COMMON n-TUPLED FIXED POINT FOR HYBRID PAIR OF MAPPINGS UNDER NEW CONTRACTIVE CONDITION

Abstract

We establish a common n-tupled fixed point theorem for hybrid pair of mappings under new contractive condition. It is to be noted that to find n-tupled coincidence point, we do not use the condition of continuity of any mapping involved. An example supporting to our result has also been cited. We improve, extend and generalize several known results.

1. Introduction and Preliminaries

Let (X, d) be a metric space and CB(X) be the set of all nonempty closed bounded subsets of X. Let D(x, A) denote the distance from x to A ⊂ X and H denote the Hausdorff metric induced by d, that is,D(x,A) = and H(A,B) = for all A,B ∈ CB(X).

The study of fixed points for multivalued contractions and non-expansive mappings using the Hausdorff metric was initiated by Markin [10]: The existence of fixed points for various multivalued contractive mappings has been studied by many authors under different conditions. For details, we refer the reader to [3, 4, 6, 7, 12] and the reference therein. The theory of multivalued mappings has application in control theory, convex optimization, differential inclusions and economics.

In [1], Bhaskar and Lakshmikantham established some coupled fixed point theorems and apply these results to study the existence and uniqueness of solution for periodic boundary value problems. Lakshmikantham and Ciric [9] proved coupled coincidence and common coupled fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces, extended and generalized the results of Bhaskar and Lakshmikantham [1],

Chandok, Sintunavarat and Kumam [2] established some coupled coincidence point and coupled common fixed point theorems for a pair of mappings having a mixed g-monotone property in partially ordered G-metric spaces. Kumam et al. [8] proved some tripled fixed point theorems in fuzzy normed spaces. Rahimi, Radenovic, Soleimani Rad [11] introduced some new definitions about quadrupled fixed point and obtained some new quadrupled fixed point results in abstract metric spaces.

Imdad, Soliman, Choudhury and Das [5] introduced the concept of n-tupled fixed point, n-tupled coincidence point and proved some n-tupled coincidence point and n-tupled fixed point results for single valued mapping.

These concepts was extended by Deshpande and Handa [4] to multivalued mappings and obtained n-tupled coincidence points and common n-tupled fixed point theorems involving hybrid pair of mappings under generalized Mizoguchi-Takahashi contraction. In [4], Deshpande and Handa introduced the following for multivalued mappings:

Definition 1.1. Let X be a nonempty set, F : Xr → 2X (a collection of all nonempty subsets of X) and g be a self-mapping on X. An element (x1, x2,…, xr) ∈ Xr is called

(1) an r−tupled fixed point of F if x1 ∈ F(x1, x2,…, xr), x2 ∈ F(x2,…, xr, x1)…, xr ∈ F(xr, x1,…, xr−1).

(2) an r-tupled coincidence point of hybrid pair {F, g} if g(x1) ∈ F(x1, x2,…, xr), g(x2) ∈ F(x2,…, xr, x1),…, g(xr) ∈ F(xr, x1,…, xr−1).

(3) a common r−tupled fixed point of hybrid pair {F, g} if x1 = g(x1) ∈ F(x1, x2,…, xr), x2 = g(x2) ∈ F(x2,…, xr, x1),…, xr = g(xr) ∈ F(xr, x1,…, xr−1).

We denote the set of r−tupled coincidence points of mappings F and g by C{F, g}. Note that if (x1, x2,…, xr) ∈ C{F, g}, then (x2,…, xr, x1),…, (xr, x1,…, xr−1) are also in C{F, g}.

Definition 1.2. Let F : Xr → 2X be a multivalued mapping and g be a self-mapping on X. The hybrid pair {F, g} is called w−compatible if g(F(x1, x2,…, xr)) ⊆ F(g(x1), g(x2),…, g(xr)) whenever (x1, x2,…, xr) ∈ C{F, g}.

