# Common fixed points of ordered g-quasicontractions and weak contractions in ordered metric spaces

## Abstract

We introduce ordered quasicontractions and g-quasicontractions in partially ordered metric spaces and prove the respective coincidence point and (common) fixed point results. An example shows that the new concepts are distinct from the existing ones. We also prove fixed point theorems for mappings satisfying so-called weak contractive conditions in the setting of partially ordered metric space. Hence, generalizations of several known results are obtained.

Mathematics Subject Classification (2010): 47H10; 47N10.

## 1 Introduction

It is well known that the Banach contraction principle has been generalized in various directions. A lot of authors have used generalized contractive conditions of the type

$d\left(fx,fy\right)\le \lambda M\left(x,y\right),$

where (X, d) is a metric space, f, g : XX, λ [0,1) and M(x, y) is the maximum of one of the sets

$\begin{array}{c}\left\{d\left(gx,gy\right)\right\},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left\{d\left(gx,gy\right),d\left(gx,fx\right),d\left(gy,fy\right)\right\},\\ \left\{d\left(gx,gy\right),d\left(gx,fx\right),d\left(gy,fy\right),\frac{1}{2}\left(d\left(fx,gy\right)+d\left(gx,fy\right)\right)\right\},\end{array}$

and alike. To obtain results about coincidence points or common fixed points of mappings f and g, in the case when f X gX, usually the technique of so-called Jungck sequence y n = fx n = gxn+1was used (see, [1, 2]).

In the case when

$M\left(x,y\right)=\text{max}\left\{d\left(gx,gy\right),d\left(gx,fx\right),d\left(gy,fy\right),d\left(gx,fy\right),d\left(gy,fx\right)\right\},$

the mapping f is called a g-quasicontraction (or simply quasicontraction if g = i X ), see [3, 4]. In this case different techniques were used to obtain common fixed point results in the cited and subsequent articles.

The existence of fixed points in partially ordered metric spaces was first investigated by Ran and Reurings , and then by Nieto and Lopez [6, 7]. Further results in this direction were proved, e.g., in .

In Section 2 of this article, we introduce ordered quasicontractions and g-quasicontractions in partially ordered metric spaces and prove the respective (common) fixed point results. An example shows that the new concepts are distinct from the existing ones, so our results are extensions of known ones.

Recently, several authors (see, e.g., [13, 1723]) have begun to use more general conditions (so-called weak contractive conditions) of the type

$\psi \left(d\left(fx,fy\right)\right)\le \psi \left(M\left(x,y\right)\right)-\phi \left(M\left(x,y\right)\right),$

where ψ and φ are so-called control functions (see definition in Section 3). The usage of Jungck sequence is here also possible, but becomes more involved. Also, some applications were obtained, in particular when dealing with differential, matrix and integral equations (see, e.g., [57, 10]).

In Section 3 of the present article, we consider weak contractive conditions in the setting of partially ordered metric space (it is then restricted to the case when gx and gy are comparable) and prove respective common fixed point theorems. These results can be considered as generalizations of theorems from  since control functions are more general than comparison functions used in these articles.

## 2 Common fixed points of ordered g-quasicontractions

Consider a partially ordered set (X, ) and two self-maps f, g : XX such that f X gX. Following  we shall say that the mapping f is g-nondecreasing (resp., g-nonincreasing) if gx gy fx fy (resp., gx gy fx fy) holds for each x, y X.

For arbitrary x0 X one can construct a so-called Jungck sequence {y n } in the following way: denote y0 = fx0 f X gX; there exists x1 X such that gx1 = y0 = fx0; now y1 = fx1 f X gX and there exists x2 X such that gx2 = y1 = fx1 and the procedure can be continued. The following properties of this sequence can be easily deduced.

Lemma 1If f is g-nondecreasing and gx0 fx0 (resp., gx0 fx0), then the sequence {y n } is nondecreasing (resp., nonincreasing) w.r.t. .

If f is g-nonincreasing and fx0is comparable with gx0, then the sequence {y n } is not monotonous, but arbitrary two of its adjacent terms are comparable.

Proof 1° It follows from gx0 fx0 = gx1 that fx0 fx1, i.e., y0 y1 and it is easy to proceed by induction. The other case follows similarly.

2° Let, e.g., gx0 fx0 = gx1. Then, by assumption, fx0 fx1, i.e., y0 y1. Now gx1 gx2 implies fx1 fx2, i.e., y1 y2 and the conclusion again follows.

