Some Fixed Point Theorems in Ordered 𝐺-Metric Spaces and Applications

Abstract

We study a number of fixed point results for the two weakly increasing mappings 𝑓 and 𝑔 with respect to partial ordering relation ⪯ in generalized metric spaces. An application to integral equation is given.

1. Introduction

The existence of fixed points in partially ordered sets has been at the center of active research. In fact, the existence of fixed point in partially ordered sets has been investigated in [1]. Moreover, Ran and Reurings [1] applied their results to matrix equations. Some generalizations of the results of [1] are given in [2–6]. In [6], O’Regan and Petruşel gave some existence results for the Fredholm and Volterra type.

The notion of 𝐺-metric space was introduced by Mustafa and Sims [7] as a generalization of the notion of metric spaces. Mustafa et al. studied many fixed point results in 𝐺-metric space [8–10] (also see [11–15]). In fact the study of common fixed points of mappings satisfying certain contractive conditions has been at the center of strong research activity. The following definition is introduced by Mustafa and Sims [7].

Definition 1.1 (see [7]). Let 𝑋 be a nonempty set and let 𝐺∶𝑋×𝑋×𝑋→𝐑⁺ be a function satisfying the following properties: (𝐺₁)𝐺(𝑥,𝑦,𝑧)=0 if 𝑥=𝑦=𝑧, (𝐺₂)0<𝐺(𝑥,𝑥,𝑦), for all 𝑥,𝑦∈𝑋 with 𝑥≠𝑦, (𝐺₃)𝐺(𝑥,𝑥,𝑦)≤𝐺(𝑥,𝑦,𝑧), for all 𝑥,𝑦,𝑧∈𝑋 with 𝑧≠𝑦, (𝐺₄)𝐺(𝑥,𝑦,𝑧)=𝐺(𝑥,𝑧,𝑦)=𝐺(𝑦,𝑧,𝑥)=⋯, symmetry in all three variables, (𝐺₅)𝐺(𝑥,𝑦,𝑧)≤𝐺(𝑥,𝑎,𝑎)+𝐺(𝑎,𝑦,𝑧), for all 𝑥,𝑦,𝑧,𝑎∈𝑋. Then the function 𝐺 is called a generalized metric, or, more specifically, a 𝐺-metric on 𝑋, and the pair (𝑋,𝐺) is called a 𝐺-metric space.

Definition 1.2 (see [7]). Let (𝑋,𝐺) be a 𝐺-metric space, and let {𝑥_𝑛} be a sequence of points of 𝑋, a point 𝑥∈𝑋 is said to be the limit of the sequence {𝑥_𝑛}, if lim_{𝑛,𝑚→+∞}𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)=0, and we say that the sequence {𝑥_𝑛} is 𝐺-convergent to 𝑥 or {𝑥_𝑛}𝐺-converges to 𝑥.

Thus, 𝑥_𝑛→𝑥 in a 𝐺-metric space (𝑋,𝐺) if for any 𝜀>0, there exists 𝑘∈𝐍 such that 𝐺(𝑥,𝑥_𝑛,𝑥_𝑚)<𝜀 for all 𝑚,𝑛≥𝑘.

Proposition 1.3 (see [7]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the following are equivalent: (1){𝑥_𝑛} is 𝐺-convergent to 𝑥; (2)𝐺(𝑥_𝑛,𝑥_𝑛,𝑥)→0 as 𝑛→+∞; (3)𝐺(𝑥_𝑛,𝑥,𝑥)→0 as 𝑛→+∞; (4)𝐺(𝑥_𝑛,𝑥_𝑚,𝑥)→0 as 𝑛,𝑚→+∞.

Definition 1.4 (see [7]). Let (𝑋,𝐺) be a 𝐺-metric space, a sequence {𝑥_𝑛} is called 𝐺-Cauchy if for every 𝜀>0, there is 𝑘∈𝐍 such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)<𝜀, for all 𝑛,𝑚,𝑙≥𝑘; that is 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑙)→0 as 𝑛,𝑚,𝑙→+∞.

Proposition 1.5 (see [7]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the following are equivalent: (1)the sequence {𝑥_𝑛} is 𝐺-Cauchy;(2)for every 𝜖>0, there is 𝑘∈𝐍 such that 𝐺(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)<𝜖, for all 𝑛,𝑚≥𝑘.

