Hybrid Fixed Point Theorems in Symmetric Spaces via Common Limit Range Property

Abstract In this paper, we point out that some recent results of Vijaywar et al. (Coincidence and common fixed point theorems for hybrid contractions in symmetric spaces, Demonstratio Math. 45 (2012), 611-620) are not true in their present form. With a view to prove corrected and improved versions of such results, we introduce the notion of common limit range property for a hybrid pair of mappings and utilize the same to obtain some coincidence and fixed point results for mappings defined on an arbitrary set with values in symmetric (semi-metric) spaces. Our results improve, generalize and extend some results of the existing literature especially due to Imdad et al., Javid and Imdad, Vijaywar et al. and some others. Some illustrative examples to highlight the realized improvements are also furnished.


Introduction and preliminaries
The classical Banach Contraction Principle is indeed the most fundamental result of metric fixed point theory which is very effectively utilized to establish the existence of solutions of nonlinear Volterra integral equations, Fredholm integral equations, nonlinear integro-differential equations in Banach spaces besides supporting the convergence of algorithms in Computational Mathematics. However, sometimes one may come across situations wherein the full force of metric requirements are not used in the proofs of certain metrical fixed point theorems. Motivated by this fact, Hicks and Rhoades [10] proved some common fixed point theorems in symmetric spaces and showed that a general probabilistic structures admits a compatible symmetric or semi-metric. Mihet [23] pointed out that Hicks and Rhoades [10] have inadvertently used triangle inequality in their results. Though the notion of weak commutativity in metric fixed point theory was introduced by Sessa [29] in 1982, yet the earliest use of a weak commutativity condition for a hybrid pair can be traced back to Itoh and Takahashi [16] wherein authors proved some coincidence point theorems in metric spaces. Kaneko and Sessa [21] extended the concept of compatibility (due to Jungck [18]) to a hybrid pair of mappings defined on metric spaces. Pathak [28] extended the concept of compatibility (due to Jungck [19]) by defining weak compatibility for hybrid pairs of mappings (including single valued case) and utilize the same to prove some coincidence and common fixed point theorems satisfying a suitable contraction condition. Naturally, compatible mappings are weakly compatible but not conversely.
This remains an established fact that the contractive conditions do not ensure the existence of fixed points unless the underlying space is assumed compact or the contractive conditions are replaced by relatively stronger conditions. Firstly, Pant [26,27] studied metrical fixed point theorems for single-valued non-compatible mappings under strict contractions. In 2004, Kamran [20] extended the notion of the property (E.A) (due to Aamri and Moutawakil [1]) to hybrid pairs of mappings and proved some coincidence and common fixed point theorems. It is observed in Imdad and Ali [13] that the property (E.A) buys the suitable required containment of the range of one mapping into the range of another up to a pair of mappings. Sintunavarat and Kumam [31] coined the idea of 'common limit range property' for single-valued mappings whose use does not demand the completeness (or closedness) of the underlying subspaces.
In 2010, Ali and Imdad [3] noticed some errors in certain results of Singh and Hashim [30] and proved some fixed point results for two pairs of hybrid mappings in symmetric (or semi-metric) spaces. Recently, Vijaywar et al. [33] proved some fixed point theorems for a pair of hybrid mappings satisfying strict contractive conditions in symmetric (semi-metric) spaces under the property (E.A). Our main purpose in this paper is twofold. Firstly, we point out that some results of Vijaywar et al. [33] are not true in their present form. Secondly, we introduce the notion of common limit range property for a pair of hybrid mappings which are defined on an arbitrary nonempty set with values in a symmetric ( semi-metric) space and utilize the same to prove corrected and improved versions of some results due to Vijaywar et al. [33] in symmetric spaces. We furnish some examples to support our main result besides deriving some related results.
The following definitions and results will be needed in the sequel.
A symmetric on a non-empty set X is a non-negative real valued function d on XˆX such that (1) dpx, yq " 0 if and only if x " y, (2) dpx, yq " dpy, xq.
Let d be a symmetric on a set X and for r ą 0 and any x P X, let Bpx, rq " ty P X : dpx, yq ă ru. A topology T pdq on X is given by U P T pdq if and only if for each x P U , Bpx, rq Ă U for some r ą 0. A symmetric d is a semi-metric if for each x P X and each r ą 0, Bpx, rq is a neighborhood of x in the topology T pdq. Note that lim nÑ8 dpx n , xq " 0 if and only if x n Ñ x in the topology T pdq.
