Comp. Appl. Math. https://doi.org/10.1007/s40314-018-0614-6

On generalized Ri’s contraction mappings and its applications Pathaithep Kumrod1 · Wutiphol Sintunavarat1

Received: 9 December 2017 / Accepted: 16 March 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Abstract In this work, we improve the Ri’s fixed point theorem in Ri (Indag Math 27:85– 93, 2016) by creating the ϕ-fixed point version of such result in metric spaces. These results generalize and extend several well-known comparable results in the literature. Several applications and some examples of our theorems are also given. Keywords Banach contraction mapping · Boyd and Wong’s contraction mapping · ϕ-fixed point Mathematics Subject Classification 54H25 · 47H10

1 Introduction and preliminaries In 1992, Banach (1922) published the well-known result called the Banach contraction principle (briefly, BCP), which states that, if (X, d) is a complete metric space and T : X → X is a mapping satisfying the contractive condition, i.e., d(T x, T y) ≤ kd(x, y) for each x, y ∈ X , where k ∈ [0, 1), then T has a unique fixed point. This principle has a lot of citations in mathematical analysis fields and a most important role in many fields. Later, sevaral mathematicians proved many generalizations of the Banach contraction principle. In the sequel, Boyd and Wong (1969) established a similar result, which was followed by replacing the constant k by some control function, which generalizes the result of Banach.

Communicated by Carlos Conca.

B

Wutiphol Sintunavarat [email protected] Pathaithep Kumrod [email protected]

1

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Rangsit, Pathum Thani 12121, Thailand

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Theorem 1.1 (Boyd–Wong’s (1969) fixed point theorem) Let (X, d) be a complete metric space and T : X → X be a mapping satisfying d(T x, T y) ≤ λ(d(x, y)) for each x, y ∈ X,

(1.1)

where P = {d(x, y)|x, y ∈ X }, P is the closure P, λ : P → [0, ∞) is an upper semicontinuous function from the right on P, and satisfies λ(t) < t for all t ∈ P/{0}. Then, T has a unique fixed point p ∈ X , and limn→∞ T n x = p for each x ∈ X . On the other hand, several authors extended the metric space into various generalized metric spaces. One of most interesting one is due to Matthews (1994), which is called a partial metric space. This generalized the concept of a metric space in the sense that the distance from a point to itself need not be equal to zero. Various authors have obtained many useful fixed point results in these spaces. Recently, Jleli et al. (2014) introduced the concept of ϕ-fixed point and also proved the ϕ-fixed point results on a metric space, which improves the famous Banach contraction principle. Furthermore, they also claimed that these ϕ-fixed point results in metric spaces can be derived from some results in partial metric spaces. Motivated by the works of Boyd and Wong (1969) and Jleli et al. (2014), Karapinar et al. (2015) established some new ϕ-fixed point theorem, which generalizes the results of both Boyd and Wong (1969) and Jleli et al. (2014). They also presented a partial metric version of the Boyd–Wong’s fixed point theorem by using some new process for setting of a control function ϕ. Quite recently, Ri (2016) proved Boyd–Wong’s fixed point theorems in the case of the condition λ, in (1.1), being upper semi-continuous and replaced by lim sups→t + λ(s) < t for all t > 0. A part of Ri (2016), Remark 2.1 shows that the condition of Boyd–Wong fixed point theorem can be implied to this condition. They also give some example to show that the converse does not hold. Theorem 1.2 [Ri’s fixed point theorem Ri (2016)] Let (X, d) be a complete metric space and T : X → X be a mapping satisfying d(T x, T y) ≤ λ(d(x, y)) for each x, y ∈ X,

(1.2)

where λ : [0, ∞) → [0, ∞) satisfies λ(0) = 0, λ(t) < t and lim sups→t + λ(s) < t for all t > 0. Then, T has a unique fixed point p ∈ X . The purpose of this paper is to generalize some ϕ-fixed point result of Karapinar et al. (2015) using a new control function introduced by Ri (2016). Also, we present a result on the existence and uniqueness of fixed points in the setting of partial metric spaces. The readers can see the relation of all focused results in Fig. 1.

