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Mixed g-monotone property and quadruple fixed point theorems in partially ordered metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 71 (2012)
Abstract
In this manuscript, we prove some quadruple coincidence and common fixed point theorems for F : X4 → X and g : X → X satisfying generalized contractions in partially ordered metric spaces. Our results unify, generalize and complement various known results from the current literature. Also, an application to matrix equations is given.
2000 Mathematics subject Classifications: 46T99; 54H25; 47H10; 54E50.
1 Introduction and preliminaries
Existence of fixed points in partially ordered metric spaces was first investigated by Turinici [1], where he extended the Banach contraction principle in partially ordered sets. In 2004, Ran and Reurings [2] presented some applications of Turinici's theorem to matrix equations. Following these initial articles, some remarkable results were reported see, e.g., [3–13].
Gnana Bhashkar and Lakshmikantham in [14] introduced the concept of a coupled fixed point of a mapping F : X × X → X and investigated some coupled fixed point theorems in partially ordered complete metric spaces. Later, Lakshmikantham and Ćirić [15] proved coupled coincidence and coupled common fixed point theorems for nonlinear mappings F : X × X → X and g : X → X in partially ordered complete metric spaces. Various results on coupled fixed point have been obtained, since then see, e.g., [6, 9, 16–33]. Recently, Berinde and Borcut [34] introduced the concept of tripled fixed point in ordered sets.
For simplicity, we denote by Xk where k ∈ ℕ. Let us recall some basic definitions.
Definition 1.1 (See [34]) Let (X, ≤) be a partially ordered set and F: X3 → X. The mapping F is said to has the mixed monotone property if for any x, y, z ∈ X
Definition 1.2 Let F : X3 → X. An element (x, y, z) is called a tripled fixed point of F if
Also, Berinde and Borcut [34] proved the following theorem:
Theorem 1.1 Let (X,≤, d) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X3 → X having the mixed monotone property. Suppose there exist j, r, l ≥ 0 with j + r + l < 1 such that
for any x, y, z ∈ X for which × ≤ u, v ≤ y and z ≤ w. Suppose either F is continuous or X has the following properties:
1. if a non-decreasing sequence x n → x, then x n ≤ x for all n,
2. if a non-increasing sequence y n → y, then y ≤ y n for all n.
If there exist x0, y0, z0 ∈ X such that x0 ≤ F (x0, y0, z0), y0 ≥ F (y0, x0, z0) and z0 ≤ F (z0, y0, x0), then there exist x, y, z ∈ X such that
that is, F has a tripled fixed point.
Recently, Aydi et al. [35] introduced the following concepts.
Definition 1.3 Let (X, ≤) be a partially ordered set. Let F : X3 → X and g : X → X. The mapping F is said to has the mixed g-monotone property if for any x, y, z ∈ X
Definition 1.4 Let F : X3 → X and g : X → X. An element (x, y, z) is called a tripled coincidence point of F and g if
(gx, gy, gz) is said a tripled point of coincidence of F and g.
Definition 1.5 Let F : X3 → X and g : X → X. An element (x, y, z) is called a tripled common fixed point of F and g if
Definition 1.6 Let X be a non-empty set. Then we say that the mappings F : X3 → X and
g : X → X are commutative if for all x, y, z ∈ X
The notion of fixed point of order N ≥ 3 was first introduced by Samet and Vetro [36]. Very recently, Karapinar used the concept of quadruple fixed point and proved some fixed point theorems on the topic [37]. Following this study, quadruple fixed point is developed and some related fixed point theorems are obtained in [38–41].
Definition 1.7 [38] Let X be a nonempty set and F : X4 → X be a given mapping. An element (x, y, z, w) ∈ X × X × X × X is called a quadruple fixed point of F if
Let (X, d) be a metric space. The mapping given by
defines a metric on X4, which will be denoted for convenience by d.
Definition 1.8 [38] Let (X, ≤) be a partially ordered set and F : X4 → X be a mapping. We say that F has the mixed monotone property if F (x, y, z, w) is monotone non-decreasing in x and z and is monotone non-increasing in y and w; that is, for any x, y, z, w ∈ X,
and
In this article, we establish some quadruple coincidence and common fixed point theorems for F : X4 → X and g : X → X satisfying nonlinear contractions in partially ordered metric spaces. Also, some interesting corollaries are derived and an application to matrix equations is given.
2 Main results
We start this section with the following definitions.
