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Common fixed points of ordered gquasicontractions and weak contractions in ordered metric spaces
Fixed Point Theory and Applicationsvolume 2012, Article number: 20 (2012)
Abstract
We introduce ordered quasicontractions and gquasicontractions 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 socalled 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
where (X, d) is a metric space, f, g : X → X, λ ∈ [0,1) and M(x, y) is the maximum of one of the sets
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 socalled Jungck sequence y_{ n }= fx_{ n }= gx_{n+1}was used (see, [1, 2]).
In the case when
the mapping f is called a gquasicontraction (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 [5], and then by Nieto and Lopez [6, 7]. Further results in this direction were proved, e.g., in [8–16].
In Section 2 of this article, we introduce ordered quasicontractions and gquasicontractions 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, 17–23]) have begun to use more general conditions (socalled weak contractive conditions) of the type
where ψ and φ are socalled 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., [5–7, 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 [9–11] since control functions are more general than comparison functions used in these articles.
2 Common fixed points of ordered gquasicontractions
Consider a partially ordered set (X, ≼) and two selfmaps f, g : X → X such that f X ⊂ gX. Following [9] we shall say that the mapping f is gnondecreasing (resp., gnonincreasing) if gx ≼ gy ⇒ fx ≼ fy (resp., gx ≼ gy ⇒ fx ≽ fy) holds for each x, y ∈ X.
For arbitrary x_{0} ∈ X one can construct a socalled Jungck sequence {y_{ n }} in the following way: denote y_{0} = fx_{0} ∈ f X ⊂ gX; there exists x_{1} ∈ X such that gx_{1} = y_{0} = fx_{0}; now y_{1} = fx_{1} ∈ f X ⊂ gX and there exists x_{2} ∈ X such that gx_{2} = y_{1} = fx_{1} and the procedure can be continued. The following properties of this sequence can be easily deduced.
Lemma 1 1° If f is gnondecreasing and gx_{0} ≼ fx_{0} (resp., gx_{0} ≽ fx_{0}), then the sequence {y_{ n }} is nondecreasing (resp., nonincreasing) w.r.t. ≼.
2° If f is gnonincreasing and fx_{0}is comparable with gx_{0}, then the sequence {y_{ n }} is not monotonous, but arbitrary two of its adjacent terms are comparable.
Proof 1° It follows from gx_{0} ≼ fx_{0} = gx_{1} that fx_{0} ≼ fx_{1}, i.e., y_{0} ≼ y_{1} and it is easy to proceed by induction. The other case follows similarly.
2° Let, e.g., gx_{0} ≼ fx_{0} = gx_{1}. Then, by assumption, fx_{0} ≽ fx_{1}, i.e., y_{0} ≽ y_{1}. Now gx_{1} ≽ gx_{2} implies fx_{1} ≼ fx_{2}, i.e., y_{1} ≼ y_{2} and the conclusion again follows.
Putting g = i_{ X }(identity map) in the previous lemma, we obtain
Corollary 1 1° If f is nondecreasing and x_{0} ≼ fx_{0}(resp. x_{0} ≽ fx_{0}), then the Picard sequence {f^{n}x_{0}} is nondecreasing (nonincreasing) w.r.t. ≼.
2° If f is nonincreasing and fx_{0}is comparable with x_{0}, then the sequence {f^{n}x_{0}} is not monotonous, but arbitrary two of its adjacent terms are comparable.
Quasicontractions and gquasicontractions in metric spaces were first studied in [3, 4]. We shall call the mapping f an ordered gquasicontraction if there exists λ ∈ (0,1) such that for each x, y ∈ X satisfying gy ≼ gx, the inequality
holds, where
Theorem 1 Let (X, d, ≼) be a partially ordered metric space and let f, g : X → X be two selfmaps on X satisfying the following conditions:
(i) fX ⊂ gX;
(ii) gX is complete;
(iii) f is gnondecreasing;
(iv) f is an ordered gquasicontraction;
(v) there exists x_{0} ∈ X such that gx_{0} ≼ fx_{0};
(vi) if {gx_{ n }} is a nondecreasing sequence that converges to some gz ∈ gX then gx_{ n }≼ gz for each n ∈ 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 x_{0} from condition (v), construct a Jungck sequence y_{ n }= fx_{ n }= gx_{n+1}. By Lemma 1, using condition (iii) we conclude that {y_{ n }} is nondecreasing. It can be proved by the DasNaik's method (as in [8]) 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
and passing to the limit when n → ∞ (and using that d(gx_{ n }, fz) ≤ d(gx_{ n }, gz) + d(gz, fz)) we deduce that
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
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 fx_{0} ≼ gx_{0}, 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 gnonincreasing;
(v') there exists x_{0} ∈ X such that fx_{0}and gx_{0}are 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 fx_{0} ≼ gx_{0} or gx_{0} ≼ fx_{0} (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 ${y}_{{n}_{k}}=f{x}_{{n}_{k}}=g{x}_{{n}_{k}+1},\phantom{\rule{2.77695pt}{0ex}}k\in \mathbb{N}$, 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
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 fiveterm set in condition (2.1) is replaced by any of the following sets
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 [5], 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
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 [3] is obtained.
Corollary 2 (a) Let (X, d, ≼) be a partially ordered complete metric space and let f : X → X be a nondecreasing selfmap such that for some λ ∈ (0,1),
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 ∈ ℕ.
If there exists x_{0} ∈ X such that x_{0} ≼ fx_{0}, then f has at least one fixed point.

