈{1,2,…,𝑚} such that both sets contain infinitely many terms of sequence {𝑥_𝑛}, since 𝑋_𝑖's are closed so that 𝑥∈𝑋_𝑗∩𝑋_𝑗+1 for some 𝑗∈{1,…,𝑚}, and thus one can easily extract a subsequence such that (𝑥_𝑛𝑘,𝑥)∈𝐸(𝐺) holds for 𝑘≥1.

Now, we establish an existence theorem for the solution of an integral equation as a consequence of our Theorem 18.

Theorem 41. Consider the integral equation 𝑥(𝑡)=^𝑏_𝑎𝑘(𝑡,𝑠,𝑥(𝑠))𝑑𝑠+𝑔(𝑡),𝑡∈[𝑎,𝑏], (35) where 𝑘∶[𝑎,𝑏]×[𝑎,𝑏]×ℝ^𝑛→ℝ^𝑛 and 𝑔∶[𝑎,𝑏]→ℝ^𝑛 is continuous. Assume that(i)𝑘(𝑡,𝑠,⋅)∶ℝ^𝑛→ℝ^𝑛 is nondecreasing for each 𝑡,𝑠∈[𝑎,𝑏],(ii)there exists a (𝑐)-comparison function 𝜑∶ℝ⁺→ℝ⁺ and a continuous function 𝑝∶[𝑎,𝑏] × [𝑎,𝑏]→ℝ⁺ such that |𝑘(𝑡,𝑠,𝑥(𝑠))−𝑘(𝑡,𝑠,𝑦(𝑠))|≤𝑝(𝑡,𝑠)𝜑(|𝑥(𝑠)−𝑦(𝑠)|) for each 𝑡,𝑠∈[𝑎,𝑏] and 𝑥≤𝑦 (i.e., 𝑥(𝑡)≤𝑦(𝑡) for all 𝑡∈[𝑎,𝑏]),(iii)sup_{𝑡∈[𝑎,𝑏]}∫^𝑏_𝑎𝑝(𝑡,𝑠)𝑑𝑠≤1,(iv)there exists 𝑥₀∈𝐶([𝑎,𝑏],ℝ^𝑛) such that 𝑥₀(𝑡)≤∫^𝑏_𝑎𝑘(𝑡,𝑠,𝑥₀(𝑠))𝑑𝑠+𝑔(𝑡) for all 𝑡∈[𝑎,𝑏]. Then, the integral equation (35) has a unique solution in the set {𝑥∈𝐶([𝑎,𝑏],ℝ^𝑛)∶𝑥(𝑡)≤𝑥₀(𝑡) or 𝑥(𝑡)≥𝑥₀(𝑡) , for all 𝑡∈[𝑎,𝑏]}.

Proof. Let (𝑋,‖⋅‖_∞), where 𝑋=𝐶([𝑎,𝑏],ℝ^𝑛), and define a mapping 𝑇∶𝐶([𝑎,𝑏],ℝ^𝑛)→𝐶([𝑎,𝑏],ℝ^𝑛) by 𝑇𝑥(𝑡)=^𝑏_𝑎𝑘(𝑡,𝑠,𝑥(𝑠))𝑑𝑠+𝑔(𝑡),𝑡∈[𝑎,𝑏]. (36) Consider a graph 𝐺 consisting of 𝑉(𝐺)∶=𝑋 and 𝐸(𝐺)={(𝑥,𝑦)∈𝑋×𝑋∶𝑥(𝑡)≤𝑦(𝑡) for all 𝑡∈[𝑎,𝑏]}. From property (i), we observe that the mapping 𝑇 is nondecreasing, thus 𝑇 preserves edges. Furthermore, 𝐺 is a (𝐶)-graph; that is, for every nondecreasing sequence {𝑥_𝑛}⊂𝑋 which converges to some 𝑧∈𝑋; then 𝑥_𝑛(𝑡)≤𝑧(𝑡) for all 𝑡∈[𝑎,𝑏]. Now, for every 𝑥,𝑦∈𝑋 with (𝑥,𝑦)∈𝐸(𝐺), we have (𝑡)(𝑡≤𝑇𝑥−𝑇𝑦)_𝑏𝑎𝑘(𝑡,𝑠,𝑥(𝑠))−𝑘𝑡,𝑠,𝑦(𝑠)𝑑𝑠≤^𝑏_𝑎𝑝(𝑡,𝑠)𝜑𝑥(𝑠)−𝑦(𝑠)𝑑𝑠≤𝜑𝑥−𝑦_∞^𝑏_𝑎𝑝(𝑡,𝑠)𝑑𝑠. (37) Hence, 𝑑(𝑇𝑥,𝑇𝑦)≤𝜑(𝑑(𝑥,𝑦)). From (iv), we have (𝑥₀,𝑇𝑥₀)∈𝐸(𝐺), so that [𝑥₀]𝐺={𝑥∈𝐶([𝑎,𝑏],ℝ^𝑛)∶𝑥(𝑡)≤𝑥₀(𝑡) or 𝑥(𝑡)≥𝑥₀(𝑡) for all 𝑡∈[𝑎,𝑏]}. The conclusion follows from Theorem 18.

Note that Theorem 41 specifies region of solution which invokes the novelty of our result.