BCP

Reduce

Reduce

Reduce

Karapinar et al.’s result

Reduce

Fig. 1 Relationship between several fixed points and ϕ-fixed point results

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Ri’s result Reduce

Reduce

Reduce

Jleli et al.’s result

Boyd and Wong’s result

Our result

On generalized Ri’s contraction mappings…

2 Preliminaries Throughout this paper, for a metric space (X, d) and a given mapping T : X → X , we denote the set of all fixed points of T by FT := {x ∈ X : T x = x} and we denote the set of all zeros of the function ϕ by Z ϕ := {x ∈ X : ϕ(x) = 0}. Furthermore, unless otherwise specified, F will be denoted by the class of all functions F : [0, ∞) → [0, ∞) satisfying the following conditions: (F1) (F2) (F3)

max{a, b} ≤ F(a, b, c) for all a, b, c ∈ [0, ∞); F(a, 0, 0) = a for all a ≥ 0; F is continuous.

Example 2.1 Let F1 , F2 , F3 , F4 , F5 : [0, ∞)3 → [0, ∞) be defined by F1 (a, b, c) = a + b + c, F2 (a, b, c) = a + b2 + c, F3 (a, b, c) = (a + b)ec , F4 (a, b, c) = max{a, b} + c, F5 (a, b, c) = a + b + b3 + c for all a, b, c ∈ [0, ∞). Then F1 , F2 , F3 , F4 , F5 ∈ F . Next, we recall the definition of a partial metric space and its properties. Definition 2.2 (Matthews 1994) Let X be a nonempty set. A mapping p : X × X → [0, ∞) is said to be a partial metric (briefly, p-metric) if and only if the following conditions hold for any x, y, z ∈ X : ( p1 ) ( p2 ) ( p3 ) ( p4 )

p(x, x) = p(x, x) ≤ p(x, y) = p(x, y) ≤

p(y, y) = p(x, y) ⇐⇒ x = y (equality); p(x, y) (small self-distances); p(y, x) (symmetry); p(x, z) + p(z, y) − p(z, z) (triangularity).

Also, the pair (X, p) is called a partial metric space. Note that a metric is evidently a partial metric. However, a partial metric on X need not be a metric on X . Here, we give some examples of a partial metric which is not a metric. Example 2.3 (Matthews 1994) Let X = [0, ∞) and the function p : X × X → [0, ∞) be defined by p(x, y) = max{x, y} for all x, y ∈ X . Then, p is a partial metric. Example 2.4 (Matthews 1994) Let X = {[a, b] : a, b ∈ R and a ≤ b} and the function p : X × X → [0, ∞) defined by p([a, b], [c, d]) = max{b, d} − min{a, c}. Then, p is a partial metric on X .

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Remark 2.5 (Matthews 1994) If p is partial metric on a nonempty set X , then the function d p : X × X → [0, ∞) defined by d p (x, y) := 2 p(x, y) − p(x, x) − p(y, y) for all x, y ∈ X

(2.1)

is a metric on X . Lemma 2.6 (Matthews 1994) Let (X, p) be a partial metric space. Then the following assertions hold: (i) {xn } is a Cauchy sequence in (X, p) if and only if {xn } is a Cauchy sequence in the metric space (X, d p ); (ii) the partial metric space (X, p) is complete if and only if the metric space (X, d p ) is complete; ((iii) for each sequence {xn } in X and x ∈ X , lim d p (xn , x) = 0 ⇐⇒ p(x, x) = lim p(xn , x) =

n→∞

n→∞

lim p(xn , xm ).

n,m→∞

3 Main results In this section, we introduce the concept of new type of mappings and prove ϕ-fixed point results for these mappings. Definition 3.1 Let (X, d) be a metric space, and F ∈ F , ϕ : X → [0, ∞) and λ : [0, ∞) → [0, ∞) be two given mappings such that λ(t) < t for t > 0, λ(0) = 0 and lim sups→t + λ(s) < t for all t > 0. We say that the mapping T : X → X is (F, λ, ϕ)-contraction if F (d(T x, T y), ϕ(T x), ϕ(T y)) ≤ λ(F(d(x, y), ϕ(x), ϕ(y)))

(3.1)

for all x, y ∈ X . Remark 3.2 Note that, in the case F(a, b, c) = a + b + c for each a, b, c ∈ [0, ∞) and ϕ(x) = 0 for each x ∈ X , the condition (3.1) can be converted directly to (1.2). Now, let us start this section with some auxiliary tools for proving the ϕ-fixed point result. Lemma 3.3 Let (X, d) be a metric space, and F ∈ F , ϕ : X → [0, ∞) and λ : [0, ∞) → [0, ∞) be two given mappings such that λ(t) < t for t > 0, λ(0) = 0 and lim sups→t + λ(s) < t for all t > 0. If T : X → X is an (F, λ, ϕ)-contraction mapping and {x n } is a Picard iteration with the initial point x0 ∈ X such that xn = xn+1 for all n ∈ N ∪ {0}, then lim d(xn , xn+1 ) = lim ϕ(xn ) = 0 for each x ∈ X.