Putting g = i X (identity map) in the previous lemma, we obtain

Corollary 1If f is nondecreasing and x0 fx0(resp. x0 fx0), then the Picard sequence {fnx0} is nondecreasing (nonincreasing) w.r.t. .

If f is nonincreasing and fx0is comparable with x0, then the sequence {fnx0} is not monotonous, but arbitrary two of its adjacent terms are comparable.

Quasicontractions and g-quasicontractions in metric spaces were first studied in [3, 4]. We shall call the mapping f an ordered g-quasicontraction if there exists λ (0,1) such that for each x, y X satisfying gy gx, the inequality

$d\left(fx,fy\right)\le \lambda \cdot M\left(x,y\right)$

holds, where

$M\left(x,y\right)=\text{max}\left\{d\left(gx,gy\right),d\left(gx,fx\right),d\left(gy,fy\right),d\left(gx,fy\right),d\left(gy,fx\right)\right\}.$
(2.1)

Theorem 1 Let (X, d, ) be a partially ordered metric space and let f, g : XX be two self-maps on X satisfying the following conditions:

(i) fX gX;

(ii) gX is complete;

(iii) f is g-nondecreasing;

(iv) f is an ordered g-quasicontraction;

(v) there exists x0 X such that gx0 fx0;

(vi) if {gx n } is a nondecreasing sequence that converges to some gz gX then gx n gz for each n and gz g(gz).

Then f and g have a coincidence point, i.e., there exists z X such that fz = gz. If, in addition, (vii) f and g are weakly compatible ([1, 2]), i.e., fx = gx implies f gx = g fx, for each x X, then they have a common fixed point.

Proof Starting with the point x0 from condition (v), construct a Jungck sequence y n = fx n = gxn+1. By Lemma 1, using condition (iii) we conclude that {y n } is nondecreasing. It can be proved by the Das-Naik's method (as in ) that {y n } is a Cauchy sequence. Since gX is complete (condition (ii)), there exists z X such that y n w = gz gX. We shall prove that fz = w.

By condition (vi), gx n gz holds. Hence, putting in the contractive condition (iv) x = x n , y = z, we obtain that

$d\left(f{x}_{n},fz\right)\le \lambda \text{max}\left\{d\left(g{x}_{n},gz\right),d\left(g{x}_{n},f{x}_{n}\right),d\left(gz,fz\right),d\left(g{x}_{n},fz\right),d\left(f{x}_{n},gz\right)\right\},$

and passing to the limit when n → ∞ (and using that d(gx n , fz) ≤ d(gx n , gz) + d(gz, fz)) we deduce that

$d\left(fz,gz\right)\le \lambda \text{max}\left\{0,0,d\left(fz,gz\right),0+d\left(gz,fz\right),0\right\}=\lambda d\left(fz,gz\right).$

It follows that fz = gz = w and f and g have a coincidence point.

In the case when condition (vii) holds, we obtain that fw = fgz = gfz = gw and it remains to prove that fw = w. By condition (vi) we have that gz ggz = gw and then d(fw, w) = d(fw, fz) ≤ λM(w, z), where

$\begin{array}{ll}\hfill M\left(w,z\right)& =\text{max}\left\{d\left(gw,gz\right),d\left(gw,fw\right),d\left(gz,fz\right),d\left(gw,fz\right),d\left(gz,fw\right)\right\}\phantom{\rule{2em}{0ex}}\\ =\text{max}\left\{d\left(fw,w\right),0,0,d\left(fw,w\right),d\left(w,fw\right)\right\}=d\left(fw,w\right).\phantom{\rule{2em}{0ex}}\end{array}$

It follows that fw = w and w is a common fixed point for f and g.

Remark 1 If condition (v) in the previous theorem is replaced by fx0 gx0, the respective Jungck sequence is nonincreasing and the conclusion of the theorem holds.

Theorem 2 Let the conditions of Theorem 1 be satisfied, except that (iii), (v), and (vi) are, respectively, replaced by:

(iii') f is g-nonincreasing;

(v') there exists x0 X such that fx0and gx0are comparable;

(vi') if {gx n } is a sequence in gX which has comparable adjacent terms and that converges to some gz gX, then there exists a subsequence$\left\{g{x}_{{n}_{k}}\right\}$of {gx n } having all the terms comparable with gz and gz is comparable with ggz.

Then all the conclusions of Theorem 1 hold.

