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Article

Coupled Fixed Point Theorems with Rational Type Contractive Condition via C-Class Functions and Inverse Ck-Class Functions

1
College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, China
2
South Sichuan Center for Applied Mathematics, Zigong 643000, China
3
School of Science and Techology, University of Sanya, Sanya 572022, China
4
Center for Mathematical Reaserch, University of Sanya, Sanya 572022, China
5
Academician Guoliang Chen Team Innovation Center, University of Sanya, Sanya 572022, China
6
Academician Chunming Rong Workstation, University of Sanya, Sanya 572022, China
7
Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj 6915136111, Iran
8
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria 0204, South Africa
9
Department of Mathematics, Scottish Church College, Urquhart Square, Kolkata 700006, India
10
Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood Road, Pretoria 0002, South Africa
11
Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak 484887, India
*
Authors to whom correspondence should be addressed.
Symmetry 2022, 14(8), 1663; https://doi.org/10.3390/sym14081663
Submission received: 23 July 2022 / Revised: 7 August 2022 / Accepted: 9 August 2022 / Published: 11 August 2022

Abstract

:
The purpose of this paper is to develop coupled fixed point theorems by C-class functions with mixed monotonicity, discuss the existence of the coupled fixed point of two mappings and obtain a coupled coincidence point theorem via inverse C k -class functions in partially ordered H-metric spaces. This improves the existing results. In addition, some instances and an application are given to illustrate the usability and validity of the results.

1. Introduction

Since the Banach Contraction Principle, many scholars have developed it by changing the contraction environment and changing the metric type, e.g., [1,2,3,4,5,6,7,8,9,10,11]. For instance, in 2004, Ran and Reurings [1] introduced partial order in metric spaces and proposed some fixed point results in such spaces. In 2006, Bhaskar and Lakshmikantham [2] presented some coupled fixed point theorems for linear contractions in partially ordered metric spaces. In 2009, Lakshmikantham and Ćirić [3] introduced some coupled fixed point results for nonlinear contraction in partially ordered metric spaces. There are a large number of researchers who have considered coupled fixed points well in a variety of contractive conditions [12,13,14,15,16]. In 2006, Mustafa and Sims [17] introduced a class of new metric spaces called G-metric spaces which is equivalent to H-metric spaces in this paper, some fixed point results in such spaces can be found in [18,19,20,21,22,23,24,25,26,27]. In 2010, Harjani et al. [28] utilized rational type contractive mapping proposed by Jaggi [29] in partially ordered metric spaces. Shortly after, Chakrabarti [30] generalized these contractive conditions in [28] and extended the results of [28] to partially ordered G-metric spaces.
On the other hand, in 2014, Ansari [31] presented C-class functions and proved some relevant fixed points of complete metric spaces. In 2018, Saleem et al. [32] presented a new notion of inverse C-class functions where the first inequality is opposite to that of C-class functions.
As far as we know, there are also some contractions that have not been studied in partially ordered complete H-metric space. Thus, inspired by [30] and C-class functions, in Section 3 we intend to study the coupled fixed point theorems and coupled coincidence point theorem in partially ordered H-metric space, which develops some results of [30]. In Section 4, motivated by inverse C-class functions [32], a class of development functions of inverse C-class functions called inverse C k -class functions is introduced. Furthermore, the existence theorem of a coupled coincidence point by using inverse C k -class functions is raised.