Definition 2.1 Let (X, ≤) be a partially ordered set. Let F : X4 → X and g : X → X. The mapping F is said to has the mixed g-monotone property if for any x, y, z, w ∈ X
Definition 2.2 Let F : X4 → X and g : X → X. An element (x, y, z, w) is called a quadruple coincidence point of F and g if
(gx, gy, gz, gw) is said a quadruple point of coincidence of F and g.
Definition 2.3 Let F : X4 → X and g : X → X. An element (x, y, z, w) is called a quadruple common fixed point of F and g if
Definition 2.4 Let X be a non-empty set. Then we say that the mappings F : X4 → X and g : X → × are commutative if for all x, y, z, w ∈ X
Let Φ be the set of all functions ϕ : [0, ∞) → [0, ∞) such that:
-
1.
ϕ(t) < t for all t ∈ (0,+∞).
-
2.
for all t ∈ (0,+∞).
For simplicity, we define the following.
Now, we state the first main result of this article.
Theorem 2.1 Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Suppose F : X4 → X and g : X → X are such that F is continuous and has the mixed g-monotone property. Assume also that there exist ϕ ∈ Φ and L ≥ 0 such that
for any x, y, z, w, u, v, h, l ∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gh and gl ≤ gw. Suppose F (X4) ⊂ g(X), g is continuous and commutes with F. If there exist x0, y0, z0, w0 ∈ X such that
then there exist x, y, z, w ∈ X such that
that is, F and g have a quadruple coincidence point.
Proof. Let x0, y0, z0, w0 ∈ X such that
Since F (X4) ⊂ g(X), then we can choose x1, y1, z1, w1 ∈ X such that
Taking into account F (X4) ⊂ g(X), by continuing this process, we can construct sequences {x n }, {y n }, {z n }, and {w n } in X such that
We shall show that
For this purpose, we use the mathematical induction. Since, gx0 ≤ F (x0, y0, z0, w0), gy0 ≥ F (y0, z0, w0, x0), gz0 ≤ F (z0, w0, x0, y0), and gw0 ≥ F (w0, x0, y0, z0), then by (4), we get
that is, (6) holds for n = 0.
We presume that (6) holds for some n > 0. As F has the mixed g-monotone property and gx n ≤ gxn+1, gyn+1≤ gy n , gz n ≤ gzn+1and gwn+1≤ gw n , we obtain
and
Thus, (6) holds for any n ∈ ℕ. Assume for some n ∈ ℕ,
then, by (5), (x n , y n , z n , w n ) is a quadruple coincidence point of F and g. From now on, assume for any n ∈ ℕ that at least
By (2) and (5), it is easy that
Due to (3) and (8), we have
and
Having in mind that ϕ (t) < t for all t > 0, so from (9)-(12) we obtain that
It follows that
Thus, {max{d(gx n , gxn+1), d(gy n , gyn+1), d(gz n , gzn+1), d(gw n , gwn+1)}} is a positive decreasing sequence. Hence, there exists r ≥ 0 such that
Suppose that r > 0. Letting n → +∞ in (13), we obtain that
It is a contradiction. We deduce that
We shall show that {gx n }, {gy n }, {gz n }, and {gw n } are Cauchy sequences in the metric space (X, d). Assume the contrary, that is, one of the sequence {gx n }, {gy n }, {gz n } or {gw n } is not a Cauchy, that is,
or
This means that there exists ε > 0, for which we can find subsequences of integers (m k ) and (n k ) with n k > m k > k such that
Further, corresponding to m k we can choose n k in such a way that it is the smallest integer with n k > m k and satisfying (17). Then
By triangular inequality and (18), we have
Thus, by (16) we obtain
Similarly, we have
and
Again by (18), we have
Letting k → + ∞ and using (16), we get
and
Using (17) and (23)-(26), we have
By (16), it is easy to see that
Now, using inequality (3), we obtain
and
From (29)-(32), we deduce that
Letting k → +∞ in (33) and having in mind (27) and (28), we get that
it is a contradiction. Thus, {gx n }, {gy n }, {gz n }, and {gw n } are Cauchy sequences in (X, d).
Since (X, d) is complete, there exist x, y, z, w ∈ X such that
From (34) and the continuity of g, we have
From (5) and the commutativity of F and g, we have
and
Now we shall show that gx = F (x, y, z, w), gy = F (y, z, w, x), gz = F (z, w, x, y), and gw = F (w, x, y, z).