Proof Regardless whether fx0 gx0 or gx0 fx0 (condition( v')), Lemma 1 implies that the adjacent terms of the Jungck sequence {y n } are comparable. This is again sufficient to imply that {y n } is a Cauchy sequence. Hence, it converges to some gz gX.

By (vi'), there exists a subsequence ${y}_{{n}_{k}}=f{x}_{{n}_{k}}=g{x}_{{n}_{k}+1},\phantom{\rule{2.77695pt}{0ex}}k\in ℕ$, having all the terms comparable with gz. Hence, we can apply the contractive condition to obtain $d\left(fz,f{x}_{{n}_{k}}\right)\le \lambda M\left(z,{x}_{{n}_{k}}\right)$ where

$\begin{array}{ll}\hfill M\left(z,{x}_{{n}_{k}}\right)& =\text{max}\left\{d\left(gz,g{x}_{{n}_{k}}\right),d\left(gz,fz\right),d\left(g{x}_{{n}_{k}},f{x}_{{n}_{k}}\right),d\left(g{x}_{{n}_{k}},fz\right),d\left(gz,f{x}_{{n}_{k}}\right)\right\}\phantom{\rule{2em}{0ex}}\\ \to \text{max}\left\{0,d\left(fz,gz\right),0,d\left(gz,fz\right),0\right\}=d\left(fz,gz\right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(k\to \infty \right),\phantom{\rule{2em}{0ex}}\end{array}$

and since $d\left(fz,gz\right)\le d\left(fz,f{x}_{{n}_{k}}\right)+d\left(f{x}_{{n}_{k}},gz\right)$ and $f{x}_{{n}_{k}}\to gz,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}k\to \infty$, it follows that fz = gz = w. The rest of conclusions follow in the same way as in Theorem 1.

Remark 2 If the five-term set in condition (2.1) is replaced by any of the following sets

$\begin{array}{c}\left\{d\left(gx,gy\right)\right\},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left\{d\left(gx,gy\right),d\left(gx,fx\right),d\left(gy,fy\right)\right\},\\ \left\{d\left(gx,gy\right),\frac{1}{2}\left(d\left(gx,fx\right)+d\left(gy,fy\right)\right),\frac{1}{2}\left(d\left(gx,fy\right)+d\left(gy,fx\right)\right)\right\},\\ \left\{d\left(gx,gy\right),d\left(gx,fx\right),d\left(gy,fy\right),\frac{1}{2}\left(d\left(gx,fy\right)+d\left(gy,fx\right)\right)\right\},\end{array}$

then conclusions of both Theorems 1 and 2 remain valid. Thus, "ordered" versions of several generalized contraction theorems proved recently are obtained. In particular, the case of the set {d(gx, gy)} gives a generalization of the results of Ran and Reurings , and Nieto and Lopez [6, 7]. Note also that in most of these cases the proof that the Jungck sequence is a Cauchy sequence is directly obtained from the famous Jungck relation

$d\left({y}_{n+1},{y}_{n}\right)\le \lambda d\left({y}_{n},{y}_{n-1}\right).$

Remark 3 Taking g = i X we obtain "ordered" variants of various fixed point theorems involving one function f. In particular, the following "ordered" version of the known result of Ćirić on quasicontractions  is obtained.

Corollary 2 (a) Let (X, d, ) be a partially ordered complete metric space and let f : XX be a nondecreasing self-map such that for some λ (0,1),

$d\left(fx,fy\right)\le \lambda \text{max}\left\{d\left(x,y\right),d\left(x,fx\right),d\left(y,fy\right),d\left(x,fy\right),d\left(y,fx\right)\right\}$
(2.2)

holds for all x, y X such that y x. Suppose also that either

(i) f is continuous or

(ii) for each nondecreasing sequence {x n } converging to some u X, x n u holds for each n .

If there exists x0 X such that x0 fx0, then f has at least one fixed point.

1. (b)

The same holds if f is nonincreasing, there exists x 0 comparable with fx 0 and (ii) is replaced by

(ii') if a sequence {x n } converging to some u X has every two adjacent terms comparable, then there exists a subsequence$\left\{{x}_{{n}_{k}}\right\}$having each term comparable with x.

Proof As an illustration, we include a direct proof of case (b) when condition (ii ') is fulfilled.

