Abstract
In this paper, we first establish some new fixed point theorems for -functions. By using these results, we can obtain some generalizations of Kannan's fixed point theorem and Chatterjea's fixed point theorem for nonlinear multivalued contractive maps in complete metric spaces. Our results generalize and improve some main results in the literature and references therein.
Mathematics Subject Classifications
47H10; 54H25
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1. Introduction
Throughout this paper, we denote by ℕ and ℝ, the sets of positive integers and real numbers, respectively. Let (X, d) be a metric space. For each x ∈ X and A ⊆ X, let d(x, A) = infy ∈ Ad(x, y). Denote by the family of all nonempty subsets of the family of all nonempty closed subsets of X and the class of all nonempty closed bounded subsets of X, respectively.
For any , define a function by
then is said to be the Hausdorff metric on induced by the metric d on X. A point x in X is a fixed point of a map T if Tx = x (when T: X → X is a single-valued map) or x ∈ Tx (when T: X → 2Xis a multivalued map). The set of fixed points of T is denoted by .
It is known that many metric fixed point theorems were motivated from the Banach contraction principle (see, e.g., [1]) that plays an important role in various fields of applied mathematical analysis. Later, Kannan [2, 3] and Chatterjea [4] established the following fixed point theorems.
Theorem K. (Kannan [2, 3]) Let (X,d) be a complete metric space and T: X → X be a selfmap. Suppose that there exists such that
Then, T has a unique fixed point in X.
Theorem C. (Chatterjea [4]) Let (X,d) be a complete metric space and T: X → X be a selfmap. Suppose that there exists such that
Then, T has a unique fixed point in X.
Let f be a real-valued function defined on ℝ. For c ∈ ℝ, we recall that
and
Definition 1.1. [5–10] A function φ: [0, ∞) → [0,1) is said to be an -function if it satisfies Mizoguchi-Takahashi's condition ( i.e., lim sups → t+ φ(s) < 1 for all t ∈ [0, ∞)).
It is obvious that if φ: [0, ∞) → [0,1) is a nondecreasing function or a nonincreasing function, then φ is an -function. So the set of -functions is a rich class. But it is worth to mention that there exist functions that are not -functions.
Example 1.1. [8] Let φ: [0, ∞) → [0, 1) be defined by
Since is not an -function.
Very recently, Du [8] first proved some characterizations of -functions.
Theorem D. [8] Let φ: [0, ∞) → [0,1) be a function. Then, the following statements are equivalent.
-
(a)
φ is an -function.
-
(b)
For each t ∈ [0, ∞), there exist and such that for all .
-
(c)
For each t ∈ [0, ∞), there exist and such that for all .
-
(d)
For each t ∈ [0, ∞), there exist and such that for all .
-
(e)
For each t ∈ [0, ∞), there exist and such that for all .
-
(f)
For any nonincreasing sequence {x n }n ∈ℕin [0, ∞), we have 0 ≤ supn ∈ℕφ(x n ) < 1.
-
(g)
φ is a function of contractive factor [10]; that is, for any strictly decreasing sequence {x n }n ∈ℕin [0, ∞), we have 0 ≤ supn ∈ℕφ(x n ) < 1.
In 2007, Berinde and Berinde [11] proved the following interesting fixed point theorem.
Theorem BB. (Berinde and Berinde [11]) Let (X,d) be a complete metric space, be a multivalued map, φ: [0, ∞) → [0,1) be an -function and L ≥ 0. Assume that
Then .
It is quite obvious that if let L = 0 in Theorem BB, then we can obtain Mizoguchi-Takahashi's fixed point theorem [12] that is a partial answer of Problem 9 in Reich [13, 14].
Theorem MT. (Mizoguchi and Takahashi [12]) Let (X,d) be a complete metric space, be a multivalued map and φ: [0, ∞) → [0,1) be an -function. Assume that
Then .
In fact, Mizoguchi-Takahashi's fixed point theorem is a generalization of Nadler's fixed point theorem, but its primitive proof is difficult. Later, Suzuki [15] give a very simple proof of Theorem MT. Recently, Du [5] established new fixed point theorems for τ0-metric (see Def. 2.1 below) and -functions to extend Berinde-Berinde's fixed point theorem. In [5], some generalizations of Kannan's fixed point theorem, Chatterjea's fixed point theorem and other new fixed point theorems for nonlinear multivalued contractive maps were given.
In this paper, we first establish some new fixed point theorems for -functions. By using these results, we can obtain some generalizations of Kannan's fixed point theorem and Chatterjea's fixed point theorem for nonlinear multivalued contractive maps in complete metric spaces. Our results generalize and improve some main results in [1–5, 7–9, 12–15] and references therein.
