Common fixed point theorems for weakly increasing mappings on ordered orbitally complete metric spaces

In this article, we prove existence results for common fixed points of two or three relatively asymptotically regular mappings satisfying the orbital continuity of one of the involved maps on ordered orbitally complete metric spaces. We furnish suitable examples to demonstrate the validity of the hypotheses of our results.

Mathematics Subject Classification (2010): 47H10; 54H25.

Browder and Petryshyn introduced the concept of asymptotic regularity of a self-map at a point in a metric space.

Definition 1 [1] A self-map on a metric space is said to be asymptotically regular at a point if .

Recall that the set is called the orbit of the self-map at the point .

Definition 2 [2] A metric space is said to be -orbitally complete if every Cauchy sequence contained in (for some x in ) converges in .

Here, it can be pointed out that every complete metric space is -orbitally complete for any , but a -orbitally complete metric space need not be complete.

Definition 3 [1] A self-map defined on a metric space is said to be orbitally continuous at a point z in if for any sequence (for some ), x_n → z as n → ∞ implies as n → ∞.

Clearly, every continuous self-mapping of a metric space is orbitally continuous, but not conversely.

Sastry et al. [3] extended the above concepts to two and three mappings and employed them to prove common fixed point results for commuting mappings. In what follows, we collect such definitions for three maps.

Definition 4 Let be three self-mappings defined on a metric space .

1. If for a point , there exits a sequence {x_n} in such that , then the set is called the orbit of at x₀.

2. The space is said to be -orbitally complete at x₀ if every Cauchy sequence in converges in .

3. The map is said to be orbitally continuous at x₀ if it is continuous on .

4. The pair is said to be asymptotically regular (in short a.r.) with respect to at x₀ if there exists a sequence {x_n} in such that , and as n → ∞.

5. If is the identity mapping on , we omit in respective definitions.

On the other hand, fixed point theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [4] who presented its applications to matrix equations. Subsequently, Nieto and López [5] extended this result for nondecreasing mappings and applied it to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Thereafter, several authors obtained many fixed point theorems in ordered metric spaces. For more details, see [6-15] and the references cited therein.

Recently, Nashine and Altun (HK Nashine and I Altun, unpublished work) proved the following ordered version of a result of Zhang [16]:

Theorem 1 Let be a complete partially ordered metric space and let be two weakly increasing mappings such that

holds for each comparable , where F, ψ : [0, +∞) → [0, +∞) are functions such that

(i) F is nondecreasing, continuous, and F(0) = 0 < F(t) for every t > 0;

(ii) ψ is nondecr easing, right continuous, and ψ(t) < t for every t > 0, and

If or is continuous, then and have a unique common fixed point.

In this article, we generalize this theorem of Nashine and Altun (HK Nashine and I Altun, unpublished work) (and, hence, some other related common fixed point results) in two directions. The first is treated in Section 3, where a pair of asymptotically regular mappings in an orbitally complete ordered metric space is considered. The existence and (under additional assumptions) uniqueness of their common fixed point is obtained under assumptions that these mappings are strictly weakly isotone increasing, one is orbitally continuous and they satisfy a generalized weakly contractive condition.

In Section 4, we consider the case of three self-mappings where the pair is -relatively asymptotically regular and relatively weakly increasing, while the contractive condition is given with the help of two control functions.

We furnish suitable examples to demonstrate the validity of the hypotheses of our results.

First, we introduce some further notation and definitions that will be used later.

If is a partially ordered set then are called comparable if x ≼ y or y ≼ x holds. A subset of is said to be well ordered if every two elements of are comparable. If is such that, for , x ≼ y implies , then the mapping is said to be nondecreasing.

Definition 5 Let be a partially ordered set and .

1. The mapping is called dominating if for each [17].

2. The pair is called weakly increasing if and for all [18,19].

3. The mapping is said to be -weakly isotone increasing if for all we have [18-20].

4. The mapping is said to be -strictly weakly isotone increasing if, for all such that , we have (HK Nashine, B Samet, and C Vetro, unpublished work).

5. Let be such that and , and denote , for . We say that and are weakly increasing with respect to if and only if for all , we have [10]:

and

Example 1 [17] Let be endowed with the usual ordering. Let be defined by . Since for all is a dominating map.

