Fixed point theorems for nonlinear weakly C -contractive mappings in metric spaces

Mathematical and Computer Modelling 54 (2011) 2816–2826

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Fixed point theorems for nonlinear weakly C -contractive mappings in metric spaces W. Shatanawi Department of Mathematics, Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan

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Article history: Received 16 January 2011 Received in revised form 28 June 2011 Accepted 30 June 2011 Keywords: Common fixed point Generalized contractions Weakly C -contraction Ordered metric spaces

abstract The purpose of this paper is to present some fixed point and coupled fixed point theorems for a nonlinear weakly C -contraction type mapping in metric and ordered metric spaces. Also, an example is given to support our results. Our results generalize several well-known results from the current literature. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The Banach contraction mapping principle [1] is a very popular tool for solving the existence of problems in many branches of mathematical analysis. Generalizations of this principle have been established in various settings (see [2–14]). Chatterjea [15] introduced the concept of C -contraction as follows. Definition   1.1 ([15]). A mapping T : X → X where (X , d) is a metric space is said to be a C -contraction if there exists α ∈ 0, 12 such that for all x, y ∈ X , the following inequality holds: d(Tx, Ty) ≤ α(d(x, Ty) + d(y, Tx)). Alber and Guerre-Delabriere [16] introduced the definition of weak φ -contraction. Definition 1.2 ([16]). A self mapping T on a metric space X is called weak φ -contraction if there exists a continuous nondecreasing function φ : [0, +∞) → [0, +∞) with φ(t ) = 0 if and only if t = 0 such that d(Tx, Ty) ≤ d(x, y) − φ(d(x, y)) for each x, y ∈ X . The notion of φ -contraction and weak φ -contraction has been studied by many authors (see [17,18,5–7,11,19,13,20]). In recent years, many results have appeared related to fixed points in a complete ordered metric space (see, for example, [21–23,12,24,8,25–27]). Chatterjea [15] proved that if X is complete, then every C -contraction has a unique fixed point. Choudhury [28] generalized the concept of C -contraction as follows.

E-mail address: [email protected]. 0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.06.069

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Definition 1.3 ([28]). A mapping T : X → X where (X , d) is a metric space is said to be weakly C -contractive if for all x, y ∈ X , the following inequality holds: d(Tx, Ty) ≤

1 2

(d(x, Ty) + d(y, Tx)) − φ(d(x, Ty), d(y, Tx))

where φ : [0, +∞)2 → [0, +∞) is a continuous function such that φ(x, y) = 0 if and only if x = y = 0. Choudhury [28] proved the following theorem. Theorem 1.1 ([28], Theorem 2.1). If X is a complete metric space, then every weakly C -contraction T has a unique fixed point (u = Tu for some u ∈ X ). Harjani et al. [29] studied Theorem 1.1 in ordered metric space. They proved the following. Theorem 1.2 ([29], Theorem 2.1). Let (X , ≼) be a partially ordered set and suppose that there exists a metric d in X such that (X , d) is a complete metric space. Let T : X → X be a continuous and nondecreasing mapping such that d(Tx, Ty) ≤

1 2

(d(x, Ty) + d(y, Tx)) − φ(d(x, Ty), d(y, Tx))

for every comparative x and y, where φ : [0, +∞)2 → [0, +∞) is a continuous function such that φ(x, y) = 0 if and only if x = y = 0. If there exists x0 ∈ X with x0 ≼ Tx0 , then T has a fixed point. Theorem 1.3 ([29], Theorem 2.2). Let (X , ≼) be a partially ordered set and suppose that there exists a metric d in X such that (X , d) is a complete metric space. Let T : X → X be a nondecreasing mapping such that d(Tx, Ty) ≤

1 2

(d(x, Ty) + d(y, Tx)) − φ(d(x, Ty), d(y, Tx))

for every comparative x and y, where φ : [0, +∞)2 → [0, +∞) is a continuous function such that φ(x, y) = 0 if and only if x = y = 0. Suppose that for a nondecreasing sequence {xn } in X with xn → x, we have xn ≼ x for all n ∈ N. If there exists x0 ∈ X with x0 ≼ Tx0 , then T has a fixed point. Bhaskar and Lakshmikantham [21] introduced the notion of the mixed monotone property and the coupled fixed point of a mapping F from X × X into X and studied coupled fixed points of such mappings in partially ordered metric spaces. Since then, many authors established coupled fixed point results (see [30,21–24,26,31–33]). Definition 1.4 ([21]). Let (X , ≼) be a partially ordered set and F : X × X → X be a mapping. We say that F has the mixed monotone property if F (x, y) is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any x, y ∈ X , x1 , x2 ∈ X ,

x1 ≼ x2 ⇒ F (x1 , y) ≼ F (x2 , y)

y1 , y2 ∈ X ,

y1 ≼ y2 ⇒ F (x, y1 ) ≽ F (x, y2 ).

and

Definition 1.5 ([21]). An element (x, y) ∈ X × X is called a coupled fixed point of a mapping F : X × X → X if F (x, y) = x and F (y, x) = y. Bhaskar and Lakshmikantham [21] proved the following results. Theorem 1.4 ([21], Theorem 2.1). Let (X , ≼) be a partially ordered set and F : X × X → X be a continuous mapping having the mixed monotone property on X . Assume there exists k ∈ [0, 1) such that d(F (x, y), F (u, v)) ≤

k 2

(d(x, u) + d(y, v)),

∀x ≼ u, y ≽ v.