Definition 1.3. Let F : Xr → 2X be a multivalued mapping and g be a self-mapping on X. The mapping g is called F− weakly commuting at some point (x1, x2,…, xr) ∈ Xr if g2(x1) ∈ F(g(x1), g(x2),…, g(xr)), g2(x2) ∈ F(g(x2),…, g(xr), g(x1)),…, g2(xr) ∈ F(g(xr), g(x1),…, g(xr−1)).

Lemma 1.1. Let (X, d) be a metric space. Then, for each a ∈ X and B ∈ CB(X), there is b0 ∈ B such that D(a, B) = d(a, b0), where D(a, B) = infb∈B d(a, b).

In this paper, we establish a common n−tupled fixed point theorem for hybrid pair of mappings satisfying new contractive condition. It is to be noted that to find n−tupled coincidence point, we do not use the condition of continuity of any mapping involved. Our result improves, extend, and generalize the results of Bhaskar and Lakshmikantham [1] and Lakshmikantham and Ciric [9]. An example is also given to validate our result.

 

2. Main Results

Let Φ denote the set of all functions φ : [0; +∞) → [0; +∞) satisfying (iφ) φ is non-decreasing, (iiφ) φ(t) < t for all t >0, (iiiφ) limr→t+ φ(r) < t for all t > 0

and Ψ denote the set of all functions ψ : [0, +∞) → [0, +∞) which satisfies (iψ) ψ is continuous, (iiψ) ψ(t) < t, for all t > 0.Note that, by (iψ) and (iiψ) we have that ψ(t) = 0 if and only if t = 0.

For simplicity, we define the following:

Theorem 2.1. Let (X, d) be a metric space. Assume F : Xr → CB(X) and g : X → X be two mappings satisfying

for all x1, x2,…, xr, y1, y2,…, yr ∈ X. where φ ∈ Φ and ψ ∈Ψ. Furthermore assume that F(Xr) ⊆ g(X) and g(X) is a complete subset of X. Then F and g have an r−tupled coincidence point. Moreover, F and g have a common r−tupled fixed point, if one of the following conditions holds:

(a) F and g are w−compatible. limi→∞ gix1 = y1, limi→∞  gix2 = y2, …, limi→∞  gixr = yr, for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X and g is continuous at y1, y2, … , yr.

(b) g is F−weakly commuting for some (x1, x2, …, xr) ∈ C{F, g}, gx1, gx2, …, gxr are fixed points of g, that is, g2x1 = gx1, g2x2 = gx2, …, g2xr = gxr.

(c) g is continuous at x1, x2, …, xr. limi→∞  giy1 = x1, limi→∞ giy2 = x2, …, limi→∞  giyr = xr for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X.

(d) g(C{F, g}) is a singleton subset of C{F, g}.

Proof. Let ∈ X be arbitrary. Then F(), …, F() are well defined. Choose ∈ F(), …, g ∈ F () because F(Xr) ⊆ g(X). Since F : Xr → CB(X), therefore by Lemma 1.1, there exist z1 ∈ F(), …, zr ∈ F() such that

Since F(Xr) ⊆ g(X), there exist ∈ such that z1 = , z2 = , …, zr = Thus

Continuing this process, we obtain sequences ⊂ X, ⊂ X, …, ⊂ X such that for all i ∈ N, we have ∈ F, ∈ F , …, ∈ F such that

Thus

Similarly

Combining them, we get

which implies, by (iiφ); that

This shows that the sequence defined by δi = is a decreasing sequence of positive numbers. Then there exists δ ≥ 0 such that

We shall prove that δ = 0. Suppose that δ > 0. Letting i → ∞ in (2.2), by using (2.3) and (iiiφ), we get

which is a contradiction. Hence

We now prove that are Cauchy sequences in (X, d). Suppose, to the contrary, that one of the sequences is not a Cauchy sequence.