Definition 1.6 (see [7]). Let (𝑋,𝐺) and (𝑋^,𝐺^) be 𝐺-metric spaces, and let 𝑓∶(𝑋,𝐺)→(𝑋^,𝐺^) be a function. Then 𝑓 is said to be 𝐺-continuous at a point 𝑎∈𝑋 if and only if for every 𝜀>0, there is 𝛿>0 such that 𝑥,𝑦∈𝑋 and 𝐺(𝑎,𝑥,𝑦)<𝛿 implies 𝐺^(𝑓(𝑎),𝑓(𝑥),𝑓(𝑦))<𝜀. A function 𝑓 is 𝐺-continuous at 𝑋 if and only if it is 𝐺-continuous at all 𝑎∈𝑋.

Proposition 1.7 (see [7]). Let (𝑋,𝐺) be a 𝐺-metric space. Then the function 𝐺(𝑥,𝑦,𝑧) is jointly continuous in all three of its variables.

Every 𝐺-metric on 𝑋 will define a metric 𝑑_𝐺 on 𝑋 by𝑑_𝐺(𝑥,𝑦)=𝐺(𝑥,𝑦,𝑦)+𝐺(𝑦,𝑥,𝑥),∀𝑥,𝑦∈𝑋.(1.1) For a symmetric 𝐺-metric space,𝑑_𝐺(𝑥,𝑦)=2𝐺(𝑥,𝑦,𝑦),∀𝑥,𝑦∈𝑋.(1.2) However, if 𝐺 is not symmetric, then the following inequality holds:32𝐺(𝑥,𝑦,𝑦)≤𝑑^𝐺(𝑥,𝑦)≤3𝐺(𝑥,𝑦,𝑦),∀𝑥,𝑦∈𝑋.(1.3)

The following are examples of 𝐺-metric spaces.

Example 1.8 (see [7]). Let (𝐑,𝑑) be the usual metric space. Define 𝐺_𝑠 by 𝐺_𝑠(𝑥,𝑦,𝑧)=𝑑(𝑥,𝑦)+𝑑(𝑦,𝑧)+𝑑(𝑥,𝑧),(1.4) for all 𝑥,𝑦,𝑧∈𝐑. Then it is clear that (𝐑,𝐺_𝑠) is a 𝐺-metric space.

Example 1.9 (see [7]). Let 𝑋={𝑎,𝑏}. Define 𝐺 on 𝑋×𝑋×𝑋 by 𝐺(𝐺𝑎,(𝑎𝑎,,𝑏𝑎),=𝑎)1=,𝐺(𝑏𝐺,𝑏(𝑎,𝑏,)𝑏,=𝑏)0=,2(1.5) and extend 𝐺 to 𝑋×𝑋×𝑋 by using the symmetry in the variables. Then it is clear that (𝑋,𝐺) is a 𝐺-metric space.

Definition 1.10 (see [7]). A 𝐺-metric space (𝑋,𝐺) is called 𝐺-complete if every 𝐺-Cauchy sequence in (𝑋,𝐺) is 𝐺-convergent in (𝑋,𝐺).

The notion of weakly increasing mappings was introduced in by Altun and Simsek [16].

Definition 1.11 (see [16]). Let (𝑋,⪯) be a partially ordered set. Two mappings 𝐹,𝐺∶𝑋→𝑋 are said to be weakly increasing if 𝐹𝑥⪯𝐺𝐹𝑥 and 𝐺𝑥⪯𝐹𝐺𝑥, for all 𝑥∈𝑋.

Two weakly increasing mappings need not be nondecreasing.

Example 1.12 (see [16]). Let 𝑋=𝐑, endowed with the usual ordering. Let 𝐹,𝐺∶𝑋→𝑋 defined by 𝑔𝐹𝑥𝑥==⎧⎪⎨⎪⎩√𝑥0,,01≤<𝑥𝑥≤<1+,∞,0,𝑥,01≤<𝑥𝑥<≤+1∞,.(1.6) Then 𝐹 and 𝐺 are weakly increasing mappings. Note that 𝐹 and 𝐺 are not nondecreasing.

The aim of this paper is to study a number of fixed point results for two weakly increasing mappings 𝑓 and 𝑔 with respect to partial ordering relation (⪯) in a generalized metric space.

