fixed point theorems in bimetric space endowed with binary ... - Core

Miskolc Mathematical Notes Vol. 16 (2015), No. 2, pp. 939–951

HU e-ISSN 1787-2413 DOI: 10.18514/MMN.2015.1263

FIXED POINT THEOREMS IN BIMETRIC SPACE ENDOWED WITH BINARY RELATION AND APPLICATIONS M.S. KHAN, M. BERZIG, AND S. CHANDOK Received 24 May, 2014 Abstract. In this paper, we obtained some fixed point results for continuous mappings satisfying a generalized contractive condition in the setting of two metrics space endowed with a binary relation. Our theorems generalize and extend several known results in the literature. As application, we establish an existence theorem for the solution of a nonlinear first order differential equation. 2010 Mathematics Subject Classification: 46T99; 41A50; 47H10; 54H25 Keywords: fixed point, generalized contraction mappings, ordered metric spaces

1. I NTRODUCTION The Banach contraction mapping is one of the pivotal results of analysis. It is very popular tool for solving existence and uniqueness problems in many different fields of mathematics. Due to its importance and applications potential, the Banach Contraction Mapping Principle has been investigated heavily by many authors. Consequently, a number of generalizations of this celebrated principle have appeared in the literature (see [1–19]). Recently, Samet and Turinici [17] introduced the notion of contractive mapping in a metric space endowed with amorphous binary relation. They showed a theorem subsumes many known results in the literature. For further study about contractive mappings in a metric space endowed with binary relation, we refer the reader to [3–7], [13], [17] and [19]. The purpose of this paper is to establish some fixed point results in the setting of two metrics space; called bimetric space; endowed with a binary relation. Our results generalized and extended many existing fixed point theorems, for generalized contractive and quasi-contractive mappings, in a metric space endowed with binary relation. Also, an application to the study the existence of solution to a nonlinear first order differential equation has been given. c 2015 Miskolc University Press

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2. P RELIMINARIES We first recall Maia’s fixed point theorem: Theorem 1 ([14]). Let .X; d; ı/ be a bimetric space. Assume that for T W X ! X , the following conditions are satisfied: (i) d.x; y/  ı.x; y/ for all x; y 2 X; (ii) X is complete with respect to d ; (iii) T is continuous with respect to d ; (iv) there exists a constant ˛ 2 Œ0; 1/ such that ı.T x; T y/  ˛ı.x; y/: Then T has a unique fixed point in X. In the sequel, let .X; d; ı/ be a bimetric space, and T W X ! X be a mapping. Denote F ix.T / WD fx  2 X W x  D T x  g: Let R be a binary relation on X , and let S be the symmetric binary relation defined by x; y 2 X; xSy ” xRy or yRx: From x0 2 X, we define the sequence fxn g by xn D T xn

1

for all n 2 N:

Definition 1. Let .X; ı/ be metric space and n 2 N [ f0g. For A  X we denote by D.A/ WD supfı.a; b/ W a; b 2 Ag. For each x0 2 X, the orbit sets of T at x0 are defined as following On .x0 / D fx0 ; x1 ; : : : ; xn g and

O1 .x0 / D fx0 ; x1 ; x2 ; : : : g:

We say that .X; ı/ is T -orbitally complete iff every ı-Cauchy sequence from O1 .x/ for some x 2 X converges in X. Definition 2 ([17]). A subset D of X is called R-directed if for every x; y 2 D, there exists ´ 2 X such that xR´ and yR´. Definition 3. A mapping T W X ! X is called R-preserving mapping if x; y 2 X;

xRy H) T xRT y:

Next, we define the set ˚ of functions ' W Œ0; C1/ ! Œ0; C1/ satisfying: (I) ' is nondecreasing; 1 X (II) ' n .t / < 1 for each t > 0, where ' n is the n-th iterate of '. nD1

Remark 1. Let ' 2 ˚. We have '.t / < t for all t > 0. Remark 2. Let ' 2 ˚. We have limn!1 ' n .t / D 0 for all t > 0.