2. Preliminaries
Let (X, d) be a metric space. Recall that a function p: X × X → [0, ∞) is called a w-distance [1, 16, 17], if the following are satisfied:
(w 1) p(x, z) ≤ p(x, y) + p(y, z) for any x, y, z ∈ X;
(w 2) for any x ∈ X,p(x, ⋅) : X → [0, ∞) is l.s.c;
(w 3) for any ε > 0, there exists δ > 0 such that p(z, x) ≤ δ and p(z, y) ≤ δ imply d(x, y) ≤ ε.
Recently, Lin and Du introduced and studied τ-functions [5, 9, 18–22]. A function p: X × X → [0, ∞) is said to be a τ-function, if the following conditions hold:
(τ 1) p(x, z) ≤ p(x, y) + p(y, z) for all x, y, z ∈ X;
(τ 2) If x ∈ X and {y n } in X with limn →∞y n = y such that p(x, y n ) ≤ M for some M = M(x) > 0, then p(x, y) ≤ M;
(τ 3) For any sequence {x n } in X with limn →∞sup{p(x n , x m ): m >n} = 0, if there exists a sequence {y n } in X such that limn →∞p(x n , y n ) = 0, then limn →∞d(x n , y n ) = 0;
(τ 4) For x, y, z ∈ X,p(x, y) = 0 and p(x, z) = 0 imply y = z.
Note that not either of the implications p(x, y) = 0 ⇔ x = y necessarily holds and p is nonsymmetric in general. It is well-known that the metric d is a w-distance and any w-distance is a τ-function, but the converse is not true; see [5, 19].
The following Lemma is essentially proved in [19]. See also [5, 8, 20, 22].
Lemma 2.1. [5, 8, 19, 20, 22] Let (X,d) be a metric space and p: X × X → [0, ∞) be any function. Then, the following hold:
-
(a)
If p satisfies (w 2), then p satisfies (τ 2);
-
(b)
If p satisfies (w 1) and (w 3), then p satisfies (τ 3);
-
(c)
Assume that p satisfies (τ 3). If {x n } is a sequence in X with limn →∞sup{p(x n ,x m ): m >n} = 0, then {x n } is a Cauchy sequence in X.
Let (X, d) be a metric space and p: X × X → [0, ∞) a τ-function. For each x ∈ X and A ⊆ X, let
Recall that a selfmap T: X → X is said to be
-
(a)
Kannan's type [2, 5, 16] if there exists , such that d(Tx, Ty) ≤ γ{d(x, Tx)+d(y, Ty)} for all x, y ∈ X;
-
(b)
Chatterjea's type [3, 5] if there exists , such that d(Tx, Ty) ≤ γ{d(x, Ty) + d(y, Tx)} for all x, y ∈ X.
Lemma 2.2. [5, 9, 21, 22] Let A be a closed subset of a metric space (X, d) and p: X × X → [0, ∞) be any function. Suppose that p satisfies (τ 3) and there exists u ∈ X such that p(u, u) = 0. Then, p(u, A) = 0 if and only if u ∈ A.
Recently, Du [5, 21] first has introduced the concepts of τ0-functions and τ0-metrics as follows.
Definition 2.1. [5, 9, 21, 22] Let (X, d) be a metric space. A function p: X × X → [0, ∞) is called a τ0-function if it is a τ-function on X with p(x, x) = 0 for all x ∈ X.
Remark 2.1. If p is a τ0-function then, from (τ 4), p(x, y) = 0 if and only if x = y.
Example 2.1. [5] Let X = ℝ with the metric d(x, y) = |x —y| and 0 <a <b. Define the function p: X × X → [0, ∞) by
Then, p is nonsymmetric, and hence, p is not a metric. It is easy to see that p is a τ0-function.
Definition 2.2. [5, 9, 21, 22] Let (X, d) be a metric space and p be a τ0-function (resp. w0-distance). For any , define a function by
where δ p (A, B) = supx ∈ Ap(x, B) and δ p (B, A) = supx ∈ Bp(x, A), then is said to be the τ0-metric (resp. w0-metric) on induced by p.
Clearly, any Hausdorff metric is a τ0-metric, but the reverse is not true. It is well-known that every τ0-metric is a metric on ; for more detail, see [5, 9, 21, 22].
Lemma 2.3. Let (X,d) be a metric space, be a multivalued map and {z n } be a sequence in X satisfying zn +1∈ Tz n , n ∈ ℕ, and {z n } converge to v in X. Then, the following statements hold.
