Common Fixed Points of Weak Contractions in Cone Metric Spaces

and Applied Analysis 3 Let two mappings f, g : X → X and two arbitrary points x, y ∈ X be given. The following four sets of vectors will be used: M4 f,g ( x, y ) { d ( gx, gy ) , d ( gx, fx ) , d ( gy, fy ) , 1 2 ( d ( gx, fy ) d ( gy, fx )) } , M3 f,g ( x, y ) { d ( gx, gy ) , 1 2 ( d ( gx, fx ) d ( gy, fy )) , 1 2 ( d ( gx, fy ) d ( gy, fx )) } , N4 f,g ( x, y ) { d ( x, y ) , d ( x, fx ) , d ( y, gy ) , 1 2 ( d ( x, gy ) d ( y, fx )) } , N3 f,g ( x, y ) { d ( x, y ) , 1 2 ( d ( x, fx ) d ( y, gy )) , 1 2 ( d ( x, gy ) d ( y, fx )) } . 2.1 If i : X → X is the identity mapping, we will writeM f x, y : M f,i x, y for k ∈ {3, 4}. Let f, g : X → X be two self-maps on a nonempty set X. Recall that a point x ∈ X is called a coincidence point of the pair f, g and y is its point of coincidence if fx gx y. The pair f, g is said to be weakly compatible if for each x ∈ X, fx gx implies fgx gfx. A classical result of Jungck states that if, two weakly compatible maps have a unique point of coincidence y, then y is their unique common fixed point. Roughly speaking, there are two types of common fixed point results with weak contractive conditions. Those of the first type use conditions with d fx, fy on the left-hand side and some element of the M-set on the right-hand one. The other use conditions with d fx, gy on the left-hand side and some element of the N-set on the right-hand side. An example of the first type is the following results in cone metric spaces that were proved by Choudhury and Metiya. Theorem 2.2 see 24 , Theorems 3.1, 3.2, and 3.3 . Let X, d be a cone metric space over a regular cone P such that d4 holds. Let f, g : X → X be such that one of the following inequalities holds for all x, y ∈ X: ψ ( d ( fx, fy )) ψdgx, gy − φdgx, gy, 2.2 ψ ( d ( fx, fy )) ψ ( 1 2 [ d ( fx, gx ) d ( fy, gy )] ) − φdgx, gy, 2.3 ψ ( d ( fx, fy )) ψ ( 1 2 [ d ( fx, gy ) d ( fy, gx )] ) − φdgx, gy, 2.4 where φ ∈ Φ and ψ ∈ Ψ are continuous. If fX ⊂ gX and gX is complete, then f and g have a unique point of coincidence in X (and so they have a unique common fixed point in X if the pair f, g is weakly compatible). On the other hand, in the case of metric spaces, the following result was proved by D − orić. 4 Abstract and Applied Analysis Theorem 2.3 see 16 , Theorem 2.1 . Let X, d be a complete metric space, ψ ∈ Ψ be continuous, and φ ∈ Φ be lower semicontinuous (here P 0, ∞ ). Let f, g : X → X be two self-maps