(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 {f^{n}x_{0}} 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
and it follows that fu = u.
Note also that instead of the completeness of X, its forbitally completeness is sufficient to obtain the conclusion of the corollary.
It is clear that the following diagram, where arrows stand for implications, is valid:
Reverse implications do not hold. An example of a quasicontraction which is not a contraction was given in [3]. 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
Consider the function f: X → X, $fx=\left\{\begin{array}{cc}2x,\hfill & 0\le x<1,\hfill \\ \frac{1}{3}\left(x+5\right)\hfill & 1\le x\le 4.\hfill \end{array}\right.$ Then
and we have to check the condition d(fx,fy) = 2x  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 a ≤ c and b ≤ d, and let d_{1} 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 d_{1}(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 x_{0}which can be a starting point of the Picard sequence {f^{n}x_{0}} this sequence converges to u.
Proof Let u and v be two fixed points of f. If these two points are comparable, then
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, f^{n}w) → 0 and d(v, f^{n}w) → 0 when n → ∞, which, taking into account that d(u, v) ≤ d(u, f^{n}w) + d(f^{n}w, v), will imply that d(u, v) = 0 and u = v. Indeed,
It follows that
Since 0 < λ < 1/2, it is μ < 1 and it follows that d(u, f^{n}w) ≤ μ^{n}d(u,w) → 0, when n → ∞. Similarly, d(v, f^{n}w) → 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 gquasicontraction.
3 Weak ordered contractions
Functions ψ, φ : [0, ∞) → [0, ∞) will be called control functions if:

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

(ii)
φ is a lower semicontinuous function with φ(t) = 0 if and only if t = 0.
Let (X, d) be a metric space and let f,g : X → X. 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
where
were used to obtain common fixed points results. We shall use here the following condition
where
Note that the respective result in the setting of ordered cone metric spaces is proved in [14].
Assertions similar to the following lemma (see, e.g., [24]) were used (and proved) in the course of proofs of several fixed point results in various articles.
Lemma 2[24]Let (X, d) be a metric space and let {y_{ n }} be a sequence in X such that the sequence {d(y_{n+1}, y_{ n })} is nonincreasing and
If {y_{2n}} 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 → ∞:
Theorem 4 Let (X, d, ≼) be a partially ordered metric space and let f, g be two selfmaps on X satisfying the following conditions:
(i) fX ⊂ gX;
(ii) gX is complete;
(iii) f is gnondecreasing;
(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 x_{0} ∈ X such that gx_{0} ≼ fx_{0};
(vi) if {gx_{ n }} is a nondecreasing sequence that converges to some gz ∈ gX then gx_{ n }≼ gz for each n ∈ 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 }= gx_{n+1}can be constructed. Consider the following two possibilities: 1° ${y}_{{n}_{0}}={y}_{{n}_{0}+1}$ for some n_{0} ∈ ℕ and 2° y_{ n }≠ y_{n+1}for each n ∈ ℕ.
1° We shall prove that in this case ${y}_{n}={y}_{{n}_{0}}$ for each n ≥ n_{0}. Using that $g{x}_{{n}_{0}+1}\underset{}{\prec}g{x}_{{n}_{0}+2}$ we obtain
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 n ≥ n_{0}. 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
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
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 gx_{n+1}≼ gx_{n+2}, we have that
where
(d (y_{ n }, y_{n+2}) ≤ d(y_{ n }, y_{n+1})+ d(y_{n+1},y_{n+2}) was used to obtain the last equality). If d (y_{n+1}, y_{n+2}) ≥ d(y_{ n }, y_{n+1}), then it follows from
and the properties of control functions that d(y_{n+1},y_{n+2}) = 0, contrary to the assumption that all terms of the sequence {y_{ n }} are distinct. Hence, d(y_{n+1},y_{n+2}) < d(y_{ n },y_{n+1}), i.e., the sequence {d(y_{ n },y_{n+1})} is decreasing. Therefore, it converges to some d* ≤ 0 when n → ∞. Then, M(x_{n+1},x_{n+2}) → d*, n → ∞.
From (3.4), passing to the (upper) limit when n → ∞, it follows that
i.e., φ(d*) = 0 and d* = 0. We conclude that d(y_{ n }, y_{n+1}) → 0, n → ∞. As a consequence, in order to prove that {y_{ n }} is a Cauchy sequence, it is enough to prove that {y_{2n}} is a Cauchy sequence.
Suppose that {y_{2n}} 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
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
where
Passing to the (upper) limit when n → ∞ in (3.5) we obtain that
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 fourterm set in condition (3.2) is replaced by any of the following sets
similar conclusions for the existence of common fixed points of mappings f and g can be obtained.
Remark 5 In his important article [22], 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. [25] have obtained some results closely related to the ones given in this section. However, their assumptions on the given mappings are different.
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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.
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Keywords
 partially ordered metric space
 gquasicontraction
 weak contraction
 coincidence point
 common fixed point