Notice that symmetric spaces are not essentially Hausdorff and also the symmetric d is not continuous in general. Therefore, in the course of proving fixed point theorems, some additional axioms are required. The following axioms are available in the papers of Aliouche [4], Galvin and Shore [7], Hicks and Rhoades [10] and Wilson [34]. pW 3 q [34] Given tx n u; x, y P X, lim nÑ8 dpx n , xq " 0 and lim nÑ8 dpx n , yq " 0 imply x " y. pW 4 q [34] Given tx n u, ty n u; x P X, lim nÑ8 dpx n , xq " 0 and lim nÑ8 dpx n , y n q " 0 imply lim nÑ8 dpy n , xq " 0. pHEq [4] Given tx n u, ty n u; x P X, lim nÑ8 dpx n , xq " 0 and lim nÑ8 dpy n , xq " 0 imply lim nÑ8 dpx n , y n q " 0. p1Cq [7] A symmetric d is said to be 1-continuous if lim nÑ8 dpx n , xq " 0 implies lim nÑ8 dpx n , yq " dpx, yq, where tx n u is a sequence in X and x, y P X. pCCq [7] A symmetric d is said to be continuous if lim nÑ8 dpx n , xq " 0 and lim nÑ8 dpy n , yq " 0 imply lim nÑ8 dpx n , y n q " dpx, yq, where tx n u, ty n u are sequences in X and x, y P X.
Here, it is observed that pCCq ùñ p1Cq, pW 4 q ùñ pW 3 q, and p1Cq ùñ pW 3 q but the converse implications are not true. In general, all other possible implications amongst pW 3 q, p1Cq, and pHEq are not true. For detailed description, we refer to Cho et al. [6] which also contains some illustrative examples. However, pCCq implies all the remaining four conditions namely: pW 3 q, pW 4 q, pHEq and p1Cq. Employing these axioms, several authors proved common fixed point theorems in the framework of symmetric spaces (e.g. [5,8,9,12,13,15,17,22,32]). With a view to obtain our results under optimal conditions, we utilize condition pW 3 q or p1Cq (along with pHEq) instead of pW 4 q.
Let pX, dq be a symmetric (or semi-metric) space. Then, on the lines of Nadler [25], we adopt (1) CLpXq " tA : A is a non-empty closed subset of X} and (2) CBpXq " tA : A is a non-empty closed and bounded subset of X}, (3) for non-empty closed and bounded subsets A, B of X and x P X, dpx, Aq " inftdpx, aq : a P Au and HpA, Bq " max tsuptdpa, Bq : a P Au, suptdpA, bq : b P Buu .
It is easy to see that pCBpXq, Hq is a semi-metric space (see [24]). It is also well known that CBpXq is a metric space under the metric H, which is known as the Hausdorff-Pompeiu metric on CBpXq provided pX, dq is a metric space.
Definition 2. [26] Let pX, dq be a symmetric (semi-metric) space with F : X Ñ CBpXq and g : X Ñ X. The pair of hybrid mappings pF, gq is said to be R-weakly commuting if, for every x P X and gF x P CBpXq, there exists some positive real number R such that HpF gx, gF xq ≤ RdpF x, gxq.
Let pX, dq be a symmetric (semi-metric) space with F : X Ñ CBpXq and g : X Ñ X. The pair of hybrid mappings pF, gq is said to be compatible if gF x P CBpXq for all x P X and lim nÑ8 HpF gx n , gF x n q " 0 whenever tx n u is a sequence in X such that lim nÑ8 gx n " t P A " lim nÑ8 F x n .
Here it may be noted that compatible mappings need not be R-weakly commuting (see [26]). Also, on coincidence points, R-weak commutativity is equivalent to commutativity and remains a necessary minimal condition for the existence of common fixed points for contractive type mappings. Definition 4. [12] Let pX, dq be a symmetric (semi-metric) space wherein d satisfies condition pW 3 q (Hausdorffness of τ pdq) with F : X Ñ CBpXq and g : X Ñ X. The pair of hybrid mappings pF, gq is said to be non-compatible if there exists at least one sequence tx n u in X such that lim nÑ8 gx n " t P A " lim nÑ8 F x n but lim nÑ8 HpF gx n , gF x n q is either non-zero or nonexistent.
