# Some integral type common fixed point theorems satisfying [phi]-contractive conditions.

1 Introduction and preliminaries

It is claimed that one of the dispensable tools of nonlinear analysis is the fixed point theory. The importance of fixed point theory arise from the application potential not only in the distinct branches of mathematics, but also various disciplines in quantitive sciences. The theory of fixed point is essential for the existence (and usually uniqueness) for solution of nonlinear differential equations, integrodifferential equations and integral equations in various abstract spaces. The renowned Banach fixed point theorem [6], the Banach contraction mapping principle, is the most impressed and earlier results in this direction, see e.g. [8, 10, 12, 13, 24, 31, 32, 33, 35, 38]. Banach [6] proved that every contraction has a unique fixed point in the context of a complete metric space. This remarkable results has been generalized in various ways in distinct abstract spaces. Following this paper, many authors have investigated the answer of the following question: Is it possible to guarantee the existence (and uniqueness) of a fixed point by replacing the weaker conditions on the set-up abstract space or on mappings. One of the remarkable answer of this question was given by Hicks and Rhoades [19]. In the mentioned paper, the authors obtained some nice results on common fixed point theorems in a semi-metric (symmetric) spaces. The notion of symmetric was obtained from metric by excluding the assumption of the subadditivity, that is, triangle inequality. Recently, Sintunavarat and Kumam [46] defined more refine notion, common limit range property. This property raze the requirement of completeness of the spaces closedness of the underlying subspaces. Very recently, Karapinar et al. [34] utilized the notion of common limit range property and established some common fixed point results for Lipschitz type mappings in context of symmetric spaces.

On the other hand, in 2002, Branciari [9] firstly established an integral type fixed point theorem for a self mapping which generalized Banach's contraction principle. Following this pioneer paper, a number of fixed point results for different integral type contraction condition have been reported by various authors. For more details, we refer the reader to e.g. [5, 4, 2, 7, 14, 37,41, 42, 48,49, 51].

In this manuscript, we obtain some common fixed point results of two pairs having the common limit range property in the setting of integral type [phi]-contraction in the framework of symmetric (semi-metric) spaces. As an extension of presented result, we state some fixed point theorems for five mappings, six mappings and for four finite families of mappings in metric spaces by using the notion of the pairwise commuting mappings which is studied by Imdad et al. [23]. We conclude with examples that supports the usefulness of our results.

We recollect the essential definitions and basic results that will be needed later on.

For a non-empty set X, a real valued function d : X x X [right arrow] [0, [infinity]) is called symmetric if

(sm1) d(x, y) = 0 [??] x = y,

(sm2) d(x, y) = d(y, x),

hold for all x, y [member of] X. For any x [member of] X, we define an open ball with respect to the corresponding topology, [T.sub.d] on X, via B(x, [epsilon]) = {y [member of] X : d(x, y) < [epsilon]} where x [member of] X and [epsilon] > 0. If for each [epsilon] > 0 and x [member of] X, the set B(x, [epsilon]) is a neighborhood of x due to the topology [T.sub.d], then a symmetric space d is a semi-metric. We say that the sequence {[x.sub.n]} converges to a point x [member of] X, denoted as [x.sub.n] [right arrow] x, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with respect to the topology [T.sub.d]. For more details about the topological properties of symmetric space (X, d) see e.g. [11].

In the proof of the existence and uniqueness of a fixed point, whether the space is Hausdorff has a critical role. On the other hand, the symmetric d may not be continuous since symmetric space is not necessarily Hausdorff. To compensate the missing of the continuity of a symmetric d and being Hausdorff of the related space, some additional axioms were suggested, see e.g. [4], [16], [19], [52].

Definition 1. Suppose that (X, d) is a symmetric space where X is a non-empty set. We suppose also that the sequences {[x.sub.n]},{[y.sub.n]}, and the points x, y in X. Then,

([W.sub.3]) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [52].

([W.sub.4]) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [52].

(HE) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [4].

(1C) We say that a symmetric d is 1-continuous if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [16].

(CC) We say that a symmetric d is continuous if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [16].

Here, it is observed that (CC) [??] (1C), ([W.sub.4]) [??] ([W.sub.3]), and (1C) [??] ([W.sub.3]). We notice that the converse of the implications above are not true (see e.g. [15]). As it is expected, (CC) implies all four conditions, ([W.sub.3]), ([W.sub.4]), (HE) and (1C). Employing these axioms, several fixed point results have appeared in framework of symmetric spaces (see [3,11,17,18, 20, 21, 22, 26, 27]).