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Definition 4. Assume that for T W X ! X, there exists ' 2 ˚ such that ı.T x; T y/  '.Mı .x; y//;

for all x; y 2 X with xy:

A mapping T is called a generalized contractive with respect to ı, if 1 1 Mı .x; y/ D maxfı.x; y/; Œı.x; T x/ C ı.y; T y/; Œı.x; T y/ C ı.y; T x/g/: 2 2 A mapping T is called a generalized quasi-contractive with respect to ı, if Mı .x; y/ D maxfı.x; y/; ı.x; T x/; ı.y; T y/; ı.x; T y/; ı.y; T x/g/: 3. M AIN RESULTS To begin with, we have the following result. Theorem 2. Let .X; d; ı/ be a bimetric space, and S be a binary relation over X. Assume that for T W X ! X, the following conditions are satisfied: (A1) d.x; y/  ı.x; y/ for all x; y 2 X; (A2) .X; d / is T -orbitally complete; (A3) T is continuous with respect to d ; (A4) T is S-preserving; (A5) there exists x0 2 X with x0 ST x0 ; (A6) T is a generalized contractive with respect to ı. Then T has a fixed point x  in X. Moreover, if in addition, F ix.T / is S-directed, then x  is the unique fixed point of T in X. Proof. From (A5), there exists x0 2 X with x0 ST x0 , and from (A4), T is Spreserving, we get xn ST xn for all n  0: (3.1) If xn D T xn , then xn is a fixed point of T . Suppose that xn ¤ T xn for all n. Since (3.1) is satisfied for all n  1, by applying the contraction condition (A6), we have ˚ 1 ı.xn ; T xn /  '. max ı.xn 1 ; xn /; Œı.xn 1 ; xn / C ı.xn ; xnC1 /; 2 1 Œı.xn 1 ; xnC1 / C ı.xn ; xn / // 2 Hence, we get ı.xn ; xnC1 /  '.maxfı.xn

1 ; xn /; ı.xn ; xnC1 /g/:

Now, we will show that fxn g is a Cauchy sequence in .X; ı/. If for some n  1, we have ı.xn 1 ; xn /  ı.xn ; xnC1 /, then from Remark 1, we get ı.xn ; xnC1 /  '.ı.xn ; xnC1 // < ı.xn ; xnC1 /; which is a contradiction. Thus, ı.xn

1 ; xn /

ı.xn ; xnC1 /  '.ı.xn

> ı.xn ; xnC1 /, and

1 ; xn //;

for all n  1:

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Hence, by induction, we obtain ı.xn ; xnC1 /  ' n .ı.x0 ; x1 //;

for all n  1:

(3.2)

Fix " > 0 and let n" be a positive integer such that X ' n .ı.x1 ; x0 // < ": nn"

Using the triangular inequality and (3.2), for some m > n > n" , we get ı.xn ; xm / 

m X1

ı.xk ; xkC1 / 

kDn

m X1

' k .ı.x1 ; x0 // 

kDn

X

' n .ı.x1 ; x0 // < ":

nn"

Consequently, the sequence fxn g is ı-Cauchy, so by (A1), fxn g is d -Cauchy too. Since from (A2), we have metric space .X; d / is T -orbitally complete, then there exists x  2 X such that lim d.xn ; x  / D 0: (3.3) n!1

From (A3), we have that T is continuous with respect to d , and, so it follows that x  D lim T xn D T . lim xn / D T x  , that is, x  is a fixed point of T . n!1

n!1

Next, suppose that F ix.T / is S-directed, and we will show that x  is the unique fixed point of T in X. Suppose that y  2 F ix.T / is another fixed point of T . Then, there exists ´0 2 X such that ´0 Sx  and ´0 y  . Define the sequence f´n g in X by ´nC1 D T ´n for all n  0. Since T is S-preserving, for all n  0, we have ´n Sx  and ´n Sy  . Applying (A6), for all n  0, we get ı.´nC1 ; x  / D ı.T ´n ; T x  /  '.M.´n ; x  // 1 1 D '.maxfı.´n ; x  /; Œı.´n ; ´nC1 / C ı.x  ; x  /; Œı.´n ; x  / C ı.x  ; ´nC1 /g/ 2 2 1 1  '.maxfı.´n ; x  /; Œı.´n ; x  / C ı.x  ; ´nC1 /; Œı.´n ; x  / C ı.x  ; ´nC1 /g/ 2 2  '.maxfı.´n ; x  /; ı.x  ; ´nC1 /g/: Now, we will show that lim ı.´n ; x  / D 0. Without the loss of generality, supn!1