satisfying the inequality ψ ( d ( fx, gy )) ≤ ψmx, y − φmx, y, 2.5 for all x, y ∈ X, wherem x, y maxN4 f,g x, y . Then f and g have a unique common fixed point in X. In this paper we generalize and unify results of Theorem 2.2 we will call the conditions that we use “weak contractive conditions of the first type” . Examples show that these generalizations are proper. Further, we extend Theorem 2.3 and some related results to the case of cone metric spaces the respective conditions will be called “weak contractive conditions of the second type” and give examples of applications of the obtained results. 3. Auxiliary Results We will make use of the following result of Choudhury and Metiya. Lemma 3.1 see 24 . Let X, d be a cone metric space over a regular cone P such that d4 holds and suppose that there exists φ ∈ Φ (see Definition 2.1). If {yn} is a sequence in X such that {d yn, yn 1 } is decreasing, then {d yn, yn 1 } converges either to θ or to r ∈ intP . Note that φ ∈ Φ is not supposed to be continuous. It is easy to show that without the existence of function φ the conclusion of Lemma 3.1 may fail to hold. The following result is a cone metric version of 21, lemma 2.1 . Lemma 3.2. Let X, d be a cone metric space over a regular cone P such that d4 holds and suppose that there exists φ ∈ Φ (see Definition 2.1). Let {yn} be a sequence in X such that {d yn, yn 1 } is decreasing w.r.t. and that lim n→∞ d ( yn, yn 1 ) θ. 3.1 If {y2n} is not a Cauchy sequence, then there exists c ∈ intP and two sequences {mk} and {nk} of positive integers such that the following five sequences tend to φ c when k → ∞: d ( y2mk , y2nk ) , d ( y2mk 1, y2nk 1 ) , d ( y2mk , y2nk 1 ) , d ( y2mk−1, y2nk ) , d ( y2mk−1, y2nk 1 ) . 3.2 Proof. Suppose that {y2n} is not a Cauchy sequence. Then there exists c ∈ intP such that for each n0 ∈ N there exist n,m ∈ N with n > m ≥ n0 and φ c −d y2m, y2n / ∈ intP . Hence, by property 2.4 of function φ, φ c d y2m, y2n holds for n > m ≥ n0. Therefore, there exist sequences {mk} and {nk} of positive integers such that nk > mk > k, d ( y2mk , y2nk ) φ c , dy2mk , y2nk−2 ) φ c 3.3 Abstract and Applied Analysis 5and Applied Analysis 5 the last inequality is obtained by taking the smallest possible nk . Now we have φ c dy2mk , y2nk ) dy2mk , y2nk−2 ) d ( y2nk−2, y2nk−1 ) d ( y2nk−1, y2nk ) φ c dy2nk−2, y2nk−1 ) d ( y2nk−1, y2nk ) . 3.4 Letting k → ∞ and using assumption 3.1 and the normality of the cone, we obtain that lim k→∞ d ( y2mk , y2nk ) φ c . 3.5