2. Preliminaries

Denote a partial order relation by ≪ on Z. For arbitrary a , b Z endowed with the partial order ≪, a b and b a are the same. Moreover, a < b means that a b and a b .
We first introduce the fundamental definitions and notions which will be used in the following section.
Definition 1
([17]). Suppose that Z and H : Z × Z × Z R + has the following property:
( H 1 ) H ( a , b , c ) = 0 if a = b = c .
( H 2 ) 0 < H ( a , a , b ) , for all a , b Z with a b .
( H 3 ) H ( a , a , b ) H ( a , b , c ) , for all a , b , c Z with c b .
( H 4 ) H ( a , b , c ) = H ( p { a , b , c } ) , where p is any permutation of a , b , c ( a , b , c are symmetrical).
( H 5 ) H ( a , b , c ) H ( a , s , s ) + H ( s , b , c ) , for all a , b , c , s Z .
Then, H is a generalized metric, or in more detail, an H-metric on Z and ( Z , H ) is an H-metric space.
Proposition 1
([17]). Suppose that ( Z , H ) is an H-metric space; then for any a , b , c , s Z , the following is obviously appropriate:
  • Supposing that H ( a , b , c ) = 0 , we have a = b = c .
  • H ( a , b , c ) H ( a , a , b ) + H ( a , a , c ) .
  • H ( a , b , b ) 2 H ( b , a , a ) .
  • H ( a , b , c ) H ( a , s , c ) + H ( s , b , c ) .
Definition 2
([17]). Assume that ( Z , H ) is an H-metric space. The sequence { a m } Z is H-convergent to a if for any arbitrary ε > 0 , there exists a positive integer M such that H ( a , a m , a m ) < ε for m M , i.e., lim m + H ( a , a m , a m ) = 0 .
Proposition 2
([17]). Suppose that ( Z , H ) is an H-metric space, it follows that for a sequence { a m } Z and a point a Z , the terms in the following are equivalent:
  • { a m } is H-convergent to a.
  • H ( a m , a m , a ) 0 when m + .
  • H ( a m , a , a ) 0 when m + .
  • H ( a m , a l , a ) 0 when m , l + .
Proposition 3
([17]). Assume that ( Z , H ) is an H-metric space. It follows that the function H ( a , b , c ) is jointly continuous with respect to three variables.
Remark 1.
It is noted that if { a s } , { b t } , { c w } are sequences in Z such that lim s + a s = a , lim t + b t = b , lim w + c w = c , then H ( a s , b t , c w ) H ( a , b , c ) when s , t , w + .
Definition 3
([17]). Suppose that ( Z , H ) is an H-metric space. We said that the sequence { a s } X is H-Cauchy if for an arbitrary ε > 0 , there is a positive integer S such that H ( a s , a t , a w ) < ε for all s , t , w S .
Proposition 4
([17]). Assume that ( Z , H ) is an H-metric space; the terms as follows are the same:
  • The sequence { a s } is H-Cauchy.
  • For arbitrary ε > 0 , there is a positive integer S satisfying that H ( a s , a t , a w ) < ε for all s , t , w > S .
Lemma 1.
Let ( Z , H ) be an H-metric space. Suppose that { a s } is a sequence in Z satisfying that G ( a s , a s + 1 , a s + 1 ) 0 when s + . Suppose that { a s } is not H-Cauchy, then there exist ε > 0 and sequences of the positive integers s ( w ) and t ( w ) with s ( w ) > t ( w ) > w satisfying that H ( a t ( w ) , a s ( w ) , a s ( w ) ) ε , H ( a t ( w ) , a s ( w ) 1 , a s ( w ) 1 ) < ε , and
lim w + H ( a t ( w ) , a s ( w ) , a s ( w ) ) = ε , lim w H ( a s ( w ) + 1 , a t ( w ) + 1 , a t ( w ) + 1 ) = ε , lim w H ( a t ( w ) + 1 , a s ( w ) + 1 , a s ( w ) + 2 ) = ε .
Proof. 
Because H ( a s , a s + 1 , a s + 1 ) 0 when s + , i.e.,
lim s H ( a s , a s + 1 , a s + 1 ) = 0 .
From the inequality ( 3 ) in Proposition 1, it follows that
lim s H ( a s + 1 , a s , a s ) = 0 .
If { a s } is not H-Cauchy, there are ε > 0 subsequences { a s ( w ) } and { a t ( w ) } of { a s } with s ( w ) > t ( w ) > w satisfying that
H ( a t ( w ) , a s ( w ) , a s ( w ) ) ε ,
for arbitrary w N . Meanwhile, for t ( w ) , we can take s ( w ) which is the smallest integer by virtue of s ( w ) > t ( w ) and (3). Therefore, it follows that
H ( a t ( w ) , a s ( w ) 1 , a s ( w ) 1 ) < ε ,
for all w N . With ( H 5 ) in Definition 1, it occurs that
ε H ( a t ( w ) , a s ( w ) , a s ( w ) ) H ( a t ( w ) , a s ( w ) 1 , a s ( w ) 1 ) + H ( a s ( w ) 1 , a s ( w ) , a s ( w ) ) ε + H ( a s ( w ) 1 , a s ( w ) , a s ( w ) ) .
Taking w in (5) and noting that H ( a s , a s + 1 , a s + 1 ) 0 , it is naturally obtained that
lim s H ( a t ( w ) , a s ( w ) , a s ( w ) ) = ε .
We can obtain
H ( a t ( w ) + 1 , a s ( w ) + 1 , a s ( w ) + 1 ) H ( a t ( w ) + 1 , a t ( w ) , a t ( w ) ) + H ( a t ( w ) , a s ( w ) , a s ( w ) ) + H ( a s ( w ) , a s ( w ) + 1 , a s ( w ) + 1 ) .
and
H ( a t ( w ) , a s ( w ) , a s ( w ) ) H ( a t ( w ) , a t ( w ) + 1 , a t ( w ) + 1 ) + H ( a t ( w ) + 1 , a s ( w ) + 1 , a s ( w ) + 1 ) + H ( a s ( w ) + 1 , a s ( w ) , a s ( w ) )
Letting w + in expressions (7) and (8) and regrading (1), (2) and (6), we derive
lim w H ( a t ( w ) + 1 , a s ( w ) + 1 , a s ( w ) + 1 ) = ε .
Similarly, it follows that
H ( a t ( w ) + 1 , a s ( w ) + 1 , a s ( w ) + 1 ) H ( a t ( w ) + 1 , a s ( w ) + 2 , a s ( w ) + 1 ) H ( a t ( w ) + 1 , a s ( w ) + 1 , a s ( w ) + 1 ) + H ( a s ( w ) + 1 , a s ( w ) + 2 , a s ( w ) + 1 )
from ( H 3 ) and ( H 5 ) . Taking w + in expressions (10) and using (1), (2) and (9), we conclude that
lim w H ( a t ( w ) + 1 , a s ( w ) + 1 , a s ( w ) + 2 ) = ϵ .
In 2014, in a partially ordered H-metric space, Chakrabarti [30] proposed coupled fixed point theorems for a mapping satisfying mixed monotone property and a nonlinear, rational type contractive condition and presented results for the existence of coupled coincidence points of two mappings via the notion of mixed f-monotonicity [3].
Definition 4
([3]). Suppose that ( Z , ) is a set endowed with partial order, S : Z × Z Z and f : Z Z . S is said to be mixed f-monotone if S is monotone f-non-decreasing with respect to the first variable and is monotone f-non-increasing with respect to the second variable, i.e., for all s , t Z , s 1 , s 2 Z , f s 1 f s 2 S ( s 1 , t ) S ( s 2 , t ) , and t 1 , t 2 Z , f t 1 f t 2 S ( s , t 1 ) S ( s , t 2 ) .
Theorem 1
([30]). Suppose that Z is a set endowed with a partial order ≪ and G is a generalized metric on H satisfying that ( Z , H ) is completed with respect to the H-metric. Assume that S : Z × Z Z is continuous on Z and mixed monotone. Meanwhile, for all ( s , t ) , ( a , b ) , ( c , d ) Z × Z with ( s , t ) ( a , b ) ( c , d ) ,
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) [ α H ( s , S ( s , t ) , S ( s , t ) ) H ( a , S ( a , b ) , S ( a , b ) ) H ( c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( s , a , c ) ] 2 + β H ( s , a , c ) ] ,
where 8 α + β < 1 . In case there exists s 0 , t 0 Z satisfying s 0 S ( s 0 , t 0 ) and t 0 S ( t 0 , s 0 ) , it can be deduced that ( s * , t * ) Z × Z is a coupled fixed point of S, i.e., ( s * , t * ) fulfills s * = S ( s * , t * ) , t * = S ( t * , s * ) .
Theorem 2
([30]). Assume that the conditions are the same as in Theorem 1, except for the following: if for each ( s , t ) , ( a , b ) Z × Z , there exists a ( c , d ) Z × Z that can be comparable to ( s , t ) and ( a , b ) , there exists uniqueness of coupled fixed points of S.
Theorem 3
([30]). Suppose that Z is a set with a partial order and H is a metric on Z satisfying that ( Z , H ) is completed with respect to the H-metric. Assume that S : Z × Z Z and f : Z Z is continuous on Z with the fact that S is mixed f monotone. Let S ( Z × Z ) f ( Z ) ; f commutes with S and for each ( s , t ) , ( a , b ) , ( c , d ) Z × Z with ( s , t ) ( a , b ) ( c , d ) and f s f a f c , f t f b f d ,
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) α [ H ( f s , S ( s , t ) , S ( a , b ) ) H ( f a , S ( a , b ) , S ( a , b ) ) H ( f c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( f s , f a , f c ) ] 2 + β H ( f s , f a , f c ) ] ,
where 8 α + β < 1 . If there is s 0 , t 0 Z such that f s 0 S ( s 0 , t 0 ) and f t 0 S ( t 0 , s 0 ) , so ( s * , t * ) Z × Z is a coupled coincidence point of S and f, i.e., ( s * , t * ) is the solution of f s * = S ( s * , t * ) , f t * = S ( t * , s * ) .
In addition, C-class functions were proposed by Ansari [31]. Improved by this kind of function, a lot of fixed point theorems can be generalized in the existing results.
Definition 5
([31]). We call a mapping G : [ 0 , + ) 2 R a C-class function if it has continuity properties and meets the terms as follows:
(1) G ( x , y ) x .
(2) G ( x , y ) = x can deduce that one of the two terms holds: (a) x = 0 ; (b) y = 0 , for all x , y [ 0 , + ) .
We utilize C to represent the set of all the C-class functions.
Example 1
([31]). Every G : [ 0 , + ) 2 R in the following belongs to C for all x , y [ 0 , + ) :
(1) G ( x , y ) = x y , G ( x , y ) = x leads to y = 0 .
(2) G ( x , y ) = n x , n > 1 , G ( x , y ) = x leads to x = 0 .
(3) G ( x , y ) = x ( 1 + y ) s ; s ( 0 , + ) , G ( x , y ) = x leads to x = 0 or y = 0 .
(4) G ( x , y ) = log ( y + a x ) / ( 1 + y ) , a > 1 , G ( x , y ) = x leads to x = 0 or y = 0 .
(5) G ( x , y ) = ln ( 1 + a x ) / 2 , a > e , G ( x , 1 ) = x leads to x = 0 .
(6) G ( x , y ) = ( x + l ) ( 1 / ( 1 + y ) s ) l , l > 1 , s ( 0 , + ) , G ( x , y ) = x leads to y = 0 .
(7) G ( x , y ) = x log y + a a , a > 1 , G ( x , y ) = x leads to x = 0 or y = 0 .
(8) G ( x , y ) = x ( 1 + x 2 + x ) ( y 1 + y ) , G ( x , y ) = x leads to y = 0 .
(9) G ( x , y ) = x β ( x ) , β : [ 0 , + ) [ 0 , 1 ) , and is continuous, G ( x , y ) = x leads to x = 0 .
(10) G ( x , y ) = x y t + y , G ( x , y ) = x leads to y = 0 .
(11) G ( x , y ) = x ϕ ( x ) , G ( x , y ) = x leads to x = 0 where ϕ : [ 0 , + ) [ 0 , + ) is continuous satisfying that ϕ ( y ) = 0 y = 0 .
(12) G ( x , y ) = x k ( x , y ) , G ( x , y ) = x leads to x = 0 where k : [ 0 , + ) × [ 0 , + ) [ 0 , + ) is continuous satisfying that k ( y , x ) < 1 for all x , y > 0 .
(13) G ( x , y ) = x ( 2 + y 1 + y ) y , G ( x , y ) = x leads to y = 0 .
(14) G ( x , y ) = ln ( 1 + x n ) n , G ( x , y ) = x leads to x = 0 .
(15) G ( x , y ) = ψ ( x ) , G ( x , y ) = x leads to x = 0 where ψ : [ 0 , + ) [ 0 , + ) is upper semi-continuous satisfying that ψ ( 0 ) = 0 , and ψ ( y ) < y for y > 0 .
(16) G ( x , y ) = x ( 1 + x ) s ; s ( 0 , + ) , G ( x , y ) = x leads to x = 0 .
(17) G ( x , y ) = θ ( s ) ; θ : R + × R + R is a generalized Mizoguchi–Takahashi-type function; G ( x , y ) = x leads to x = 0 .
(18) G ( x , y ) = x β ( 1 / 2 ) 0 + e s s + y d s , where β is the Euler Gamma function.
Furthermore, Saleem et al. [32] presented a new notion of inverse C-class functions in the following.
Definition 6.
A mapping G : [ 0 , + ) 2 R is said to be an inverse C-class function in cases where it is continuous and the terms as follows hold:
(i)
G ( x , y ) x for all x , y [ 0 , + ) .
( i i )
G ( x , y ) = x yields that one of the two terms holds: (a) x = 0 ; (b) y = 0 .
C i n v denotes the set of all the inverse C-class functions. Some examples are given in [32].
Example 2.
All the mappings G : [ 0 , + ) 2 R belong to C inv , for all x , y [ 0 , + ) :
  • G ( x , y ) = x + y , G ( x , y ) = x leads to y = 0 .
  • G ( x , y ) = t x , for some t ( 1 , + ) , G ( x , y ) = x leads to x = 0 .
  • G ( x , y ) = x ( 1 + y ) s , for some s ( 0 , + ) , G ( x , y ) = x leads to s = 0 or y = 0 .
  • G ( x , y ) = log a [ ( y + a x ) ( 1 + y ) ] , for some a > 1 , G ( x , y ) = x leads to y = 0 .
  • G ( x , y ) = φ ( x ) , G ( x , y ) = x leads to x = 0 , where φ : [ 0 , + ) [ 0 , + ) is an upper semi-continuous function satisfying that φ ( 0 ) = 0 and φ ( x ) > x , for any x > 0 .
As far as we know, there are also some contractions that have not been studied in partially ordered complete H-metric space. Thus, inspired by [30] and C-class functions, in Section 3, we intend to study the coupled fixed point theorems and coupled coincidence point theorem in partially ordered H-metric space, which develops some results of [30]. In Section 4, the existence theorem of a coupled coincidence point by use of inverse C k -class functions arises. In Section 5, an application is given to justify the main results.