If there exists (x0 , y0 ) ∈ X × X such that x0 ≼ F (x0 , y0 ) and y0 ≽ F (y0 , x0 ), then F has a coupled fixed point. Theorem 1.5 ([21], Theorem 2.2). Let (X , ≼, d) be a partially ordered complete metric space and F : X × X → X be a mapping having the mixed monotone property on X . Assume that X has the following properties. 1. If (xn ) is a nondecreasing sequence in X which converges to x, then xn ≼ x∀n ∈ N, and 2. If (yn ) is a nonincreasing sequence in X which converges to y, then yn ≽ y∀n ∈ N.

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Suppose that there exists k ∈ [0, 1) such that d(F (x, y), F (u, v)) ≤

k 2

(d(x, u) + d(y, v)),

∀x ≼ u, y ≽ v.

If there exists (x0 , y0 ) ∈ X × X such that x0 ≼ F (x0 , y0 ) and y0 ≽ F (y0 , x0 ), then F has a coupled fixed point. The aim of this paper is to establish some fixed point and coupled fixed point theorems for nonlinear weakly C -contraction type mappings in metric and ordered metric spaces. Our results generalize Theorems 1.1–1.5. 2. Main results Khan et al. [34] introduced the concept of altering distance function as follows. Definition 2.1. The function φ : [0, +∞) → [0, +∞) is called an altering distance function, if the following properties are satisfied. 1. φ is continuous and nondecreasing. 2. φ(t ) = 0 if and only if t = 0. The methods of proof of the following result are based on those given by Doric [17, Theorem 2.1]. Theorem 2.1. Let (X , ≼, d) be an ordered complete metric space. Let f : X → X be a continuous nondecreasing mapping. Suppose that for comparable x, y, we have

ψ(d(fx, fy)) ≤ ψ



1 2

 (d(x, fy) + d(y, fx)) − φ(d(x, fy), d(y, fx))

(1)

where 1. ψ : [0, +∞) → [0, +∞) is an altering distance function. 2. φ : [0, +∞) × [0, +∞) → [0, +∞) is a continuous function with φ(t , s) = 0 if and only if t = s = 0. If there exists x0 ∈ X such that x0 ≼ fx0 , then f has a fixed point. Proof. If x0 = fx0 , then x0 is a fixed point of f . Now, suppose that x0 ≺ fx0 , we choose x1 ∈ X such that fx0 = x1 . since f is a nondecreasing function, we have x0 ≺ x1 = fx0 ≼ fx1 . Again, let x2 = fx1 . Then we have x0 ≺ x1 = fx0 ≼ x2 = fx1 ≼ fx2 . Continuing this process, we can construct a sequence (xn ) in X such that xn+1 = fxn with x 0 ≺ x 1 ≼ x 2 ≼ · · · ≼ x n ≼ x n +1 ≼ · · · . Since xn and xn+1 are comparable, by inequality (1), we have

ψ(d(xn , xn+1 )) = ψ(d(fxn−1 , fxn ))   1 ≤ψ (d(xn−1 , fxn ) + d(xn , fxn−1 )) − φ(d(xn−1 , fxn ), d(xn , fxn−1 )) 2   1 =ψ d(xn−1 , xn+1 ) − φ(d(xn−1 , xn+1 ), 0) 2   1 ≤ψ (d(xn−1 , xn+1 )) . 2

Since ψ is a nondecreasing function, we get that d(xn , xn+1 ) ≤

1 2

(d(xn−1 , xn+1 )).

(2)

By triangular inequality, we have d(xn , xn+1 ) ≤

≤

1 2 1 2

(d(xn−1 , xn+1 )) (d(xn−1 , xn ) + d(xn , xn+1 )).

Thus we have d(xn , xn+1 ) ≤ d(xn−1 , xn ) ∀ n ∈ N∗ .

(3)

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So, by (3) we get that {d(xn , xn+1 ) : n ∈ N} is a nonincreasing sequence. Hence there is r ≥ 0 such that lim d(xn , xn+1 ) = r .

(4)

n→+∞

By (2), we have d(xn , xn+1 ) ≤

1 2

d(xn−1 , xn+1 ) ≤

1 2

(d(xn−1 , xn ) + d(xn , xn+1 )).

(5)

Letting n → +∞ and using (4), we get r ≤ lim

n→+∞

1 2

d(xn−1 , xn+1 ) ≤

1 2

(r + r ).

(6)

Hence lim d(xn−1 , xn+1 ) = 2r .

(7)

n→+∞

Using the continuity of ψ, φ and (7), we get

ψ(r ) ≤ ψ



1 2

 (2r ) − φ(2r , 0),

which implies that φ(2r , 0) = 0 and hence r = 0. Next, we show that, (xn ) is a Cauchy sequence in X . Suppose to the contrary that (xn ) is not a Cauchy sequence. Then there exists ϵ > 0 for which we can find two subsequences (xm(i) ) and (xn(i) ) of (xn ) such that n(i) is the smallest index for which n(i) > m(i) > i,

d(xm(i) , xn(i) ) ≥ ϵ.