Remark 1(1) None of two weakly increasing mappings need be nondecreasing. There exist some examples to illustrate this fact in [21].

(2) If are weakly increasing, then is -weakly isotone increasing.

(3) can be -strictly weakly isotone increasing, while some of these two mappings can be not strictly increasing (see the following example).

(4) If is the identity mapping ( for all ), then and are weakly increasing with respect to if and only if they are weakly increasing mappings.

Example 2 Let be endowed with the usual ordering and define as

Clearly, we have for all , and so, is -strictly weakly isotone increasing; is not strictly increasing.

Definition 6 [22,23]. Let be a metric space and .

1. If w = fx = gx, for some , then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g.If w = x, then x is a common fixed point of f and g.

2. The mappings f and g are said to be compatible if lim_n→∞ d(fgx_n, gfx_n) = 0, whenever {x_n} is a sequence in such that lim_n→∞ fx_n = lim_n→∞gx_n = t for some .

Definition 7 Let be a nonempty set. Then is called an ordered metric space if

(i) is a metric space,

(ii) is a partially ordered set.

The space is called regular if the following hypothesis holds: if {z_n} is a nondecreasing sequence in with respect to ≼ such that as n → ∞, then z_n ≼ z.

In this section, we improve the results of Nashine and Altun (HK Nashine and I Altun, unpublished work) by considering the following:

1. a pair of asymptotically regular mappings;

2. orbital continuity of one of the involved maps;

3. strictly weakly isotone increasing condition;

4. generalized weakly contractive condition, and

5. an ordered orbitally complete metric space.

We will denote by and Ψ the set of functions F, ψ : [0, +∞) → [0, +∞), respectively, such that:

(i) F is nondecreasing, continuous, and F(0) = 0 < F(t) for every t > 0;

(ii) ψ is nondecreasing, right continuous, and ψ(0) = 0.

The first main result of this section is as follows:

Theorem 2 Let be an ordered metric space. Let be two mappings satisfying

for all (for some x₀) such that x and y are comparable, where , ψ ∈ Ψ and

We assume the following hypotheses:

(i) is a.r. at x₀;

(ii) is -orbitally complete at x₀;

(iii) or is -orbitally continuous at x₀;

(iv) is -strictly weakly isotone increasing;

(v) there exists an such that .

Then and have a common fixed point. Moreover, the set of common fixed points of in is well ordered if and only if it is a singleton.

Proof First of all we show that, if or has a fixed point, then it is a common fixed point of and . Indeed, let z be a fixed point of . Now assume . If we use the inequality (3.1), for x = y = z, we have

wherefrom , which is a contradiction. Thus and so z is a common fixed point of and . Analogously, one can observe that if z is a fixed point of , then it is a common fixed point of and .

Since is a.r. at x₀ in , there exists a sequence {x_n} in such that

and

If or for some n₀, then the proof is finished. So assume x_n ≠ x_n+1 for all n.

Since is -strictly weakly isotone increasing, we have

and continuing this process we get

Next, we claim that {x_n} is a Cauchy sequence in the metric space . We proceed by negation and suppose that {x_n} is not a Cauchy sequence. That is, there exists ε > 0 such that d(x_n,x_m) ≥ ε for infinitely many values of m and n with m < n. This assures that there exist two sequences {m(k)}, {n(k)} of natural numbers, with m(k) < n(k), such that for each k ∈ ℕ

It is not restrictive to suppose that n(k) is the least positive integer exceeding m(k) and satisfying (3.6). We have

and letting k → ∞, we have d(x_2m(k), x_2n(k)+1) → ε. We note that

and thus as k → ∞. We have

and so letting k → ∞, we have . Therefore, we have

and letting k → ∞ in the above equation, F being continuous and ψ right continuous, we get

a contradiction. Therefore, {x_n} is a Cauchy sequence in . Since is -orbitally complete at x₀, there exists with lim_n→∞ x_n = z.

If or is orbitally continuous, then clearly

Theorem 3 Let and satisfy all the conditions of Theorem 2, except that condition (iii) is substituted by

(iii') is regular.