pose that ı.´n ; x  / > 0 for all n. Assume that for some n, we have ı.´n ; x  /  ı.x  ; ´nC1 /. Hence, ı.´nC1 ; x  /  '.ı.x  ; ´nC1 // < ı.x  ; ´nC1 /; which is a contradiction. Then, for all n  0, we have ı.´n ; x  / > ı.x  ; ´nC1 /. Consequently, for all n, we obtain ı.´nC1 ; x  /  '.ı.x  ; ´n //:

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By induction, for all n, we get ı.´n ; x  /  ' n .ı.x  ; ´0 //: Which implies that lim ı.´n ; x  / D 0. Similarly, we can prove that n!1

lim ı.´n ; y  / D 0:

n!1

Hence, x  D y  .



To prove our next result, we need the following lemmas which will be used in the sequel. Lemma 1. Let n 2 N, .X; ı/ be a metric space, and R a transitive binary relation over X . Assume that for T W X ! X, the following conditions are satisfied: (a1) there exists x0 2 X such that x0 RT x0 ; (a2) T is R-preserving mapping; (a3) T is generalized quasi-contractive with respect to ı. Then, ı.xi ; xj /  '.D.On .x0 /// for all i; j 2 f1; : : : ; ng. Proof. From (a1) there exists x0 2 X such that x0 Rx1 . Hence, by (a2) we get xk RxkC1 for all k. Since, R is transitive, then xi

1 Rxj 1

for all 1  i < j  n:

(3.4)

Without loss of generality, we can suppose that 1  i < j  n for which we have xi 1 , xi , xj 1 , xj 2 On .x0 /. Now, using (a3) and (3.4), we get ı.T xi

1 ; T xj 1 /

' maxfı.xi


;

1 ; xj 1 /; ı.xi 1 ; xi /; ı.xj 1 ; xj /; ı.xi 1 ; xj /; ı.xi ; xj 1 /g

which implies that ı.xi ; xj /  '.D.On .x0 ///:



Remark 3. From the previous lemma and Remark 1, we have ı.xi ; xj /  '.D.On .x0 /// < D.On .x0 // for all 1  i < j  n; which implies the existence of an integer k  n such that ı.x0 ; T k x0 / D D.On .x0 //. Lemma 2. Let .X; ı/ be a metric space, and R a transitive binary relation over X. Assume that for T W X ! X, the following conditions are satisfied: (b1) there exists x0 2 X such that x0 RT x0 ; (b2) T is R-preserving mapping; (b3) T is generalized quasi-contractive with respect to ı. Then, 1 X D.O1 .x0 //  ' ` .ı.x0 ; T x0 //: `D0

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Proof. At first, we note that we have D.O1 .x0 //  D.O2 .x0 //  : : : ; which implies that D.O1 .x// D supfD.On .x// W n 2 Ng. By (b1), there exists x0 2 X such that x0 Rx1 , then (3.4) holds. Let n 2 N so by Lemma 1, there exists T k x0 2 On .x0 / for 1  k  n, such that ı.x0 ; T k x0 / D D.On .x0 //. Using the triangular inequality, (3.4) and (b3), we get D.On .x0 // D ı.x0 ; T k x0 /  ı.x0 ; T x0 / C ı.T x0 ; T k x0 /  ı.x0 ; T x0 / C '.D.On .x0 ///; that is, .id '/.D.On .x0 ///  ı.x0 ; T x0 / where id W Œ0; C1/ ! Œ0; C1/ is the identity function. It follows from the monotony k k P P of ' that ' ` is monotone too for all k 2 N. Thus, by applying ' ` to both side `D0