Definition 2. Suppose that (X, d) is a symmetric (semi-metric) space. Let A and S be two self mappings on X. We say that the pair (A, S) is

1. commuting if ASx = SAx, for all x [member of] X,

2. weakly commuting if d(ASx, SAx) [less than or equal to] d(Ax, Sx), for all x [member of] X, [44],

3. compatible if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII][29], under the assumption that {[x.sub.n]} is a sequence in X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for some t [member of] X,

4. non-compatible if there exists a sequence {[x.sub.n]} in X for some t [member of] X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are either non-zero or non-existent,[40],

5. weakly compatible if self-mappings A and S commute at their coincidence points, (ASu = SAu whenever Au = Su, for some u [member of] X), [30].

For more details on systematic comparisons and illustrations of above described notions, we refer to Singh and Tomar [45] and Murthy [39].

Definition 3. [2] Suppose that (X, d) is a symmetric (semi-metric) space. We said that a pair self-mappings (A, S) on X, satisfy the property (E.A) if there exists a sequence {[t.sub.n]} and t in X such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

Definition 4. [36] Suppose that (X, d) is a symmetric (semi-metric) space and A, S, B, T be self mappings on X. Pairs (A, S) and (B, T) of self mappings are said to satisfy the common property (E.A), if there exist two sequences {[t.sub.n]} and {[s.sub.n]} in X, and some t [member of] X such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Definition 5. [46] Suppose that (X, d) is a symmetric (semi-metric) space and A, S are two self mappings on X. We say that a pair (A, S) is said to satisfy the common limit range of S property, ([CLR.sub.s]) property, if there exists a sequence {[t.sub.n]} in X such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where t [member of] S(X).

Hence it is assured that a pair (A, S) satisfying the property (E.A) along with closedness of the subspace S(X) always enjoys the ([CLR.sub.s]) property (see [21, Examples 2.16-2.17]).

Definition 6. [34] Suppose that (X, d) is a symmetric (semi-metric) space and A, S, B, T be self mappings on X. Pairs (A, S) and (B, T) of self mappings are said to satisfy the common limit in the range of S and T property, ([CLR.sub.ST]) property for short, if there exist two sequences {[t.sub.n]} and {[s.sub.n]} in X such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where t [member of] S(X) [intersection] T(X).

Therefore, common limit range property implies the common property (E.A). On the other hand, the converse of this implication is not true in general (see e.g. [34, Example 5]).

Definition 7. [23] Suppose that (X, d) is a symmetric (semi-metric) space. Let [{[A.sub.i]}.sup.m.sub.i=1] and [{[S.sub.k]}.sup.n.sub.k=1] be Two families of self mappings on X. Two families [{[A.sub.i]}.sup.m.sub.i=1] and [{[S.sub.k]}.sup.n.sub.k=1] of self-mappings are called pairwise commuting if

1. [A.sub.j][A.sub.k] = [A.sub.k][A.sub.j] for all k, j [member of] {1,2,..., m},

2. [S.sub.j][S.sub.k] = [S.sub.k][S.sub.j] for all k, j[member of]{1,2,..., n},

3. [A.sub.j][S.sub.k] = [S.sub.k][A.sub.j] for all j [member of] {1,2,..., m} and k [member of] {1,2,..., n}.

2 Main Results

We start to this section by recalling the following auxiliary functions. Let [PHI] be the set of all functions [phi] such that [phi] : [R.sup.+] [right arrow] [R.sup.+] with the conditions 0 < [phi](t) < t for each t > 0 and [phi](0) = 0. Let A be set of all functions [phi] such that [phi] : [R.sup.+] [right arrow] [R.sup.+] is a summable and non-negative Lebesgue-integrable mapping such that for all [member of] > 0

[[integral].sup.[member of].sub.0] [phi](t)dt > 0. (2.1)

We, first, prove the following auxiliary result.

Lemma 1. Let X be a non-empty set and (X, d) be a symmetric (semi-metric) space (X, d) satisfying the condition (CC). Suppose that self-mappings A, B, S and T satisfy the conditions below:

1. either the pair (A, S) satisfies the ([CLR.sub.S]) property or the pair (B, T) satisfies the ([CLR.sub.T]) property,

2. A(X) [subset] T(X) (or B(X) [subset] S(X)),

3. T(X) (or S(X)) is a closed subset of X,

4. {[By.sub.n]} converges for every sequence {[y.sub.n]} in X such that {[Ty.sub.n]} converges (or {[Ax.sub.n]} converges for every sequence {[x.sub.n]} in X such that {[Sx.sub.n]} converges),

5. there exists [phi] [member of] [PHI] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

for all x, y [member of] X, 0 [less than or equal to] a [less than or equal to] 1

where [phi] [member of] [LAMBDA] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then, the pairs (A, S) and (B, T) share the ([CLR.sub.ST]) property.