Then the same conclusions as in Theorem 2 hold.

Proof Following the proof of Theorem 2, we have that {x_n} is a Cauchy sequence in which is -orbitally complete at x₀. Then, there exists such that

Now suppose that . From regularity of , we have for all n ∈ ℕ. Hence, we can apply the considered contractive condition. Then, setting and y = z in (3.1), we obtain:

where

Letting n → ∞ in the above inequality and using the continuity of F and right continuity of ψ, we have

a contradiction. Therefore, and thus . Hence, z is a common fixed point of and .

Corollary 1 Let be an ordered metric space. Let be a mapping satisfying

for all (for some x₀) such that x and y are comparable, where , ψ ∈ Ψ and

We assume the following hypotheses:

(i) is a.r. at some point x₀;

(ii) is -orbitally complete at x₀;

(iii) is orbitally continuous at x₀ or is regular.

Also suppose that for all such that and there exists an such that . Then has a fixed point. Moreover, the set of fixed points of in is well ordered if and only if it is a singleton.

We also state a corollary of Theorem 2 involving a contraction of integral type.

Corollary 2 Let and satisfy the conditions of Theorem 2, except that condition (3.1) is replaced by the following: there exists a positive Lebesgue integrable function u on ℝ₊ such that for each ε > 0 and that

Then, and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.

We present an example showing how our results can be used.

Example 3 Let , where and B = (1, +∞), be equipped with Euclidean metric d and the order ≼ given by

Consider the mappings given by

It is easy to check that and satisfy conditions (i)-(v) of Theorem 2 with . Take defined by

and ψ ∈ Ψ, given as . In order to check the contractive condition (3.1), take with, say x ≺ y, i.e., x > y (the case x = y is trivial). Then and for some m, n ∈ ℕ, m > n. We get that and

Hence, (3.1) is fulfilled. Applying Theorem 2, we conclude that and have a (unique) common fixed point (z = 0).

Note that and do not satisfy the contractive condition for arbitrary .

In this section, we improve and generalize the results of Nashine and Altun (HK Nashine and I Altun, unpublished work) by taking into account the following for three maps :

1. is a pair of asymptotically regular mappings with respect to ;

2. orbital continuity of one of the involved maps;

3. is a pair of weakly increasing mappings with respect to ;

4. is a pair of dominating maps;

5. is a pair of compatible maps, and

6. the basic space is an ordered orbitally complete metric space.

We will denote by Φ the set of functions φ : [0 + ∞) → [0, +∞), such that φ is right continuous, φ(0) = 0 and φ(t) < t for every t > 0.

The first result of this section is the following.

Theorem 4 Let be a regular ordered metric space and let and be self-maps on satisfying

for all (for some x₀) such that and are comparable, where , φ ∈ Φ and

We assume the following hypotheses:

(i) is a.r. with respect to at ;

(ii) is ()-orbitally complete at x₀;

(iii) and are weakly increasing with respect to ;

(iv) and are dominating maps;

(v) is monotone and orbitally continuous at x₀.

Assume either

(a) and are compatible; or

(b) and are compatible.

Then and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.

Proof Since is a.r. with respect to at x₀ in , there exists a sequence {x_n} in such that

and

holds. We claim that

To this aim, we will use the increasing property with respect to satisfied by the mappings and . From (4.3), we have

Since , then , and we get

Again,

Since , we get

Hence, by induction, (4.5) holds. Therefore, we can apply (4.1) for x = x_p and y = x_q for all p and q.

Now, we assert that is a Cauchy sequence in the metric space . We proceed by negation and suppose that is not Cauchy. Then, there exists ε > 0 for which we can find two sequences of positive integers {m(k)} and {n(k)} such that for all positive integers k,

From (4.6) and using the triangular inequality, we get

Letting k → ∞ in the above inequality and using (4.4), we obtain

Again, the triangular inequality gives us

Letting k → ∞ in the above inequality and using (4.4) and (4.7), we get:

On the other hand, we have

Letting k → ∞ in the above inequality and using (4.4), (4.7) and properties of , we have

Applying (4.1), we get:

One can check easily that for k large enough, we have:

where d_k ≥ 0 and d_k → 0 as k → ∞. From (4.10), for k large enough, we have

Letting k → ∞ in (4.11) and using properties of F and φ, we have

Combining (4.9) and (4.12), we get F(ε) < F(ε), a contradiction.