`D0

of the previous inequality, we get k X

`

' ..id

'/.D.On .x0 //// 

`D0

k X

' ` .ı.x0 ; T x0 //

`D0

which implies D.On .x0 //

' kC1 .D.On .x0 /// 

k X

' ` .ı.x0 ; T x0 //

`D0

Since get

1 P

' ` .ı.x0 ; T x0 // < 1, then by letting k ! 1 in the above inequality, we

`D0

D.On .x0 // 

1 X

' ` .ı.x0 ; T x0 //;

for all n  1:

`D0

Hence, we get D.O1 .x0 // 

1 X

' ` .ı.x0 ; T x0 //:



`D0

Now, we are ready to state our second main result. Theorem 3. Let .X; d; ı/ be a bimetric space, and R a transitive binary relation over X . Assume that for T W X ! X, the following conditions are satisfied: (B1) d.x; y/  ı.x; y/ for all x; y 2 X; (B2) .X; d / is T -orbitally complete; (B3) T is continuous with respect to d ; (B4) T is R-preserving;

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(B5) there exists x0 2 X with x0 RT x0 ; (B6) T is a generalized quasi-contractive with respect to ı. Then T has a fixed point x  in X. Moreover, if in addition, R is symmetric and F ix.T / is R-directed, then x  is the unique fixed point of T in X. Proof. Let x0 2 X such that x0 RT x0 . Define a sequence fxn g in X by xnC1 D T xn , for all n  0. Since T is an R-preserving, then xn RxnC1 for all n. Let n and m .n < m/ be any positive integers. From (B6) and Lemma 1 it follows ı.T n x0 ; T m x0 / D ı.T T n

1

x0 ; T m

nC1

Tn

1

x0 /  '.D.Om

From Remark 1, there exists an integer k1 , 1  k1  m D.Om

nC1 .T

n 1

x0 // D ı.T n

1

nC1 .T

n 1

x0 ///:

n C 1, such that

x; T k T n

1

x0 /:

Using Lemma 1 again, we get ı.T n

1

x0 ; T k T n

1

x0 / D ı.T T n

2

x0 ; T k1 C1 T n

 '.D.Ok1 C1 .T  '.D.Om

n 2

nC2 .T

2

x0 /

x0 ///

n 2

x0 ///

Combining, the above inequalities, we get ı.T n x0 ; T m x0 /  '.D.Om

nC1 .T

n 1

x0 ///  ' 2 .D.Om

nC2 .T

n 2

x0 ///:

Continue this process, we obtain ı.T n x0 ; T m x0 /  '.D.Om

nC1 .T

n 1

x0 /// 


 ' n .D.Om .x0 ///:

By Lemma 2, we get n

m

n

ı.T x0 ; T x0 /  ' .

1 X

' ` .ı.x0 ; T x0 ///:

`D0

Since lim

n!1

' n .s/ D 0 for all s

 0, it follows that the sequence fT n x0 g is a ı-Cauchy

sequence. Therefore, by (B1) the sequence fT n x0 g is a d -Cauchy sequence too. Now, since the metric space .X; d / is T -orbitally complete, we deduce that the sequence fT n x0 g converges to some x  in X. From (B3), we have that T is continuous with respect to d , and, so it follows that x  D lim T xn D T . lim xn / D T x  , that n!1

n!1

is, x  is a fixed point of T . Now, suppose that F ix.T / is R-directed, and we claim that the fixed point is unique. Let x  and y  two fixed points of T . Suppose that x  ¤ y  . Since F ix.T / is R-directed, then there exists ´ 2 X such that ´Rx  and ´Ry  , which implies by the transitivity of R, that x  Ry  . Then, we apply the contraction condition (B6), we get ı.x  ; y  / D ı.T x  ; T y  /  '.maxfı.x  ; y  /; ı.x  ; T x  /; ı.y  ; T y  /; ı.x  ; T y  /; ı.y  ; T x  /g/

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M.S. KHAN, M. BERZIG, AND S. CHANDOK

which implies, by Remark 1, that ı.x  ; y  /  '.ı.x  ; y  // < ı.x  ; y  /: which is a contradiction. Consequently, x  D y  , and our claim holds.