Then there exists an ε > 0 for which we can find subsequences of of of such that

We can choose i(k) to be the smallest positive integer satisfying (2:5). Then

By (2.5), (2.6) and triangle inequality, we have Letting k → ∞ in the above inequality and using (2.4), we get

By triangle inequality, we have Thus

Since , therefore by (2.1) and by triangle inequality, we have

Thus

Similarly Combining them, we get

By (2.8) and (2.9), we get

Letting k → ∞ in the above inequality, by using (2.4), (2.7), (A), (iψ), (iiψ) and (iiiφ), we get

which is a contradiction. This shows that are Cauchy sequences in g(X). Since g(X) is complete, thus there exist x1, x2, …, xr ∈ X such that

Now, since therefore by using condition (2.1), we get

Letting i → ∞ in the above inequality, by using (2.10), (A), (iψ), (iiψ) and (iiiφ), we get D(gx1, F(x1, x2, …, xr)) ≤ φ(t) + 0 = 0 + 0 = 0.

Thus D(gx1, F(x1, x2, …, xr)) = 0.

Similarly D(gx2, F(x2, …, xr, x1)) = 0, …, D(gxr, F(xr, x1, …, xr−1)) = 0,

which implies that gx1 ∈ F(x1, x2, …, xr), …, gxr ∈ F(xr, x1, …, xr−1),that is, (x1, x2, …, xr) is an r−tupled coincidence point of F and g.

Suppose now that (a) holds. Assume that for some (x1, x2, …, xr) ∈ C{F, g},

Since g is continuous at y1, y2, …, yr, we have, by (2.11), that y1, y2, …, yr are fixed points of g, that is,

As F and g are w−compatible, so for all i ≥ 1,

By using (2.1) and (2.13), we obtain D(gix1, F(y1, y2, …, yr)) ≤ H(F(gi−1x1, gi−1x2, …, gi−1xr), F(y1, y2, …, yr)) ≤ φ [max {d(gix1, gy1), d(gix2, gy2), …, d(gixr, gyr)}] + ψ [M{gi−1x1, gi−1x2, …, gi−1xr, y1, y2, …, yr}].

On taking limit as i → ∞ in the above inequality, by using (2.11), (2.12), (A), (iψ), (iiψ) and (iiiφ), we get D(gy1, F(y1, y2,…, yr)) ≤ φ(t) + 0 = 0 + 0 = 0,

which implies that D(gy1, F(y1, y2,…, yr)) = 0.

Similarly D(gy2, F(y2,…, yr, y1)) = 0,…, D(gyr, F(yr, y1,…, yr−1)) = 0.Thus

Thus, by (2.12) and (2.14), we get y1 = gy1 ∈ F(y1, y2,…, yr), …, yr = gyr ∈ F(yr, y1,…, yr−1),that is, (y1, y2,…, yr) is a common r−tupled fixed point of F and g.

Suppose now that (b) holds. Assume that for some (x1, x2,…, xr) ∈ C{F, g}, g is F−weakly commuting, that is, g2x1 ∈ F(gx1, gx2,…, gxr), g2x2 ∈ F(gx2, …, gxr, gx1),…, g2xr ∈ F(gxr, gx1, …, gxr−1) and g2x1 = gx1, g2x2 = gx2,…, g2xr = gxr. Thus gx1 = g2x1 ∈ F(gx1, gx2,…, gxr), gx2 = g2x2 ∈ F(gx2,…, gxr, gx1),…, gxr = g2xr ∈ F(gxr, gx1,…, gxr−1), that is, (gx1, gx2,…, gxr) is a common r−tupled fixed point of F and g.

Suppose now that (c) holds. Assume that for some (x1, x2,…, xr) ∈ C{F, g} and for some y1, y2,…, yr ∈ X, limi→∞  giy1 = x1, limi→∞  giy2 = x2,…, limi→∞  giyr = xr. Since g is continuous at x1, x2,…, xr. We have that x1, x2,…, xr are fixed points of g, that is, gx1 = x1, gx2 = x2,…, gxr = xr. Since (x1, x2,…, xr) ∈ C{F, g}, therefore, we obtain x1 = gx1 ∈ F(x1, x2,…, xr), x2 = gx2 ∈ F(x2,…, xr, x1),…, xr = gxr ∈ F(xr, x1,…, xr−1), that is, (x1, x2,…, xr) is a common r−tupled fixed point of F and g.