(8)

This means that d(xm(i) , xn(i)−1 ) < ϵ.

(9)

From (8), (9) and triangular inequality, we get

ϵ ≤ ≤ ≤ <

d(xm(i) , xn(i) ) d(xm(i) , xm(i)+1 ) + d(xm(i)+1 , xn(i)−1 ) + d(xn(i)−1 , xn(i) ) d(xm(i) , xm(i)+1 ) + d(xm(i)+1 , xn(i) ) + 2d(xn(i)−1 , xn(i) ) 2d(xm(i) , xm(i)+1 ) + ϵ + 3d(xn(i)−1 , xn(i) ).

Using (4) and letting i → +∞, we get lim d(xm(i) , xn(i) ) = lim d(xm(i)+1 , xn(i)−1 ) = d(xm(i)+1 , xn(i) ) = ϵ.

i→+∞

(10)

i→+∞

From (1), we have

ψ(d(xm(i)+1 , xn(i) )) = ψ(d(fxm(i) , fxn(i)−1 ))   1 ≤ψ (d(xm(i) , fxn(i)−1 ) + d(xn(i)−1 , fxm(i) )) − φ(d(xm(i) , fxn(i)−1 ), d(xn(i)−1 , fxm(i) )) 2   1 =ψ (d(xm(i) , xn(i) ) + d(xn(i)−1 , xm(i)+1 )) − φ(d(xm(i) , xn(i) ), d(xn(i)−1 , xm(i)+1 )). 2

Letting i → +∞, using (10) and the continuity of φ and ψ , we get

ψ(ϵ) ≤ ψ(ϵ) − φ(ϵ, ϵ). Therefore, we get φ(ϵ, ϵ) = 0. Hence ϵ = 0, a contradiction. Thus (xn ) is a Cauchy sequence in X . Thus there is u ∈ X such that lim xn = u.

n→+∞

Since f is continuous and xn → u, we obtain xn+1 = fxn → fu. By the uniqueness of the limit, we get fu = u.



Theorem 2.1 is still valid if f is not necessarily continuous. Theorem 2.2. Suppose that X , f , ψ and φ are as in Theorem 2.1 except the continuity of f . Suppose that for a nondecreasing sequence (xn ) in X with xn → x ∈ X , we have xn ≼ x for all n ∈ N. If there exists x0 ∈ X such that x0 ≼ fx0 , then f has a fixed point.

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Proof. As in the proof of Theorem 2.1, we have a Cauchy sequence (xn ) in X . Since X is complete, there is u ∈ X such that lim xn = u.

n→+∞

Since (xn ) is a nondecreasing sequence, we have by hypothesis xn ≼ u for all n ∈ N. Thus by (1), we have

ψ(d(fu, xn+1 )) = ψ(d(fu, fxn ))   1 ≤ψ (d(u, fxn ) + d(xn , fu)) − φ(d(u, fxn ), d(xn , fu)) 2   1 ≤ψ (d(u, fxn ) + d(xn , fu)) 2   1 =ψ (d(u, xn+1 ) + d(xn , fu)) . 2

Letting n → +∞ and using the continuity of ψ , we get

ψ(fu, u) ≤ ψ



1 2

 d(u, fu) .

Since ψ is nondecreasing, we get that d(fu, u) ≤

1 2

d(fu, u).

Thus d(fu, u) = 0 and hence u = fu.



Remark 2.1. If ψ = i[0,+∞] (the identity function) in Theorem 2.1, then we get Theorem 1.2. Remark 2.2. If ψ = i[0,+∞] in Theorem 2.2, then we obtain Theorem 1.3. We now consider the case of two mappings; the technique used in the proof is due to Doric [17, Theorem 2.1]. Theorem 2.3. Let (X , d) be a complete metric space. Let f , g : X → X be two mappings. Suppose that for x, y ∈ X , we have

ψ(d(fx, gy)) ≤ ψ



1 2



(d(x, gy) + d(y, fx)) − φ(d(x, gy), d(y, fx))

(11)

where ψ and φ are as in Theorem 2.1. Then f and g have a unique common fixed point, that is, there exists u ∈ X such that u = fu = gu. Proof. Let x0 ∈ X , we choose x1 ∈ X such that fx0 = x1 . Also, we choose x2 ∈ X such that gx1 = x2 . Continuing this process, we construct a sequence (xn ) in X such that x2n+1 = fx2n and x2n+2 = gx2n+1 . By inequality (11), we have

ψ(d(x2n+1 , x2n+2 )) = ψ(d(fx2n , gx2n+1 ))   1 (d(x2n , gx2n+1 ) + d(x2n+1 , fx2n )) − φ(d(x2n , gx2n+1 ), d(x2n+1 , fx2n )) ≤ψ 2   1 =ψ d(x2n , x2n+2 ) − φ(d(x2n , x2n+2 ), 0) 2   1 ≤ψ d(x2n , x2n+2 ) . 2

As ψ is a nondecreasing function, we get d(x2n+1 , x2n+2 ) ≤

1 2

(d(x2n , x2n+2 )).