3. Main Results

In the section, assume a set Z endowed with a partial order ≪. The partial order on Z × Z is induced by demanding that for any ( a , b ) , ( s , t ) Z × Z , ( a , b ) ( s , t ) a s , b t .
Before starting the main results, let us introduce an auxiliary function used in the main result. Ψ s denotes the set of all functions ψ : [ 0 , + ) [ 0 , + ) which meets terms in the following [31]:
(1) ψ is continuous on [ 0 , + ) .
(2) ψ ( 0 ) 0 .
(3) ψ ( x ) > 0 for all x > 0 .
Now, list the main results. Firstly, a rational type contractive condition is presented in a partially ordered complete H-metric space and a coupled fixed point theorem is obtained.
Theorem 4.
Assume that Z is a set endowed with a partial order ≪ and we define H as a generalized metric on it. Moreover, ( Z , H ) is completed with regard to H-metric. Suppose S : Z × Z Z is continuous on Z with the mixed monotone property. For all ( s , t ) , ( a , b ) , ( c , d ) Z × Z with ( s , t ) ( a , b ) ( c , d ) ,
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) G ( 1 8 α + β [ α H ( s , S ( s , t ) , S ( s , t ) ) H ( a , S ( a , b ) , S ( a , b ) ) H ( c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( s , a , c ) ] 2 + β H ( s , a , c ) ] , ψ ( 1 8 α + β [ α H ( s , S ( s , t ) , S ( s , t ) ) H ( a , S ( a , b ) , S ( a , b ) ) H ( c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( s , a , c ) ] 2 + β H ( s , a , c ) ] ) ) ,
where G C , ψ Ψ s and 8 α + β > 0 , α , β 0 . If there exists s 0 , t 0 Z satisfying that s 0 S ( s 0 , t 0 ) and t 0 S ( t 0 , s 0 ) , then S has a coupled fixed point ( s * , t * ) Z × Z ., i.e., ( s * , t * ) is the solution of s * = S ( s * , t * ) , t * = S ( t * , s * ) .
Proof. 
Assume that there exist s 0 , t 0 Z such that s 0 S ( s 0 , t 0 ) and t 0 S ( t 0 , s 0 ) . Define s w + 1 = S ( s w , t w ) , t w + 1 = S ( t w , s w ) , w N { 0 } . Because G is mixed monotone, and meanwhile is from the mathematical induction, it yields that
s 0 s 1 s 2 s w s w + 1 .
and
t 0 t 1 t 2 t w t w + 1 .
Taking s = s w , a = b = s w 1 , t = t w , c = d = t w 1 in inequality (14), we have,
H ( s w + 1 , s w , s w ) = H ( S ( s w , t w ) , S ( s w 1 , t w 1 ) , S ( s w 1 , t w 1 ) ) G ( 1 8 α + β [ α H ( s w , s w + 1 , s w + 1 ) H ( s w 1 , s w , s w ) H ( s w 1 , s w , s w ) 1 + [ H ( s w , s w 1 , s w 1 ) ] 2 + β H ( s w , s w 1 , s w 1 ) ] , ψ ( 1 8 α + β [ α H ( s w , s w + 1 , s w + 1 ) H ( s w 1 , s w , s w ) H ( s w 1 , s w , s w ) 1 + [ H ( s w , s w 1 , s w 1 ) ] 2 + β H ( s w , s w 1 , s w 1 ) ] ) ) 1 8 α + β [ α H ( s w , s w + 1 , s w + 1 ) H ( s w 1 , s w , s w ) H ( s w 1 , s w , s w ) 1 + [ H ( s w , s w 1 , s w 1 ) ] 2 + β H ( s w , s w 1 , s w 1 ) ] .
Now, by inequality ( 3 ) in Proposition 1, it follows that
H ( s w + 1 , s w , s w ) 1 8 α + β [ α H ( s w , s w + 1 , s w + 1 ) H ( s w , s w 1 , s w 1 ) H ( s w , s w 1 , s w 1 ) 1 + [ H ( s w , s w 1 , s w 1 ) ] 2 + β H ( s w , s w 1 , s w 1 ) ] 1 8 α + β [ 4 α H ( s w , s w + 1 , s w + 1 ) + β H ( s w , s w 1 , s w 1 ) ] 1 8 α + β [ 8 α H ( s w + 1 , s w , s w ) + β H ( s w , s w 1 , s w 1 ) ] .
and
H ( s w + 1 , s w , s w ) 1 8 α + β [ 4 α H ( s w + 1 , s w , s w ) + β H ( s w , s w 1 , s w 1 ) ] .
Reconsidering and rewriting inequality (16), it follows that
H ( s w + 1 , s w , s w ) H ( s w , s w 1 , s w 1 ) H ( s 1 , s 0 , s 0 ) .
In the same manner, we can see that
H ( t w + 1 , t w , t w ) H ( t w , t w 1 , t w 1 ) H ( t 1 , t 0 , t 0 ) .
Hence, the sequence { H ( s w + 1 , s w , s w ) } is positive decreasing and bounded below. Thus, the sequence { H ( s w + 1 , s w , s w ) } converges to r 0 . We shall prove that r = 0 . Letting w + in (17), we have
r 1 8 α + β ( 4 α r + β r ) < r .
Thus r = 0 ; that is,
lim w + H ( s w + 1 , s w , s w ) = 0 .
Moreover, by Proposition 1, we derive that
lim w + H ( s w , s w + 1 , s w + 1 ) = 0 .
From inequality (21) and ( H 3 ) , there exists W N satisfying that for all w > W ,
H ( s w + 2 , s w + 1 , s w ) = H ( S ( s w + 1 , t w + 1 ) , S ( s w , t w ) , S ( s w 1 , t w 1 ) ) G ( 1 8 α + β [ α H ( s w + 1 , s w + 2 , s w + 2 ) H ( s w , s w + 1 , s w + 1 ) H ( s w 1 , s w , s w ) 1 + [ H ( s w + 1 , s w , s w 1 ) ] 2 + β H ( s w + 1 , s w , s w 1 ) ] , ψ ( 1 8 α + β [ α H ( s w + 1 , s w + 2 , s w + 2 ) H ( s w , s w + 1 , s w + 1 ) H ( s w 1 , s w n , s w ) 1 + [ H ( s w + 1 , s w , s w 1 ) ] 2 ] + β H ( s w + 1 , s w , s w 1 ) ] ) ) 1 8 α + β [ α H ( s w + 1 , s w + 2 , s w + 2 ) H ( s w , s w + 1 , s w + 1 ) H ( s w 1 , s w , s w ) 1 + [ H ( s w + 1 , s w , s w 1 ) ] 2 + β H ( s w + 1 , s w , s w 1 ) ] 1 8 α + β [ 4 α H ( ( s w + 2 , s w + 1 , s w ) + β H ( s w + 1 , s w , s w 1 ) ] 1 8 α + β [ 8 α H ( s w + 2 , s w + 1 , s w ) + β H ( s w + 1 , s w , s w 1 ) ] .
i.e., for w > W ,
H ( s w + 2 , s w + 1 , s w ) H ( s w + 1 , s w , s w 1 ) .
Hence, the sequence { H ( s w + 2 , s w + 1 , s w ) } is positive decreasing and bounded below. Thus, the sequence { H ( s w + 2 , s w + 1 , s w ) } converges to p 0 . We shall prove that p = 0 . Letting w + in (22), together with (21) and continuity of ψ , we have
p G ( β p 8 α + β , ψ ( β p 8 α + β ) ) β p 8 α + β .
Hence, p = 0 , which gives us that
lim w + H ( s w , s w + 1 , s w + 2 ) = 0 .
Next, we shall show that { s w } is an H-Cauchy sequence.
From Lemma 1, (20) and (21), we derive that
lim v + H ( s u ( v ) , s w ( v ) , s w ( v ) ) = ε ,
lim v H ( s u ( v ) + 1 , s w ( v ) + 1 , s w ( v ) + 2 ) = ε .
Again, from ( H 3 ) and ( H 5 ) , it follows that
H ( s u ( v ) , s w ( v ) , s w ( v ) + 1 ) H ( s u ( v ) , s w ( v ) , s w ( v ) ) + H ( s w ( v ) , s w ( v ) , s w ( v ) + 1 ) H ( s u ( v ) , s w ( v ) , s w ( v ) ) + H ( s w ( v ) , s w ( v ) + 1 , s w ( v ) + 2 ) .
Hence, taking v + and in virtue of the Sandwich Theorem and (23), (24), we have
lim v + H ( s u ( v ) , s w ( v ) , s w ( v ) + 1 ) = ε .
Since s u ( v ) s w ( v ) s w ( v ) + 1 , putting s = s u ( v ) , t = t u ( v ) , a = s w ( v ) , b = t w ( v ) and c = s w ( v ) + 1 , d = t w ( v ) + 1 in (14), for all v 0 , it follows that
H ( s u ( v ) + 1 , s w ( v ) + 1 , s w ( v ) + 2 ) = H ( S ( s u ( v ) , t u ( v ) ) , S ( s w ( v ) , t w ( v ) ) , S ( s w ( v ) + 1 , t w ( v ) + 1 ) G ( 1 8 α + β [ α H ( s u ( v ) , s u ( v ) + 1 , s u ( v ) + 1 ) H ( s w ( v ) , s w ( v ) + 1 , s w ( v ) + 1 ) H ( s w ( v ) + 1 , s w ( v ) + 2 , s w ( v ) + 2 ) 1 + [ H ( s u ( v ) , s w ( v ) , s w ( v ) + 1 ) ] 2 ] + β H ( s u ( v ) , s w ( v ) , s w ( v ) + 1 ) , ψ ( 1 8 α + β [ α H ( s u ( v ) , s u ( v ) + 1 , s u ( v ) + 1 ) H ( s w ( v ) , s w ( v ) + 1 , s w ( v ) + 1 ) H ( s w ( v ) + 1 , s w ( v ) + 2 , s w ( v ) + 2 ) 1 + [ H ( s u ( v ) , s w ( v ) , s w ( v ) + 1 ) ] 2 ] + β H ( s u ( v ) , s w ( v ) , s w ( v ) + 1 ) ] ) .
Letting u ( v ) + , w ( v ) + in (28), from (25) and (27), we have
ε G ( β ε 8 α + β , ψ ( β ε 8 α + β ) ) β ε 8 α + β .
Thus, ε = 0 , this contracts with the assumption ε > 0 . So { s w } is an H-Cauchy sequence in Z.
Sequentially, the same reasoning applies to the sequence { t w } ; we can also prove that { t w } is an H-Cauchy sequence in Z. Since ( Z , H ) is completed with regard to H-metric, it leads to that there exist points s * , t * Z satisfying that s w s * and t w t * as w + .
In the following, it suffices to show that ( s * , t * ) is a coupled fixed point of S. Since S is continuous on Z and H as a metric is continuous in each variable, it follows that
H ( S ( s * , t * ) , s * , s * ) = H ( lim w + S ( s w , t w ) , s * , s * ) = H ( lim w + s w + 1 , s * , s * ) = lim w + H ( s w + 1 , s * , s * ) = 0 ,
which infers that S ( s * , t * ) = s * . Similar to the above proof, note that H ( S ( t * , s * ) , t * , t * ) and repeating the arguments used to derive (29), we can show S ( t * , s * ) = t * . This shows us that S has a coupled fixed point ( s * , t * ) . □
Remark 2.
Taking G ( s , t ) = ( 8 α + β ) s with 0 < 8 α + β < 1 in Theorem 4, it easily leads to Theorem 1. In addition, if we take the special function in Example 1 and its other functions about the function G, we can also obtain many results.
Example 3.
Suppose that Z = [ 0 , + ) and define H : Z × Z × Z R +
H ( s , t , r ) = 0 , if s = t = r , max { s , t , r } , otherwise .
We can derive that ( Z , H ) is completed with regard to H-metric. At the same time, a partial order ≪ is defined on Z by the following: for any s , t Z , s t if s t . Simultaneously, define S : Z × Z Z