2. Main Results

Theorem 2.1. Let (𝑋,⪯) be a partially ordered set and suppose that there exists 𝐺-metric in 𝑋 such that (𝑋,𝐺) is 𝐺-complete. Let 𝑓,𝑔∶𝑋→𝑋 be two weakly increasing mappings with respect to ⪯. Suppose there exist nonnegative real numbers 𝑎, 𝑏, and 𝑐 with 𝑎+2𝑏+2𝑐<1 such that 𝐺(𝑓𝑥,𝑔𝑦,𝑔𝑦)≤+𝑎𝐺𝑐[(𝐺𝑥,(𝑦𝑥,,𝑦𝑔)𝑦+,𝑔𝑏𝑦[)𝐺+(𝐺𝑥,(𝑓𝑦𝑥,,𝑓𝑓𝑥𝑥,)𝑓+𝑥)]𝐺,(𝑦,𝑔𝑦,𝑔𝑦)](2.1)𝐺(𝑔𝑥,𝑓𝑦,𝑓𝑦)≤+𝑎𝐺𝑐[(𝐺𝑥,(𝑦𝑥,,𝑦𝑓)𝑦+,𝑓𝑏𝑦[)𝐺+(𝑥𝐺,(𝑔𝑦𝑥,,𝑔𝑔𝑥𝑥,)𝑔+𝑥)𝐺],(𝑦,𝑓𝑦,𝑓𝑦)](2.2) for all comparative 𝑥,𝑦∈𝑋. If 𝑓 or 𝑔 is continuous, then 𝑓 and 𝑔 have a common fixed point 𝑢∈𝑋.