Introduction
The idea to use an ordered Banach space instead of the set of real numbers, as the codomain for a metric, goes back to the mid-20th century see, e.g., Kurepa 1 , Kreȋn and Rutman 2 , Kantorovič 3 .Fixed point theory in K-metric and K-normed spaces was developed by Perov 4 , Vandergraft 5 , and others.For more details we refer the reader to survey papers of Zabrejko 6 and Proinov 7 .In 2007, Huang and Zhang 8 reintroduced such spaces under the name of cone metric spaces and gave definitions of convergent and Cauchy sequences in the terms of interior points of the underlying cone, proving some fixed point theorems in such spaces.After that, fixed-points in cone metric spaces have been a subject of intensive research see 9 for a survey of these results, and also 10-12 .Fixed point results under so-called weak contractive conditions were first obtained in 13, 14 .They were generalized by various authors see, e.g., 15 , in particular, using a pair of control functions ϕ and ψ 16-21 .Note, however, that it was shown in 22 that in a certain Let two mappings f, g : X → X and two arbitrary points x, y ∈ X be given.The following four sets of vectors will be used:

2.1
If i : X → X is the identity mapping, we will write M k f x, y : M k f,i x, y for k ∈ {3, 4}.Let f, g : X → X be two self-maps on a nonempty set X. Recall that a point x ∈ X is called a coincidence point of the pair f, g and y is its point of coincidence if fx gx y.The pair f, g is said to be weakly compatible if for each x ∈ X, fx gx implies fgx gfx.A classical result of Jungck states that if, two weakly compatible maps have a unique point of coincidence y, then y is their unique common fixed point.
Roughly speaking, there are two types of common fixed point results with weak contractive conditions.Those of the first type use conditions with d fx, fy on the left-hand side and some element of the M-set on the right-hand one.The other use conditions with d fx, gy on the left-hand side and some element of the N-set on the right-hand side.An example of the first type is the following results in cone metric spaces that were proved by Choudhury and Metiya.Theorem 2.2 see 24 , Theorems 3.1, 3.2, and 3.3 .Let X, d be a cone metric space over a regular cone P such that d 4 holds.Let f, g : X → X be such that one of the following inequalities holds for all x, y ∈ X: On the other hand, in the case of metric spaces, the following result was proved by D − orić.Theorem 2.3 see 16 , Theorem 2.1 .Let X, d be a complete metric space, ψ ∈ Ψ be continuous, and ϕ ∈ Φ be lower semicontinuous (here P 0, ∞ ).Let f, g : X → X be two self-maps satisfying the inequality for all x, y ∈ X, where m x, y max N 4 f,g x, y .Then f and g have a unique common fixed point in X.
In this paper we generalize and unify results of Theorem 2.2 we will call the conditions that we use "weak contractive conditions of the first type" .Examples show that these generalizations are proper.Further, we extend Theorem 2.3 and some related results to the case of cone metric spaces the respective conditions will be called "weak contractive conditions of the second type" and give examples of applications of the obtained results.

Auxiliary Results
We will make use of the following result of Choudhury and Metiya.Lemma 3.1 see 24 .Let X, d be a cone metric space over a regular cone P such that d 4 holds and suppose that there exists ϕ ∈ Φ (see Definition 2.1).If {y n } is a sequence in X such that {d y n , y n 1 } is decreasing, then {d y n , y n 1 } converges either to θ or to r ∈ intP .
Note that ϕ ∈ Φ is not supposed to be continuous.It is easy to show that without the existence of function ϕ the conclusion of Lemma 3.1 may fail to hold.
The following result is a cone metric version of The other three limits can be obtained similarly.

Weak Contractions of the First Type in Cone Metric Spaces
Theorem 4.1.Let X, d be a cone metric space over a regular cone P such that d 4 holds and suppose that there exists a continuous function ϕ ∈ Φ.Let f, g : X → X be two selfmaps such that fX ⊂ gX and let one of these subsets of X be complete.Suppose that for all x, y ∈ X there exists for some continuous ψ ∈ Ψ see Definition 2.1 .The proof is essentially the same, and so, for the sake of simplicity, we stay within the given version.The same remark applies to all other results in the rest of the paper.See also paper 22 where it is shown that practically each weak contractive condition with function ψ can be replaced by an equivalent condition without ψ.
Proof.Starting from arbitrary x 1 ∈ X and using the assumption fX ⊂ gX, construct a Jungck sequence {y n } satisfying y n fx n gx n 1 for n ∈ N. If y n 0 y n 0 1 for some n 0 ∈ N, then gx n 0 1 y n 0 y n 0 1 fx n 0 1 and f and g have a point of coincidence.Suppose, further, that y n / y n 1 for n ∈ N. Putting x x n 1 , y x n in 4.2 we obtain that where