S ( s , t ) = 1 , if s t , 0 , otherwise .
If s , t , a , b , c , d Z such that d b t s a c with s 0 , a 0 , c 0 ; then it follows that c a s t b d . The left side of (12) turns out to be H ( 0 , 0 , 0 ) = 0 . The right side of (12) becomes
α H ( s , 0 , 0 ) H ( a , 0 , 0 ) H ( c , 0 , 0 ) 1 + [ H ( s , a , c ) ] 2 + β H ( s , a , c ) = α s a c 1 + s 2 + β s > 0 with α = 1 16 , β = 1 3 .
If s 0 = 0 and t 0 = 1 , then s 0 G ( s 0 , t 0 ) and t 0 G ( t 0 , s 0 ) . So all of Theorem 4’s conditions are met. It is obvious that S has a coupled fixed point ( 0 , 1 ) . In the meantime, S has another coupled fixed point ( 1 , 0 ) .
Next, we give the proof of the uniqueness of the coupled fixed point of S in Theorem 4.
Theorem 5.
Assume that the conditions are the same as in Theorem 4, except that, if for each ( s , t ) , ( a , b ) Z × Z , there exists a ( c , d ) Z × Z that can be comparable to ( s , t ) and ( a , b ) , there exists uniqueness of coupled fixed points of S.
Proof. 
Assume that ( s * , t * ) , ( s , t ) Z × Z are coupled fixed points.
Case I. Suppose that ( s * , t * ) and ( s , t ) are comparable,
H ( s * , s , s ) = H ( S ( s * , t * ) , S ( s , t ) , S ( s , t ) ) G ( 1 8 α + β [ α H ( s * , S ( s * , t * ) , S ( s * , t * ) ) [ H ( s , S ( s , t ) , S ( s , t ) ) ] 2 1 + [ H ( s * , s , s ) ] 2 + β H ( s * , s , s ) ] , ψ ( 1 8 α + β [ α H ( s * , S ( s * , t * ) , S ( s * , t * ) ) [ H ( s , S ( s , t ) , S ( s , t ) ) ] 2 1 + [ H ( s * , s , s ) ] 2 + β H ( s * , s , s ) ) ) = G ( 1 8 α + β [ α H ( s * , s * , s * ) [ H ( s , s , s ) ] 2 1 + [ H ( s * , s , s ) ] 2 + β H ( s * , s , s ) ] , ψ ( 1 8 α + β [ α H ( s * , s * , s * ) [ H ( s , s , s ) ] 2 1 + [ H ( s * , s , s ) ] 2 + β H ( s * , s , s ) ] ) ) = G ( β 8 α + β H ( s * , s , s ) , ψ ( β 8 α + β H ( s * , s , s ) ) ) β 8 α + β H ( s * , s , s ) H ( s * , s , s ) ;
therefore one of the following two terms holds: (a) β 8 α + β H ( s * , s , s ) = 0 ; (b) ψ ( β 8 α + β H ( s * , s , s ) ) = 0 . Then H ( s * , s , s ) = 0 . So we must have s * = s . Similar to the above proof, take H ( S ( t * , s * ) , S ( t , s ) , S ( t , s ) ) into account; it follows that t * = t . This shows ( s * , t * ) = ( s , t ) ; then the uniqueness of the coupled fixed points is verified.
Case II. Suppose that ( s * , t * ) cannot be comparable to ( s , t ) ; according to the factors of this theorem, there exists ( a , b ) Z × Z comparable to ( s * , t * ) and ( s , t ) . In cases where there exists a positive integer u 0 satisfying S u 0 ( a , b ) = ( s * , t * ) , then
S u 0 ( a , b ) = ( s * , t * ) , S u 0 + 1 ( a , b ) = S ( s * , t * ) = s * , S u 0 + 2 ( a , b ) = S 2 ( s * , t * ) = S ( S ( s * , t * ) , S ( t * , s * ) ) = S ( s * , t * ) = s * .
Continuing this process as above, then S u ( a , b ) = s * for u u 0 + 1 ; therefore S u ( a , b ) s * when u + .
On the other hand, if there does not exist u 0 , then for each u 1 ,
H ( S u ( a , b ) , s * , s * ) = H ( S u ( a , b ) , S u ( s * , t * ) , S u ( s * , t * ) ) G ( 1 8 α + β [ α H ( S u 1 ( a , b ) , S u ( a , b ) , S u ( a , b ) ) [ H ( S u 1 ( s * , t * ) , S u ( s * , t * ) , S u ( s * , t * ) ) ] 2 1 + [ H ( S u 1 ( a , b ) , S u 1 ( s * , t * ) , S u 1 ( s * , t * ) ) ] 2 + β H ( S u 1 ( a , b ) , S u 1 ( s * , t * ) , S u 1 ( s * , t * ) ) ] , ψ ( 1 8 α + β [ α H ( S u 1 ( a , b ) , S u ( a , b ) , S u ( a , b ) ) [ H ( S u 1 ( s * , t * ) , S u ( s * , t * ) , S u ( s * , t * ) ) ] 2 1 + [ H ( S u 1 ( a , b ) , S u 1 ( s * , t * ) , S u 1 ( s * , t * ) ) ] 2 + β H ( S u 1 ( a , b ) , S u 1 ( s * , t * ) , S u 1 ( s * , t * ) ) ] ) ) ,
regarding the fact that S u ( s , t ) ) = S ( S u 1 ( s , t ) , S u 1 ( t , s ) ) for each ( s , t ) Z × Z . Because S has a coupled fixed point ( s * , t * ) , it follows that S u ( s * , t * ) = s * for any u 1 and by virtue of (31), it can be inferred that
H ( S u ( a , b ) , s * , s * ) = H ( S u ( a , b ) , S u ( s * , t * ) , S u ( s * , t * ) ) G ( 1 8 α + β [ α H ( S u 1 ( a , b ) , S u ( a , b ) , S u ( a , b ) ) [ H ( s * , s * , s * ) ] 2 1 + [ H ( S u 1 ( a , b ) , s * , s * ) ] 2 + β H ( S u 1 ( a , b ) , s * , s * ) ] , ψ ( 1 8 α + β [ α H ( S u 1 ( a , b ) , S u ( a , b ) , S u ( a , b ) ) [ H ( s * , s * , s * ) ] 2 1 + [ H ( S u 1 ( a , b ) , s * , s * ) ] 2 + β H ( S u 1 ( a , b ) , s * , s * ) ] ) ) = G ( β 8 α + β H ( S u 1 ( a , b ) , s * , s * ) , ψ ( β 8 α + β H ( S u 1 ( a , b ) , s * , s * ) ) ) β 8 α + β H ( S u 1 ( a , b ) , s * , s * ) .
Reconsidering and rewriting the above expression, it can be concluded that
H ( S u ( a , b ) , s * , s * ) H ( S u 1 ( a , b ) , s * , s * ) .
Take r u = H ( S u ( a , b ) , s * , s * ) , which is positive decreasing and bounded below. Hence, r u = H ( S u ( a , b ) , s * , s * ) r 0 when u + . Taking u + in the equality (32), we have that
r G ( β 8 α + β r , ψ ( β 8 α + β r ) ) β 8 α + β r < r .
So β 8 α + β r = 0 or ψ ( β 8 α + β r ) = 0 . Thus r = 0 . This proves S u ( a , b ) s * when u + . Analogous to the above, it is easy to show that S u ( b , a ) t * when u + . Replacing s * by s and t * by t and rewriting the above description, it follows that S u ( a , b ) s and S u ( b , a ) t when u + . Therefore, s * = s and t * = t . So the ( s * , t * ) = ( s , t ) and the uniqueness of the coupled fixed point is verified. □
Remark 3.
Taking G ( s , t ) = ( 8 α + β ) s with 0 < 8 α + β < 1 in Theorem 5 easily leads to Theorem 2.
Taking special functions in Theorem 5 easily leads to the following corollaries.
Corollary 1.
Assume that Z is a set endowed with a partial order ≪ and we define H as a generalized metric on it. Moreover, ( Z , H ) is completed with regard to H-metric. Suppose S : Z × Z Z is continuous on Z with the mixed monotone property. For all ( s , t ) , ( a , b ) , ( c , d ) Z × Z with ( s , t ) ( a , b ) ( c , d ) ,
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) 1 8 α + β [ α H ( s , S ( s , t ) , S ( s , t ) ) H ( a , S ( a , b ) , S ( a , b ) ) H ( c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( s , a , c ) ] 2 + β H ( s , a , c ) ] ψ ( 1 8 α + β [ α H ( s , S ( s , t ) , S ( s , t ) ) H ( a , S ( a , b ) , S ( a , b ) ) H ( c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( s , a , c ) ] 2 + β H ( s , a , c ) ] ) ,
where ψ Ψ s and 8 α + β > 0 , α , β 0 . If there exists s 0 , t 0 Z satisfying that s 0 S ( s 0 , t 0 ) and t 0 S ( t 0 , s 0 ) , then S has a coupled fixed point ( s * , t * ) Z × Z ., i.e., ( s * , t * ) is the solution of s * = S ( s * , t * ) , t * = S ( t * , s * ) . Furthermore, if for each ( s , t ) , ( a , b ) Z × Z , there exists a ( c , d ) Z × Z that can be comparable to ( s , t ) and ( a , b ) , there exists uniqueness of coupled fixed points of S.
Proof. 
Take G ( s , t ) = s t in Theorem 5; this result obviously holds. □
Corollary 2.
Assume that Z is a set endowed with a partial order ≪ and we define H as a generalized metric on it. Moreover, ( Z , H ) is completed with regard to H-metric. Suppose S : Z × Z Z is continuous on Z with the mixed monotone property. For all ( s , t ) , ( a , b ) , ( c , d ) Z × Z with ( s , t ) ( a , b ) ( c , d ) ,
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) α 8 α + β H ( s , S ( s , t ) , S ( s , t ) ) H ( a , S ( a , b ) , S ( a , b ) ) H ( c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( s , a , c ) ] 2 + β 8 α + β H ( s , a , c ) ,
where 8 α + β > 0 , α , β 0 . If there exists s 0 , t 0 Z satisfying that s 0 S ( s 0 , t 0 ) and t 0 S ( t 0 , s 0 ) , then S has a coupled fixed point ( s * , t * ) Z × Z ., i.e., ( s * , t * ) is the solution of s * = S ( s * , t * ) , t * = S ( t * , s * ) . Furthermore, if for each ( s , t ) , ( a , b ) Z × Z , there exists a ( c , d ) Z × Z that can be comparable to ( s , t ) and ( a , b ) , there exists uniqueness of coupled fixed points of S.
Proof. 
Take G ( s , t ) = s in Theorem 5; this result obviously holds. □
Corollary 3.
Assume that Z is a set endowed with a partial order ≪ and we define H as a generalized metric on it. Moreover, ( Z , H ) is completed with regard to H-metric. Suppose S : Z × Z Z is continuous on Z with the mixed monotone property. For all ( s , t ) , ( a , b ) , ( c , d ) Z × Z with ( s , t ) ( a , b ) ( c , d ) ,