Finally, suppose that (d) holds. Let g(C{F, g}) = {(x1, x1,…, x1)}. Then {x1} = {gx1} = F(x1, x1,…, x1). Hence (x1, x1,…, x1) is a common r−tupled fixed point of F and g.

Example 2.1. Suppose that X = [0; 1], equipped with the metric d : X × X → [0, +∞) defined as d(x, y) = max{x, y} and d(x, x) = 0 for all x, y ∈ X. Let F : Xr →CB(X) be defined as

and g : X → X be defined as g(x) = x2, for all x ∈ X.

Define φ : [0, +∞) → [0, +∞) by

and ψ : [0, +∞) → [0, +∞) by

Now, for all x1, x2, …, xr, y1, y2, …, yr ∈ X with x1, x2,…, xr, y1, y2 …, yr ∈ [0, 1).

But If (x1)2 + (x2)2 + … + (xr)2 < (y1)2 + (y2)2 + … + (yr)2, then

Similarly, we obtain the same result for (y1)2 + (y2)2 + … + (yr)2 < (x1)2 + (x2)2 + … + (xr)2. Thus the contractive condition (2.1) is satisfied for all x1, x2, …, xr, y1, y2, …, yr ∈ X with x1, x2, …, xr, y1, y2, …, yr ∈ [0; 1). Again, for all x1, x2, …, xr, y1, y2, …, yr ∈ X with x1, x2, …, xr ∈ [0; 1) and y1, y2, …, yr = 1, we have

Thus the contractive condition (2.1) is satisfied for all x1, x2, …, xr, y1, y2, …, yr ∈ X with x1, x2, …, xr ∈ [0, 1) and y1, y2, …, yr = 1. Similarly, we can see that the contractive condition (2.1) is satisfied for all x1, x2, …, xr, y1, y2, …, yr ∈ X with x1, x2, …, xr, y1, y2, …, yr = 1. Hence, the hybrid pair {F, g} satisfy the contractive condition (2.1), for all x1, x2, …, xr, y1, y2, …, yr ∈ X. In addition, all the other conditions of Theorem 2.1 are satisfied and z = (0, 0, …, 0) is a common r−tupled fixed point of hybrid pair {F, g}. The function F : Xr → CB(X) involved in this example is not continuous on Xr.

Corollary 2.2. Let (X, d) be a metric space. Assume F : Xr → CB(X) and g : X → X be two mappings satisfying

for all  x1, x2, …, xr, y1, y2, …, yr ∈ X, where φ ∈ Φ and ψ ∈ Ψ. Furthermore assume that F(Xr) ⊆ g(X) and g(X) is a complete subset of X. Then F and g have an r−tupled coincidence point. Moreover, F and g have a common r−tupled fixed point, if one of the following conditions holds:

(a) F and g are w−compatible. limi→∞ gix1 = y1, limi→∞ gix2 = y2, …, limi→∞ gixr = yr, for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X and g is continuous at y1, y2, …, yr.

(b) g is F−weakly commuting for some (x1, x2, …, xr) ∈ C{F, g}, gx1, gx2, …, gxr are fixed points of g, that is, g2x1 = gx1, g2x2 = gx2, …, g2xr = gxr.

(c) g is continuous at x1, x2, …, xr. limi→∞ giy1 = x1, limi→∞ giy2 = x2, …, limi→∞  giyr = xr for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X.

(d) g(C{F, g}) is a singleton subset of C{F, g}.

Proof. It suffices to remark thatThen, we apply Theorem 2.1, since φ is non-decreasing.