Proof. By inequality (2.2), we have 𝐺(𝑔𝑦,𝑓𝑥,𝑓𝑥)≤+𝑎𝐺𝑐[(𝐺𝑦,(𝑥𝑦,,𝑥𝑓)𝑥+,𝑓𝑏𝑥[)𝐺+(𝑦𝐺,𝑔(𝑥𝑦,,𝑔𝑔𝑦𝑦,)𝑔+𝑦)𝐺].(𝑥,𝑓𝑥,𝑓𝑥)](2.3) If 𝑋 is a symmetric 𝐺-metric space, then by adding inequalities (2.1) and (2.3), we obtain 𝐺(𝑓𝑥≤,+𝑔𝑎𝑦[2,𝐺𝑐𝑔[𝑦(𝐺𝑥),+(𝑦𝑥,𝐺,𝑦𝑔)(𝑦𝑔+,𝑦𝑔,𝐺𝑦𝑓)𝑥(+𝑦,,𝑓𝐺𝑥𝑥,)(𝑥𝑦),]𝑓+𝑥,2𝑓𝑏𝑥[)𝐺],(𝑥,𝑓𝑥,𝑓𝑥)+𝐺(𝑦,𝑔𝑦,𝑔𝑦)](2.4) which further implies that 𝑑_𝐺(𝑓𝑥,𝑓𝑦)≤𝑎𝑑_𝐺(𝑥,𝑦)+𝑏𝑑_𝐺(𝑥,𝑓𝑥)+𝑑_𝐺(𝑦,𝑔𝑦)+𝑐𝑑_𝐺(𝑥,𝑔𝑦)+𝑑_𝐺(𝑦,𝑓𝑥),(2.5) for all 𝑥,𝑦∈𝑋 with 0≤𝑎+2𝑏+2𝑐<1 and the fixed point of 𝑓 and 𝑔 follows from [2].
Now if 𝑋 is not a symmetric 𝐺-metric space. Then by the definition of metric (𝑋,𝑑_𝐺) and inequalities (2.1) and (2.3), we obtain𝑑_𝐺(𝑓𝑥,𝑔𝑦)=≤≤+𝑎𝑎𝐺𝑑[2(𝐺_𝐺𝑐𝑓𝑥[(𝐺𝑥,𝑔,(𝑦𝑦𝑥,,,𝑦𝑔𝑔)𝑦𝑦)+,+𝑔𝐺𝑦𝐺)(+𝑥(𝑔,𝐺𝑥𝑦,,(𝑦𝑓𝑦)𝑥,]𝑓,+𝑓𝑥𝑥,2𝑓)𝑏𝑥[)𝐺](𝑥,𝑓𝑥,𝑓𝑥)+𝐺(𝑦,𝑔𝑦,𝑔𝑦)](𝑥,𝑦)+2𝑏23𝑑^𝐺(𝑥,𝑓𝑥)+23𝑑^𝐺+2𝑐2(𝑦,𝑔𝑦)3𝑑^𝐺(𝑥,𝑔𝑦)+23𝑑^𝐺=𝑎𝑑_𝐺(𝑦,𝑓𝑥)(𝑥,𝑦)+43𝑏𝑑^𝐺(𝑥,𝑓𝑥)+𝑑_𝐺(𝑦,𝑔𝑦)+43𝑐𝑑^𝐺(𝑥,𝑔𝑦)+𝑑_𝐺(𝑦,𝑓𝑥),(2.6) for all 𝑥∈𝑋. Here, the contractivity factor 𝑎+(8/3)𝑏+(8/3)𝑐 may not be less than 1.
Therefore metric gives no information. In this case, for given 𝑥₀∈𝑋, choose 𝑥₁∈𝑋 such that 𝑥₁=𝑓𝑥₀. Again choose 𝑥₂∈𝑋 such that 𝑔𝑥₁=𝑥₂. Also, we choose 𝑥₃∈𝑋 such that 𝑥₃=𝑓𝑥₂. Continuing as above process, we can construct a sequence {𝑥_𝑛} in 𝑋 such that 𝑥_2𝑛+1=𝑓𝑥_2𝑛, 𝑛∈𝐍∪{𝟎} and 𝑥_2𝑛+2=𝑔𝑥_2𝑛+1, 𝑛∈𝐍∪{𝟎}. Since 𝑓 and 𝑔 are weakly increasing with respect to ⪯, we have 𝑥₁=𝑓𝑥₀⪯𝑔𝑓𝑥₀=𝑔𝑥₁=𝑥₂⪯𝑓𝑔𝑥₁=𝑓𝑥₂=𝑥₃⪯𝑔𝑓𝑥₂=𝑔𝑥₃=𝑥₄⪯⋯.(2.7) Thus from (2.1), we have 𝐺𝑥_2𝑛+1,𝑥_2𝑛+2,𝑥_2𝑛+2=≤𝑎𝐺𝐺𝑓𝑥𝑥_2𝑛2𝑛,𝑔𝑥_2𝑛+1,𝑔𝑥_2𝑛+1+𝑏𝐺𝑥,𝑥_2𝑛+12𝑛,𝑥_2𝑛+1+𝑐𝐺𝑥_2𝑛,𝑓𝑥_2𝑛,𝑓𝑥_2𝑛+𝐺𝑥_2𝑛+1,𝑔𝑥_2𝑛+1,𝑔𝑥_2𝑛+1=𝑎𝐺𝑥_2𝑛,𝑔𝑥_2𝑛+1,𝑔𝑥_2𝑛+1+𝐺𝑥_2𝑛+1,𝑓𝑥_2𝑛,𝑓𝑥_2𝑛+𝑏𝐺𝑥,𝑥_2𝑛+12𝑛,𝑥_2𝑛+1+𝑐𝐺𝑥_2𝑛,𝑥_2𝑛+1,𝑥_2𝑛+1+𝐺𝑥_2𝑛+1,𝑥_2𝑛+2,𝑥_2𝑛+2=(𝑎+𝑏)𝐺𝑥,𝑥_2𝑛+22𝑛,𝑔𝑥_2𝑛+2+𝐺𝑥_2𝑛+1,𝑥_2𝑛+1,𝑥_2𝑛+1+𝑐𝐺𝑥_2𝑛,𝑥_2𝑛+1,𝑥_2𝑛+1+𝑏𝐺𝑥_2𝑛+1,𝑥_2𝑛+2,𝑥_2𝑛+2,𝑥_2𝑛+2,𝑥_2𝑛+2.(2.8) By (𝐺₅), we have 𝐺𝑥_2𝑛+1,𝑥_2𝑛+2,𝑥_2𝑛+2≤𝑎+𝑏+𝑐1−𝑏−𝑐𝐺𝑥^2𝑛,𝑥_2𝑛+1,𝑥_2𝑛+1.(2.9) Also, we have 𝐺𝑥_2𝑛,𝑥_2𝑛+1,𝑥_2𝑛+1=≤𝐺𝑎𝐺𝑔𝑥𝑥_{2𝑛−12𝑛−1},𝑓𝑥_2𝑛,𝑓𝑥_2𝑛+𝑏𝐺𝑥,𝑥_{2𝑛2𝑛−1},𝑥_2𝑛+𝑐𝐺𝑥_2𝑛−1,𝑔𝑥_2𝑛−1,𝑔𝑥_2𝑛−1+𝐺𝑥_2𝑛,𝑓𝑥_2𝑛,𝑓𝑥_2𝑛=𝑎𝐺𝑥_2𝑛−1,𝑓𝑥_2𝑛,𝑓𝑥_2𝑛+𝐺𝑥_2𝑛,𝑔𝑥_2𝑛−1,𝑔𝑥_2𝑛−1+𝑏𝐺𝑥,𝑥_{2𝑛2𝑛−1},𝑥_2𝑛+𝑐𝐺𝑥_2𝑛−1,𝑥_2𝑛,𝑥_2𝑛+𝐺𝑥_2𝑛,𝑥_2𝑛+1,𝑥_2𝑛+1=(𝑎+𝑏)𝐺𝑥,𝑥_{2𝑛+12𝑛−1},𝑔𝑥_2𝑛+1+𝐺𝑥_2𝑛,𝑥_2𝑛,𝑥_2𝑛+𝑐𝐺𝑥_2𝑛−1,𝑥_2𝑛,𝑥_2𝑛+𝑏𝐺𝑥_2𝑛,𝑥_2𝑛+1,𝑥_2𝑛+1,𝑥_2𝑛+1,𝑥_2𝑛+1.(2.10) By (𝐺₅), we get 𝐺𝑥_2𝑛,𝑥_2𝑛+1,𝑥_2𝑛+1≤𝑎+𝑏+𝑐1−𝑏−𝑐𝐺𝑥^2𝑛−1,𝑥_2𝑛,𝑥_2𝑛.(2.11) Let 𝑘=𝑎+𝑏+𝑐1−𝑏−𝑐.(2.12) Then by (2.9) and (2.11), we have 𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1≤𝑘𝐺𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,∀𝑛∈𝐍.(2.13) Thus, if 𝑥₀=𝑥₁, we get 𝐺(𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1)=0 for each 𝑛∈𝐍. Hence 𝑥_𝑛=𝑥₀ for each 𝑛∈𝐍. Therefore {𝑥_𝑛} is 𝐺-Cauchy. So we may assume that 𝑥₀≠𝑥₁. Let 𝑛,𝑚∈𝐍 with 𝑚>𝑛. By axiom (𝐺₅) of the definition of 𝐺-metric space, we have 𝐺𝑥_𝑛,𝑥_𝑚,𝑥_𝑚≤𝐺𝑥_𝑛,𝑥_𝑛+1,𝑥_𝑛+1+𝐺𝑥_𝑛+1,𝑥_𝑛+2,𝑥_𝑛+2+⋯+𝐺𝑥_𝑚−1,𝑥_𝑚,𝑥_𝑚.(2.14) By (2.13), we get 𝐺𝑥_𝑛,𝑥_𝑚,𝑥_𝑚≤≤𝑘^𝑛𝑘𝐺_𝑛𝑥₀,𝑥₁,𝑥₁+𝑘^𝑛+1𝐺𝑥₀,𝑥₁,𝑥₁+⋯+𝑘^𝑚−1𝐺𝑥₀,𝑥₁,𝑥₁1−𝑘𝐺𝑥⁰,𝑥₁,𝑥₁.(2.15) On taking limit 𝑚,𝑛→∞, we have lim_{𝑚,𝑛→∞}𝐺𝑥_𝑛,𝑥_𝑚,𝑥_𝑚=0.(2.16) So we conclude that (𝑥_𝑛) is a Cauchy sequence in 𝑋. Since 𝑋 is 𝐺-complete, then it yields that (𝑥_𝑛) and hence any subsequence of (𝑥_𝑛) converges to some 𝑢∈𝑋. So that, the subsequences (𝑥_2𝑛+1)=(𝑓𝑥_2𝑛) and (𝑥_2𝑛+2)=(𝑔𝑥_2𝑛+1) converge to 𝑢. First suppose that 𝑓 is 𝐺-continuous. Since (𝑥_2𝑛) converges to 𝑢, we get (𝑓𝑥_2𝑛) converges 𝑓𝑢. By the uniqueness of limit we get 𝑓𝑢=𝑢. Claim: 𝑔𝑢=𝑢.
Since 𝑢⪯𝑢, by inequality (2.1), we have 𝐺(𝑢,𝑔𝑢,𝑔𝑢)=≤≤+𝐺𝑎(𝑏𝐺𝑐(+𝑓[(𝐺𝑢𝑢𝑐,,)(𝑢𝑔𝑢𝐺,𝑢,𝑢,𝑔()𝑔𝑢𝑢+𝑢,,)𝑔𝑔𝑏𝑢𝑢[),𝐺𝑔+𝑢()𝐺𝑢.,(𝑓𝑢𝑢,,𝑓𝑓𝑢𝑢,)𝑓+𝑢)𝐺](𝑢,𝑔𝑢,𝑔𝑢)](2.17) Since 𝑏+𝑐<1, we get 𝐺(𝑢,𝑔𝑢,𝑔𝑢)=0. Hence 𝑔𝑢=𝑢. If 𝑔 is 𝐺-continuous, by similar argument as above we show that 𝑔 and 𝑓 have a common fixed point.