-
(a)
If T is closed (that is, GrT = {(x, y) ∈ X × X: y ∈ Tx}, the graph of T, is closed in X × X), then .
-
(b)
Let p be a function satisfying (τ 3) and p(v, v) = 0. If limn →∞p(z n , zn +1) = 0 and the map f: X → [0, ∞) defined by f(x) = p(x, Tx) is l.s.c., then .
-
(c)
If the map g: X → [0, ∞) defined by g(x) = d(x, Tx) is l.s.c., then .
-
(d)
Let p be a function satisfying (τ 3). If limn →∞p(z n , Tv) = 0 and lim n →∞sup{p(z n , z m ): m >n} = 0, then .
Proof.
-
(a)
Since T is closed, zn +1∈ Tz n , n ∈ ℕ and z n → v as n → ∞, we have v ∈ Tv. So .
-
(b)
Since z n → v as n → ∞, by the lower semicontinuity of f, we obtain
which implies p(v, Tv) = 0. By Lemma 2.2, we get .
-
(c)
Since {z n } is convergent in X, limn →∞d(z n , z n +1) = 0. Since
we have d(v,Tv) = 0 and hence .
-
(d)
Since limn →∞sup{p(z n , z m ): m >n} = 0 and limn →∞p(z n , Tv) = 0, there exists {a n } ⊂ {z n } with limn →∞sup{p(a n , a m ): m >n} = 0 and {b n } ⊂ Tv such that limn →∞p(a n , b n ) = 0. By (τ 3), limn →∞d(a n , b n ) = 0. Since a n → v as n → ∞ and d(b n ,v) ≤ d(b n ,a n ) + d(a n ,v), it implies b n → v as n → ∞. By the closedness of Tv, we have v ∈ Tv or . □
In this paper, we first introduce the concepts of capable maps as follows.
Definition 2.3. Let (X, d) be a metric space and be a multivalued map. We say that T is capable if T satisfies one of the following conditions:
(D1) T is closed;
(D2) the map f: X → [0, ∞) defined by f(x) = p(x, Tx) is l.s.c;
(D3) the map g: X → [0, ∞) defined by g(x) = d(x, Tx) is l.s.c;
(D4) for each sequence {x n } in X with xn +1∈ Tx n , n ∈ ℕ and limn →∞x n = v, we have limn →∞p(x n , Tv) = 0;
(D5) inf{p(x, z) + p(x,Tx) : x ∈ X} > 0 for every .
Remark 2.2.
-
(1)
Let (X, ||⋅||) be a Banach space. If is u.s.c, then T is a capable map since it is closed (for more detail, see [5, 23]).
-
(2)
Let (X, d) be a metric space and be u.s.c. Since the function f: X → [0, ∞) defined by f(x) = d(x,Tx) is l.s.c. (see, e.g., [24, Lemma 3.1] and [25, Lemma 2]), T is a capable map.
-
(3)
Let (X, d) be a metric space and be a generalized multivalued (φ, L)-weak contraction [11], that is, there exists an -function φ and L ≥ 0 such that
Then, T is a capable map. Indeed, let {x n } in X with xn +1∈ Tx n , n ∈ ℕ and limn →∞x n = v.
Then
which means that T satisfies (D4).
-
(4)
Let (X, d) be a metric space and T: X → X is a single-valued map of Kannan's type, then T is a capable map since (D5) holds; for more detail, see [[16], Corollary 3].
3. Fixed point theorems of generalized Chatterjea's type and others
Below, unless otherwise specified, let (X, d) be a complete metric space, p be a τ0-function and be a τ0-metric on induced by p.
In this section, we will establish some fixed point theorems of generalized Chatterjea's type.
Theorem 3.1. Let be a capable map. Suppose that there exists an -function φ: [0, ∞) → [0,1) such that for each x ∈ X,
Then .