Let u be the limit of the Picard sequence {fnx0} and let $\left\{{f}^{{n}_{k}}{x}_{0}\right\}$ be a subsequence having all the terms comparable with u. Then we can apply contractivity condition to obtain

Using that $d\left(fu,u\right)\le d\left(fu,{f}^{{n}_{k}+1}{x}_{0}\right)+d\left({f}^{{n}_{k}+1}{x}_{0},u\right)$ and passing to the limit when k → ∞ we get

$d\left(fu,u\right)\le \lambda \text{max}\left\{0,d\left(u,fu\right),0,0,d\left(u,fu\right)\right\}=\lambda d\left(u,fu\right),$

and it follows that fu = u.

Note also that instead of the completeness of X, its f-orbitally completeness is sufficient to obtain the conclusion of the corollary.

It is clear that the following diagram, where arrows stand for implications, is valid:

$\begin{array}{ccc}\text{contraction}\hfill & \to \hfill & \text{quasicontraction}\hfill \\ ↓\hfill & ↓\hfill \\ \text{ordered}\phantom{\rule{2.77695pt}{0ex}}\text{contraction}\hfill & \to \hfill & \text{ordered}\phantom{\rule{2.77695pt}{0ex}}\text{quasicontraction}\hfill \end{array}$

Reverse implications do not hold. An example of a quasicontraction which is not a contraction was given in . We shall present an example of an ordered quasicontraction which is not a quasicontraction.

Example 1 Let X = [0,4], equipped with the usual metric and ordered by

$x\underset{}{\preccurlyeq }y⇔\left(x=y\right)\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\left(x,y\in \left[1,4\right]\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}x\le y\right).$

Consider the function f: XX, $fx=\left\{\begin{array}{cc}2x,\hfill & 0\le x<1,\hfill \\ \frac{1}{3}\left(x+5\right)\hfill & 1\le x\le 4.\hfill \end{array}\right\$ Then

$\begin{array}{ll}\hfill M\left(x,y\right)& =\underset{0\le x<1}{\text{max}}\left\{d\left(x,y\right),d\left(x,fx\right),d\left(y,fy\right),d\left(x,fy\right),d\left(y,fy\right)\right\}\phantom{\rule{2em}{0ex}}\\ =\underset{0\le x<1.}{\text{max}}\left\{\left|x-y\right|,\left|x\right|,\left|y\right|,\left|x-2y\right|,\left|y-2x\right|\right\},\phantom{\rule{2em}{0ex}}\end{array}$

and we have to check the condition d(fx,fy) = 2|x - y| ≤ λM(x, y). Taking in each of the possible five inequalities that (x → 1- and y = 0) or (x = 0 and y → 1-) we obtain that λ cannot be in the interval [0,1). Hence, f is not a quasicontraction. But it is an ordered contraction, and so an ordered quasicontraction, since d(fx, fy) ≤ λd(x, y) holds for $\lambda \in \left[\frac{1}{3},1\right)$ and all x, y X such that y x.

### 2.1 Uniqueness of the fixed point

The following simple example shows that conditions of theorems in the previous section are not sufficient for the uniqueness of fixed points (resp., common fixed points).

Example 2 Let X = {(1, 0), (0,1)}, let (a, b) (c, d) if and only if ac and bd, and let d1 be the Euclidean metric. The function f((x, y)) = (x, y) is continuous. The only comparable pairs of points in X are (x, x) for x X and then (2.2) reduces to d1(fx, fx) = 0, and is trivially fulfilled. However, f has two fixed points (1,0) and (0,1).

We shall give a sufficient condition for the uniqueness of the fixed point in the case of an ordered quasicontraction.

Theorem 3 Let all the conditions of Corollary 2 be satisfied with λ (0, 1/2) in the contractivity condition (2.2) and let for all x, y X there exists z X comparable with both of them. Then:

1. f has a unique fixed point u;

2. for arbitrary x0which can be a starting point of the Picard sequence {fnx0} this sequence converges to u.

Proof Let u and v be two fixed points of f. If these two points are comparable, then

$\begin{array}{ll}\hfill d\left(u,v\right)& =d\left(fu,fv\right)\phantom{\rule{2em}{0ex}}\\ \le \lambda \text{max}\left\{d\left(u,v\right),d\left(u,fu\right),d\left(v,fv\right),d\left(u,fv\right),d\left(v,fu\right)\right\}=\lambda d\left(u,v\right),\phantom{\rule{2em}{0ex}}\end{array}$

implying u = v.