Theorem 2.2. Let (𝑋,⪯) be a partially ordered set and suppose that there exists 𝐺-metric in 𝑋 such that (𝑋,𝐺) is 𝐺-complete. Let 𝑓,𝑔∶𝑋→𝑋 be two weakly increasing mappings with respect to ⪯. Suppose there exist nonnegative real numbers 𝑎, 𝑏, and 𝑐 with 𝑎+2𝑏+2𝑐<1 such that 𝐺𝐺((𝑔𝑓𝑥𝑥,,𝑓𝑔𝑦𝑦,,𝑓𝑔𝑦𝑦))≤≤++𝑎𝑎𝐺𝐺𝑐𝑐[[((𝐺𝐺𝑥𝑥,,((𝑦𝑦𝑥𝑥,,,,𝑦𝑦𝑔𝑓))𝑦𝑦++,,𝑔𝑓𝑏𝑏𝑦𝑦[[))𝐺𝐺++((𝐺𝑥𝑥𝐺,,((𝑓𝑔𝑦𝑦𝑥𝑥,,,,𝑓𝑔𝑔𝑓𝑥𝑥𝑥𝑥,,))𝑓𝑔++𝑥𝑥))𝐺]]𝐺,,((𝑦𝑦,,𝑓𝑔𝑦𝑦,,𝑓𝑔𝑦𝑦))]](2.18) for all comparative 𝑥,𝑦∈𝑋. Assume that 𝑋 has the following property: (𝑃)If (𝑥_𝑛) is an increasing sequence converges to 𝑥 in 𝑋, then 𝑥_𝑛⪯𝑥 for all 𝑛∈𝐍.Then 𝑓 and 𝑔 have a common fixed point 𝑢∈𝑋.