n→∞

n→∞

(3.2)

Proof For simplicity, we denote an := F(d(xn , xn+1 ), ϕ(xn ), ϕ(xn+1 )), n ∈ N. From the contractive condition (3.1) and the condition λ(t) < t for each t > 0, we deduce that an+2 ≤ λ(an+1 ) < an+1 ≤ λ(an ) < an

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On generalized Ri’s contraction mappings…

for all n ∈ N. It implies that {an } and {λ(an )} is a strictly decreasing sequence of non-negative real number and bounded below. This yields that limn→∞ an exist. Assume that a := lim an ≥ 0. n→∞

We will claim that a = 0. Suppose for the sake of contradiction that a > 0 and an = a + εn , where εn > 0. If lim sups→t + λ(s) < t for all t > 0, then we have lim suptn →a + < a whenever tn ↓ a + as n → +∞. This implies that 0
≤ lim λ(an ) n→+∞

≤ lim sup λ(s) n→+∞s∈(a,a

=

lim

n+1 )

sup λ(s)

εn+1 →+0s∈(a,a+εn+1 )

≤ lim sup λ(an ) ε→+0 s∈(a,a+ε)

< a, which is a contradiction. Thus, we must have limn→+0 an = 0. Now using the condition (F1), we get 0 ≤ ϕ(xn ) ≤ max{d(xn , xn+1 ), ϕ(xn )} ≤ F(d(xn , xn+1 ), ϕ(xn ), ϕ(xn+1 )) = an , and 0 ≤ d(xn , xn+1 ) ≤ max{d(xn , xn+1 ), ϕ(xn )} ≤ F(d(xn , xn+1 ), ϕ(xn ), ϕ(xn+1 )) = an for all n ∈ N. Letting the limit as n → ∞ in the above two inequalities, we obtain lim d(xn , xn+1 ) = lim ϕ(xn ) = 0 for each x ∈ X.

n→∞

n→∞



This completes the proof.

Lemma 3.4 Let (X, d) be a metric space, and F ∈ F , ϕ : X → [0, ∞) and λ : [0, ∞) → [0, ∞) be two given mappings such that λ(t) < t for t > 0, λ(0) = 0 and lim sups→t + λ(s) < t for all t > 0. If T : X → X is an (F, λ, ϕ)-contraction mapping and {x n } is a Picard iteration with the initial point x0 ∈ X such that xn = xn+1 for all n ∈ N ∪ {0}, then {xn } is a Cauchy sequence. Proof Suppose that {xn } is not a Cauchy sequence in (X, d). Then, there exist ε > 0 and two subsequence {xn k } and {xm k } of {xn } such that m k > n k > k and d(xm k , xn k ) ≥ ε for all n ∈ N. Then, we can choose m k to be assumed that d(xm k −1 , xn k ) < ε. For each k ∈ N, we obtain ε ≤ d(xm k , xn k ) ≤ d(xm k , xm k−1 ) + d(xm k−1 , xn k ) ≤ d(xm k , xm k−1 ) + ε. It follows from Lemma 3.3 that lim d(xm k , xn k ) = ε.

(3.3)

k→∞

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By using Lemma 3.3, (3.3), (F2) and the continuity of F, we get lim F(d(xm k , xn k ), ϕ(xm k ), ϕ(xn k )) = F(ε, 0, 0) = ε.

k→∞

(3.4)

Then by using the contractive condition (3.1), (F1) and the triangle inequality, we obtain, for all k ∈ N : d(xm k , xn k ) ≤ d(xm k , xm k +1 ) + d(xm k +1 , xn k +1 ) + d(xn k+1 , xn k ) ≤ d(xm k , xm k +1 ) + max{d(xm k +1 , xn k +1 ), ϕ(xm k +1 )} + d(xn k+1 , xn k ) ≤ d(xm k , xm k +1 ) + F(d(xm k +1 , xn k +1 ), ϕ(xm k +1 ), ϕ(xn k +1 )) + d(xn k+1 , xn k ) ≤ d(xm k , xm k +1 ) + λ(F(d(xm k , xn k ), ϕ(xm k ), ϕ(xn k ))) + d(xn k+1 , xn k ). Letting k → ∞ in the above inequality, using (3.2), (3.4), lim sups→t + λ(s) < t for all t > 0, and the continuity of F, we get ε = lim d(xm k , xn k ) k→∞

≤ lim supλ(F(d(xm k , xn k ), ϕ(xm k ), ϕ(xn k ))) k→∞

≤ lim sup λ(s)

ε →+0s∈(ε,ε+ε )

< ε, which is a contradiction. Then, {xn } is a Cauchy sequence in X .