4.5
The case u d y n , y n 1 is impossible, since it would imply u u − ϕ u and u θ using properties of the function ϕ , which is already excluded.In all other cases we get that d y n 1 , y n d y n , y n−1 , and, more precisely, We have proved that the sequence {d y n , y n 1 } is decreasing w.r.t. and so Lemma 3.1 implies that it converges to some r, where either r θ or r ∈ intP .But, if r ∈ intP , then 4.6 implies that also u x n 1 , x n → r as n → ∞.Hence, passing to the limit in 4.4 we get that r r − ϕ r and r θ, a contradiction.Thus, r lim n → ∞ d y n , y n 1 θ.Let us prove that {y n } is a Cauchy sequence in X. Suppose that it is not.It follows from monotonicity of the sequence {d y n , y n 1 } and lim n → ∞ d y n , y n 1 θ that neither {y 2n } is a Cauchy sequence.Lemma 3.2 implies that there exist sequences {m k } and {n k } of positive integers such that the sequences 3.2 all tend to ϕ c for some c 0. Using 4.6 and putting θ and gp fp q; hence q is a point of coincidence for the pair f, g .
To prove that this point of coincidence is unique, assume that there is another q 1 ∈ X such that q 1 fp 1 gp 1 for some p 1 ∈ X.Then where In both cases we get that d q, q 1 θ, that is, the point of coincidence is unique.
Obviously, the theorem in 24, Theorem 3.1 Theorem 2.2 with condition 2.2 is a special case of Theorem 4.1.
Remark 4.3.The previous theorem can be modified so that continuity of ϕ is substituted by its lower semicontinuity; however, in this case it has to be assumed that the cone P is strongly minihedral.For details see 10 .The same applies to other assertions to the end of the paper.
The following example shows that there are cases when the existence of a common fixed point can be deduced using Theorem 4.1, but cannot be obtained using the theorem in 24, Theorems 3.1, 3.2, and 3.3 Theorem 2.2 with either of the conditions 2.2 , 2.3 , or 2.4 .d x, y |x−y|, α|x−y| , where α > 0 is fixed.d is obviously a cone metric satisfying property d 4 and the cone P is regular even minihedral .Function ϕ : intP ∪{θ} → intP ∪{θ} defined by ϕ θ θ and ϕ t 1 , t 2 1/4 t 1 , 1/4 t 2 for t 1 > 0, t 2 > 0 belongs to the respective class Φ.Consider the mappings f, g : X → X defined by: gx x for x ∈ X, fx 3/4 for x ∈ 0, 1 and fx 1/4 for x ∈ 3/2, 2 .We will show that, taking, for example, ψ t t, neither of conditions 2.2 , 2.3 , 2.4 is satisfied; hence neither of Theorems 3.1, 3.2, and 3.3 from 24 can be used to conclude that there exists a common fixed point of f and g which is obviously p 3/4 .In the next theorem the set M 3 f,g x, y is used instead of M 4 f,g x, y .The proof is essentially the same as for Theorem 4.1 and so is omitted.Theorem 4.5.Let X, d be a cone metric space over a regular cone P such that d 4 holds and suppose that there exists a continuous function ϕ ∈ Φ.Let f, g : X → X be two selfmaps such that fX ⊂ gX and let one of these subsets of X be complete.Suppose that for all x, y ∈ X there exists u u x, y ∈ M 3 f,g x, y , 4.12 such that d fx, fy u x, y − ϕ u x, y 4.13 holds true.Then f and g have a unique point of coincidence.If, moreover, the pair f, g is weakly compatible, then f and g have a unique common fixed point.
In the following theorem we unify Theorems 3.1, 3.2, and 3.3 of 24 Theorem 2.2 with conditions 2.2 , 2.3 , or 2.4 .Theorem 4.6.Let X, d be a cone metric space over a regular cone P such that d 4 holds and suppose that there exists a continuous function ϕ ∈ Φ.Let f, g : X → X be two selfmaps such that fX ⊂ gX and let one of these subsets of X be complete.Suppose that for all x, y ∈ X there exists

4.17
In each of the three possible cases it is easy to obtain that {d y n 1 , y n } is a decreasing sequence and that d y n 1 , y n u x n 1 , x n d y n , y n−1 .