Definition 5. [11] Let Y be a non-empty subset of X, F : Y Ñ 2 X and g : Y Ñ X. The pair of hybrid mappings pF, gq is said to be quasicoincidentally commuting if gx P F x (for x P X with F x, gx P Y ) implies gF x is contained in F gx. Definition 6. [11] Let Y be a non-empty subset of X, F : Y Ñ 2 X and g : Y Ñ X. The mapping g is said to be coincidentally idempotent with respect to mapping F , if gx P F x with gx P Y imply ggx " gx, that is, g is idempotent at coincidence points of the pair pF, gq. Definition 7. [2] Let pX, dq be a symmetric (semi-metric space) wherein d satisfies condition pW 3 q whereas Y be an arbitrary non-empty set with F : Y Ñ CBpXq and g : Y Ñ X. Then the pair of hybrid mappings pF, gq is said to satisfy the property (E.A) if there exists a sequence tx n u in Y , for some t P X and A P CBpXq such that

Main result
Firstly, we introduce the notion of common limit range property with respect to mapping g (briefly, (CLRg) property) as follows: Definition 8. Let pX, dq be a semi-metric space wherein d satisfies condition pW 3 q whereas Y be an arbitrary non-empty set with F : Y Ñ CBpXq and g : Y Ñ X. Then the pair of hybrid mappings pF, gq is said to satisfy the (CLRg) property if there exists a sequence tx n u in Y , for some u P X and A P CBpXq such that Now, we present some examples demonstrating the preceeding definition.
Example 1. Let us consider X " r0, 1s with the symmetric dpx, yq " px´yq 2 . Define F : X Ñ CBpXq and g : X Ñ X as follows: If we consider the sequence tx n u " 1 2´1 n ( nPN , then one can verify that the pair pF, gq enjoys the (CLRg) property as Example 2. Consider X " r0, 1s with the symmetric dpx, yq " px´yq 2 . Define F : X Ñ CBpXq and g : X Ñ X by 2´1 n˙" lim nÑ8 gˆ1´1 2`1 n˙" Example 3. Consider X " r0, 1s with the symmetric dpx, yq " px´yq 2 . Define F : X Ñ CBpXq and g : X Ñ X by If we consider tx n u " 1 2´1 n ( nPN , then we find Similar verification can be carried in respect of the other possible sequences. Thus in all, the pair pF, gq doesn't satisfy the property (E.A) as well as (CLRg) property.
Remark 1. If the pair pF, gq satisfies the property (E.A) along with the closedness of gpXq, then the pair also satisfies the (CLRg) property.
For the sake of completeness, we state the following theorem due to Vijaywar et al. [33] proved for a pair of hybrid mappings defined on a symmetric (semi-metric) space pX, dq.
If gpXq is a d-closed (τ pdq-closed) subset of X, then F and g have a coincidence point.
Unfortunately, the preceeding theorem is not true in it's present form as authors use the continuity of the symmetric d but fail to mention the same. To substantiate the claim, we furnish an example of a discontinuous symmetric d which demonstrates that Theorem 1 is not valid in it's present form even for single valued pair of mappings. , if x " 0, y " 0, 0, if x " y,
Notice that, in the foregoing example, all the conditions of Theorem 1 are satisfied but f and g have no coincidence point.
However a more general result can be obtained by using common limit range property under additional conditions p1Cq and pHEq.
Then F and g have a coincidence point. In particular, if Y Ă X and the pair of mappings pF, gq is quasi-coincidentally commuting and coincidentally idempotent, then the pair pF, gq has a common fixed point.
Proof. Firstly, one needs to note that a sequence tx n u in a potent semimetric space pX, dq converges to a point x in τ pdq iff dpx n , xq Ñ 0. To substantiate this, suppose x n Ñ x and let ą 0. Since Spx, q is a neighbourhood of x, there exists U P τ pdq such that x P U Ă Spx, q. Since x n Ñ x, there is a m P N (the natural number) such that x n P U Ă Spx, q for n ≥ m so that dpx n , xq ă for n ≥ m, that is, dpx n , xq Ñ 0. The converse part is obvious in view of the definition of τ pdq.