Theorem 3.5 Let (X, d) be a complete metric space, and F ∈ F , ϕ : X → [0, ∞) and λ : [0, ∞) → [0, ∞) be two given mappings such that λ(t) < t for t > 0, λ(0) = 0 and lim sups→t + λ(s) < t for all t > 0. If T : X → X is an (F, λ, ϕ)-contraction mapping and ϕ is lower semi-continuous, then FT ⊂ Z ϕ and T has a unique ϕ-fixed point. Moreover, the Picard iteration {xn }, which is defined by xn = T xn−1 for all n ∈ N, where x0 ∈ X , converges to the unique ϕ-fixed point of T . Proof To prove FT ⊂ Z ϕ , let x be a fixed point of T . By using (3.1) with y = x, we obtain F(0, ϕ(T x), ϕ(T x)) ≤ λ(F(0, ϕ(x), ϕ(x))). Suppose that ϕ(x) = 0. From condition (F1) and λ(t) < t for each t > 0, we have λ(F(0, ϕ(x), ϕ(x))) < F(0, ϕ(x), ϕ(x)). This implies that F(0, ϕ(x), ϕ(x)) < F(0, ϕ(x), ϕ(x)), which is a contradiction. Hence, ϕ(x) = 0. Let x0 ∈ X be an arbitrary point and let {xn } be the Picard iteration defined by xn = T n x0 , n ∈ N. If there exists some n ∗ ∈ N such that xn ∗ = xn ∗ +1 , there is nothing to prove. We may assume that xn = xn+1 for all n ∈ N. From Lemma 3.4, we get {xn } is a Cauchy sequence in (X, d). As X is complete, there exists p ∈ X such that xn → p as n → ∞. Since ϕ is lower semi-continuous, from Lemma 3.3, we get 0 ≤ ϕ( p) ≤ lim supϕ(xn ) = 0 n→∞

123

(3.5)

On generalized Ri’s contraction mappings…

and so ϕ( p) = 0.

(3.6)

Next, we claim that p is a fixed point of T . Since T is an (F, λ, ϕ)-contraction mapping and λ(t) < t for each t > 0, we obtain   F d(T n+1 x, T p), ϕ(T n+1 x), ϕ(T p) ≤ λ(F(d(T n x, p), ϕ(T n x), ϕ( p))) ≤ F(d(T n x, p), ϕ(T n x), ϕ( p)) for all n ∈ N. Letting n → ∞ in the above inequality, using (3.2), (3.5), (3.6), (F2) , and the continuity of F, we get F(d( p, T p), 0, ϕ(T p)) ≤ F(0, 0, 0) = 0, which implies by condition (F1) that d( p, T p) = 0.

(3.7)

Therefore, from (3.6) and (3.7), we get that p is a ϕ-fixed point of T . Finally, we need to verify that the ϕ-fixed point p is unique. Let q be a ϕ-fixed point of T such that p = q. From condition (F1) and the contractive condition (3.1), we obtain d( p, q) = F(d( p, q), 0, 0) = F(d(T p, T q), 0, 0) ≤ λ(F(d( p, q), 0, 0)) = λ(d( p, q)) < d( p, q), which is a contradiction, since both sides of this strict inequality are equal. It yields that p = q. This finishes the proof.