4.18
Hence, all these three terms tend to some r ∈ intP ∪ {θ}.Passing to the limit in relation 4. 16 we get that r r − ϕ r , wherefrom it follows that r θ.That {y n } is a Cauchy sequence can be proved using Lemma 3.2 similarly as in the proof of Theorem 4.1.Hence, there exists p ∈ X such that y n fx n gx n 1 → gp when n → ∞.Let us prove that gp is a point of coincidence of the pair f, g . Putting We show that, however, condition 4.15 is satisfied and so Theorem 4.6 can be used to conclude that there exists a common fixed point of f and g which is obviously p 0 note that this can also be done using condition 2.2 .Indeed, take u u x, y d gx, gy d x, y ∈ M 3 f,g x, y .In order to prove inequality 4.15 it is enough to consider the first coordinates of respective vectors, that is, we have to prove that holds for all x, y ∈ 0, 1 .But, it is an easy consequence of |x − y| ≤ x y.
Finally, we state proof can be deduced similarly as for the previous theorems the following cone metric version of 18, Theorems 3.1 and 4.1 see also 21, Theorem 3.6 .Theorem 4.8.Let X, d be a cone metric space over a regular cone P such that d 4 holds and suppose that there exists a continuous function ϕ ∈ Φ.Let f : X → X be a selfmap such that for all x, y ∈ X there exist Note that in this case fixed point of f need not be unique.It is enough to consider the identity mapping f i X and take v θ.

Weak Contractions of the Second Type in Cone Metric Spaces
In this section we consider weak contractions which we have called "of the second type" see the end of Section 2 .
Theorem 5.1.Let X, d be a complete cone metric space over a regular cone P such that d 4 holds and suppose that there exists a continuous function ϕ ∈ Φ.Let f, g : X → X be two mappings such that for all x, y ∈ X there exists u x, y ∈ N 4 f,g x, y 5.1 such that d fx, gy u x, y − ϕ u x, y .

5.2
Then f and g have a unique common fixed point.
Proof.Let us prove first that the common fixed point of f and g is unique if it exists .Suppose that p / q are two distinct common fixed points of f and g.Then 5.2 implies that d p, q d fp, gq u p, q − ϕ u p, q , 5.3 where u p, q ∈ N 4 f,g p, q {d p, q , θ, θ, d p, q } {θ, d p, q }.Checking both possible cases and using the properties of function ϕ, we readily obtain that d p, q θ, that is, p q.In order to prove the existence of a common fixed point, proceed this time constructing a Jungck sequence by x 2n 1 fx 2n , x 2n 2 gx 2n 1 , for arbitrary x 0 ∈ X.Consider the two possible cases.
Suppose that x n x n 1 for some n ∈ N. Then x n 1 x n 2 and it follows that the sequence is eventually constant, and so convergent.Indeed, let, for example, n 2k in the case n 2k 1 the proof is similar .Then, putting x x 2k , y x 2k 1 in 5.2 , we get that there exists Consider the three possible cases: and by the properties of function ϕ that x 2k x 2k 1 .
Suppose now that x n / x n 1 for all n ∈ N. Putting x x 2n , y x 2n−1 in 5.2 , we get that there exists

5.7
Now, from 5.11 and the obtained limits, we have that lim } is a decreasing sequence which by the regularity of cone P tends to some r ∈ intP ∪ {θ}.In order to prove that r θ, put x x n 1 and y x n in 5.23 to obtain where

5.31
On the other hand, 5.24 implies that

5.32
In the case when D E, passing to the limit when n → ∞, we obtain that lim n → ∞ Θ 5 f x n 1 , x n r; the same conclusion is obtained if D < E or D > E .Hence, passing to the limit in 5.29 , we get that r r − ϕ r , wherefrom r θ.
As in some previous proofs, in order to obtain that {x n } is a Cauchy sequence, suppose that it is not the case and using Lemma 3. implying that ϕ c θ because A > 0 .Thus, the sequence {x n } converges to some z in the complete metric space X.In order to prove that fz z, suppose the contrary and put x x n and y z in 5. 24  The proof that the fixed point of f is unique is standard.
In a similar way one can obtain a version of the previous theorem containing two selfmaps f and g see 21, Theorem 5.2 .
At the end, we again state a cone metric version of a result from 18, Theorems 3.2 and 4.2 see also 21, Theorem 3.7 .
Theorem 5.5.Let X, d be a cone metric space over a regular cone P such that d 4 holds and suppose that there exists a continuous function ϕ ∈ Φ.Let f, g : X → X be two selfmaps such that for all x, y ∈ X there exist u x, y ∈ N 4 f x, y , v x, y ∈ d x, y , d x, fx , d y, gy , 5.39 such that d fx, gy u x, y − ϕ v x, y .Then f and g have a common fixed point.
Here also common fixed point of f and g need not be unique.