Proof. As in the proof of Theorem 3.1, we construct an increasing sequence (𝑥_𝑛) in 𝑋 such that 𝑥_2𝑛+1=𝑓𝑥_2𝑛 and 𝑥_2𝑛+2=𝑔𝑥_2𝑛+1. Also, we can show (𝑥_𝑛) is 𝐺-Cauchy. Since 𝑋 is 𝐺-complete, there is 𝑢∈𝑋 such that (𝑥_𝑛) is converges to 𝑢∈𝑋. Thus (𝑥_2𝑛),(𝑥_2𝑛+1),(𝑓𝑥_2𝑛) and (𝑔𝑥_2𝑛+1) converge to 𝑢. Since 𝑋 satisfies property (𝑃), we get that 𝑥_𝑛⪯𝑢, for all 𝑛∈𝐍. Thus 𝑥_2𝑛 and 𝑢 are comparative. Hence by inequality (2.1), we have 𝐺𝑓𝑥_2𝑛,𝑔𝑢,𝑔𝑢≤+𝑎𝐺𝑐𝐺𝑥_2𝑛𝑥,𝑢_2𝑛,𝑢+𝑏𝐺𝑥_2𝑛,𝑓𝑥_2𝑛,𝑓𝑥_2𝑛+𝐺(𝑢,𝑔𝑢,𝑔𝑢),𝑔𝑢,𝑔𝑢+𝐺𝑢,𝑓𝑥_2𝑛,𝑓𝑥_2𝑛.(2.19) On letting 𝑛→+∞, we get 𝐺(𝑢,𝑔𝑢,𝑔𝑢)≤(𝑏+𝑐)𝐺(𝑢,𝑔𝑢,𝑔𝑢).(2.20) Since 𝑏+𝑐<1, we get 𝐺(𝑢,𝑔𝑢,𝑔𝑢)=0. Hence 𝑔𝑢=𝑢. By similar argument, we may show that 𝑢=𝑓𝑢.