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) [ α 8 α + β H ( s , S ( s , t ) , S ( s , t ) ) H ( a , S ( a , b ) , S ( a , b ) ) H ( c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( s , a , c ) ] 2 + β 8 α + β H ( s , a , c ) ] μ ( α 8 α + β H ( s , S ( s , t ) , S ( s , t ) ) H ( a , S ( a , b ) , S ( a , b ) ) H ( c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( s , a , c ) ] 2 + β 8 α + β H ( s , a , c ) ) ,
where μ : [ 0 , + ) [ 0 , 1 ) , and is continuous and 8 α + β > 0 , α , β 0 . If there exists s 0 , t 0 Z satisfying that s 0 S ( s 0 , t 0 ) and t 0 S ( t 0 , s 0 ) , then S has a coupled fixed point ( s * , t * ) Z × Z ., i.e., ( s * , t * ) is the solution of s * = S ( s * , t * ) , t * = S ( t * , s * ) . Furthermore, if for each ( s , t ) , ( a , b ) Z × Z , there exists a ( c , d ) Z × Z that can be comparable to ( s , t ) and ( a , b ) , there exists uniqueness of coupled fixed points of S.
Proof. 
Take G ( s , t ) = s μ ( s ) in Theorem 5; this result obviously holds. □
In the following, we offer the proof of the existence of a coupled coincidence point for two maps S : Z × Z Z and f : Z Z .
Theorem 6.
Suppose that Z is a set endowed with a partial order, H is a metric on Z such that ( Z , H ) is completed with respect to H-metric. Assume that S : Z × Z Z and f : Z Z are continuous on Z satisfying that S is mixed f-monotone. If S ( Z × Z ) f ( Z ) , f commutes with S, also for ( s , t ) , ( a , b ) , ( c , d ) Z × Z with ( s , t ) ( a , b ) ( c , d ) and f s f a f c , f t f b f d ,
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) G ( 1 8 α + β [ α H ( f s , S ( s , t ) , S ( s , t ) ) H ( f a , S ( a , b ) , S ( a , b ) ) H ( f c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( f s , f a , f c ) ] 2 + β H ( f s , f a , f c ) ] , ψ ( 1 8 α + β [ α H ( f s , S ( s , t ) , S ( s , t ) ) H ( f a , S ( a , b ) , S ( a , b ) ) H ( f c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( f s , f a , f c ) ] 2 + β H ( f s , f a , f c ) ] ) )
where G C , ψ Ψ u and 8 α + β > 0 , α , β 0 . Supposing that there is s 0 , t 0 Z satisfying that f s 0 S ( s 0 , t 0 ) and f t 0 S ( t 0 , s 0 ) , so ( s * , t * ) Z × Z is a coupled coincidence point of S and f, i.e., ( s * , t * ) is the solution of f s * = S ( s * , t * ) , f t * = S ( t * , s * ) .
Proof. 
In virtue of S ( Z × Z ) f ( Z ) , taking s 1 , t 1 Z satisfying that f s 1 = S ( s 0 , t 0 ) , f t 1 = S ( t 0 , s 0 ) . Going on with this, s 2 , t 2 Z can be found with f s 2 = S ( s 1 , t 1 ) , f t 2 = S ( t 1 , s 1 ) . Since S is mixed g monotone, it follows that f s 0 f s 1 f s 2 and f t 2 f t 1 f t 0 . In general, it can be shown that for u 0 ,
f s u = S ( s u 1 , t u 1 ) f s u + 1 = S ( s u , t u ) , f t u + 1 = S ( t u , s u ) f t u = S ( t u 1 , s u 1 ) .
Setting s = s u , a = c = s u 1 , t = t u , b = d = t u 1 in inequality (35), an analysis analogous to that in Theorem 4 shows
H ( f s u + 1 , f s u , f s u ) G ( 1 8 α + β [ 4 α H ( f s u , f s u + 1 , f s u + 1 ) [ H ( f s u 1 , f s u , f s u ) ] 2 1 + [ H ( f s u 1 , f s u , f s u ) ] 2 + β H ( f s u , f s u 1 , f s u 1 ) ] , ψ ( 1 8 α + β [ 4 α H ( f s u , f s u + 1 , f s u + 1 ) [ H ( f s u 1 , f s u , f s u ) ] 2 1 + [ H ( f s u 1 , f s u , f s u ) ] 2 + β H ( f s u , f s u 1 , f s u 1 ) ] ) ) 1 8 α + β [ 4 α H ( f s u + 1 , f s u , f s u ) + β H ( f s u , f s u 1 , f s u 1 ) ] 1 8 α + β [ 8 α H ( f s u + 1 , f s u , f s u ) + β H ( f s u , f s u 1 , f s u 1 ) ] .
and
H ( f s u + 1 , f s u , f s u ) 1 8 α + β [ 4 α H ( f s u + 1 , f s u , f s u ) + β H ( f s u , f s u 1 , f s u 1 ) ] .
Reconsidering and rewriting (36), it follows that
H ( f s u + 1 , f s u , f s u ) H ( f s u , f s u 1 , f s u 1 ) H ( f s 1 , f s 0 , f s 0 ) .
In the same manner, we also obtain
H ( f t u + 1 , f t u , f t u ) H ( f t u , f t u 1 , f t u 1 ) H ( f t 1 , f t 0 , f t 0 ) .
Hence, the sequence { H ( f s u + 1 , f s u , f s u ) } is positive decreasing and bounded below. Thus, the sequence { H ( f s u + 1 , f s n , f s n ) } converges to r 0 . We shall prove that r = 0 .
Taking n + in (37), we have
r 1 8 α + β ( 4 α r + β r ) < r .
So r = 0 , i.e.,
lim u + H ( f s u + 1 , f s u , f s u ) = 0 .
Further, making use of Proposition 1, we derive that
lim u + H ( f s u , f s u + 1 , f s u + 1 ) = 0 .
An analysis analogous to that in Theorem 4 infers that { f s u } is Cauchy with regard to H-metric in ( Z , H ) .
Since ( Z , H ) is completed with regard to H-metric, it leads to a point s * Z satisfying f s u s * when u + .
Keeping on as before, we can also show further that { f t u } is Cauchy with regard to H-metric in ( Z , H ) and ( Z , H ) is completed with regard to H-metric; there exists a point t * Z such that f t u t * when u + .
At last, it suffices to show that S has a coupled coincident point ( s * , t * ) . Because S and f commute, it follows that
f ( f s u + 1 ) = f ( S ( s u , t u ) ) = S ( f s u , f t u ) .
f ( f t u + 1 ) = f ( S ( t u , s u ) ) = S ( f t u , f s u ) .
Setting limits when u + in (42), (43) and S and f are, respectively, continuous on Z × Z and Z, we obtain
f s * = lim u + f ( f s u + 1 ) = lim u + f ( S ( s u , t u ) ) = lim u + S ( f s u , f t u ) .
f t * = lim u + f ( f t u + 1 ) = lim u + f ( S ( t u , s u ) ) = lim u + S ( f t u , f s u ) .
Because H is continuous with respect to its variables, it follows that
H ( S ( s * , t * ) , f s * , f s * ) = H ( lim u + S ( f s u , f t u ) , f s * , f s * ) = H ( f s * , f s * , f s * ) = 0 .
Therefore f s * = S ( s * , t * ) . Moreover, it leads to f t * = S ( t * , s * ) . It shows that S and f have coupled coincidence point ( s * , t * ) . □
Remark 4.
Take G ( s , t ) = ( 8 α + β ) s with 0 < 8 α + β < 1 in Theorem 6; it is easy to obtain Theorem 3. In addition, with the same idea as the above corollaries, by taking special functions G and ψ in Theorem 6, we can also obtain some coupled coincidence point results of the two mappings.
Example 4.
Let Z = [ 0 , 1 ] endowed with the natural ordering of real numbers and H ( s , t , r ) = max { s , t , r } ; obviously, ( Z , H ) is completed with respect to H-metric. Definite functions S and g are by
S ( s , t ) = s 12 + t 2 , if s t , 0 , otherwise . , f ( s ) = 2 s 3 .
Obviously, there exist s 0 = t 0 = 0 such that f s 0 S ( s 0 , t 0 ) and f t 0 S ( t 0 , s 0 ) ; furthermore, S and f are continuous functions, S ( Z , Z ) g ( Z ) , S is mixed f-monotone and f commutes with S. Take G ( s , t ) = 1 8 α + β s , where α 0 and β = 7 8 . Now, we show S and f satisfy condition (35). Indeed, we take s , t , a , b , c , d with s a c and t b d .
Case I : If s t , a b and c d , we have
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) = H ( s 12 + t 2 , a 12 + b 2 , c 12 + d 2 ) = max { s 12 + 6 t 12 , a 12 + 6 b 12 , c 12 + 6 d 12 } 7 c 12 7 8 max { 2 s 3 , 2 a 3 , 2 c 3 } = 7 8 H ( f s , f a , f c ) α H ( f s , S ( s , t ) , S ( s , t ) ) H ( f a , S ( a , b ) , S ( a , b ) ) H ( f c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( f s , f a , f c ) ] 2 + 7 8 H ( f s , f a , f c ) .
Case II : If s t , a b and c < d , we obtain
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) = H ( s 12 + t 2 , a 12 + b 2 , 0 ) = max { s 12 + 6 t 12 , a 12 + 6 b 12 , 0 } 7 c 12 7 8 max { 2 s 3 , 2 a 3 , 2 c 3 } = 7 8 H ( f s , f a , f c ) α H ( f s , S ( s , t ) , S ( s , t ) ) H ( f a , S ( a , b ) , S ( a , b ) ) H ( f c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( f s , f a , f c ) ] 2 + 7 8 H ( f s , f a , f c ) .
Case III : If s t , a < b and c < d , we obtain
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) = H ( s 12 + t 2 , 0 , 0 ) = max { s 12 + 6 t 12 , 0 , 0 } 7 c 12 7 8 max { 2 s 3 , 2 a 3 , 2 c 3 } = 7 8 H ( f s , f a , f c ) α H ( f s , S ( s , t ) , S ( s , t ) ) H ( f a , S ( a , b ) , S ( a , b ) ) H ( f c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( f s , f a , f c ) ] 2 + 7 8 H ( f s , f a , f c ) .
Case IV : If s < t , a < b and c < d , we obtain
H ( S ( s , t ) , S ( a , b ) , S ( c , d ) ) = H ( 0 , 0 , 0 ) = max { 0 , 0 , 0 } 7 8 H ( f s , f a , f c ) α H ( f s , S ( s , t ) , S ( s , t ) ) H ( f a , S ( a , b ) , S ( a , b ) ) H ( f c , S ( c , d ) , S ( c , d ) ) 1 + [ H ( f s , f a , f c ) ] 2 + 7 8 H ( f s , f a , f c ) .
Similarly, the other cases follow immediately. Therefore, all the conditions of Theorem 6 are satisfied, so S and f have a coupled coincidence point.