4 . 6 Indeed
, the right-hand inequality is trivial in the case when u d y n , y n−1 , and in the case u 1/2 d y n−1 , y n 1 , then d y n 1 , y n d y n , y n−1 and u 1/2 d y n−1 , y n 1/2 d y n , y n 1 1/2 d y n−1 , y n 1/2 d y n−1 , y n d y n−1 , y n .
y , d y, fy , 4.22 such that d fx, fy u x, y − ϕ v x, y .Then f has a fixed point.
As usual, form a Jungck sequence by y n fx n gx n 1 .If y n 0 y n 0 1 , then it can be proved as in Theorem 4.1 that f, g has a point of coincidence.Assume that y n / y n 1 for all n ∈ N. Then holds true.Then f and g have a unique point of coincidence.If, moreover, the pair f, g is weakly compatible, then f and g have a unique common fixed point.Proof.
Passing to the limit more precisely, considering one of the inequalities that holds for infinitely many n as in the proof of Theorem 4.1 we get either d gp, fp θ or d gp, fp 1/2 d gp, fp and both are possible only if fp gp.Hence gp is a point of coincidence of f, g .The proof that the point of coincidence is unique is essentially the same as in Theorem 4.1.The next example illustrates how Theorem 4.6 can be used to prove the existence of a common fixed point, while either Theorem 3.2 or 3.3 of 24 cannot.
This shows that {x 2n } is a Cauchy sequence and hence {x n } is a Cauchy sequence.Since the space X, d is complete, there exists p ∈ X such that lim n → ∞ x n p. Then also x 2n 1 fx 2n → p and x 2n gx 2n−1 → p n → ∞ .Putting x x 2n and y p in 5.2 , we get d fx 2n , gp u − ϕ u , where , where u ∈ {θ, d p, fp , 1/2 d p, fp } and in each of the possible three cases it easily follows that fp p.Hence, p is a common fixed point of f and g.Let X, d be a complete cone metric space over a regular cone P such that d 4 holds and suppose that there exists a continuous function ϕ ∈ Φ.Let f : X → X be such that for all x, y ∈ X there exists Adapting an example from 16 we give an example when Theorem 5.1 modified to use a function ψ ∈ Ψ according to Remark 4.2 can be used to deduce the existence of a common fixed point.Example 5.3.Let X 0, 1 , and let E R 2 , P { x, y ∈ E : x ≥ 0, y ≥ 0} and d x, y |x − y|, α|x−y| be as in Examples 4.4 and 4.7.Take ϕ ∈ Φ defined by ϕ θ θ and ϕ t 1 , t 2 t 1 , t 2 for t 1 , t 2 > 0; take ψ ∈ Ψ defined by ψ t 1 , t 2 3t 1 , 3t 2 for t 1 , t 2 ≥ 0 they satisfy the conditions of Definition 2.1 .Consider the mappings f, g : X → X given as fx 1/3 x and gx 0.The next is a kind of Hardy-Rogers-type result with weak condition.It can be considered as a cone metric version of results from 19, 21 .For the sake of simplicity we take only one mapping f : X → X and for x, y ∈ X denote Starting with arbitrary x 0 ∈ X construct the Picard sequence by x n 1 fx n .
reduced to the first coordinates of respective vectors, has the form which was checked to be true in 16 .Hence, the existence of a common fixed point p 0 of mappings f and g follows from Theorem 5.1.whereA>0, B, C, D, E ≥ 0, A B C D E ≤ 1.Theorem 5.4.Let X, d be a complete cone metric space over a regular cone P such that d 4 holds and suppose that there exists a continuous function ϕ ∈ Φ.Let f : X → X and suppose that 2A B C D E / 2−B−C−D−E ≤ 1.It follows that {d x n 1 , x n . It follows that