Suppose that u and v are incomparable and that w X is comparable with both of them. We shall prove that d(u, fnw) → 0 and d(v, fnw) → 0 when n → ∞, which, taking into account that d(u, v) ≤ d(u, fnw) + d(fnw, v), will imply that d(u, v) = 0 and u = v. Indeed,

It follows that

$d\left(u,{f}^{n}w\right)\le \frac{\lambda }{1-\lambda }d\left(u,{f}^{n-1}w\right)=\mu d\left(u,{f}^{n-1}w\right).$

Since 0 < λ < 1/2, it is μ < 1 and it follows that d(u, fnw) ≤ μnd(u,w) → 0, when n → ∞. Similarly, d(v, fnw) → 0 and the proof is complete in the case, when f is nondecreasing. The proof is similar when f is nonincreasing.

It is an open question whether the previous theorem is true for λ [1/2,1). Also, an open problem is to find sufficient conditions for the uniqueness of the common fixed point in the case of an ordered g-quasicontraction.

## 3 Weak ordered contractions

Functions ψ, φ : [0, ∞) → [0, ∞) will be called control functions if:

1. (i)

ψ is a continuous nondecreasing function with ψ(t) = 0 if and only if t = 0,

2. (ii)

φ is a lower semi-continuous function with φ(t) = 0 if and only if t = 0.

Let (X, d) be a metric space and let f,g : XX. In the articles [20, 21] (in the setting of metric spaces) and [12, 13] (in the setting of ordered metric spaces) contractive conditions of the form

$\psi \left(d\left(fx,gy\right)\right)\le \psi \left(M\left(x,y\right)\right)-\phi \left(M\left(x,y\right)\right),$

where

$M\left(x,y\right)=\text{max}\left\{d\left(x,y\right),d\left(x,fx\right),d\left(y,gy\right),\frac{1}{2}\left(d\left(x,gy\right)+d\left(y,fx\right)\right)\right\},$

were used to obtain common fixed points results. We shall use here the following condition

$\psi \left(d\left(fx,fy\right)\right)\le \psi \left(M\left(x,y\right)\right)-\phi \left(M\left(x,y\right)\right),$
(3.1)

where

$M\left(x,y\right)=\text{max}\left\{d\left(gx,gy\right),d\left(gx,fx\right),d\left(gy,fy\right),\frac{1}{2}\left(d\left(gx,fy\right)+d\left(gy,fx\right)\right)\right\}.$
(3.2)

Note that the respective result in the setting of ordered cone metric spaces is proved in .

Assertions similar to the following lemma (see, e.g., ) were used (and proved) in the course of proofs of several fixed point results in various articles.

Lemma 2Let (X, d) be a metric space and let {y n } be a sequence in X such that the sequence {d(yn+1, y n )} is nonincreasing and

$\underset{n\to \infty }{\text{lim}}d\left({y}_{n+1},{y}_{n}\right)=0.$

If {y2n} is not a Cauchy sequence, then there exist ε > 0 and two sequences {m k } and {n k } of positive integers such that the following four sequences tend to ε, when k → ∞:

$d\left({y}_{2{m}_{k}},{y}_{2{n}_{k}}\right),\phantom{\rule{1em}{0ex}}d\left({y}_{2{m}_{k}},{y}_{2{n}_{k}+1}\right),\phantom{\rule{1em}{0ex}}d\left({y}_{2{m}_{k}-1},{y}_{2{n}_{k}}\right),\phantom{\rule{1em}{0ex}}d\left({y}_{2{m}_{k}-1},{y}_{2{n}_{k}+1}\right).$
(3.3)

Theorem 4 Let (X, d, ) be a partially ordered metric space and let f, g be two self-maps on X satisfying the following conditions:

(i) fX gX;

(ii) gX is complete;

(iii) f is g-nondecreasing;

(iv) f and g satisfy condition (3.1) for each x, y X such that gy gx, where (ψ,φ) is a pair of control functions and M(x,y) is defined by (3.2);

(v) there exists x0 X such that gx0 fx0;

(vi) if {gx n } is a nondecreasing sequence that converges to some gz gX then gx n gz for each n and gz g(gz).

Then f and g have a coincidence point. If in addition,

(vii) f and g are weakly compatible,

then they have a common fixed point.

Proof As in the proof of Theorem 1, a nondecreasing Jungck sequence y n with y n = fx n = gxn+1can be constructed. Consider the following two possibilities: 1° ${y}_{{n}_{0}}={y}_{{n}_{0}+1}$ for some n0 and 2° y n yn+1for each n .