4. Coupled Coincidence Point Theorem via Inverse C k -Class Functions

In the section, if we carry out some changes in the first inequality of Definition 8, we obtain another kind of inverse C-class function, which can be regarded as the generalization of the inverse C-class functions.
Definition 7.
A mapping G : [ 0 , + ) 2 R is said to be an inverse C k -class function in cases where it is continuous and the terms as follows hold:
(i)
G ( x , y ) l x for all x , y [ 0 , + ) and some l 1 .
( i i )
G ( x , y ) = l x yields that one of the two terms holds: (a) x = 0 ; (b) y = 0 .
C inv denotes the set of all the inverse C k -class functions. Every inverse C-class function is equivalent to an inverse C k -class function when k = 1 , but the scope of an inverse C k -class function must be larger than that of an inverse C-class function.
Example 5.
A mapping G : [ 0 , + ) 2 R is a function satisfying that G ( x , y ) = l x log a ( 1 2 l a + y ) for all x , y [ 0 , + ) , l 1 and a > 1 . Then, obviously, G is an inverse C k -class function when l 1 , because G ( x , y ) l x when l 2 , a > 1 and y 0 and G ( x , y ) = l x y = 0 when l = 2 . Yet it is not an inverse C-class function, because G ( x , y ) x does not hold if l = 1 , y 0 and G ( x , y ) = x does not imply y = 0 since we can set l = 1 , y = 1 2 a > 2 .
Example 6.
We define a mapping G : [ 0 , + ) R by G ( x , y ) = 2 x + y for all x , y [ 0 , + ) . So, it is obvious that G is an inverse C k -class function for k = 2 , but it is not an inverse C-class function.
Example 7.
All the mappings G : [ 0 , + ) 2 R belong to C inv - k , for all x , y [ 0 , + ) :
  • G ( x , y ) = k x + l y , G ( x , y ) = k x leads to y = 0 for some k 1 and l > 0 .
  • G ( x , y ) = k n x , G ( x , y ) = k x leads to x = 0 for n ( 1 , + ) and some k 1 .
  • G ( x , y ) = k x ( 1 + k y ) r , G ( x , y ) = k x leads to x = 0 or y = 0 for r ( 0 , + ) and some k 1 .
  • G ( x , y ) = log a [ ( k y + a k x ) ( 1 + k y ) ] , G ( x , y ) = k s leads to y = 0 for some a > 1 and some k 1 .
  • G ( x , y ) = k φ ( s ) , G ( x , y ) = k x leads to x = 0 , where φ : [ 0 , + ) [ 0 , + ) is an upper semi-continuous function satisfying that φ ( 0 ) = 0 and φ ( x ) > x , for x > 0 and some k 1 .
The purpose of this section is to give the existence theorem of a coupled coincidence point by use of inverse C k -class functions.
Singh et al. [33], also introduce new notions which are called g-compatible of type ( E ) and f-compatible of type ( E ) . We all know that f and g are self-mappings on Z. Inspired by Singh’s work, we define f-compatible of type ( E ) and S-compatible of type ( E ) where f : Z Z and S : Z × Z Z .
Definition 8.
Two maps f : Z Z and S : Z × Z Z in a metric space ( Z , d ) are called f-compatible of type ( E ) if lim u + f f s u = lim u + f S ( s u , s u ) = S ( y , y ) , whenever { s u } is a sequence in Z with the fact that lim u + f s u = lim u + S ( s u , s u ) = y for some y Z . Similarly, two self-mappings f and S of a metric space ( Z , d ) are called S-compatible of type ( E ) if lim u + S ( S ( s u , s u ) , S ( s u , s u ) ) = lim u + S ( f s u , f s u ) = f y , whenever { s u } is a sequence in Z satisfying that lim u + f s u = lim u + S ( s u , s u ) = y for some y Z .
Theorem 7.
Suppose that Z is a set endowed with a partial order; H is a metric on Z satisfying that ( Z , H ) is completed with respect to H-metric. Assume that S : Z × Z Z and f : Z Z are, respectively, continuous on Z × Z and Z satisfying that ( f , S ) is S-compatible of type ( E ) and S is mixed f-monotone. Let S ( Z × Z ) f ( Z ) ; f commutes with S and for ( s , t ) , ( a , b ) , ( c , d ) Z × Z with ( s , t ) ( a , b ) ( c , d ) and f s f a f c , f t f b f d ,
k H ( f s , f a , f c ) G ( 1 2 α + β [ 4 α H ( f s , S ( s , t ) , S ( s , t ) ) ( 1 + H ( f a , S ( a , b ) , S ( a , b ) ) ) ( 1 + H ( f c , S ( c , d ) , S ( c , d ) ) ) [ 1 + H ( f s , f a , f c ) ] 2 + β H ( f s , f a , f c ) ] , ψ ( 1 2 α + β [ 4 α H ( f s , S ( s , t ) , S ( s , t ) ) ( 1 + H ( f a , S ( a , b ) , S ( a , b ) ) ) ( 1 + H ( f c , S ( c , d ) , S ( c , d ) ) ) [ 1 + H ( f s , f a , f c ) ] 2 + β H ( f s , f a , f c ) ] ) )
where G C inv - k , ψ Ψ u and 2 α + β > 0 , α , β 0 . If there exists s 0 , t 0 Z satisfying that f s 0 S ( s 0 , t 0 ) and f t 0 S ( t 0 , s 0 ) , then ( x * , y * ) X × X is a a coupled coincidence point of S and f, i.e., ( s * , t * ) is the solution of f s * = S ( s * , t * ) , f t * = S ( t * , s * ) .
Proof. 
Because S ( Z × Z ) f ( Z ) , take s 1 , t 1 Z satisfying that f s 1 = S ( s 0 , t 0 ) , f t 1 = S ( t 0 , s 0 ) . Continuing this way, s 2 , t 2 Z can be found satisfying that f s 2 = S ( s 1 , t 1 ) , f t 2 = S ( t 1 , s 1 ) . Since S is mixed g monotone, it can be inferred that f s 0 f s 1 f s 2 , f t 2 f t 1 f t 0 . Inductively, it can be shown that for u 0 ,
f s u = S ( s u 1 , t u 1 ) f s u + 1 = S ( s u , t u ) , f t u + 1 = S ( t u , s u f t u = S ( t u 1 , s u 1 ) .
Taking s = s u , a = c = s u 1 , t = t u , b = d = t u 1 in inequality (47), we have,
k H ( f s u , f s u 1 , f s u 1 ) G ( 1 2 α + β [ 4 α H ( f s u , f s u + 1 , f s u + 1 ) [ 1 + H ( f s u 1 , f s u , f s u ) ] 2 [ 1 + H ( f s u , f s u 1 , f s u 1 ) ] 2 + β H ( f s u , f s u 1 , f s u 1 ) ] , ψ ( 1 2 α + β [ 4 α H ( f s u , f s u + 1 , f s u + 1 ) [ 1 + H ( f s u 1 , f s u , f s u ) ] 2 [ 1 + H ( f s u , f s u 1 , f s u 1 ) ] 2 + β H ( f s u , f s u 1 , f s u 1 ) ] ) ) k 1 2 α + β [ 4 α H ( f s u + 1 , f s u , f s u ) + β H ( f s u , f s u 1 , f s u 1 ) ] , k 1 2 α + β [ 2 α H ( f s u + 1 , f s u , f s u ) + β H ( f s u , f s u 1 , f s u 1 ) ] .
and
k H ( f s u , f s u 1 , f s u 1 ) G ( 1 2 α + β [ 4 α H ( f s u , f s u + 1 , f s u + 1 ) [ 1 + H ( f s u 1 , f s u , f s u ) ] 2 [ 1 + H ( f s u , f s u 1 , f s u 1 ) ] 2 + β H ( f s u , f s u 1 , f s u 1 ) ] , ψ ( 1 2 α + β [ 4 α H ( f s u , f s u + 1 , f s u + 1 ) [ 1 + H ( f s u 1 , f s u , f s u ) ] 2 [ 1 + H ( f s u , f s u 1 , f s u 1 ) ] 2 + β H ( f s u , f s u 1 , f s u 1 ) ] ) ) k 1 2 α + β [ 4 α H ( f s u + 1 , f s u , f s u ) + β H ( f s u , f s u 1 , f s u 1 ) ] .
Reconsidering and rewriting (48), it follows that
H ( f s u + 1 , f s u , f s u ) H ( f s u , f s u 1 , f s u 1 ) H ( f s 1 , f s 0 , f s 0 ) .
In the same manner, we can see that
H ( f t u + 1 , f t u , f t u ) H ( f t u , f t u 1 , f t u 1 ) H ( f t 1 , f t 0 , f t 0 ) .
Hence, the sequence { H ( f s u + 1 , f s u , f s u ) } is positive decreasing and bounded below. Thus, the sequence { H ( f s u + 1 , f s u , f s u ) } converges to r 0 . It suffices to prove that r = 0 .
Taking u + in (49), it follows that
k r k 2 α + β ( 4 α r + β r ) > k r .
Therefore, r = 0 ,i.e.,
lim u + H ( f s u + 1 , f s u , f s u ) = 0 .
Furthermore, making use of Proposition 1, we deduce that
lim u + H ( f s u , f s u + 1 , f s u + 1 ) = 0 .
An analysis analogous to that in Theorem 4 yields that { f s u } is Cauchy with respect to H metric in ( Z , H ) .
Because ( Z , H ) is completed with respect to H-metric, it is obvious that there is a point s * X satisfying that f s u s * when u + .
Going on as before, we can also show further that { f t u } is Cauchy with respect to H-metric in ( Z , H ) . Since ( Z , H ) is completed with respect to H-metric, there exists a point t * Z satisfying that f t u t * when u + .
At last, it suffices to prove that f and S have a coupled coincident point ( s * , t * ) .
Because S and f commute, it follows that
f ( f s u + 1 ) = f ( S ( s u , t u ) ) = S ( f s u , f t u ) ,
f ( f t u + 1 ) = f ( S ( t u , s u ) ) = S ( f t u , f s u ) .
Setting limits when u + in (54), (55), since S and f are, respectively, continuous on Z × Z and Z, we obtain
f x * = lim u + f ( f s u + 1 ) = lim u + f ( S ( s u , t u ) ) = lim u + S ( f s u , f t u )
f t * = lim u + f ( f t u + 1 ) = lim u + f ( S ( t u , s u ) ) = lim u + S ( f t u , f s u )
Sequentially, since H is continuous with respect to all its variables, it follows that
H ( S ( s * , t * ) , f s * , f s * ) = H ( lim u + S ( f s u , f t u ) , f s * , f s * ) = H ( f s * , f s * , f s * ) = 0 .
Therefore, f s * = S ( s * , t * ) . Analogously, f t * = S ( t * , s * ) . This shows that ( s * , t * ) is a coupled coincidence point. □
Remark 5. ( i ) The contractive condition (47) is different from (35) in Theorem 6. At first, the direction of inequality changes from ≤ to ≥. In fact, that is the key difference between G C and G C i n v k . Secondly, the left-hand side of the inequality becomes k H ( f s , f a , f c ) which is different from H ( S ( s , t ) , S ( a , b ) , S ( c , d ) in Theorem 6. This difference makes the proof of the Cauchy sequence f s u easier.
( i i ) There is an open question. Why not use the S-compatibility of type ( E ) of the pair ( f , S ) . There is also the question whether we can replace the mixed f-monotonicity of ( f , S ) with S-compatibility of type ( E ) of ( f , S ) . In other words, it seems that the proof of Theorem 7 could pass without the S-compatibility of type ( E ) of ( f , S ) .