The following example shows that Theorem 3.5 is more applicable than many other fixed point results. Example 3.6 Let X = [0, ∞) and d : X × X → R be defined by d(x, y) = |x − y| for all x, y ∈ X . Then, (X, d) is a complete metric space. Assume that T : X → X and λ : [0, ∞) → [0, ∞) are defined by ⎧ 4x ⎧ 2 4 0 ≤ t < 1, ⎪ ⎨ x , 0 ≤ x < 13 , ⎨ 5 2 4 , t = 1, T x = 0, 13 ≤ x ≤ 1, and λ(t) = 17  ⎩ 1 ⎪ 1 1 1 ⎩ 4 sin t−1 + 2 , t > 1. 3x , x > 1, 4 , 1 and λ is not upper semi-continuous from Obviously, T is not continuous at x = 13 the right; hence, the Banach contraction principle, the Boyd–Wong fixed point result and the main result of Karapinar et al. (2015) are not applicable. Also, the Ri fixed point result cannot be applied in this case. Indeed, for x = 1 and y = 2, we get

1 2 ≮ = λ(1) = λ(d(1, 2)) = λ(d(x, y)). 6 17 Next, we will show that Theorem 3.5 can be applied in this example. Now, we need the following functions, which are essential: ϕ : X → [0, ∞) and F : [0, ∞)3 → [0, ∞) are defined by d(T x, T y) = |T 1 − T 2| =

ϕ(x) = x 2 , x ∈ X and F(a, b, c) = a + b + c, a, b, c ≥ 0. It is easy to see that F ∈ F , λ(t) < t for t > 0, λ(0) = 0, lim sups→t + λ(s) < t for all t > 0 and ϕ is lower semi-continuous.

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Now, we claim that the mapping T satisfies the contractive condition (3.1). Suppose that x, y ∈ X . We have to consider the following cases:

4 4 Case 1 If (x, y) ∈ 0, 13 × 0, 13 , then we get F(d(T x, T y), ϕ(T x), ϕ(T y)) = d(T x, T y) + ϕ(T x) + ϕ(T y) = |T x − T y| + (T x)2 + (T y)2 = |x 2 − y 2 | + x 4 + y 4 16 2 8 16 2 |x − y| + x + y < 13 169 169 4 ≤ (|x − y| + x 2 + y 2 ) 5 = λ(|x − y| + x 2 + y 2 ) = λ(d(x, y) + ϕ(x) + ϕ(y)) = λ(F(d(x, y), ϕ(x), ϕ(y))). Case 2 If (x, y) ∈



4 13 , 1



, the claim is obvious.

Case 3 If (x, y) ∈ (1, ∞) × (1, ∞), then we get F(d(T x, T y), ϕ(T x), ϕ(T y)) = d(T x, T y) + ϕ(T x) + ϕ(T y) = |T x − T y| + (T x)2 + (T y)2 1 1 1 1 − + = + 2 3x 3y 9x 2 9y 1 < 2

 1 1 1 ≤ sin + 4 (|x − y| + x 2 + y 2 ) − 1 2 = λ(|x − y| + x 2 + y 2 ) = λ(d(x, y) + ϕ(x) + ϕ(y)) = λ(F(d(x, y), ϕ(x), ϕ(y))).

4 4  4  4 Case 4 Let (x, y) ∈ 0, 13 × 13 , 1 ∪ 13 , 1 × 0, 13 . Without loss of generality, we can

4

4  assume that x ∈ 0, 13 and y ∈ 13 , 1 . These cases were divided into two subcases: Subcase 4.1 If 0 ≤ |x − y| + x 2 + y 2 < 1, then we get F(d(T x, T y), ϕ(T x), ϕ(T y)) = d(T x, T y) + ϕ(T x) + ϕ(T y) = |T x − T y| + (T x)2 + (T y)2 = x2 + x4 < 0.135 3 < (|x − y| + x 2 + y 2 ) 4 = λ(|x − y| + x 2 + y 2 ) = λ(d(x, y) + ϕ(x) + ϕ(y)) = λ(F(d(x, y), ϕ(x), ϕ(y))).

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On generalized Ri’s contraction mappings…

Subcase 4.2 If |x − y| + x 2 + y 2 ≥ 1, then we get F(d(T x, T y), ϕ(T x), ϕ(T y)) = d(T x, T y) + ϕ(T x) + ϕ(T y) = |T x − T y| + (T x)2 + (T y)2 = x2 + x4 2 < 17  1 1 1 + < sin 4 (|x − y| + x 2 + y 2 ) − 1 2 = λ(|x − y| + x 2 + y 2 ) = λ(d(x, y) + ϕ(x) + ϕ(y)) = λ(F(d(x, y), ϕ(x), ϕ(y))).