(12)

since d(x2n , x2n+2 ) ≤ d(x2n , x2n+1 ) + d(x2n+1 , x2n+2 ), we have d(x2n+1 , x2n+2 ) ≤ d(x2n , x2n+1 ).

(13)

Similarly, one can show that d(x2n+2 , x2n+3 ) ≤ d(x2n+1 , x2n+2 ).

(14)

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From (13) and (14), we have d(xn , xn+1 ) ≤ d(xn−1 , xn ) ∀ n ∈ N∗ .

(15)

So, by (15) we get that {d(xn , xn+1 ) : n ∈ N} is a nonincreasing sequence. Hence there is r ≥ 0 such that lim d(xn , xn+1 ) = r .

(16)

n→+∞

By (12) we have 1

d(x2n+1 , x2n+2 ) ≤

1

(d(x2n , x2n+1 ) + d(x2n+1 , x2n+2 )). 2 2 Letting n → +∞ and using (16), we get that r ≤ lim

n→+∞

1 2

d(x2n , x2n+2 ) ≤

d(x2n , x2n+2 ) ≤

1 2

(r + r )

(17)

(18)

Hence lim d(x2n , x2n+2 ) = 2r .

n→+∞

(19)

Using the continuity of ψ, φ and (19), we get that

ψ(r ) ≤ ψ



1 2

 (2r ) − φ(2r , 0),

which implies that φ(2r , 0) = 0 and hence r = 0. To prove that (xn ) is a Cauchy sequence in X . It is sufficient to show that (x2n ) is a Cauchy sequence. Suppose to the contrary, that (x2n ) is not a Cauchy sequence. Then there exists ϵ > 0 for which we can find two subsequences (x2m(i) ) and (x2n(i) ) of (xn ) such that n(i) is the smallest index for which n(i) > m(i) > i,

d(x2m(i) , x2n(i) ) ≥ ϵ.

(20)

This means that d(x2m(i) , x2n(i)−2 ) < ϵ.

(21)

From (20), (21) and triangular inequality, we get that

ϵ ≤ d(x2m(i) , x2n(i) ) ≤ d(x2m(i) , x2n(i)−2 ) + d(x2n(i)−2 , x2n(i)−1 ) + d(x2n(i)−1 , x2n(i) ) < ϵ + d(x2n(i)−2 , x2n(i)−1 ) + d(x2n(i)−1 , x2n(i) ). On letting i → +∞ in the above inequality and using (16), we have lim d(x2m(i) , x2n(i) ) = ϵ.

i→+∞

(22)

Also,

ϵ ≤ d(x2m(i) , x2n(i) ) ≤ d(x2m(i) , x2m(i)−1 ) + d(x2m(i)−1 , x2n(i) ) ≤ 2d(x2m(i) , x2m(i)−1 ) + d(x2m(i) , x2n(i) ). Using (16), (22) and letting i → +∞, we get that lim d(x2m(i) , x2n(i) ) = lim d(x2m(i)−1 , x2n(i) ) = ϵ.

i→+∞

i→+∞

(23)

On other hand, we have d(x2m(i) , x2n(i) ) ≤ d(x2m(i) , x2n(i)+1 ) + d(x2n(i)+1 , x2n(i) )

≤ d(x2m(i) , x2n(i) ) + 2d(x2n(i)+1 , x2n(i) ). Letting i → +∞, we get that lim d(x2m(i) , x2n(i)+1 ) = ϵ.

i→+∞

Also, we have d(x2m(i)−1 , x2n(i) ) ≤ d(x2m(i)−1 , x2n(i)+1 ) + d(x2n(i)+1 , x2n(i) )

≤ d(x2m(i)−1 , x2n(i) ) + 2d(x2n(i)+1 , x2n(i) ).

(24)

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Using (16), (23) and letting i → +∞, we get that lim d(x2m(i)−1 , x2n(i)+1 ) = ϵ.

(25)

i→+∞

From (11), we have

ψ(d(x2n(i)+1 , x2m(i) )) = ψ(d(fx2n(i) , gx2m(i)−1 ))   1 (d(x2n(i) , gx2m(i)−1 ) + d(x2m(i)−1 , fx2n(i) )) − φ(d(x2n(i) , gx2m(i)−1 ), d(x2m(i)−1 , fx2n(i) )) ≤ψ 2   1 =ψ (d(x2n(i) , x2m(i) ) + d(x2m(i)−1 , x2n(i)+1 )) − φ(d(x2n(i) , x2m(i) ), d(x2m(i)−1 , x2n(i)+1 )). 2

Letting i → +∞, using (23)–(25), and the continuity of φ and ψ , we get that

ψ(ϵ) ≤ ψ(ϵ) − φ(ϵ, ϵ). Therefore, we get that φ(ϵ, ϵ) = 0. Hence ϵ = 0, a contradiction. Thus (x2n ) is a Cauchy sequence in X and hence (xn ) is a Cauchy sequence. Thus there is u ∈ X such that lim xn = u.