5. Application

In this section, Theorem 5 is used to guarantee the existence and uniqueness of solutions to the following integral equations:
x ( t ) = a b ( K 1 ( t , s ) + K 2 ( t , s ) ) ( f 1 ( s , x ( s ) ) + f 2 ( s , x ( s ) ) ) d s + h ( t ) ,
where f i : [ a , b ] × R R , K 1 : [ a , b ] × [ a , b ] [ 0 , + ) , K 2 : [ a , b ] × [ a , b ] ( , 0 ] and h ( t ) : [ a , b ] R are continuous for all i = 1 , 2 . Let H : C ( [ a , b ] , R ) × C ( [ a , b ] , R ) × C ( [ a , b ] , R ) R be a function defined by
H ( x , y , z ) = max { ( sup t [ a , b ] | x ( t ) y ( t ) | , sup t [ a , b ] | x ( t ) z ( t ) | , sup t [ a , b ] | y ( t ) z ( t ) | ) }
where C ( [ a , b ] , R ) by the set of all continuous real-value functions defined on [ a , b ] . It is easy to see that ( C ( [ a , b ] , R ) , H ) is completed with regard to H-metric. Let S : C ( [ a , b ] , R ) × C ( [ a , b ] , R ) C ( [ a , b ] , R ) be a mapping defined by
S ( x ( t ) , y ( t ) ) = a b ( K 1 ( t , s ) + K 2 ( t , s ) ) ( f 1 ( s , x ( s ) ) + f 2 ( s , y ( s ) ) ) d s + h ( t ) , for all t [ a , b ] .
Suppose that the following conditions hold:
( i ) There exist two numbers λ > 0 and α , β 0 with 8 α + β > 0 such that
0 f 2 ( t , u ( t ) ) f 2 ( t , v ( t ) ) f 1 ( t , x ( t ) ) f 1 ( t , y ( t ) ) λ β 8 α + β ( x y ) ,
for all x y , u v , x , y , u , v R ;
( i i ) sup t [ a , b ] | a b K 1 ( t , s ) + K 2 ( t , s ) d s | k λ where k 0 , 1 ;
( i i i ) f i and g are increasing functions for all i = 1 , 2 ;
( i v ) There exists a couple lower-upper solution ( x 0 , y 0 ) of Equation (59); that is,
x 0 ( t ) a b ( K 1 ( t , s ) + K 2 ( t , s ) ) ( f 1 ( s , x 0 ( s ) ) + f 2 ( s , y 0 ( s ) ) ) d s + h ( t )
and
y 0 ( t ) a b ( K 1 ( t , s ) + K 2 ( t , s ) ) ( f 1 ( s , y 0 ( s ) ) + f 2 ( s , x 0 ( s ) ) ) d s + h ( t ) .
Now we prove the existence of a solution of the mentioned integral equations (59).
Theorem 8.
Under conditions (i)–(iv), (59) has a solution in C ( [ a , b ] , R ) .
Proof. 
Consider a partial order on C ( [ a , b ] , R ) ; that is, x y if and only if x y . Assume that x , y , u , v , w , z C ( [ a , b ] , R ) such that x ( t ) u ( t ) w ( t ) and y ( t ) v ( t ) z ( t ) , for all t [ a , b ] . By ( i ) and ( i i ) ; we obtain
| S ( x , y ) S ( u , v ) | = | a b ( K 1 ( t , s ) + K 2 ( t , s ) ) ( f 1 ( s , x ( s ) ) + f 2 ( s , y ( s ) ) ) d s + h ( t ) a b ( K 1 ( t , s ) + K 2 ( t , s ) ) ( f 1 ( s , u ( s ) ) + f 2 ( s , v ( s ) ) ) d s + h ( t ) | = | a b ( K 1 ( t , s ) + K 2 ( t , s ) ( f 1 ( s , x ( s ) ) f 1 ( s , u ( s ) ) ) d s + a b ( K 1 ( t , s ) + K 2 ( t , s ) ( ( f 2 ( s , y ( s ) ) f 2 ( s , v ( s ) ) ) d s | = | a b ( K 1 ( t , s ) + K 2 ( t , s ) ( f 1 ( s , u ( s ) ) f 1 ( s , x ( s ) ) ) d s a b ( K 1 ( t , s ) + K 2 ( t , s ) ( ( f 2 ( s , y ( s ) ) f 2 ( s , v ( s ) ) ) d s | | a b ( K 1 ( t , s ) + K 2 ( t , s ) ) ( f 1 ( s , u ( s ) ) f 1 ( s , x ( s ) ) ) d s | sup t [ a , b ] | a b K 1 ( t , s ) + K 2 ( t , s ) d s | sup t [ a , b ] ( λ β 8 α + β [ u ( t ) x ( t ) ] ) k β 8 α + β sup t [ a , b ] [ u ( t ) x ( t ) ] .
Analogously, we can obtain
| S ( u , v ) S ( w , z ) | k β 8 α + β sup t [ a , b ] [ w ( t ) u ( t ) ]
and
| S ( x , y ) S ( w , z ) | k β 8 α + β sup t [ a , b ] [ w ( t ) x ( t ) ] .
Since
H ( S ( x , y ) , S ( u , v ) , S ( w , z ) ) = max { ( sup t [ a , b ] | S ( x , y ) S ( u , v ) | , sup t [ a , b ] | S ( x , y ) S ( w , z ) | , sup t [ a , b ] | S ( u , v ) S ( w , z ) | ) } max { k β 8 α + β sup t [ a , b ] [ u ( t ) x ( t ) ] , k β 8 α + β sup t [ a , b ] [ w ( t ) x ( t ) ] , k β 8 α + β sup t [ a , b ] [ w ( t ) u ( t ) ] } = k β 8 α + β max { sup t [ a , b ] [ u ( t ) x ( t ) ] , sup t [ a , b ] [ w ( t ) x ( t ) ] , sup t [ a , b ] [ w ( t ) u ( t ) ] } = k β 8 α + β H ( x , u , w ) 1 8 α + β [ α H ( x , S ( x , y ) , S ( x , y ) ) H ( u , S ( u , v ) , S ( u , v ) ) H ( w , S ( w , z ) , S ( w , z ) ) 1 + [ H ( x , u , w ) ] 2 + β H ( x , u , w ) ] 1 k 8 α + β [ α H ( x , S ( x , y ) , S ( x , y ) ) H ( u , S ( u , v ) , S ( u , v ) ) H ( w , S ( w , z ) , S ( w , z ) ) 1 + [ H ( x , u , w ) ] 2 + β H ( x , u , w ) ]
Let ψ ( x ) = ( 1 k ) x and G ( x , y ) = x y ; we can obtain
H ( S ( x , y ) , S ( u , v ) , S ( w , z ) ) G ( 1 8 α + β [ α H ( x , S ( x , y ) , S ( x , y ) ) H ( u , S ( u , v ) , S ( u , v ) ) H ( w , S ( w , z ) , S ( w , z ) ) 1 + [ H ( x , u , w ) ] 2 + β H ( x , u , w ) ] , ψ ( 1 8 α + β [ α H ( x , S ( x , y ) , S ( x , y ) ) H ( u , S ( u , v ) , S ( u , v ) ) H ( w , S ( w , z ) , S ( w , z ) ) 1 + [ H ( x , u , w ) ] 2 + β H ( x , u , w ) ] ) ) ,
Thus, S satisfies condition (14). It is obvious that S is continuous with the mixed monotone property and there exist x 0 ( t ) , y 0 ( t ) such that x 0 ( t ) S ( x 0 ( t ) , y 0 ( t ) ) and y 0 ( t ) S ( y 0 ( t ) , x 0 ( t ) ) by ( i i i ) and ( i v ) . S satisfies all the conditions of Theorem 5; then, S has a unique couple fixed point. Assume that ( x , y ) is the couple fixed point of S; we can deduce that ( y , x ) is also a couple fixed point of S. x = y due to the uniqueness of the couple fixed point of S. That is (59) has a unique solution for C ( [ a , b ] , R ) . □

6. Conclusions

In conclusion, a rational type contractive condition is proposed in an H-metric space and some coupled fixed point theorems for mappings which are mixed monotone are given. The uniqueness of the coupled fixed point is offered. The existence of coupled coincidence points of two mappings are also obtained. In addition, the coupled coincidence point theorems by virtue of inverse C k -class functions are proved in the context of mixed f-monotonicity of ( f , S ) . Furthermore, Some examples and an application are given to justify the main results. It is well known that the fixed point theorem plays an important role in ensuring the existence and uniqueness of solutions concerning some equations; we consider possible future research directions.
There are some works in the future:
(i)
Discuss the possibility of applying coupled fixed point results to the problems of differential equations, for example, fuzzy and set differential equations and discontinuous differential equations
(ii)
Discuss the properties of the C-class functions for coupled fixed point problems which include data dependence, well-posedness, Ulam–Hyers stability, limit shadowing property.