Proof. Suppose that (A, S) satisfies the ([CLR.sub.S]) property with respect to mapping S, that is, there exists a sequence {[x.sub.n]} in X and t [member of] S(X) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Regarding the condition (CC), we obtain that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] So, for any sequence {[x.sub.n] } in X, there exists another sequence {[y.sub.n] } in X with [Ax.sub.n] = [Ty.sub.n], owing to the fact that A(X) [subset] T(X). Hence, t [member of] S(X) [intersection] T(X) since T(X) is closed. Consequently, we derive that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By (4), the sequence {[By.sub.n]} converges. Let the sequence {[By.sub.n]} converges to z([not equal to] t) as n [right arrow] [infinity]. Now we need to show that z = t.

Again by (CC), we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] On using inequality (2.2) with x = [x.sub.n], y = [y.sub.n], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

where

L([x.sub.n],[y.sub.n]) = max {d([Sx.sub.n], [Ty.sub.n]),d([Sx.sub.n], [By.sub.n]),d([By.sub.n], [Ty.sub.n])} = max {d{[Sx.sub.n], [Ax.sub.n]), d{[Sx.sub.n], [By.sub.n]), d{[By.sub.n], [Ax.sub.n])},

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking limit as n [right arrow] [infinity] in (2.3), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence (2.4) implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is a contradiction. Therefore, [[integral].sup.d(t,z).sub.0] [phi](t)dt = 0. In view of (2.1), we obtain d(t, z) = 0, i.e., [By.sub.n] [right arrow] t as n [right arrow] [infinity]. Hence the pairs (A, S) and (B, T) satisfy the ([CLR.sub.ST]) property. This completes the proof.

Now, we state the first theorem of this manuscript as follows.

Theorem 1. Let (X, d) be a symmetric space. Suppose that the self-mappings A, B, S and T defined on X satisfying the hypothesis (4) of Lemma 1 with (1C) and (HE). Each pair (A, S) and (B, T) have a coincidence point each if the pairs (A, S) and (B, T) satisfy the ([CLR.sub.ST]) property. Furthermore, we conclude that the self-mappings A, B, S and T have a unique common fixed point if (A, S) and (B, T) are weakly compatible.

Proof. Let (A, S) and (B, T) be pairs of self-mappings on X and satisfy the ([CLR.sub.ST]) property. Thus, we have sequences {[x.sub.n]} and {[y.sub.n]} in X such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where t [member of] S(X) [intersection] T(X). Since t [member of] S(X), there exists a point u [member of] X such that Su = t. Hence, we derive that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now we assert that Au = t. Suppose that Au [not equal to] t. So, by using (2.2) with x = u and y = [y.sub.n], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

where

L(u,[y.sub.n]) = max {d(Su, [Ty.sub.n]), d(Su, [By.sub.n]),d([By.sub.n], [Ty.sub.n])}

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Regarding (1C) and (HE) together with letting n [right arrow] [infinity] in (2.5), we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (2.6), we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a contradiction. Hence, by taking (2.1) into account, we find d(Au, t) = 0 and so t = Au = Su.

Since t [member of] T(X), there exists a point [upsilon] [member of] X such that T[upsilon] = t. We shall show that B[upsilon] [not equal to] T[upsilon]. Suppose that B[upsilon] = T[upsilon]. Then, by using the inequality (2.2) with x = u, y = [upsilon], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

where

L(u, v) = max {d(Su, T[upsilon]),d(Su, B[upsilon]),d(B[upsilon], T[upsilon])} = max {d(t, t), d(t, B[upsilon]), d(B[upsilon], t)} = d(t, B[upsilon])

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence (2.7) implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a contradiction. Thus, we have [[integral].sup.d(t,B[upsilon]).sub.0] [psi](t)dt = 0. On account of (2.1), we derive that B[upsilon] = T[upsilon] = t.