Hence, we deduce that is a Cauchy sequence in . Since is -orbitally complete at x₀, there exists some such that

We will prove that z is a common fixed point of the three mappings and .

We have

and

Suppose that (a) holds, i.e., and are compatible. Then, using condition (v),

From (4.13) and the orbitally continuity of , we have also

Now, using (iv), and since is monotone, and are comparable. Thus, we can apply (4.1) to obtain

where

Letting n → ∞ in (4.18), using (4.13)-(4.17), we obtain

unless

Now, and as n → ∞, so by the assumption we have x_2n+1 ≼ z and and are comparable. Hence (4.1) gives

Passing to the limit as n → ∞ in the above inequality and using (4.19), it follows that

which holds unless

Similarly, and as n → ∞, implies that , hence and are comparable. From (4.1) we get

Passing to the limit as n → ∞, we have

which gives that

Therefore, , hence z is a common fixed point of and .

Similarly, the result follows when condition (b) holds.

Now, suppose that the set of common fixed points of and in is well ordered. We claim that there is a unique common fixed point of and in . Assume to the contrary that and but u ≠ v. By supposition, we can replace x by u and y by v in (4.1) to obtain

a contradiction. Hence, u = v. The converse is trivial.

We obtain the following corollaries from Theorem 4.

Corollary 3 Let be a regular ordered metric space and let and be self-maps on satisfying

for all (for some x₀) such that x and y are comparable, where , φ ∈ Φ and

We assume the following hypotheses:

(i) is a.r. at some point ;

(ii) is -orbitally complete at x₀;

(iii) and are weakly increasing;

(iv) and are dominating maps.

Then and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.

Corollary 4 Let be a regular ordered metric space and let and be self-maps on satisfying

for all (for some x₀) such that and are comparable, where , φ ∈ Φ and

We assume the following hypotheses:

(i) is a.r. with respect to at ;

(ii) is -orbitally complete at x₀;

(iii) is weakly increasing with respect to ;

(iv) is a dominating map;

(v) is monotone and orbitally continuous at x₀.

Then and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.

Corollary 5 Let be a regular ordered metric space and let be a self-map on satisfying for all such that x and y are comparable,

where , φ ∈ Φ and

We assume the following hypotheses:

(i) is a.r. at some point x₀ of ;

(ii) is -orbitally complete at x₀;

(iii) for all ;

(iv) is a dominating map.

Then has a fixed point. Moreover, the set of fixed points of in is well ordered if and only if it is a singleton.

We also state a corollary of Theorem 4 involving a contraction of integral type.

Corollary 6 Let and satisfy the conditions of Theorem 4, except that condition (4.1) is replaced by the following: there exists a positive Lebesgue integrable function u on ℝ₊ such that for each ε > 0 and that

Then, and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.

Example 4 Let the set be equipped with the usual metric d and the order defined by

Consider the following self-mappings on :

Take . Then it is easy to show that

and , and all the conditions (i)-(v) and (a)-(b) of Theorem 4 are fulfilled (condition (iii) on . Take and of the form F(t) = kt, k > 0. Then contractive condition (4.1) takes the form

for . Using substitution y = tx, t ≥ 0, the last inequality reduces to

and can be checked by discussion on possible values for t ≥ 0. Hence, all the conditions of Theorem 4 are satisfied and have a unique common fixed point in (which is 0).

Remark 2 It was shown by examples in [24] that (in similar situations):

(1) if the contractive condition is satisfied just on , there might not exist a (common) fixed point;

(2) under the given hypotheses (common) fixed point might not be unique in the whole space .

The authors declare that they have no competing interests.

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

The authors are highly indebted to the referees for their careful reading of the manuscript and valuable suggestions. H-S Ding acknowledges the support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province (20114BAB211002), the Jiangxi Provincial Education Department (GJJ12173), and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University. Z. Kadelburg is thankful to the Ministry of Science and Technological Development of Serbia.

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