Suppose that the pair pF, gq enjoys the (CLRg) property, there exists a sequence tx n u in Y , for some u P X and A P CLpXq such that lim nÑ8 gx n " gu P A " lim nÑ8 F x n . Now we show that gu P F u. If not, then using inequality (2.2), one obtains HpF x n , F uq ă max " dpgx n , guq, k 2 rdpgx n , F x n q`dpgu, F uqs , k 2 rdpgu, F x n q`dpgx n , F uqs * .
Since gu P A, the above inequality implies dpgu, F uq ≤ HpA, F uq which is a contradiction. Hence gu P F u which shows that the pair pF, gq has a point of coincidence. Since Y Ă X and u is a point of coincidence of the pair pF, gq, using the quasi-coincidentally commuting property of pF, gq and the coincidentally idempotent property of g with respect to F , one can have gu P F u and ggu " gu. Therefore gu " ggu P gpF uq Ă F pguq which shows that gu is a common fixed point of the pair pF, gq.
Example 5. Consider X " Y " r0, 1s equipped with the symmetric defined by dpx, yq " px´yq 2 for all x, y P X which satisfies conditions p1Cq and pHEq. Define the mappings F : X Ñ CLpXq and g : X Ñ X as follows: Consider a sequence tx n u " 1 2´1 n ( nPN , one can see that the pair pF, gq enjoys the (CLRg) property, By a routine calculation one can show that the contractive condition (2.2) holds for every x ‰ y P X. It is pointed out that gpXq "`1 3 , 2 is not a closed (τ pdq-closed) subset of X. Also the pair pF, gq is quasi-coincidentally commuting at x " 1 2 , that is, g`1 2˘P F`1 2˘a nd gF`1 2˘"`1 3 , 1 . Thus, all conditions of Theorem 2 are satisfied and 1 2 " g`1 2˘P F`1 2˘. Corollary 1. Let pX, dq be a symmetric (semi-metric) space wherein d satisfies conditions p1Cq and pHEq while Y is an arbitrary non-empty set with F : Y Ñ CLpXq and g : Y Ñ X. Suppose that the hybrid pair pF, gq enjoys the property (E.A) and satisfies inequality (2.2). If f pY q is a closed (τ pdq-closed) subset of X, then the pair pF, gq has a coincidence point.
In particular, if Y Ă X and the pair of mappings pF, gq is quasi-coincidentally commuting and coincidentally idempotent, then the pair pF, gq has a common fixed point.
Proof. The proof of this corollary easily follows in view of Remark 1.
Since the class of compatible as well as non-compatible mappings are contained in the class of mappings pairs satisfying the property (E.A), therefore we have the following.
Corollary 2. Let pX, dq be a symmetric (semi-metric) space wherein d satisfies conditions p1Cq and pHEq while Y is an arbitrary non-empty set with F : Y Ñ CLpXq and g : Y Ñ X. Suppose that the hybrid pair pF, gq is compatible or non-compatible and satisfies inequality (2.2). If f pY q is a closed (τ pdq-closed) subset of X, then the pair pF, gq has a coincidence point.
In particular, if Y Ă X and the pair of mappings pF, gq is quasi-coincidentally commuting and coincidentally idempotent, then the pair pF, gq has a common fixed point.
Our next theorem involves a function φ : R`Ñ R`which satisfies the following properties: (1) φ is upper semi-continuous on R`and (2) 0 ă φptq ă t for each t P R`.
Theorem 3. Let pX, dq be a symmetric (semi-metric) space wherein d satisfies conditions p1Cq and pHEq while Y is an arbitrary non-empty set with F : Y Ñ CLpXq and g : Y Ñ X. Suppose that p1q the hybrid pair pF, gq enjoys the pCLRgq property and p2q for all x ‰ y P Y and 0 ă k ≤ 2, dpgx, gyq, k 2 rdpgx, F xq`dpgy, F yqs , k 2 rdpgy, F xq`dpgx, F yqs * .
Then F and g have coincidence point.
In particular, if Y Ă X and the pair of mappings pF, gq is quasi-coincidentally commuting and coincidentally idempotent, then the pair pF, gq has a common fixed point.