By letting n → +∞ in (36) - (39), by (34), (35) and the continuity of F , we obtain
and
We have proved that F and g have a quadruple coincidence point. This completes the proof of Theorem 2.1.
In the following theorem, we omit the continuity hypothesis of F. We need the following definition.
Definition 2.5 Let (X, ≤) be a partially ordered metric set and d be a metric on X. We say that (X, d, ≤) is regular if the following conditions hold:
(i) if non-decreasing sequence a n → a, then a n ≤ a for all n,
(ii) if non-increasing sequence b n → b, then b ≤ b n for all n.
Theorem 2.2 Let (X, ≤) be a partially ordered set and d be a metric on X such that (X, d, ≤) is regular. Suppose F : X4 → X and g : X → X are such that F has the mixed g-monotone property. Assume that there exist ϕ ∈ Φ and L ≥ 0 such that
for any x, y, z, w, u, v, h, l ∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gh, and gl ≤ gw. Also, suppose F (X4) ⊂ g(X) and (g(X), d) is a complete metric space. If there exist x0, y0, z0, w0 ∈ X such that gx0 ≤ F (x0, y0, z0, w0), gy0 ≥ F (y0, z0, w0, x0), gz0 ≤ F (z0, w0, x0, y0) and gw0 ≥ F (w0, x0, y0, z0), then there exist x, y, z, w ∈ X such that
that is, F and g have a quadruple coincidence point.
Proof. Proceeding exactly as in Theorem 2.1, we have that {gx n }, {gy n }, {gz n }, and {gw n } are Cauchy sequences in the complete metric space (g(X), d). Then, there exist x, y, z, w ∈ X such that
Since {gx n }, {gz n } are non-decreasing and {gy n }, {gw n } are non-increasing, then since (X, d, ≤) is regular we have
for all n. If gx n = gx, gy n = gy, gz n = gz, and gw n = gw for some n ≥ 0, then gx = gx n ≤ gxn+1≤ gx = gx n , gy ≤ gyn+1≤ gy n = gy, gz = gz n ≤ gzn+1≤ gz = gz n , and gw ≤ gwn+1≤ gw n = gw, which implies that
and
that is, (x n , y n , z n , w n ) is a quadruple coincidence point of F and g. Then, we suppose that (gx n , gy n , gz n , gw n ) ≠ (gx, gy, gz, gw) for all n ≥ 0. By (3), consider now
Taking n → ∞ and using (44), the quantity M(x n , y n , z n , w n , x, y, z, w) tends to 0 and so the right-hand side of (45) tends to 0, hence we get that d(gx, F (x, y, z, w)) = 0. Thus, gx = F (x, y, z, w). Analogously, one finds
Thus, we proved that F and g have a quartet coincidence point. This completes the proof of Theorem 2.2.
Corollary 2.1 Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Suppose F : X4 → X and g : X → X are such that F is continuous and has the mixed g-monotone property. Assume also that there exist ϕ ∈ Φ a non-decreasing function and L ≥ 0 such that
for any x, y, z, w, u, v, h, l,∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Suppose F (X4) ⊂ g(X), g is continuous and commutes with F .
If there exist x0, y0, z0, w0 ∈ X such that gx0 ≤ F (x0, y0, z0, w0), gy0 ≥ F (y0, z0, w0, x0), gz0 ≤ F (z0, w0, x0, y0), and gw0 ≥ F (w0, x0, y0, z0), then there exist x, y, z, w ∈ X such that
Proof. It suffices to remark that
Then, we apply Theorem 2.1, since ϕ is assumed to be non-decreasing.
Similarly, as an easy consequence of Theorem 2.2 we have the following corollary.
Corollary 2.2 Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d, ≤) is regular. Suppose F : X4 → X and g : X → X are such that F has the mixed g-monotone property. Assume also that there exist ϕ ∈ Φ a non-decreasing function and L ≥ 0 such that
for any x, y, z, w, u, v, h, l ∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Also, suppose F (X4) ⊂ g(X) and (g(X), d) is a complete metric space.
If there exist x0, y0, z0, w0 ∈ X such that gx0 ≤ F (x0, y0, z0, w0), gy0 ≥ F (y0, z0, w0, x0), gz0 ≤ F (z0, w0, x0, y0), and gw0 ≥ F (w0, x0, y0, z0), then there exist x, y, z, w ∈ X such that