Proof. Let κ: [0, ∞) → [0,1) be defined by . Then
Let x1 ∈ X and x2 ∈ Tx1. If x1 = x2, then and we are done. Otherwise, if x2 ≠ x1, by Remark 2.1, we have p(x1,x2) > 0. If x1 ∈ Tx2, then it follows from (3.1) that
which implies p(x2,Tx2) = 0. Since p is a τ0-function and Tx2 is closed in X, by Lemma 2.2, x2 ∈ Tx2 and . If x1 ∉ Tx2, then p(x1,Tx2) > 0 and, by (3.1), there exists x3 ∈ Tx2 such that
By induction, we can obtain a sequence {x n } in X satisfying xn +1∈ Tx n , n ∈ ℕ, p(x n , xn +1) > 0
and
By (3.2), we get
Since 0 <κ(t) < 1 for all for all n ∈ ℕ. So the sequence {p(x n , xn +1)} is strictly decreasing in [0, ∞). Since φ is an -function, by applying (g) of Theorem D, we have
Hence, it follows that
Let λ:= supn ∈ℕκ(p(x n , xn +1)) and take . Then λ, c ∈ (0,1). We claim that {x n } is a Cauchy sequence in X. Indeed, by (3.3), we have
It implies from (3.4) that
We have limn →∞sup{p(x n ,x m ): m >n} = 0. Indeed, let . For m, n ∈ ℕ with m >n, we have
Since c ∈ (0,1), limn →∞αn = 0 and, by (3.5), we get
Applying (c) of Lemma 2.1, {x n } is a Cauchy sequence in X. By the completeness of X, there exists v ∈ X such that x n → v as n → ∞. From (τ 2) and (3.5), we have
Now, we verify that . Applying Lemma 2.3, we know that if T satisfies one of the conditions (D1), (D2), (D3) and (D4).
Finally, assume (D5) holds. On the contrary, suppose that v ∉ Tv. Then, by (3.5) and (3.7), we have
a contradiction. Therefore . The proof is completed. □
Here, we give a simple example illustrating Theorem 3.1.
Example 3.1. Let X = [0,1] with the metric d(x,y) = |x — y| for x,y ∈ X. Then, (X,d) is a complete metric space. Let be defined by
and φ: [0, ∞) → [0,1) be defined by
Then, φ is an -function and .
On the other hand, one can easily see that
So f(x): = d(x,Tx) is l.s.c., and hence, T is a capable map. Moreover, it is not hard to verify that for each x ∈ X,
Therefore, all the assumptions of Theorem 3.1 are satisfied, and we also show that from Theorem 3.1.
Theorem 3.2. Let be a capable map and φ: [0, ∞) → [0,1) be an -function. Let k ∈ ℝ with k ≥ 2. Suppose that for each x ∈ X
Then .
Proof. Since k ≥ 2, (3.9) implies (3.1). Therefore, the conclusion follows from Theorem 3.1. □
The following result is immediate from the definition of and Theorem 3.1.
Theorem 3.3. Let be a capable map. Suppose that there exists an -function φ: [0, ∞) → [0,1) such that for each x ∈ X,
Then .
Theorem 3.4. Let be a capable map. Suppose that there exist two -functions φ, τ: [0, ∞) → [0,1) such that
Then .
Proof. For each x ∈ X, let y ∈ Tx be arbitrary. Since p(y,Tx) = 0, we have . Therefore, the conclusion follows from Theorem 3.3. □
Theorem 3.5. Let be a capable map. Suppose that there exists an -function φ: [0, ∞) → [0,1) such that
Then .
Proof. Let τ = φ. Then, the conclusion follows from Theorem 3.4. □
Theorem 3.6. Let T: X → X be a selfmap. Suppose that there exists an -function φ: [0, ∞) → [0,1) such that
Then, T has a unique fixed point in X.
Proof. Let p ≡ d. Then, (3.11) and (3.10) are identical. We prove that T is a capable map. In fact, it suffices to show that (D5) holds. Assume that there exists w ∈ X with w ≠ Tw and inf {d(x,w) + d(x,Tx): x ∈ X} = 0. Then, there exists a sequence {x n } in X such that limn →∞(d(x n , w) + d(x n ,Tx n )) = 0. It follows that d(x n ,w) → 0 and d(x n ,Tx n ) → 0 and hence d(w,Tx n ) → 0 or Tx n → w as n → ∞. By hypothesis, we have
for all n ∈ ℕ. Letting n → ∞ in (3.12), since φ is an -function and d(x n ,w) → 0, we have d(w,Tw) <d(w,Tw), which is a contradiction. So (D5) holds and hence T is a capable map. Applying Theorem 3.5, . Suppose that there exists with u ≠ v. Then, by (3.11), we have
a contradiction. Hence, is a singleton set. □
Applying Theorem 3.6, we obtain the following primitive Chatterjea's fixed point theorem [3].
Corollary 3.1. [3] Let T: X → X be a selfmap. Suppose that there exists such that
Then, T has a unique fixed point in X.
Proof. Define φ: [0, ∞) → [0,1) by φ(t) = 2γ. Then, φ is an -function. So (3.13) implies (3.11), and the conclusion is immediate from Theorem 3.6. □
Corollary 3.2. Let be a capable map. Suppose that there exist such that
Then .
Proof. Let φ, τ: [0, ∞) → [0,1) be defined by φ(t) = 2α and τ(t) = 2β for all t ∈ [0, ∞). Then, φ and τ are -functions, and the conclusion follows from Theorem 3.4. □
The following conclusion is immediate from Corollary 3.2 with α = β = γ.