n→+∞

By (11), we get that

ψ(d(x2n+1 , gu)) = ψ(d(fx2n , gu))   1 ≤ψ (d(x2n , gu) + d(u, fx2n )) − φ(d(x2n , gu), d(u, fx2n )) 2   1 (d(x2n , gu) + d(u, x2n+1 )) − φ(d(x2n , gu), d(u, x2n+1 )). =ψ 2

Letting n → +∞, we get that

ψ(d(u, gu)) ≤ ψ



1 2

   1 d(u, gu) − φ(d(u, gu), 0) ≤ ψ d(u, gu) . 2

Since ψ is an altering distance map, we get that d(u, gu) = 0 and hence gu = u. Again by (11), we get that

ψ(d(fu, u)) = ψ(d(fu, gu))   1 ≤ψ (d(u, gu) + d(u, fu)) − φ(d(u, gu), d(u, fu)) 2   1 =ψ (d(u, fu)) − φ(0, d(u, fu)) 2   1 ≤ψ (d(u, fu)) . 2

As ψ is nondecreasing, we get d(fu, u) ≤

1 2

d(u, fu),

and hence d(u, fu) = 0. Thus u = fu. Therefore u = fu = gu, that is, u is a common fixed point of f and g. To prove the uniqueness of the common fixed point, let q be any other common fixed point of f and g. By (11), we have d(u, q) = d(fu, gq)

 (d(u, gq) + d(q, fu)) − φ(d(u, gq), d(q, fu)) 2   1 =ψ (d(u, q) + d(q, u)) − φ(d(u, q), d(q, u)) ≤ψ



1

2

= ψ(d(u, q)) − φ(d(u, q), d(q, u)). Therefore φ(d(u, q), d(q, u)) = 0. Thus d(u, q) = 0, and hence u = q.



W. Shatanawi / Mathematical and Computer Modelling 54 (2011) 2816–2826

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Corollary 2.1. Let (X , d) be a complete metric space. Let f : X → X be a mapping. Suppose that for x, y ∈ X , we have

ψ(d(fx, fy)) ≤ ψ



1 2

 (d(x, fy) + d(y, fx)) − φ(d(x, fy), d(y, fx))

where 1. ψ : [0, +∞) → [0, +∞) is an altering distance function, 2. φ : [0, +∞) × [0, +∞) → [0, +∞) is a continuous function with φ(t , s) = 0 if and only if t = s = 0. Then f has a unique fixed point. Proof. Follows from Theorem 2.3 by taking f = g.



Corollary 2.2. Let (X , d) be a complete metric space. Let f , g : X → X be two mappings. Suppose that for x, y ∈ X , we have d(fx, gy) ≤

1 2

(d(x, gy) + d(y, fx)) − φ(d(x, gy), d(y, fx))

where

φ : [0, +∞) × [0, +∞) → [0, +∞) is a continuous function with φ(t , s) = 0 if and only if t = s = 0. Then f and g have a unique common fixed point. Proof. Follows from Theorem 2.3 by taking ψ = i[0,+∞] .



Remark 2.3. Set f = g in Corollary 2.2 to obtain Theorem 1.1. Example 2.1. Let X = [0, 1]. Let d be the complete metric on X defined by d(x, y) = |x − y|. Set fx = x ∈ X . Define

1 2 x 4

and gx = 0 for all

ψ : [0, +∞) → [0, +∞) and φ : [0, +∞) × [0, +∞) → [0, +∞) by

ψ(t ) = t 2 and φ(t , s) =

1 8

(t + s)2 .

Then for x, y ∈ X , we have

   1 2 ψ(d(fx, gy)) = ψ d x ,0 4   1 2 1 4 1 =ψ x = x ≤ x2 4

16

8

 2  1 1 2  x + y − x  ≤ 8 4  2   2     1 1  1 1  = x + y − x2  − x + y − x2  4 4 8 4   1 =ψ (d(x, gy) + d(y, fx)) − φ(d(x, gy), d(y, fx)). 

2

Thus by Theorem 2.3, f and g have a unique common fixed point 0. Now, we present a coupled fixed point on an ordered metric space for a nonlinear weakly C -contraction type mapping. Theorem 2.4. Let (X , ≼) be a partially ordered set and d be a metric on X such that (X , d) is a complete metric space. Let F : X × X → X be a continuous mapping having the mixed monotone property on X . Assume that for x, y, u, v ∈ X with x ≼ u and y ≽ v , we have

ψ(d(F (x, y), F (u, v))) ≤ ψ



1 2

 (d(x, u) + d(y, v)) − φ(d(x, u), d(y, v))

where ψ and φ are as in Theorem 2.1. If there exists (x0 , y0 ) ∈ X × X such that x0 ≼ F (x0 , y0 ) and y0 ≽ F (y0 , x0 ), then F has a coupled fixed point.