4

4 Case 5 Let (x, y) ∈ 0, 13 × (1, ∞) ∪ (1, ∞) × 0, 13 . Without loss of generality, we can

4 assume that x ∈ 0, 13 and y ∈ (1, ∞). In these cases, we get F(d(T x, T y), ϕ(T x), ϕ(T y)) = d(T x, T y) + ϕ(T x) + ϕ(T y) = |T x − T y| + (T x)2 + (T y)2 1 1 + x4 + 2 = x 2 − 3y 9y

 1 1 1 < sin + 2 2 4 (|x − y| + x + y ) − 1 2 = λ(|x − y| + x 2 + y 2 ) = λ(d(x, y) + ϕ(x) + ϕ(y)) = λ(F(d(x, y), ϕ(x), ϕ(y))).

4  Case 6 Let (x, y) ∈ × (1, ∞) ∪ (1, ∞) × 13 , 1 . Without loss of generality, we

4 can assume that x ∈ 13 , 1 and y ∈ (1, ∞). In these cases, we get

4 13 , 1



F(d(T x, T y), ϕ(T x), ϕ(T y)) = d(T x, T y) + ϕ(T x) + ϕ(T y) = |T x − T y| + (T x)2 + (T y)2 1 1 = + 2 3y 9y

 1 1 1 < sin + 4 (|x − y| + x 2 + y 2 ) − 1 2 = λ(|x − y| + x 2 + y 2 ) = λ(d(x, y) + ϕ(x) + ϕ(y)) = λ(F(d(x, y), ϕ(x), ϕ(y))). From all cases, the condition (3.1) holds for all x, y ∈ X . In Figs. 2, 3 and 4, the reader can see some 3D surfaces in MATLAB to guarantee the validity of the comparison of the left hand side (L.H.S.) and the right hand side (R.H.S.) of (3.1). Therefore, all the required hypotheses of Theorem 3.5 are fulfilled. Thus, we deduce the existence and uniqueness of the ϕ-fixed point of T . In this case, a point 0 is a fixed point of T.

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Fig. 2 The value of the comparison of the L.H.S. and the R.H.S. of (3.1) in Case 1 and Case 3 1

L.H.S. R.H.S.

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Fig. 3 The value of the comparison of the L.H.S. and the R.H.S. of (3.1) in Subcase 4.1 and Subcase 4.2 0.7

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Fig. 4 The value of the comparison of the L.H.S. and the R.H.S. of (3.1) in Case 5 and Case 6

Remark 3.7 Taking ϕ(x) = 0 for all x ∈ X and F(a, b, c) = a + b + c for each a, b, c ∈ [0, ∞) in Theorem 3.5, we deduce immediately Ri’s fixed point result (Theorem 1.2, Ri 2016). Remark 3.8 (Ri 2016) If λ : [0, ∞) → [0, ∞) is an upper semi-continuous function from the right such that λ(t) < t for all t > 0, then lim sups→t + λ(s) < t for all t > 0.

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On generalized Ri’s contraction mappings…

Using Remark 3.8, we obtain immediately Corollary 3.9. Corollary 3.9 (Karapinar et al. 2015) Let (X, d) be a complete metric space, F ∈ F , ϕ : X → [0, ∞) and λ : [0, ∞) → [0, ∞) be two given mappings such that λ(t) < t for t > 0 and an upper semi-continuous function from the right such that λ(t) < t for all t > 0. If T : X → X satisfying F(d(T x, T y), ϕ(T x), ϕ(T y)) ≤ λ(F(d(x, y)), ϕ(x), ϕ(y)) and ϕ is lower semi-continuous, then FT ⊂ Z ϕ and T has a unique ϕ-fixed point. Moreover, the Picard iteration {xn }, which is defined by xn = T xn−1 for all n ∈ N, where x0 ∈ X , converges to the unique ϕ-fixed point of T . Remark 3.10 If we take λ(t) := kt for all t ∈ [0, ∞), where k ∈ [0, 1), then Corollary 3.9 reduces to Jleli et al.’s fixed point result (Theorem 2.1 in Jleli et al. 2014).

4 Application to the fixed point results in partial metric spaces In this section, we will use Theorem 3.5 to show that we can derive fixed point results in partial metric spaces. Theorem 4.1 Let (X, p) be a complete partial metric space and λ : [0, ∞) → [0, ∞) be a given mapping such that λ(t) < t, λ(0) = 0, lim sups→t + λ(s) < t. If T : X → X is a mapping satisfying the following condition: p(T x, T y) ≤ λ( p(x, y)) for each x, y ∈ X,

(4.1)

then T has a unique fixed point z ∈ X and p(z, z) = 0. Moreover, the Picard iteration {x n }, which is defined by xn = T xn−1 for all n ∈ N, where x0 ∈ X , converges to the unique fixed point of T . Proof Setting the metric d p on X, it is defined by d p (x, y) = 2 p(x, y) − p(x, x) − p(y, y) for each x, y ∈ X.