Corollary 3.3. Let be a capable map. Suppose that there exists such that
Then .
Remark 3.1.
-
(a)
Corollary 3.2 and Corollary 3.3 are equivalent. Indeed, it suffices to prove that Corollary 3.2 implies Corollary 3.3. Suppose all assumptions of Corollary 3.2 are satisfied. Let γ:= max {α, β}. Then and (3.14) implies (3.15), and the conclusion of Corollary 3.3 follows from Corollary 3.2.
-
(b)
Theorems 3.1-3.4 and Corollaries 3.1 and 3.2 all generalize and improve [5, Theorem 3.4] and the primitive Chatterjea's fixed point theorem [3].
4. Fixed point theorems of generalized Kannan's type and others
The following result is given essentially in [5, Theorem 2.1].
Theorem 4.1. Let be a capable map. Suppose that there exists an -function φ: [0, ∞) → [0,1) such that for each x ∈ X,
Then .
Applying Theorem 4.1, we establish the following new fixed point theorem.
Theorem 4.2. Let be a capable map. Suppose that there exist two -functions φ, τ: [0, ∞) → [0,1) such that for each x ∈ X,
Then .
Proof. Notice that for each x ∈ X, if y ∈ Tx, then (4.2) implies
and hence
Applying Theorem 4.1, we can get the thesis. □
The following conclusion is immediate from Theorem 4.2.
Theorem 4.3. Let be a capable map. Suppose that there exist two -functions φ, τ: [0, ∞) → [0,1) such that
Then .
Theorem 4.4. Let be a capable map. Suppose that there exists an -function φ: [0, ∞) → [0,1) such that for each x ∈ X,
Then .
Theorem 4.5. Let be a capable map. Suppose that there exists an -function φ: [0, ∞) → [0,1) such that
Then .
Theorem 4.6. Let T: X → X be a selfmap. Suppose that there exists an -function φ: [0, ∞) → [0,1) such that
Then, T has a unique fixed point in X.
Proof. Let p ≡ d. Then, (4.3) and (4.4) are identical. We prove that T is a capable map. In fact, it suffices to show that (D5) holds. Assume that there exists w ∈ X with w ≠ Tw and inf {d(x, w) + d(x,Tx): x ∈ X} = 0. Then, there exists a sequence {x n } in X such that limn→∞ (d(x n , w) + d(x n ,Tx n )) = 0. It follows that d(x n ,w) → 0 and d(x n ,Tx n ) → 0 and hence d(w,Tx n ) → 0 or Tx n → w as n → ∞. By hypothesis, we have
for all n ∈ ℕ. Since d(x n ,w) → 0 as n → ∞ and φ is an -function, limn →∞φ(d(x n ,w)) < 1. Letting n → ∞ in (4.5), since Tx n → w and d(x n ,Tx n ) → 0 as n → ∞, we have 2d(w,Tw) <d(w, Tw), which is a contradiction. So (D5) holds and hence T is a capable map. Applying Theorem 4.5, . Suppose that there exists with u ≠ v. Then, by (4.4), we have
a contradiction. Hence, is a singleton set. □
Applying Theorem 4.6, we obtain the primitive Kannan's fixed point theorem [2].
Corollary 4.1. Let T: X → X be a selfmap. Suppose that there exists such that
Then .
Corollary 4.2. Let T: X → X be a selfmap. Suppose that there exist such that
Then .
Remark 4.1. Corollary 4.1 and Corollary 4.2 are indeed equivalent.
Corollary 4.3. Let be a capable map. Suppose that there exist such that
Then .
Corollary 4.4. Let be a capable map. Suppose that there exists such that
Then .
Remark 4.2.
-
(a)
Corollary 4.3 and Corollary 4.4 are indeed equivalent.
-
(b)
Theorems 4.1-4.6 and Corollaries 4.1-4.4 all generalize and improve [5, Theorem 2.6] and the primitive Kannan's fixed point theorem [2].
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Acknowledgements
The first author was supported by the Natural Science Foundation of Yunnan Province (2010ZC152) and the Scientific Research Foundation from Yunnan Province Education Committee (08Y0338); the second author was supported partially by grant no. NSC 100-2115-M-017-001 of the National Science Council of the Republic of China.
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He, Z., Du, WS. & Lin, IJ. The existence of fixed points for new nonlinear multivalued maps and their applications. Fixed Point Theory Appl 2011, 84 (2011). https://doi.org/10.1186/1687-1812-2011-84
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DOI: https://doi.org/10.1186/1687-1812-2011-84