Corollary 2.3. Let (𝑋,⪯) be a partially ordered set, and suppose that (𝑋,𝐺) is a 𝐺-complete metric space. Let 𝑓∶𝑋→𝑋 be a continuous mapping such that 𝑓𝑥⪯𝑓(𝑓𝑥), for all 𝑥∈𝑋. Suppose there exist nonnegative real numbers 𝑎, 𝑏 and 𝑐 with 𝑎+2𝑏+2𝑐<1 such that 𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑦)≤+𝑎𝐺𝑐[(𝐺𝑥,(𝑦𝑥,,𝑦𝑓)𝑦+,𝑓𝑏𝑦[)𝐺+(𝑥𝐺,(𝑓𝑦𝑥,,𝑓𝑓𝑥𝑥,)𝑓+𝑥)𝐺],(𝑦,𝑓𝑦,𝑓𝑦)](2.21) for all comparative 𝑥,𝑦∈𝑋. Then 𝑓 has a fixed point 𝑢∈𝑋.

Proof. It follows from Theorem 2.1 by taking 𝑔=𝑓.

Corollary 2.4. Let (𝑋,⪯) be a partially ordered set and suppose that there exists 𝐺-metric in 𝑋 such that (𝑋,𝐺) is 𝐺-complete. Let 𝑓∶𝑋→𝑋 be a mapping such that 𝑓𝑥⪯𝑓(𝑓𝑥) for all 𝑥∈𝑋. Suppose there exist nonnegative real numbers 𝑎, 𝑏 and 𝑐 with 𝑎+2𝑏+2𝑐<1 such that 𝐺(𝑓𝑥,𝑓𝑦,𝑓𝑦)≤+𝑎𝐺𝑐[(𝐺𝑥,(𝑦𝑥,,𝑦𝑓)𝑦+,𝑓𝑏𝑦[)𝐺+(𝑥𝐺,(𝑓𝑦𝑥,,𝑓𝑓𝑥𝑥,)𝑓+𝑥)𝐺](𝑦,𝑓𝑦,𝑓𝑦)](2.22) for all comparative 𝑥,𝑦∈𝑋. Assume that 𝑋 has the following property: (𝑃)If (𝑥_𝑛) is an increasing sequence converges to 𝑥 in 𝑋, then 𝑥_𝑛⪯𝑥 for all 𝑛∈𝐍.Then 𝑓 has fixed point 𝑢∈𝑋.

Proof. It follows from Theorem 2.2 by taking 𝑔=𝑓.

3. Application

Consider the integral equation:𝑢(𝑡)=^𝑇0𝐾(𝑡,𝑠,𝑢(𝑠))𝑑𝑠+𝑔(𝑡),𝑡∈[0,𝑇],(3.1) where 𝑇>0. The aim of this section is to give an existence theorem for a solution of the above integral equation using Corollary 2.4 This section is related to those [16–19].

Let 𝑋=𝐶([0,𝑇]) be the set of all continuous functions defined on [0,𝑇]. Define 𝐺∶𝑋×𝑋×𝑋→𝐑⁺(3.2) by 𝐺(𝑥,𝑦,𝑧)=sup_{[]𝑡∈0,𝑇}||𝑥(𝑡)−𝑦(𝑡)||+sup_{[]𝑡∈0,𝑇}||𝑥(𝑡)−𝑧(𝑡)||+sup_{[]𝑡∈0,𝑇}||𝑦(𝑡)−𝑧(𝑡)||.(3.3) Then (𝑋,𝐺) is a 𝐺-complete metric space. Define an ordered relation ⪯ on 𝑋 by 𝑥⪯𝑦iff𝑥(𝑡)≤𝑦(𝑡),∀𝑡∈[0,𝑇].(3.4) Then (𝑋,⪯) is a partially ordered set. The purpose of this section is to give an existence theorem for solution of integral equation on (3.1). This section is inspired in [17–19].