1° We shall prove that in this case ${y}_{n}={y}_{{n}_{0}}$ for each nn0. Using that $g{x}_{{n}_{0}+1}\underset{}{\prec }g{x}_{{n}_{0}+2}$ we obtain

$\begin{array}{ll}\hfill \psi \left(d\left({y}_{{n}_{0}+1},{y}_{{n}_{0}+2}\right)\right)& =\psi \left(d\left(f{x}_{{n}_{0}+1},f{x}_{{n}_{0}+2}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(M\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right)-\phi \left(M\left({x}_{{n}_{0}+1},{x}_{{n}_{0}+2}\right)\right),\phantom{\rule{2em}{0ex}}\end{array}$

where

Hence $\psi \left(d\left({y}_{{n}_{0}+1},{y}_{{n}_{0}+2}\right)\right)\le \psi \left(d\left({y}_{{n}_{0}+1},{y}_{{n}_{0}+2}\right)\right)-\phi \left(d\left({y}_{{n}_{0}+1},{y}_{{n}_{0}+2}\right)\right)$ which, by the properties of control functions, implies that $d\left({y}_{{n}_{0}+1},{y}_{{n}_{0}+2}\right)=0$ and ${y}_{{n}_{0}+1}={y}_{{n}_{0}+2}$. By induction, ${y}_{n}={y}_{{n}_{0}}$ for each nn0. It follows that $g{x}_{{n}_{0}+1}=f{x}_{{n}_{0}+1}$ and ${x}_{{n}_{0}+1}$ is a coincidence point of f and g.

Suppose that condition (vii) holds. Then also

$f{y}_{{n}_{0}}=fg{x}_{{n}_{0}+1}=gf{x}_{{n}_{0}+1}=g{y}_{{n}_{0}},$

and if we prove that $f{y}_{{n}_{0}}={y}_{{n}_{0}},{y}_{{n}_{0}}$ will be a common fixed point for f and g. Using condition (vi) with $z={y}_{{n}_{0}}$ we obtain that $g{x}_{{n}_{0}+1}\underset{}{\prec }g{y}_{{n}_{0}}$ and we can apply condition (iv) to obtain

$\psi \left(d\left(f{x}_{{n}_{0}+1},f{y}_{{n}_{0}}\right)\right)\le \psi \left(M\left({x}_{{n}_{0}+1},{y}_{{n}_{0}}\right)\right)-\phi \left(M\left({x}_{{n}_{0}+1},{y}_{{n}_{0}}\right)\right),$

where

Thus, $\psi \left(d\left({y}_{{n}_{0}},f{y}_{{n}_{0}}\right)\right)\le \psi \left(d\left({y}_{{n}_{0}},f{y}_{{n}_{0}}\right)\right)-\phi \left(d\left({y}_{{n}_{0}},f{y}_{{n}_{0}}\right)\right)$ and again using properties of control functions it follows that $f{y}_{{n}_{0}}={y}_{{n}_{0}}$. This completes the proof of the theorem in the first case.

2° We shall prove that in this case {y n } is a Cauchy sequence. Since gxn+1 gxn+2, we have that

$\begin{array}{ll}\hfill \psi \left(d\left({y}_{n+1},{y}_{n+2}\right)\right)& =\psi \left(d\left(f{x}_{n+1},f{x}_{n+2}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(M\left({x}_{n+1},{x}_{n+2}\right)\right)-\phi \left(M\left({x}_{n+1},{x}_{n+2}\right)\right),\phantom{\rule{2em}{0ex}}\end{array}$

where

(d (y n , yn+2) ≤ d(y n , yn+1)+ d(yn+1,yn+2) was used to obtain the last equality). If d (yn+1, yn+2) ≥ d(y n , yn+1), then it follows from

$\psi \left(d\left({y}_{n+1},{y}_{n+2}\right)\right)\le \psi \left(d\left({y}_{n+1},{y}_{n+2}\right)\right)-\phi \left(d\left({y}_{n+1},{y}_{n+2}\right)\right)$
(3.4)

and the properties of control functions that d(yn+1,yn+2) = 0, contrary to the assumption that all terms of the sequence {y n } are distinct. Hence, d(yn+1,yn+2) < d(y n ,yn+1), i.e., the sequence {d(y n ,yn+1)} is decreasing. Therefore, it converges to some d* ≤ 0 when n → ∞. Then, M(xn+1,xn+2) → d*, n → ∞.