We derive that At = ASu = SAu = St owing to the fact that Au = Su and the self mappings A and S are weakly compatible. At this point, we shall prove that t is a common fixed of the self-mappings A and S. We assume that At [not equal to] t. By using the inequality (2.2) with x = t, y = [upsilon], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

where

L(t, [upsilon]) = max {d(St, T[upsilon]), d(St, B[upsilon]),d(B[upsilon], T[upsilon])} = max {d(At, t), d(At, t),d(t, t)} = d(At, t)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From (2.8), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a contradiction. Consequently, we have At = t = St, that is, t is a common fixed point of the pair (A, S).

Since the pair (B, T) is weakly compatible, the equality yields that Bt = BTw = TBw = Tt. If not, then using inequality (2.2) with x = u, y = t, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

where

L(u, t) = max {d(Su, Tt), d(Su, Bt),d(Bt, Tt)} = max {d(t, Bt),d(t, Bt), d(Bt, Bt)} = d(t, B[upsilon])

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence (2.9) implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a contradiction. Therefore, Bt = t = Tt which shows that t is a common fixed point of the pair (B, T). Hence t is a common fixed point of the self-mappings A, S, B, T.

We use the method of reductio the absurdum to prove the uniqueness. Suppose, on the contrary, that there is another common fixed point t ([not equal to] t) of the self-mappings A, B, S, T. Hence, by replacing x = t and y = t' in the inequality (2.2), we observe that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.10)

where

L(t, t) = max{d(St, Tt'), d(St, Bt'), d(Bt', Tt')} = max{d(t, t'), d(t', t'), d(t', t')| = d(t, t')

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a result, (2.8) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, we obtain that t = t', a contradiction.

Remark 1. Theorem 1 improves the corresponding results contained in Tiwari et al. [48, Theorem 3.1] as completeness (or closedness) of the underlying subspaces are not required.

Now, we give an illustrative example.

Example 1. Let X = [2,11) and the symmetric (semi-metric) d be defined as d(x,y) = [e.sup.[absolute value of x-y]] - 1 for all x, y [member of] X. We also assume that (1C) and (HE) are satisfied. Define the self mappings A, B, S and T and [phi] : [R.sup.+] [right arrow] [R.sup.+] by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we have A(X) = {2,8} [subset not equal to] [2,3) [union] {9} = T(X) and B(X) = {2,9} [subset not equal to] [2, 7) = S(X). Also define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we choose two sequences as {[x.sub.n]} = [{7 + 1/n}.sub.n[member of]N] {[y.sub.n]} = {2} (or {[x.sub.n]} = {2},{[y.sub.n]} = [{7 + 1/n}.sub.n[member of]N]), then pairs (A, S) and (B, T) satisfy the ([CLR.sub.ST]) property:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 2[member of]S(X) [intersection] T(X). By elementary calculation, we derive the inequality (2.2) easily. Hence, we can conclude that all the conditions of Theorem 1 holds. Moreover, we observe that 2 is a unique common fixed point of the self-mappings A, S, B, T. Notice that the self-mappings A, B, S, T are discontinuous at point 2. We also emphasize that the subspaces S(X) and T(X) are not closed subspaces of X. Consequently, the main result of Tiwari et al. [48, Theorem 3.1] is not applicable here.

Corollary 1. Let (X, d) be a symmetric space and A, B, S, T be a self-mappings on X satisfying all the hypotheses of Lemma 1 with (CC), then the self-mappings A, B, S, T have a unique common fixed point if (A, S) and (B, T) are weakly compatible.

Proof. Owing to Lemma 1, it follows that (A, S) and (B, T) satisfy the ([CLR.sub.ST]) property. Consequently, the conditions of Theorem 1 are satisfied, and the self-mappings A, B, S, T have a unique common fixed point under the assumption that both the pairs of selfmappings (A, S) and (B,T) are weakly compatible.

It is pointed out that Example 1 cannot be obtained using Corollary 1, since conditions (2) and (3) of Lemma 1 are not fulfilled. We present another example, showing the situation where the conclusion can be reached using Corollary 1.

Example 2. Let X = [2,24) and the symmetric (semi-metric) d be defined as d(x, y) = [e.sup.[absolute value of x-y]] - 1 for all x, y[member of]X. Assume also that the condition (CC) is satisfied. Define [phi]: [R.sup.+][right arrow][R.sup.+] as in Example 1 and the self mappings A, B, S and T by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we observe that A(X) = {2,16} [subset] [2,17] = T(X) and B(X) = {2,4} [subset] [2,5] = S(X). It is evident that the pairs (A, S) and (B, T) satisfy the ([CLR.sub.ST]) property Indeed, we have [{[x.sub.n] = 9 + 1/n}.sub.m[member of]N]/ {[y.sub.n] = 2} or {[x.sub.n] = 2}, [{[x.sub.n] = 9 + 1/n}.sub.m[member of]N], i-e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where 2[member of]S(X) [intersection] T(X). Also all the conditions of Corollary 1 can be easily verified. It is noted here that 2 is a a unique common fixed point of (A, S) and (B, T).