Theorem 3.1. Suppose the following hypotheses hold: (1)𝐾∶[0,𝑇]×[0,𝑇]×𝐑→𝐑 and 𝑔∶𝐑→𝐑 are continuous, (2)for each 𝑡,𝑠∈[0,𝑇], one has 𝐾(𝑡,𝑠,𝑢(𝑡))≤𝐾𝑡,𝑠,^𝑇0𝐾(𝑠,𝜏,𝑢(𝜏))𝑑𝜏+𝑔(𝑠),(3.5)(3)there exists a continuous function 𝐺∶[0,𝑇]×[0,𝑇]→[0,+∞] such that ||𝐾(𝑡,𝑠,𝑢)−𝐾(𝑡,𝑠,𝑣)||≤𝐺(𝑡,𝑠)|𝑢−𝑣|,(3.6) for each comparable 𝑢,𝑣∈𝐑 and each 𝑡,𝑠∈[0,𝑇], (4)sup_{𝑡∈[0,𝑇]}∫^𝑇0𝐺(𝑡,𝑠)𝑑𝑠≤𝑟 for some 𝑟<1. Then the integral equation (3.1) has a solution 𝑢∈𝐶([0,𝑇]).

Proof. Define 𝑆∶𝐶([0,𝑇])→𝐶([0,𝑇]) by 𝑆𝑥(𝑡)=^𝑇0𝐾(𝑡,𝑠,𝑥(𝑠))𝑑𝑠+𝑔(𝑡),𝑡∈[0,𝑇].(3.7) Now, we have 𝑆𝑥(𝑡)=≤^𝑇0𝐾(𝑡,𝑠,𝑥(𝑠))𝑑𝑠+𝑔(𝑡)_𝑇0=𝐾𝑡,𝑠,^𝑇0𝐾(𝑠,𝜏,𝑥(𝜏))𝑑𝜏+𝑔(𝑠)𝑑𝑠+𝑔(𝑡)_𝑇0=𝑆(𝑆𝐾𝑥((𝑡,𝑡)𝑠),.𝑆𝑥(𝑠))𝑑𝑠+𝑔(𝑡)(3.8) Thus, we have 𝑆𝑥⪯𝑆(𝑆𝑥), for all 𝑥∈𝐶([0,𝑇]).
For 𝑥,𝑦∈𝐶([0,𝑇]) with 𝑥⪯𝑦, we have 𝐺(𝑆𝑥,𝑆𝑦,𝑆𝑦)==22ssuupp_{𝑡∈[0,𝑇]𝑡∈[0,𝑇]}||𝑆𝑥(𝑡)−𝑆𝑦(𝑡)||||||^𝑇0≤2sup_{𝑡∈[0,𝑇]}𝐾(𝑡,𝑠,𝑥(𝑠))−𝐾(𝑡,𝑠,𝑦(𝑠))𝑑𝑠||||^𝑇0≤2sup_{[]𝑡∈0,𝑇}||𝐾(𝑡,𝑠,𝑥(𝑠))−𝐾(𝑡,𝑠,𝑦(𝑠))||𝑑𝑠^𝑇0≤2sup_{𝑡∈[0,𝑇]}𝐺(𝑡,𝑠)||𝑥(𝑠)−𝑦(𝑠)||𝑑𝑠||𝑥(𝑡)−𝑦(𝑡)||sup_{[]𝑡∈0,𝑇}^𝑇0=𝐺(𝑥,𝑦,𝑦)sup_{[]𝑡∈0,𝑇}𝐺(𝑡,𝑠)𝑑𝑠^𝑇0𝐺(𝑡,𝑠)𝑑𝑠.(3.9) By using hypotheses (4), there is 𝑟∈[0,1) such that sup_{𝑡∈[0,𝑇]}^𝑇0𝐺(𝑡,𝑠)𝑑𝑠<𝑟.(3.10) Thus, we have 𝐺(𝑆𝑥,𝑆𝑦,𝑆𝑦)≤𝑟𝐺(𝑥,𝑦,𝑦). Thus all the required hypotheses of Corollary 2.4 are satisfied. Thus there exist a solution 𝑢∈𝐶([0,𝑇]) of the integral equation (3.1).

Acknowledgments

The author would like to thank the editor of the paper and the referees for their precise remarks to correct and improve the paper. Also, the author would like to thank the hashemite university for the financial assistant of this paper.