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Proof. Let x0 , y0 ∈ X be such that x0 ≼ F (x0 , y0 ) and y0 ≽ F (y0 , x0 ). Let x1 = F (x0 , y0 ) and y1 = F (y0 , x0 ). Then x0 ≼ x1 and y0 ≽ y1 . Again, let x2 = F (x1 , y1 ) and y2 = F (y1 , x1 ). Since F has the mixed monotone property, then we have x1 ≼ x2 and y1 ≽ y2 . Continuing in this way, we construct two sequences (xn ) and (yn ) in X such that xn+1 = F (xn , yn ), yn+1 = F (yn , xn ), x0 ≼ x1 ≼ x2 ≼ · · · and y0 ≽ y1 ≽ y2 ≽ · · ·. For n ∈ N, we have

ψ(d(xn+1 , xn+2 )) = ψ(d(F (xn , yn ), F (xn+1 , yn+1 )))   1 ≤ψ (d(xn , xn+1 ) + d(yn , yn+1 )) − φ(d(xn , xn+1 ), d(yn , yn+1 )) 2

≤ ψ(max{d(xn , xn+1 ), d(yn , yn+1 )}) − φ(d(xn , xn+1 ), d(yn , yn+1 )).

(26)

ψ(d(yn+1 , yn+2 )) ≤ ψ(max{d(xn , xn+1 ), d(yn , yn+1 )}) − φ(d(yn , yn+1 ), d(xn , xn+1 )).

(27)

Similarly, we have

Since ψ is a nondecreasing function, then by (26) and (27), we have

ψ(max{d(xn+1 , xn+2 ), d(yn+1 , yn+2 )}) = max{ψ(d(xn+1 , xn+2 )), ψ(d(yn+1 , yn+2 ))} ≤ ψ(max{d(xn , xn+1 ), d(yn , yn+1 )}) − min{φ(d(xn , xn+1 ), d(yn , yn+1 )), φ(d(yn , yn+1 ), d(xn , xn+1 ))}.

(28)

Since φ(t , s) ≥ 0 for all t , s ∈ [0, +∞) and ψ is a nondecreasing function, we conclude that (max{d(xn+1 , xn+2 ), d(yn+1 , yn+2 )}) is a nondecreasing sequence. Thus there is r ≥ 0 such that lim max{d(xn+1 , xn+2 ), d(yn+1 , yn+2 )} = r .

n→+∞

Letting n → +∞ in (28), we get that

ψ(r ) ≤ ψ(r ) − lim min{φ(d(xn , xn+1 ), d(yn , yn+1 )), φ(d(yn , yn+1 ), d(xn , xn+1 ))}. n→+∞

Thus, we have lim φ(d(xn , xn+1 ), d(yn , yn+1 )) = 0

n→+∞

lim φ(d(yn , yn+1 ), d(xn , xn+1 )) = 0.

or

n→+∞

In both cases, we get that limn→+∞ d(xn , xn+1 ) = 0 and limn→+∞ d(yn , yn+1 ) = 0. Hence r = 0. Now, we show that, (xn ) and (yn ) are Cauchy sequences in X . Suppose to the contrary, that (xn ) or (yn ) is not a Cauchy sequence; that is, (max{d(xn , xm ), d(yn , ym )}) is not a Cauchy sequence in X . Then there exists ϵ > 0 for which we can find two subsequences (xm(i) ) and (xn(i) ) of (xn ) such that n(i) is the smallest index for which n(i) > m(i) > i,

max{d(xm(i) , xn(i) ), d(ym(i) , yn(i) )} ≥ ϵ.

(29)

This means that max{d(xm(i) , xn(i)−1 ), d(ym(i) , yn(i)−1 )} < ϵ.

(30)

By triangular inequality and (30), we have d(xm(i) , xn(i) ) ≤ d(xm(i) , xm(i)+1 ) + d(xm(i)+1 , xn(i) )

≤ 2d(xm(i) , xm(i)+1 ) + d(xm(i) , xn(i)−1 ) + d(xn(i)−1 , xn(i) ) < 2d(xm(i) , xm(i)+1 ) + ϵ + d(xn(i)−1 , xn(i) ). Letting i → +∞, we get lim d(xm(i) , xn(i) ) ≤ lim d(xm(i)+1 , xn(i) ) ≤ lim d(xm(i) , xn(i)−1 ) ≤ ϵ.

i→+∞

i→+∞

i→+∞

(31)

Similarly, we have lim d(ym(i) , yn(i) ) ≤ lim d(ym(i)+1 , yn(i) ) ≤ lim d(ym(i) , yn(i)−1 ) ≤ ϵ.

i→+∞

i→+∞

i→+∞

(32)

By (29), (31) and (32), we have lim max{d(xm(i) , xn(i) ), d(ym(i) , yn(i) )} = lim max{d(xm(i)+1 , xn(i) ), d(ym(i)+1 , yn(i) )}

i→+∞

i→+∞

= lim max{d(xm(i) , xn(i)−1 ), d(ym(i) , yn(i)−1 )} = ϵ. i→+∞

(33)

W. Shatanawi / Mathematical and Computer Modelling 54 (2011) 2816–2826

2825

Since xm(i) ≼ xn(i)−1 and ym(i) ≽ yn(i)−1 , we have

ψ(d(xm(i)+1 , xn(i) )) = ψ(F (xm(i) , ym(i) ), F (xn(i)−1 , yn(i)−1 ))   1 (d(xm(i) , xn(i)−1 ) + d(ym(i) , yn(i)−1 )) − φ(d(xm(i) , xn(i)−1 ), d(ym(i) , yn(i)−1 )) ≤ψ 2