Notice that this example is not applicable for Theorem 1 since S(X), T(X) are closed subsets of X which demonstrates the situational utility of Corollary 1 over Theorem 1.

The conclusion of Lemma 1, Theorem 1 and Corollary 1 remains true for a suitable choice of a = 1.

Corollary 2. Suppose that (X, d) is a symmetric space and self-mappings A, B, S, T on X satisfy the hypothesis (4) of Lemma 1 with (1C) and (HE). Assume that

1. the pairs of self-mappings (A, S) and (B, T) satisfies the ([CLR.sub.ST]) property,

2. there exists [phi] [member of] [PHI] such that

[[integral].sup.d(Ax,By).sub.0] [phi](t)dt[less than or equal to][phi]([[integral].sup.L(x,y).sub.0][phi](t)dt) (2.11)

where [phi] [member of] [LAMBDA] and L(x,y) = max {d(Sx, Ty),d(Sx, By),d(By, Ty)}, for all x, y [member of] X.

Then, both pairs (A, S) and (B, T) have a coincidence point. Furthermore, if the pairs of (A, S) and (B, T) are weakly compatible, then the self-mappings A, B, S, T have a unique common fixed point.

On taking [phi](t) = 1 in Theorem 1, we have the following natural result:

Corollary 3. Suppose that (X, d) is a symmetric space and self-mappings A, B, S, T on X satisfy the hypothesis (4) of Lemma 1 with the conditions (1C) and (HE). Suppose also that

1. the pairs (A, S) and (B, T) satisfy the ([CLR.sub.ST]) property,

2. there exists [phi] [member of] [PHI] such that

d(Ax, By) [less than or equal to] [phi] (aL(x,y) + (1 - a)M(x,y)), (2.12)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x, y [member of] X, 0 [less than or equal to] a [less than or equal to] 1. Then, both pairs (A, S) and (B, T) have a coincidence point. Furthermore, if the pairs (A, S) and (B, T) are weakly compatible, then the self-mappings A, B, S, T have a unique common fixed point.

By suitable choice of self-mappings, we derived the following.

Corollary 4. Suppose that (X, d) is a symmetric space and self-mappings A, B, S, T on X satisfy the hypothesis (4) of Lemma 1 with (1C) and (HE). Suppose that

1. the pair (A, S) enjoys the ([CLR.sub.S]) property,

2. there exists [phi] [member of] [PHI] such that

[[integral].sup.d(Ax,Ay).sub.0][phi](t)dt[less than or equal to][phi] ([[integral].sup.aL(x, y)+(1-a)M(x,y).sub.0][phi](t)dt), (2.13)

where [phi] [member of] [LAMBDA] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x, y [member of] X, 0 [less than or equal to] a [less than or equal to] 1. Then (A, S) has a coincidence point each. Moreover, A and S have a unique common fixed point provided the pair (A, S) is weakly compatible.

We present the following result as an application of Theorem 1.

Theorem 2. Suppose that (X, d) is a symmetric space and families of self-mappings [{[A.sub.i]}.sup.m.sub.i=1], [{[B.sub.j]}.sup.m.sub.r=1], [{[S.sub.k]}.sup.p.sub.k=1] and [{[T.sub.l]}.sup.q.sub.l=1] on X with A = [A.sub.1][A.sub.2] ... [A.sub.m],B = [B.sub.1][B.sub.2] ... [B.sub.n], S = [S.sub.1][S.sub.2] ... [S.sub.p] and T = [T.sub.1][T.sub.2] ... [T.sub.q] satisfy the conditions (1C), (HE) and also (2.2)(2.1). Suppose that the pairs (A, S) and (B, T) satisfy the ([CLR.sub.ST]) property, then both of the pairs (A, S) and (B, T) have a coincidence point.

Moreover [{[A.sub.i]}.sup.m.sub.i=1], [{[B.sub.j]}.sup.m.sub.r=1], [{[S.sub.k]}.sup.p.sub.k=1] and [{[T.sub.l]}.sup.q.sub.l=1] have a unique common fixed point if the families ({[A.sub.i]}, {[S.sub.k]}) and ({[B.sub.r]}, {[T.sub.h]}) commute pairwise wherein i [member of] {1,2,...,m},k [member of] {1,2,...,p},j [member of] {1,2,...,n} and l [member of] {1,2,...,q}.