If we put g = I (the identity mapping) in the Theorem 2.1, we get the following result:

Corollary 2.3. Let (X, d) be a complete metric space, F : Xr → CB(X) be a mapping satisfying H(F(x1, x2, …, xr), F(y1, y2, …, yr)) ≤ φ [max {d(x1, y1), …, d(xr, yr)}] + ψ [m(x1, …, xr, y1, …, yr)],for all x1, x2, …, xr, y1, y2, …, yr ∈ X, where φ ∈ Φ and ψ ∈ Ψ. Then F has an r−tupled fixed point.

If we put g = I (the identity mapping) in the Corollary 2.2, we get the following result:

Corollary 2.4. Let (X, d) be a complete metric space, F : Xr → CB(X) be a mapping satisfying for all x1, x2, …, xr, y1, y2, …, yr ∈ X, where φ ∈ Φ and ψ ∈ Ψ. Then F has an r−tupled fixed point.

If we put ψ(t) = 0 in Theorem 2.1, we get the following result:

Corollary 2.5. Let (X, d) be a metric space. Assume F : Xr → CB(X) and g : X → X be two mappings satisfying H(F(x1, x2, …, xr), F(y1, y2, …, yr)) ≤ φ [max {d(gx1, gy1), d(gx2, gy2), …, d(gxr, gyr)}], for all x1, x2, …, xr, y1, y2, …, yr ∈ X, where φ ∈ Φ. Furthermore assume that F(Xr) ⊆ g(X) and g(X) is a complete subset of X. Then F and g have an r−tupled coincidence point. Moreover, F and g have a common r−tupled fixed point, if one of the following conditions holds:

(a) F and g are w−compatible. limi→∞ gix1 = y1, limi→∞ gix2 = y2, …, limi→∞ gixr = yr, for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X and g is continuous at y1, y2, …, yr.

(b) g is F−weakly commuting for some (x1, x2, …, xr) ∈ C{F, g}, gx1, gx2, …, gxr are fixed points of g, that is, g2x1 = gx1, g2x2 = gx2, …, g2xr = gxr.

(c) g is continuous at x1, x2, …, xr. limi→∞ giy1 = x1, limi→∞ giy2 = x2, …, limi→∞  giyr = xr for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X.

(d) g(C{F, g}) is a singleton subset of C{F, g}.

If we put ψ(t) = 0 in Corollary 2.2, we get the following result:

Corollary 2.6. Let (X, d) be a metric space. Assume F : Xr → CB(X) and g : X → X be two mappings satisfying for all x1, x2, …, xr, y1, y2, …, yr ∈ X, where φ ∈ Φ. Furthermore assume that F(Xr) ⊆ g(X) and g(X) is a complete subset of X: Then F and g have an r−tupled coincidence point. Moreover, F and g have a common r−tupled fixed point, if one of the following conditions holds:

(a) F and g are w−compatible. limi→∞ gix1 = y1, limi→∞ gix2 = y2, …, limi→∞ gixr = yr, for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X and g is continuous at y1, y2, …, yr.

(b) g is F−weakly commuting for some (x1, x2, …, xr) ∈ C{F, g}, gx1, gx2, …, gxr are fixed points of g, that is, g2x1 = gx1, g2x2 = gx2, …, g2xr = gxr.

(c) g is continuous at x1, x2, …, xr. limi→∞ giy1 = x1, limi→∞ giy2 = x2, …, limi→∞  giyr = xr for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X.

(d) g(C{F, g}) is a singleton subset of C{F, g}.

If we put g = I (the identity mapping) in the Corollary 2.5, we get the following result:

Corollary 2.7. Let (X, d) be a complete metric space, F : Xr → CB(X) be a mapping satisfying H(F(x1, x2, …, xr), F(y1, y2, …, yr)) ≤ φ [max {d(x1, y1), d(x2, y2), …, d(xr, yr)}],for all x1, x2, …, xr, y1, y2, …, yr ∈ X, where φ ∈ Φ. Then F has an r−tupled fixed point.