(3.5) 

The following results are an immediate consequences of Theorems 2 and 3. Corollary 1. Theorem 1 is a particular case of Theorem 2. Corollary 2. Let .X; / be a partially ordered set and .X; d; ı/ be a bimetric space. Assume that for T W X ! X , the following conditions are satisfied: (C1) d.x; y/  ı.x; y/ for all x; y 2 X; (C2) X is complete with respect to d ; (C3) T is continuous with respect to d ; (C4) T is monotone nondecreasing mapping; (C5) there exists x0 2 X with x0  T x0 ; (C6) there exists ' 2 ˚ such that ı.T x; T y/  '.Mı .x; y// for all x; y 2 X with x  y: Then T has a unique fixed point in X. Corollary 3. Let .X; / be a partially ordered set and .X; d / be a complete metric space. Assume that for T W X ! X , the following conditions are satisfied: (D1) T is continuous; (D2) T is monotone nondecreasing mapping; (D3) there exists x0 2 X with x0  T x0 ; (D4) there exists a constant ˛ 2 Œ0; 1/ such that d.T x; T y/  ˛Md .x; y/;

for all x; y 2 X with x  y:

Then T has a unique fixed point in X. Corollary 4. Let .X; d / be a complete metric space and T W X ! X be continuous mapping. Suppose there exists ' 2 ˚ such that d.T x; T y/  ' .max fd.x; y/; d.x; T x/; d.y; T y/; d.x; T y/; d.y; T x/g/ ; 8x; y 2 X: Then T has a unique fixed point in X. 4. A PPLICATION TO C AUCHY PROBLEM In this section, we study the Cauchy problem for a class of nonlinear differential equations, using the results obtained in the previous section. Theorem 4. Let .X; / is a partially ordered set and suppose that there exists a metric d on X such that .X; d / is a complete metric space. Let T W X ! X be a

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947

nondecreasing and continuous mappings with respect to d and consider the following power series 1 X n d.T n x; T n y/: (4.1) nD0

Suppose there exists  2 .0; 1/, for which the power series (4.1) converges for all x; y 2 X . For such  we define a function ı by ı.x; y/ D

1 X

n d.T n x; T n y/:

(4.2)

nD0

Suppose that also that ı.x; y/ 

1 d.x; y/ 1 2

for all x  y:

(4.3)

Then, (i) ı is a metric satisfies d.x; y/  ı.x; y/ for all x; y 2 X ; (ii) T has a unique fixed point in X. Proof. The function ı is a metric. Indeed, If x D y, then ı.x; x/ D 0. On the other 1 P hand, if ı.x; y/ D 0, so n d.T n ; x; T n y/ D 0 which implies that x D y since nD0

d.T n ; x; T n y/  0 for all n. Next, it is easy to prove that ı.x; y/ D ı.y; x/ and that ı satisfies also the triangle inequality. Now, we shall prove (ii). From hypothesis, there exists some  2 .0; 1/ such that 1 X

ı.x; y/ D

n d.T n x; T n y/

nD0

D d.x; y/ C 

1 X

n d.T nC1 x; T nC1 y/

0

D d.x; y/ C ı.T x; T y/

(4.4)

On the other hand, from condition (4.3) we have ı.x; y/ 

1 d.x; y/ 1 2

for all x  y:

Therefore, from (4.4) we get ı.T x; T y/  ı.x; y/

for all x  y:

All conditions of Theorem 2 are satisfied, then we conclude (ii).