Proof The proof can be treated by following the lines in [20]

Now, we indicate that Theorem 2 can be used to derive common fixed point theorems for any finite number of mappings. As a sample for five mappings, we can derive the following by setting one family of two members while the remaining three of single members:

Corollary 5. Suppose that (X, d) is a symmetric space and self-mappings A, B, S, R, T on X satisfy the hypothesis (4) of Lemma 1 with the conditions (1C) and (HE). Suppose also that

1. the pairs (A, SR) and (B, T) satisfy the ([CLR.sub.(SR)(T)]) property,

2. there exists [phi] [member of] [PHI] such that

[[integral].sup.d(Ax,By).sub.0] [phi](t)dt[less than or equal to][phi]([[integral].sup.aL(x,y)+(1-a) M(x,y).sub.0][phi](t)dt), (2.14)

where [phi] [member of] [LAMBDA] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x, y [member of] X, 0 [less than or equal to] a [less than or equal to] 1. Then, the pairs (A, SR) and (B, T) have a coincidence point. Furthermore, the self-mapings A, B, R, S, T have a unique common fixed point if (A, SR) and (B, T) commute pairwise, that is, AS = SA, AR = RA, SR = RS, BT = TB.

Similarly, we can derive a common fixed point theorem for six mappings by setting two families of two members while the rest two of single members:

Corollary 6. Let A, B, H, R, S and T be self mappings of a symmetric (semi-metric) space (X, d) satisfying (1C) and (HE). Suppose that

1. the pairs (A, SR) and (B, TH) share the ([CLR.sub.(SR)(TH)]) property,

2. there exists [phi][member of][PHI] such that

[[integral].sup.d(Ax,By).sub.0] [phi](t)dt[less than or equal to][phi]([[integral].sup.aL(x,y)+ (1-a)M(x,y).sub.0][phi](t)dt) (2.15)

where [phi] [member of] [LAMBDA] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x, y [member of] X, 0 [less than or equal to] a [less than or equal to] 1. Then, both of the pairs (A, SR) and (B, TH) have a points of coincidence. Furthermore, the self-mappings A, B, H, R, S and T have a unique common fixed point if both of the pairs (A, SR) and (B, TH) commute pairwise, that is, AS = SA, AR = RA, SR = RS, BT = TB, BH = HB and TH = HT.

By setting [A.sub.1] = [A.sub.2] = ... = [A.sub.m] = A, [B.sub.1] = [B.sub.2] = ... = [B.sub.n] = B, [S.sub.1] = [S.sub.2] = ... = [S.sub.p] = S and [T.sub.1] = [T.sub.2] = ... = [T.sub.q] = T in Theorem 2, we deduce the following:

Corollary 7. Suppose that (X, d) is a symmetric space and self-mappings A, B, S, T on X satisfy the hypothesis (4) of Lemma 1 with the conditions (1C) and (HE). Suppose also that

1. the pairs ([A.sup.m], [S.sup.p]) and ([B.sup.n], [T.sup.q]) share the ([CLR.sub.SpiTq]) property,

2. there exists [phi] [member of] [PHI] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.16)

where [phi] [member of] [LAMBDA] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all x, y [member of] X, m, n, p, q are fixed positive integers. If AS = SA and BT = TB, then the self-mappings A, B, S, T have a unique common fixed point.

Remark 2. Corollary 7 is a weaker generalization of Theorem 1 as the commutativity requirements (that is, AS = SA and BT = TB) in this corollary are relatively stronger as compared to weak compatibility in Theorem 1.

Acknowledgement

The authors would like to thank Professor M. Imdad for the reprint of his valuable paper [Common fixed point theorems in symmetric spaces employing a new implicit function and common property (E.A). Bull. Belg. Math. Soc. Simon Stevin 16,421-433 (2009)].

Received by the editors in September 2013.

Communicated by F. Bastin.

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Near Nehru Training Centre, H. No. 274, Nai Basti B-14, Bijnor-246701, Uttar Pradesh, India.

email: sun.gkv@gmail.com

Department of Mathematics, Atilim University, 06836, Incek, Ankara, Turkey.

Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, Saudi Arabia.

email:erdalkarapinar@yahoo.com, ekarapinar@atilim.edu.tr