(4.2)

As an immediate Lemma 2.6, we obtain the completeness of the metric space (X, d p ). Let d (x,y) d : X × X → [0, ∞) be defined by d(x, y) = p 2 for all x, y ∈ X . Then, (X, d) is a complete metric space. Define mappings ϕ : X → [0, ∞) and F : [0, ∞)3 → [0, ∞) by ϕ(x) =

p(x, x) for all x ∈ X 2

and F(a, b, c) = a + b + c

for all a, b, c ∈ [0, ∞).

It is easy to see that ϕ is lower semi-continuous and F belongs to F . For each x, y ∈ X , we have 2 p(T x, T y) p(T x, T x) p(T y, T y) p(T x, T x) p(T y, T y) p(T x, T y) = − − + + 2 2 2 2 2 d p (T x, T y) p(T x, T x) p(T y, T y) + + = 2 2 2 = d(T x, T y) + ϕ(T x) + ϕ(T y) = F(d(T x, T y), ϕ(T x), ϕ(T y))

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P. Kumrod, W. Sintunavarat

and

2 p(x, y) p(x, x) p(y, y) p(x, x) p(y, y) − − + + 2 2 2 2 2 

d p (x, y) p(x, x) p(y, y) + + =λ 2 2 2 = λ (d(x, y) + ϕ(x) + ϕ(y))



λ( p(x, y)) = λ

= λ (F(d(x, y), ϕ(x), ϕ(y))) . From the above two equalities and (4.1), we obtain F(d(T x, T y), ϕ(x), ϕ(y)) ≤ λ(F(d(x, y), ϕ(x), ϕ(y))) for each x, y ∈ X. Then all conditions of Theorem 3.5 are satisfied and so T has a unique ϕ-fixed point z ∈ X . It yields that z is a unique fixed point of T and p(z, z) = 0. This completes the proof.

From Remark 3.8, we get immediately Boyd–Wong’s fixed point theorem in the setting of partial metric spaces. Corollary 4.2 Let (X, p) be a complete partial metric space and T : X → X be a mapping satisfying the following condition: p(T x, T y) ≤ λ( p(x, y)) for each x, y ∈ X, where λ : [0, ∞) → [0, ∞) is an upper semi-continuous function from the right and λ(t) < t. Then, T has a unique fixed point z ∈ X and p(z, z) = 0. Moreover, the Picard iteration {x n }, which is defined by xn = T xn−1 for all n ∈ N, where x0 ∈ X , converges to a fixed point of T. The following corollary immediately follows from Corollary 4.2 by taking λ(t) := kt for all t ∈ [0, ∞), where k ∈ [0, 1). Corollary 4.3 (Matthews 1994) Let (X, p) be a complete partial metric space and T : X → X be a mapping satisfying the following condition: ∃k ∈ [0, 1) such that p(T x, T y) ≤ kp(x, y) for all x, y ∈ X. Then, T has a unique fixed point z ∈ X and p(z, z) = 0. Moreover, the Picard iteration {x n }, which is defined by xn = T xn−1 for all n ∈ N, where x0 ∈ X , converges to a fixed point of T. Acknowledgements The second author would like to thank the Thailand Research Fund and Office of the Higher Education Commission, Grant no. MRG5980242, for financial support during the preparation of this manuscript.

References Banach S (1922) Sur les oprations dans les ensembles abstraits et leurs applications aux quations intgrales. Fund Math 3:133–181 Boyd DW, Wong JSW (1969) On nonlinear contractions. Proc Am. Math Soc 20:458–464 Jleli M, Samet B, Vetro C (2014) Fixed point theory in partial metric spaces via ϕ-fixed point’s concept in metric spaces. J Inequal Appl 2014:426 Karapinar E, O’Regan D, Samet B (2015) On the existence of fixed points that belong to the zero set of a certain function. Fixed Point Theory Appl 2015:152 Matthews S (1994) Partial metric topology. Ann N Y Acad Sci 728:183–197 Ri SI (2016) A new fixed point theorem in the fractal space. Indag Math 27:85–93

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