Proof. If the pair pF, gq satisfies the (CLRg) property, then there exists a sequence tx n u in Y , for some u P X and A P CLpXq such that lim nÑ8 gx n " gu P A " lim nÑ8 F x n . Now we assert that gu P F u. Suppose that gu R F u, then using inequalities (2.3) and (2.4), one obtains (2.5) HpF x n , F uq ≤ φpmpx n , uqq, where mpx n , uq " max " dpgx n , guq, k 2 rdpgx n , F x n q`dpgu, F uqs , k 2 rdpgu, F x n q`dpgx n , F uqs * .
Taking limit as n Ñ 8 in (2.5) and using conditions p1Cq and pHEq, we have lim nÑ8 HpF x n , F uq ≤ φ˜lim nÑ8 max # dpgx n , guq, k 2 rdpgx n , F x n q`dpgu, F uqs , Since gu P A, we get dpgu, F uq ≤ HpA, F uq ≤ φˆk 2 dpgu, F uq˙ă dpgu, F uq, which is a contradiction. Hence gu P F u, which shows that the pair pF, gq has a point of coincidence. Since Y Ă X and u is a point of coincidence of the pair pF, gq, using the quasi-coincidentally commuting property of pF, gq and the coincidentally idempotent property of g with respect to F , one can have gu P F u and ggu " gu. Therefore gu " ggu P gpF uq Ă F pguq, which shows that gu is a common fixed point of the pair pF, gq.
Corollary 3. Let pX, dq be a symmetric (semi-metric) space wherein d satisfies conditions p1Cq and pHEq while Y is an arbitrary non-empty set with F : Y Ñ CLpXq and g : Y Ñ X. Suppose that p1q the hybrid pair pF, gq enjoys the (CLRg) property and p2q for all x ‰ y P Y , HpF x, F yq ≤ φ pmax tdpgx, gyq, dpgx, F xq, dpgy, F yq, dpgy, F xq, dpgx, F yquq .
Then F and g have a coincidence point.
In particular, if Y Ă X and the pair of mappings pF, gq is quasi-coincidentally commuting and coincidentally idempotent, then the pair pF, gq has a common fixed point.
Our next result remains true for a pair of hybrid mappings in metric spaces.
Theorem 4. Let pX, dq be a metric space while Y is an arbitrary non-empty set with F : Y Ñ CLpXq and g : Y Ñ X. Suppose that p1q the hybrid pair pF, gq enjoys the (CLRg) property and p2q for all x ‰ y P Y and 0 ă k ă 2,
Then F and g have a coincidence point. In particular, if Y Ă X and the pair of mappings pF, gq is quasi-coincidentally commuting and coincidentally idempotent, then the pair pF, gq has a common fixed point.
Proof. The proof of this theorem can be completed on the lines of the proof of Theorem 2, hence the details are avoided. Now, we utilize a relatively weaker condition pW 3 q instead of condition p1Cq to prove our next result.
Theorem 5. Let pX, dq be a semi-metric (symmetric) space wherein d satisfies conditions pW 3 q and pHEq while Y is an arbitrary non-empty set with F : Y Ñ CLpXq and g : Y Ñ X. Suppose that p1q the hybrid pair pF, gq enjoys the (CLRg) property and p2q for all x ‰ y P Y , (2.8) HpF x, F yq ă max tdpgx, gyq, mintdpgx, F xq, dpgy, F yqu, mintdpgy, F xq`dpgx, F yquu . Then F and g have a coincidence point.
In particular, if Y Ă X and the pair of mappings pF, gq is quasi-coincidentally commuting and coincidentally idempotent, then the pair pF, gq has a common fixed point.
Proof. In view of (1), there exists a sequence tx n u in Y , for some u P X and A P CLpXq such that lim nÑ8 gx n " gu P A " lim nÑ8 F x n . Now we show that gu P F u. If not, then using inequality (2.8), one obtains HpF x n , F uq ă maxtdpgx n , guq, mintdpgx n , F x n q, dpgu, F uqu, mintdpgu, F x n q`dpgx n , F uquu.
On letting n Ñ 8 and making use of conditions pW 3 q and pHEq, we get lim nÑ8 HpF x n , F uq " 0 implying thereby Hpgu, F uq " 0, that is, gu P F u. Hence u is a coincidence point of the pair pF, gq. The rest of the proof run on the lines of the proof of Theorem 2. This concludes the proof. Remark 2. The results similar to Theorem 3 can be proved under the contractive condition (2.8). Here, we avoid the detailed description.