≤ ψ(max{d(xn(i)−1 , xm(i) ), d(yn(i)−1 , ym(i) )}) − φ(d(xn(i)−1 , xm(i) ), d(yn(i)−1 , ym(i) )). Similarly, we have

ψ(d(yn(i) , ym(i)+1 )) = ψ(F (yn(i)−1 , xn(i)−1 ), F (ym(i) , xm(i) ))   1 (d(yn(i)−1 , ym(i) ) + d(xn(i)−1 , xm(i) )) − φ(d(yn(i)−1 , ym(i) ), d(xn(i)−1 , xm(i) )) ≤ψ 2

≤ ψ(max{d(xn(i)−1 , xm(i) ), d(yn(i)−1 , ym(i) )}) − φ(d(yn(i)−1 , ym(i) ), d(xn(i)−1 , xm(i) )). Hence

ψ(max{d(xn(i) , xm(i)+1 ), d(yn(i) , ym(i)+1 )}) = max{ψ(d(xn(i) , xm(i)+1 )), ψ(d(yn(i) , ym(i)+1 ))} ≤ ψ(max{d(xn(i)−1 , xm(i) ), d(yn(i)−1 , ym(i) )}) − min{φ(d(xn(i)−1 , xm(i) ), d(yn(i)−1 , ym(i) )), φ(d(yn(i)−1 , ym(i) ), d(xn(i)−1 , xm(i) ))}. Letting i → +∞, and using (33) and the continuity of ψ and φ , we get that lim φ(d(xn(i)−1 , xm(i) ), d(yn(i)−1 , ym(i) )) = 0

i→+∞

or lim φ(d(yn(i)−1 , ym(i) ), d(xn(i)−1 , ym(i) )) = 0.

i→+∞

In both cases, we have lim d(xn(i)−1 , xm(i) ) = 0

i→+∞

and

lim d(yn(i)−1 , ym(i) ) = 0.

i→+∞

Hence lim max{d(xn(i)−1 , xm(i) ), d(yn(i)−1 , ym(i) )} = 0.

i→+∞

By (33), we get that r = 0, a contradiction. Thus both (xn ) and (yn ) are Cauchy sequences in X . Since X is complete, then there are x, y ∈ X such that xn → x and yn → y. On the other hand xn+1 = F (xn , yn ) → F (x, y) and yn+1 = F (yn , xn ) → F (y, x). By the uniqueness of limit, we conclude that x = F (x, y) and y = F (y, x). Thus (x, y) is a coupled fixed point of F .  Theorem 2.4 is still valid if F is not necessarily continuous. Theorem 2.5. Suppose that X , F , ψ and φ are as in Theorem 2.4 except the continuity of F . Suppose that for a nondecreasing sequence (xn ) in X with xn → x, we have xn ≼ x∀ n ∈ N and for a nonincreasing (yn ) in X with yn → y, we have yn ≽ y∀ n ∈ N. If there exists (x0 , y0 ) ∈ X × X such that x0 ≼ F (x0 , y0 ) and y0 ≽ F (y0 , x0 ), then F has a coupled fixed point. Proof. As in the proof of Theorem 2.4, we have (xn ), a nondecreasing sequence in X which converges to x ∈ X , and (yn ), a nonincreasing sequence in X which converges to y ∈ X . By hypotheses, we have xn ≼ x∀ n ∈ N and yn ≽ y∀ n ∈ N. Hence

ψ(d(xn+1 , F (x, y))) = ψ(d(F (xn , yn ), F (x, y)))   1 ≤ψ (d(xn , x) + d(yn , y)) − φ(d(xn , x), d(yn , y)). 2

Letting n → +∞, we get that ψ(x, F (x, y)) = 0 and hence x = F (x, y). Similarly, we may show that y = F (y, x). Thus (x, y) is a coupled fixed point of F .  Corollary 2.3. Let (X , ≼) be a partially ordered set and d be a metric on X such that (X , d) is a complete metric space. Let F : X × X → X be a continuous mapping having the mixed monotone property on X . Assume that for x, y, u, v ∈ X with x ≼ u and y ≽ v , we have d(F (x, y), F (u, v)) ≤

1 2

(d(x, u) + d(y, v)) − φ(d(x, u), d(y, v))

where φ is as in Theorem 2.1. If there exists (x0 , y0 ) ∈ X such that x0 ≼ F (x0 , y0 ) and y0 ≽ F (y0 , x0 ), then F has a coupled fixed point. Proof. Take ψ = i[0,+∞] in Theorem 2.4.