From (3.4), passing to the (upper) limit when n → ∞, it follows that

$\psi \left(d*\right)\le \psi \left(d*\right)-\underset{n\to \infty }{\mathrm{lim}\mathrm{inf}}\phi \left(d*\right)\le \psi \left(d*\right)-\phi \left(d*\right),$

i.e., φ(d*) = 0 and d* = 0. We conclude that d(y n , yn+1) → 0, n → ∞. As a consequence, in order to prove that {y n } is a Cauchy sequence, it is enough to prove that {y2n} is a Cauchy sequence.

Suppose that {y2n} is not a Cauchy sequence. Then, by Lemma 2, there exist ε > 0 and two sequences {m k } and {n k } of positive integers such that the sequences (3.3) tend to ε when k → ∞. Now, by definition of M(x,y), we obtain that $M\left({x}_{2{m}_{k}},{x}_{2{n}_{k}+1}\right)\to \epsilon ,\phantom{\rule{2.77695pt}{0ex}}k\to \infty$. Indeed,

Now, putting $x={x}_{2{m}_{k}}$ and $y={x}_{2{n}_{k}+1}$ into the contractive condition (iv) (which is possible since $g{x}_{2{n}_{k}+1}={y}_{2{n}_{k}}$ and $gx=g{x}_{2{m}_{k}}={y}_{2{m}_{k}-1}$ are comparable), we get that

$\begin{array}{ll}\hfill \psi \left(d\left({y}_{2{m}_{k}},{y}_{2{n}_{k}+1}\right)\right)& =\psi \left(d\left(f{x}_{2{m}_{k}},f{x}_{2{n}_{k}+1}\right)\right)\phantom{\rule{2em}{0ex}}\\ \le \psi \left(M\left({x}_{2{m}_{k}},{x}_{2{n}_{k}+1}\right)\right)-\phi \left(M\left({x}_{2{m}_{k}},{x}_{2{n}_{k}+1}\right)\right).\phantom{\rule{2em}{0ex}}\end{array}$

Passing to the limit when k → ∞ and using properties of control functions, we obtain that ψ(ε) ≤ ψ(ε)- φ(ε), which is in contradiction with ε > 0.

The proof that {y n } is a Cauchy sequence is complete. By the assumption (ii), there exists z X such that gx n gz, when n → ∞. We shall prove that fz = gz.

Condition (vi) implies that gx n gz and we can apply contractive condition to obtain

$\psi \left(d\left(f{x}_{n},fz\right)\right)\le \psi \left(M\left({x}_{n},z\right)\right)-\phi \left(M\left({x}_{n},z\right)\right),$
(3.5)

where

$\begin{array}{ll}\hfill M\left({x}_{n},z\right)& =\text{max}\left\{d\left(g{x}_{n},gz\right),d\left(g{x}_{n},f{x}_{n}\right),d\left(gz,fz\right),\frac{1}{2}\left(d\left(g{x}_{n},fz\right)+d\left(gz,f{x}_{n}\right)\right)\right\}\phantom{\rule{2em}{0ex}}\\ \to \text{max}\left\{0,0,d\left(gz,fz\right),\frac{1}{2}d\left(gz,fz\right)\right\}=d\left(gz,fz\right).\phantom{\rule{2em}{0ex}}\end{array}$

Passing to the (upper) limit when n → ∞ in (3.5) we obtain that

$\psi \left(d\left(gz,fz\right)\right)\le \psi \left(d\left(gz,fz\right)\right)-\phi \left(d\left(gz,fz\right)\right),$

wherefrom gz = fz follows. Hence, z is a coincidence point of f and g.

If condition (vii) is fulfilled, put w = fz = gz, and obtain that fw = fgz = gfz = gw. Using that gz ggz = gw it can be proved that fw = gw = w in a similar way as it was done in the case 1°. The theorem is proved completely.

Remark 4 If the four-term set in condition (3.2) is replaced by any of the following sets

$\begin{array}{c}\left\{d\left(gx,gy\right)\right\},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left\{d\left(gx,gy\right),d\left(gx,fx\right),d\left(gy,fy\right)\right\},\\ \left\{d\left(gx,gy\right),\frac{1}{2}\left(d\left(gx,fx\right),d\left(gy,fy\right)\right),\frac{1}{2}\left(d\left(gx,fy\right)+d\left(gy,fx\right)\right)\right\},\end{array}$

similar conclusions for the existence of common fixed points of mappings f and g can be obtained.

Remark 5 In his important article , Jachymski showed that some fixed point results based on weak contractive conditions involving functions ψ and φ can be reduced to their counterparts using just one function, say φ'. We note that our results do not fall into this category, since conditions of our Theorem 4 are not covered by the conditions of [22, Theorem 7].