If we put g = I (the identity mapping) in the Corollary 2.6, we get the following result:

Corollary 2.8. Let (X, d) be a complete metric space, F : Xr → CB(X) be a mapping satisfying for all x1, x2, …, xr, y1, y2, …, yr ∈ X, where φ ∈ Φ. Then F has an r−tupled fixed point.

If we put φ(t) = kt where 0 < k < 1 in Corollary 2.5, we get the following result:

Corollary 2.9. Let (X, d) be a metric space. Assume F : Xr → CB(X) and g : X → X be two mappings satisfyingH(F(x1, x2, …, xr), F(y1, y2, …, yr)) ≤ k max {d(gx1, gy1), d(gx2, gy2), …, d(gxr, gyr)},for all x1, x2, …, xr, y1, y2, …, yr ∈ X, where 0 < k < 1. Furthermore assume that F(Xr) ⊆ g(X) and g(X) is a complete subset of X. Then F and g have an r−tupled coincidence point. Moreover, F and g have a common r−tupled fixed point, if one of the following conditions holds:

(a) F and g are w−compatible. limi→∞ gix1 = y1, limi→∞ gix2 = y2, …, limi→∞ gixr = yr, for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X and g is continuous at y1, y2, …, yr.

(b) g is F−weakly commuting for some (x1, x2, …, xr) ∈ C{F, g}, gx1, gx2, …, gxr are fixed points of g, that is, g2x1 = gx1, g2x2 = gx2, …, g2xr = gxr.

(c) g is continuous at x1, x2, …, xr. limi→∞ giy1 = x1, limi→∞ giy2 = x2, …, limi→∞  giyr = xr for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X.

(d) g(C{F, g}) is a singleton subset of C{F, g}.

If we put φ(t) = kt where 0 < k < 1 in Corollary 2.6, we get the following result:

Corollary 2.10. Let (X, d) be a metric space. Assume F : Xr → CB(X) and g : X → X be two mappings satisfyingfor all x1, x2, …, xr, y1, y2, …, yr ∈ X, where 0 < k < 1. Furthermore assume that F(Xr) ⊆ g(X) and g(X) is a complete subset of X. Then F and g have an r−tupled coincidence point. Moreover, F and g have a common r−tupled fixed point, if one of the following conditions holds:

(a) F and g are w−compatible. limi→∞ gix1 = y1, limi→∞ gix2 = y2, …, limi→∞ gixr = yr, for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X and g is continuous at y1, y2, …, yr.

(b) g is F−weakly commuting for some (x1, x2, …, xr) ∈ C{F, g}, gx1, gx2, …, gxr are fixed points of g, that is, g2x1 = gx1, g2x2 = gx2, …, g2xr = gxr.

(c) g is continuous at x1, x2, …, xr. limi→∞ giy1 = x1, limi→∞ giy2 = x2, …, limi→∞  giyr = xr for some (x1, x2, …, xr) ∈ C{F, g} and for some y1, y2, …, yr ∈ X.

(d) g(C{F, g}) is a singleton subset of C{F, g}.

If we put g = I (the identity mapping) in the Corollary 2.9, we get the following result:

Corollary 2.11. Let (X, d) be a complete metric space, F : Xr → CB(X) be a mapping satisfying H(F(x1, x2, …, xr), F(y1, y2, …, yr)) ≤ k max {d(x1, y1), d(x2, y2), …, d(xr, yr)},for all x1, x2, …, xr, y1, y2, …, yr ∈ X, where 0 < k < 1. Then F has an r−tupled fixed point.

If we put g = I (the identity mapping) in the Corollary 2.10, we get the following result:

Corollary 2.12. Let (X, d) be a complete metric space, F : Xr → CB(X) be a mapping satisfying for all x1, x2, …, xr, y1, y2, …, yr ∈ X, where 0 < k < 1. Then F has an r−tupled fixed point.