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Theorem 5. Let .X; / is a partially ordered set and suppose that there exist a metric d on X such that .X; d / is a complete metric space. Let T W X ! X be a self mappings. Suppose that the condition (4.3) holds, further suppose that there exists a constant L  1 and k  1 such that d.T k x; T k y/  Ld.x; y/ for all x  y: Then the power series 1 X SD n d.T n x; T n y/;

(4.5)

(4.6)

nD0

converges in X for all x; y 2 X with x  y. Proof. From hypothesis, there exists some L  1 and k  1 such that: d.T k x; T k y/  Ld.x; y/

for all x  y:

By induction we get d.T nk x; T nk y/  Ln d.x; y/

for all n  1 and x  y:

(4.7)

Hence, we get SD

1 X

nk d.T nk x; T nk y/ C nkC1 d.T nkC1 x; T nkC1 y/ C : : :

nD0

C nkC.k 

1 X

1/

d.T nkC.k

1/

x; T nkC.k

1/

y/

nk Ln d.x; y/ C nkC1 LnC1 d.T x; T y/ C : : :

nD0

C nkC.k 

k X1

p

2p

 L

pD0

1 X

1/

LnC.k

1/

d.T .k

1/

x; T .k

1/

y/

.k L/n d.x; y/

nD0

for all x  y. We can choose 0 < k < series (4.6) for all x; y 2 X with x  y.

1 L

to obtain the convergence of the power 

Now, consider the nonlinear differential equation  0 x .t / D f .t; x.t //; t 2 Œa; b x.t0 / D x0 ;

(4.8)

where a; b; t0 2 R and f W Œa; b  R ! R. Let X D C.Œa; b; R/ denotes the space of all continuous R-valued functions on Œa; b. We endow this space with the metric d given by d.u; v/ D sup ju.t / v.t /j; for all u; v 2 X: t2Œa;b

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It is well known that .X; d / is a complete metric space. We define an order relation  on X by u; v 2 X;

u  v ” u.t /  v.t /; for all t 2 Œa; b:

Consider the mapping T W C.Œa; b; R/ ! C.Œa; b; R/ defined by Z t T x.t / D x0 C f .s; x.s///dsI t 2 Œa; b; t0

x

for all x 2 C.Œa; b; R/. Clearly, 2 C.Œa; b; R/ is a solution of (4.8) if and only if  x is a fixed point of T . Furthermore, we consider the following assumptions: (H1) f W Œa; b  R ! R is continuous; (H2) f W Œa; b  R ! R is nondecreasing with respect to the second variable; (H3) jf .t; x.t // f .t; y.t //j  Ljx.t / y.t /j for all x.t /  y.t / and t 2 Œa; b. It is worth noting that condition (H3) is weaker compared to those used by Maia for studying Cauchy problem in [14], that is, f is L-Lipschitzien function on the whole space. The next result shows that T satisfies the conditions of Corollary 2. Let dr .x; y/ D supat r jx.t / y.t /j where r 2 Œa; b. From condition (H3) it follows Ln .b a/n dr .x; y/ for all x  y; n  0: (4.9) dr .T n x; T n y/  nŠ Indeed, obviously (4.9) holds for n D 0. Suppose that (4.9) holds for some integer n, then using (H2) and (H3), we get ˇZ t ˇ ˇ ˇ nC1 nC1 n n dr .T x; T y/ D sup ˇˇ .f .s; T x.s// f .s; T y.s/// ds ˇˇ at r t0 Z r  Ldr .T n x.s/; T n y.s// ds a Z LnC1 r LnC1 .r a/nC1  .s a/n ds dr .x; y/ D dr .x; y/ nŠ a .n C 1/Š This implies that (4.9) holds. Next, by taking r D b in (4.9), we obtain Ln .b a/n d.x; y/ for all x  y and n  0: (4.10) nŠ Theorem 6. Suppose that (H1)–(H3) hold. Then (4.8) has at least one solution x  2 C.Œa; b; R/. d.T n x; T n y/ 

Proof. We shall show the existence of  2 .0; 1/ such that (4.3) holds. First, we have Ln .b a/n d.T n x; T n y/  d.x; y/ for all x  y; n  0: nŠ

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M.S. KHAN, M. BERZIG, AND S. CHANDOK

Thus, we get ı.x; y/  exp.L.b If we take  D Since, we have

p

1

exp.L.a

a//d.x; y/

for all x  y:

b//, then the condition (4.3) holds.