Remark 6 Very recently, Shatanawi et al.  have obtained some results closely related to the ones given in this section. However, their assumptions on the given mappings are different.

## References

1. Jungck G: Commuting mappings and fixed points. Am Math Monthly 1976, 83: 261–263. 10.2307/2318216

2. Jungck G: Compatible mappings and common fixed points. Int J Math Sci 1986, 9: 771–779. 10.1155/S0161171286000935

3. Ćirić LjB: A generalization of Banach's contraction principle. Proc Am Math Soc 1974, 45: 267–273.

4. Das KM, Naik KV: Common fixed point theorems for commuting maps on metric spaces. Proc Am Math Soc 1979, 77: 369–373.

5. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc Am Math Soc 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4

6. Nieto JJ, Lopez RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5

7. Nieto JJ, Lopez RR: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math Sinica English Ser 2007, 23(12):2205–2212. 10.1007/s10114-005-0769-0

8. O'Regan D, Shahzad N: Invariant approximations for generalized contractions. Numer Funct Anal Optim 2005, 26(4–5):565–575. 10.1080/NFA-200067306

9. Ćirić Lj, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl 2008.

10. O'Regan D, Petrusel A: Fixed point theorems for generalized contractions in ordered metric spaces. J Math Anal Appl 2008, 341: 1241–1252. 10.1016/j.jmaa.2007.11.026

11. Agarwal RP, El-Gebeily MA, O'Regan D: Generalized contractions in partially ordered metric spaces. Appl Anal 2008, 87: 109–116. 10.1080/00036810701556151

12. Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal 2009, 71: 3402–3410.

13. Radenović S, Kadelburg Z: Generalized weak contractions in partially ordered metric spaces. Comput Math Appl 2010, 60: 1776–1783. 10.1016/j.camwa.2010.07.008

14. Kadelburg Z, Pavlović M, Radenović S: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput Math Appl 2010, 59: 3148–3159. 10.1016/j.camwa.2010.02.039

15. Karapinar E: Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput Math Appl 2010, 59: 3656–3668. 10.1016/j.camwa.2010.03.062

16. Karapinar E, Berinde V: Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces. Banach J Math Anal 2012, 6: 74–89.

17. Alber YaI, Guerre-Delabriere S: Principle of weakly contractive maps in Hilbert spaces. In New results in operator theory Advances and Applications. Volume 98. Edited by: Gohberg, I, Lyubich, Yu. Birkhäuser Verlag, Basel; 1997:7–22.

18. Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal 2001, 47: 2683–2693. 10.1016/S0362-546X(01)00388-1

19. Berinde V: Approximating fixed points of weak φ-contractions. Fixed Point Theory 2003, 4: 131–142.

20. Zhang Q, Song Y: Fixed point theory for generalized φ-weak contractions. Appl Math Lett 2009, 22: 75–78. 10.1016/j.aml.2008.02.007

21. Dorić D: Common fixed point for generalized ( ψ, φ )-weak contractions. Appl Math Lett 2009, 22: 1896–1900. 10.1016/j.aml.2009.08.001

22. Jachymski J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal TMA 2011, 74: 768–774. 10.1016/j.na.2010.09.025

23. Aydi H, Karapinar E, Shatanawi W: Coupled fixed point results for ( ψ, φ )-weakly contractive condition in ordered partial metric spaces. Comput Math Appl 2011, 62: 4449–4460. 10.1016/j.camwa.2011.10.021

24. Radenović S, Kadelburg Z, Jandrlić D, Jandrlić A: Some results on weak contraction maps. Bull Iran Math Soc 2011, 192(1):111–31. available online since 30 March 2011

25. Shatanawi W, Mustafa Z, Tahat N: Some coincidence point theorems for nonlinear contractions in ordered metric spaces. Fixed Point Theory Appl 2011, 2011: 68. 10.1186/1687-1812-2011-68

## Acknowledgements

The authors thank the referees for their valuable comments that helped us to improve the text. The authors are thankful to the Ministry of Science and Technological Development of Serbia.

## Author information

Authors

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Golubović, Z., Kadelburg, Z. & Radenović, S. Common fixed points of ordered g-quasicontractions and weak contractions in ordered metric spaces. Fixed Point Theory Appl 2012, 20 (2012). https://doi.org/10.1186/1687-1812-2012-20

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/1687-1812-2012-20

### Keywords

• partially ordered metric space
• g-quasicontraction
• weak contraction
• coincidence point
• common fixed point 