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Corollary 2.4. Suppose that X , F and φ are as in Corollary 2.3 except the continuity of F . Suppose that for a nondecreasing sequence (xn ) in X with xn → x, we have xn ≼ x∀ n ∈ N and for a nonincreasing (yn ) in X with yn → y, we have yn ≽ y∀ n ∈ N. If there exists (x0 , y0 ) ∈ X × X such that x0 ≼ F (x0 , y0 ) and y0 ≽ F (y0 , x0 ), then F has a coupled fixed point. Proof. Take ψ = i[0,+∞] in Theorem 2.5. Remark 2.4. Take φ(s, t ) =

1

Remark 2.5. Take φ(s, t ) =

1

2

2

− −

 k 2 k 2





(s + t ) in Corollary 2.3 to get Theorem 1.4. (s + t ) in Corollary 2.4 to obtain Theorem 1.5.

Acknowledgments The author are grateful to the referees for their valuable comments and suggestions. References [1] S. Banach, Surles operations dans les ensembles et leur application aux equation sitegrales, Fund. Math. 3 (1922) 133–181. [2] R.P. Agarwal, M.A. El-Gebeily, D. O’regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. 87 (1) (2008) 109–116. [3] I. Altun, H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl. 2010 (2010) Article ID 621492, 17 pages. [4] C.Di. Bari, P. Vetro, φ -paris and common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo 57 (2008) 279–285. doi:10.1007/512215-0080020-9. [5] P.N. Dutta, B.S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl. (2008) Article ID 406368, 8 pages. [6] J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71 (7–8) (2008) 3403–3410. [7] J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal. 72 (3–4) (2010) 1188–1197. [8] J.J. Nieto, R. Rodŕiguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223–239. [9] J.J. Nieto, R. Rodŕiguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica 23 (12) (2007) 2205–2212. [10] J.J. Nieto, R.L. Pouso, R. Rodŕiguez-López, Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc. 135 (2007) 2505–2517. [11] H.K. Nashine, B. Samet, Fixed point results for mappings satisfying (ψ, φ)-weakly contractive condition in partially ordered metric spaces, Nonlinear Anal. 74 (2010) 2201–2209. [12] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to metrix equations, Proc. Amer. Math. Soc. 132 (5) (2004) 1435–1443. [13] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. 47 (2001) 2683–2693. [14] W. Shatanawi, Fixed point theory for contractive mappings satisfying Φ -maps in G-metric spaces, Fixed Point Theory Appl., vol. 2010, Article ID 181650, 9 pages doi:10.1155/2010/181650. [15] S.K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci. 25 (1972) 727–730. [16] Ya.I. Alber, S. Guerre-Delabriere, Principles of weakly contractive maps in Hilbert spaces, in: I. Gohberg, Yu. Lyubich (Eds.), New Results in Operator Theory, in: Advances and Appl., vol. 98, Birkhäuser, Basel, 1997, pp. 7–22. [17] D. Doric, Common fixed point for generalized (ψ, φ)-weak contraction, Appl. Math. Lett. 22 (2009) 1896–1900. [18] D.W. Boyd, T.S.W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969) 458–464. [19] O. Popescu, Fixed points for (ψ, φ)-weak contractions, Appl. Math. Lett. 24 (2011) 1–4. [20] Q. Zhang, Y. Song, Fixed point theory for generalized φ -weak contraction, Appl. Math. Lett. 22 (2009) 75–78. [21] T.G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379–1393. [22] B.S. Choudhury, A. Kundu, A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal. 73 (2010) 2524–2531. [23] V. Lakshmikantham, Lj.B. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009) 4341–4349. [24] B. Samet, Coupled fixed point theorems for a generalized Meir–Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010) 4508–4517. [25] R. Saadati, S.M. Vaezpour, P. Vetro, B.E. Rhoades, Fixed point theorems in generalized partially ordered G-metric spaces, Math. Comput. Modelling 52 (2010) 797–801. [26] W. Shatanawi, Partially ordered cone metric spaces and coupled fixed point results, Comput. Math. Appl. 60 (2010) 2508–2515. [27] W. Shatanawi, Some fixed point theorems in ordered G-metric spaces and applications, Abstr. Appl. Anal. 2011 (2011) Article ID 126205, 11 pages doi:10.1155/2011/126205. [28] B.S. Choudhury, Unique fixed point theorem for weak C -contractive mappings, Kathmandu Univ. J. Sci. Eng. Tech. 5 (1) (2009) 6–13. [29] J. Harjani, B. López, K. Sadarangani, Fixed point theorems for weakly C -contractive mappings in ordered metric spaces, Comput. Math. Appl. 61 (2011) 790–796. [30] M. Abbas, A.R. Khan, T. Nazir, Coupled common fixed point results in two generalized metric spaces, Appl. Math. Comput. 217 (2011) 6328–6336. [31] H. Aydi, B. Damjanovic, B. Samet, W. Shatanawi, Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces, Math. Comput. Modelling (2011) doi:10.1016/j.mcm.2011.05.059. [32] H.K. Nashine, W. Shatanawi, Coupled common fixed point theorems for a pair of commuting mappings in partially ordered complete metric spaces, Comput. Math. Appl. (2011) doi:10.1016/j.camwa.2011.06.042. [33] W. Shatanawi, Some common coupled fixed point results in cone metric spaces, Int. J. Math. Anal. 4 (2010) 2381–2388. [34] M.S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distancces between the points, Bull. Aust. Math. Soc. 30 (1984) 1–9.