d.T n x; T n y/  exp.L.b

a//d.x; y/

for all x  y;

n  0;

then the condition (4.5) holds too. We deduce by using Corollary 2 that T has a unique fixed point x  2 C.Œa; b; R/.  ACKNOWLEDGEMENT The authors are thankful to Professor M. Turinici and the learned referees for their valuable kind suggestions. R EFERENCES [1] R. Agarwal, M. El-Gebeily, and D. O’Regan, “Generalized contractions in partially ordered metric spaces,” Appl. Anal., vol. 87, no. 1, pp. 109–116, 2008, doi: 10.1080/00036810701556151. [2] V. Berinde, “General constructive fixed point theorems for ciric-type almost contractions in metric spaces,” Carpathian J. Math, vol. 24, no. 2, pp. 10–19, 2008. [3] M. Berzig, “Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications,” J. Fixed Point Theory Appl., vol. 12, no. 1-2, pp. 221–238, 2012, doi: 10.1007/s11784-013-0094-7. [4] M. Berzig, S. Chandok, and M. Khan, “Generalized krasnoselskii fixed point theorem involving auxiliary functions in bimetric spaces and application to two-point boundary value problem,” Appl. Math. Comput., vol. 248, pp. 323–327, 2014, doi: 10.1016/j.amc.2014.09.096. [5] M. Berzig and E. Karapınar, “Fixed point results for-contractive mappings for a generalized altering distance,” Fixed Point Theory Appl., vol. 2013, no. 1, pp. 1–18, 2013. [6] M. Berzig, E. Karapınar, and A.-F. Rold´an-L´opez-de Hierro, “Discussion on generalized-.˛ , ˇ'/-contractive mappings via generalized altering distance function and related fixed point theorems,” Abstr. Appl. Anal., vol. 2014, p. 12 pages, 2014. [7] M. Berzig and M.-D. Rus, “Fixed point theorems for ˛-contraction mappings of meir–keeler type and applications,” Nonlinear Anal. Model. Control, vol. 19, pp. 178–198, 2013. [8] S. Chandok and S. Dinu, “Common fixed points for weak -contractive mappings in ordered metric spaces with applications,” Abstr. Appl. Anal., vol. 2013, p. 7 pages, 2013, doi: 10.1155/2013/879084. [9] S. Chandok, E. Karapinar, and M. Khan, “Existence and uniqueness of common coupled fixed point results via auxiliary functions,” Bull. Iran. Math. Soc, vol. 40, no. 1, pp. 199–215, 2014. [10] S. Chandok, M. Khan, and M. Abbas, “Common fixed-point theorems for nonlinear weakly contractive mappings,” Ukrainian Math. J., vol. 66, no. 4, pp. 594–601, 2014, doi: 10.1007/s11253014-0956-1. [11] S. Chandok, M. Khan, and T. Narang, “Fixed point theorem in partially ordered metric spaces for generalized contraction mappings,” Azerb. J. Math., vol. 5, no. 1, pp. 89–96, 2015. ´ c, “A generalization of banach’s contraction principle,” Proc. Amer. Math. Soc., vol. 45, [12] L. B. Ciri´ no. 2, pp. 267–273, 1974. [13] P. Kumam and A. Rold´an L´opez de Hierro, “On existence and uniqueness of g-best proximity points under ('; ; ˛; g)-contractivity conditions and consequences,” Abstr. Appl. Anal., vol. 2014, p. 14 pages, 2014.

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Authors’ addresses M.S. Khan Department of Mathematics and Statistics College of Science Sultan Qaboos University, POBox 36, PCode 123 Al-Khod, Muscat Sultanate of Oman. E-mail address: [email protected] M. Berzig ´ Ecole Nationale Sup´erieure d’Ing´enieurs de Tunis, Tunis University, 5 Avenue Taha Hussein, BP, 56, Bab Manara, Tunis, Tunisie. E-mail address: [email protected], [email protected] S. Chandok Department of Mathematics, Khalsa College of Engineering and Technology, Punjab Technical University, footnotesize Ranjit Avenue, C-Block, Amritsar-143001, Punjab, India. E-mail address: [email protected]