Author Contributions

Conceptualization, X.L. and M.Z.; formal analysis, X.L. and M.Z.; investigation, X.L., M.Z., A.H.A., K.C. and M.A.; writing—original draft preparation, X.L., M.Z. and K.C.; writing—review and editing, A.H.A., M.A. and L.R. All authors have read and agreed to the published version of the manuscript.

Funding

This work is partially supported by National Natural Science Foundation of China (Grant No.11872043), Central Government Funds of Guiding Local Scientific and Technological Development for Sichuan Province (Grant No.2021ZYD0017), Zigong Science and Technology Program (Grant No.2020YGJC03), 2021 Innovation and Entrepreneurship Training Program for College Students of Sichuan University of Science and Engineering (Grant No.cx2021150).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

The authors thank the anonymous reviewers for their excellent comments, suggestions, and ideas that helped improve this article.

Conflicts of Interest

The authors declare that they have no competing interests.

References

  1. Ran, A.C.M.; Reurings, M.C.B. A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132, 1435–1443. [Google Scholar] [CrossRef]
  2. Bhaskar, T.G.; Lakshmikantham, V. Fixed point theorems in partially ordered metric spaces and applications. Nonl. Anal. 2006, 65, 1379–1393. [Google Scholar] [CrossRef]
  3. Lakshmikantham, V.; Ćirić, L. Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonl. Anal. 2009, 70, 4341–4349. [Google Scholar] [CrossRef]
  4. Harjani, J.; Sadarangani, K. Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonl. Anal. 2010, 72, 1188–1197. [Google Scholar] [CrossRef]
  5. Chen, C.M. Common fixed-point theorems in complete generalized metric spaces. J. Appl. Math. 2012, 2012, 945915. [Google Scholar] [CrossRef]
  6. Shahi, P.; Rathour, L.; Mishra, V.N. Expansive fixed point theorems for tri-simulation functions. J. Eng. Exact Sci. 2022, 8, 14303-01e. [Google Scholar] [CrossRef]
  7. Sanatee, A.G.; Rathour, L.; Mishra, V.N.; Dewangan, V. Some fixed point theorems in regular modular metric spaces and application to Caratheodory’s type anti-periodic boundary value problem. J. Anal. 2022, 1–14. [Google Scholar] [CrossRef]
  8. Pathak, V.K.; Mishra, L.N. Application of fixed point theorem to solvability for non-linear fractional hadamard functional integral equations. Mathematics 2022, 10, 2400. [Google Scholar] [CrossRef]
  9. Mishra, L.N.; Tiwari, S.K.; Mishra, V.N.; Tiwari, S.K. Unique fixed point theorems for generalized contractive mappings in partial metric spaces. J. Funct. Spaces 2015, 2015, 600–612. [Google Scholar] [CrossRef] [Green Version]
  10. Mishra, L.N.; Tiwari, S.K.; Mishra, V.N. Fixed point theorems for generalized weakly S-contractive mappings in partial metric spaces. J. Appl. Anal. Comput. 2015, 5, 600–612. [Google Scholar]
  11. Mishra, V.N.; Sánchez Ruiz, L.M.; Gautam, P.; Verma, S. Interpolative Reich-Rus-Ćirić and Hardy-Rogers contraction on quasi-partial b-metric space and related fixed point results. Mathematics 2020, 8, 1598. [Google Scholar] [CrossRef]
  12. Ćirić, L.; Olatinwo, M.O.; Gopal, D.; Akinbo, G. Coupled fixed point theorems for mappings satisfying a contractive condition of rational type on a partially ordered metric space. Adv. Fixed Point Theory 2012, 2, 1–8. [Google Scholar]
  13. Berinde, V. Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Non. Anal. 2011, 74, 7347–7355. [Google Scholar] [CrossRef] [Green Version]
  14. Choudhury, B.S.; Maity, P. Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54, 73–79. [Google Scholar] [CrossRef]
  15. Mustafa, Z.; Roshan, J.R.; Parvaneh, V. Coupled coincidence point results for (ψ,ϕ)-weakly contractive mappings in partially ordered Gb-metric spaces. Fixed Point Theory A. 2013, 2013, 206. [Google Scholar] [CrossRef] [Green Version]
  16. Kutbi, M.A.; Hussain, N.; Rezaei Roshan, J.; Parvaneh, V. Coupled and tripled Coincidence Point Results with Application to fredholm integral equations. Abstr. Appl. Anal. 2014, 2014, 568718. [Google Scholar] [CrossRef]
  17. Mustafa, Z.; Sims, B. A new approach to generalized metric spaces. J. Nonlinear Convex A. 2006, 7, 289–296. [Google Scholar]
  18. Saadati, R.; Vaezpoura, S.M.; Vetro, P.; Rhoadesc, B.E. Fixed point theorems in generalized partially ordered G-metric spaces. Math. Comput. Model. 2010, 52, 797–801. [Google Scholar] [CrossRef]
  19. Chakrabarti, K. Fixed point theorems in G-metric spaces with W maps. Math. Sci. Lett. 2013, 2, 29–35. [Google Scholar] [CrossRef]
  20. Karapínar, E.; Agarwal, R.P. Further fixed point results on G-metric spaces. Fixed Point Theory A. 2013, 2013, 154. [Google Scholar] [CrossRef] [Green Version]
  21. Mustafa, Z.; Rezaei Roshan, J.; Parvaneh, V. Existence of tripled coincidence point in ordered Gb-metric spaces and applications to system of integral equations. J. Inequal. Appl. 2013, 2013, 453. [Google Scholar] [CrossRef] [Green Version]
  22. Parvaneh, V.; Razani, A.; Rezaei Roshan, J. Common fixed points of six mappings in partially ordered G-metric spaces. Math. Sci. 2013, 7, 8. [Google Scholar] [CrossRef] [Green Version]
  23. Razani, A.; Parvaneh, V. On generalized weakly G-contractive mappings in partially ordered G-metric spaces. Abstr. Appl. Anal. 2012, 2012, 701910. [Google Scholar] [CrossRef] [Green Version]
  24. Mustafa, Z.; Parvaneh, V.; Abbas, M.; Rezaei Roshan, J. Some coincidence point results for generalized (ψ,ϕ)-weakly contractive mappings in ordered G-metric spaces. Fixed Point Theory A 2013, 2013, 326. [Google Scholar] [CrossRef] [Green Version]
  25. Hussain, N.; Parvaneh, V.; Rezaei Roshan, J. Fixed point results for G-α-contractive maps with application to boundary value problems. Sci. World J. 2014, 2014, 585964. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  26. Latif, A.; Hussain, N.; Rezaei Roshan, J.; Parvaneh, V. A unification of G-metric, Partial metric and b-metric spaces. Abstr. Appl. Anal. 2014, 2014, 180698. [Google Scholar]
  27. Ćirić, L.; Alsulami, S.M.; Parvaneh, V.; Roshan, J.R. Some fixed point results in ordered Gp-metric spaces. Fixed Point Theory A 2013, 2013, 317. [Google Scholar] [CrossRef] [Green Version]
  28. Harjani, J.; López, B.; Sadarangani, K. A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space. Abstr. Appl. Anal. 2010, 2010. [Google Scholar] [CrossRef]
  29. Jaggi, D.S. Some unique fixed point theorems. Indian J. Pure. Ap. Mat. 1977, 8, 223–230. [Google Scholar]
  30. Chakrabarti, K. Coupled fixed point theorems with rational type contractive condition in a partially ordered G-metric space. J. Math. 2014, 2014, 785357. [Google Scholar] [CrossRef] [Green Version]
  31. Ansari, A.H. Note on “φ-ψ-contractive type mappings and related fixed point”. In Proceedings of the the 2nd Regional Conference on Mathematics and Applications, Payame Noor University, Tehran, Iran, September 2014; pp. 377–380. [Google Scholar]
  32. Saleem, N.; Ansari, A.H.; Jain, M.K. Some fixed point theorems of inverse C-class funtion under weak semi compatibility. J. Fixed Point Theory 2018, 2018, 23. [Google Scholar]
  33. Singh, M.R.; Singh, Y.M. On various types of compatible maps and common fixed point theorems for non-continuous maps. Hacettepe J. Math. Stat. 2011, 40, 503–513. [Google Scholar]
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Liu, X.; Zhou, M.; Ansari, A.H.; Chakrabarti, K.; Abbas, M.; Rathour, L. Coupled Fixed Point Theorems with Rational Type Contractive Condition via C-Class Functions and Inverse Ck-Class Functions. Symmetry 2022, 14, 1663. https://doi.org/10.3390/sym14081663

AMA Style

Liu X, Zhou M, Ansari AH, Chakrabarti K, Abbas M, Rathour L. Coupled Fixed Point Theorems with Rational Type Contractive Condition via C-Class Functions and Inverse Ck-Class Functions. Symmetry. 2022; 14(8):1663. https://doi.org/10.3390/sym14081663

Chicago/Turabian Style

Liu, Xiaolan, Mi Zhou, Arslan H. Ansari, Kalyan Chakrabarti, Mujahid Abbas, and Laxmi Rathour. 2022. "Coupled Fixed Point Theorems with Rational Type Contractive Condition via C-Class Functions and Inverse Ck-Class Functions" Symmetry 14, no. 8: 1663. https://doi.org/10.3390/sym14081663

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