Fixed point results for multivalued contractive maps

Using the concept of u-distance, we prove a fixed point theorem for multivalued contractive maps. We also establish a multivalued version of the Caristi's fixed point theorem and common fixed point result. Consequently, several known fixed point results are either improved or generalized including the corresponding fixed point results of Caristi, Mizoguchi-Takahashi, Kada et al., Suzuki-Takahashi, Suzuki, and Ume.

Mathematics Subject Classification (2000): 47H10; 54H25.

Let X be a complete metric space with metric d. We denote the collection of nonempty subsets of X, nonempty closed subsets of X and nonempty closed bounded subsets of X by 2^X, Cl(X), CB(X), respectively Let H be the Hausdorff metric with respect to d, that is,

for every A, B ∈ CB(X), where d(x, B) = inf_y∈B d(x, y).

A point x ∈ X is called a fixed point of T : X → 2^X if x ∈ T(x). A point x ∈ X is called a common fixed point of f : X → X and T if f(x) = x ∈ T(x).

A sequence {x_n} in X is called an orbit of T at x₀ ∈ X if x_n ∈ T(x_n-1) for all n ≥ 1. A map ϕ : X → ℝ is called lower semicontinuous if for any sequence {x_n} ⊂ X with x_n → x ∈ X imply that .

The well known Banach contraction principle, which asserts that "each single-valued contraction selfmap on a complete metric space has a unique fixed point" has been generalized in many different directions. Among these generalizations, the following Caristi's fixed point theorem [1] may be the most valuable one and has extensive applications in the field of mathematics.

Theorem 1.1. Let X be a complete metric space and let ψ : X → (-∞, ∞] be a proper, lower semicontinuous bounded below function. Let f be a single-valued selfmap of X. If for each x ∈ X

then f has a fixed point.

Investigations on the existence of fixed points for multivalued maps in the setting of metric spaces was initiated by Nadler [2]. Using the concept of Hausdorff metric, he generalized Banach contraction principle which states that each multivalued contraction map T : X → CB(X) has fixed point provided X is complete. Since then, many authors have used the Hausdorff metric to obtain fixed point results for multivalued maps. For example, see [3-6], and references therein.

Kada et al. [7] introduced the notion of w-distance on a metric space as follows:

A function ω : X × X → ℝ₊ is called w-distance on X if it satisfies the following for x, y, z ∈ X:

(w₁) ω(x, z) ≤ ω(x, y) + ω(y, z);

(w₂) the map ω(x,.) : X → ℝ₊ is lower semicontinuous; i.e., for {y_n} in X with y_n → y ∈ X, ;

(w₃) for any ε > 0, there exists δ > 0 such that ω(z, x) ≤ δ and ω(z, y) ≤ δ imply d(x, y) ≤ ε.

Note that, in general for x, y ∈ X, ω(x, y) ≠ ω(y, x), and ω(x, y) = 0 ⇔ x = y does not necessarily hold. Clearly, the metric d is a w-distance on X. Examples and properties of a w-distance can be found in [7,8]. For single valued maps, Kada et al. [7] improved several classical results including the Caristi's fixed point theorem by replacing the involved metric with a generalized distance. Using this generalized distance, Suzuki and Takahashi [9] have introduced notions of single-valued and multivalued weakly contractive maps and proved fixed point results for such maps. Consequently, they generalized the Banach contraction principle and Nadler's fixed point result. Recent fixed point results concerning w-distance can be found [4,8,10-13].

Recently, Susuki [14] generalized the concept of w-distance by introducing the following notion of τ-distance on metric space (X, d).

A function p : X × X → ℝ₊ is a τ-distance on X if it satisfies the following conditions for any x, y, z ∈ X:

(τ₁) p(x, z) ≤ p(x, y) + p(y, z);

(τ₂) η(x, 0) = 0 and η(x, t) ≥ t for all x ∈ X and t ≥ 0, and η is concave and continuous in its second variable;

(τ₃) lim_n x_n = x and lim_n sup{η(z_n, p(z_n, x_m)) : m ≥ n} = 0 imply p(u, x) ≤ lim_n inf p(u, x_n) for all u ∈ X;

(τ₄) lim_n sup{p(x_n, y_m) : m ≥ n} = 0 and lim_n η(x_n, t_n) = 0 imply lim_n η(y_n, t_n) = 0;

(τ₅) lim_n η(z_n, p(z_n, x_n)) = 0 and lim_n η(z_n, p(z_n, y_n)) = 0 imply lim_n d(x_n, y_n) = 0.

Examples and properties of τ-distance are given in [14]. In [14], Suzuki improved several classical results including the Caristi's fixed point theorem for single-valued maps with respect to τ-distance.

In the literature, several other kinds of distances and various versions of known results are appeared. For example, see [15-19], and references therein. Most recently, Ume [20] generalized the notion of τ-distance by introducing u-distance as follows:

A function p : X × X → ℝ₊ is called u-distance on X if there exists a function θ : X × X × ℝ₊ × ℝ₊ → ℝ₊ such that the following hold for x, y, z ∈ X:

(u₁) p(x, z) ≤ p(x, y) + p(y, z).

(u₂) θ(x, y, 0, 0) = 0 and θ(x, y, s, t) ≥ min{s, t} for each s, t ∈ ℝ₊, and for every ε > 0, there exists δ > 0 such that | s - s₀ | < δ, | t - t₀ | < δ, s, s₀, t, t₀ ∈ R₊ and y ∈ X imply

(u₃) lim_n→∞ x_n = x

imply

(u₄)

imply

or

imply

(u₅)

imply

or

imply

Remark 1.1. [20] (a) Suppose that θ from X × X × ℝ₊ × ℝ₊ into ℝ₊ is a mapping satisfying (u2) ~ (u5). Then there exists a mapping η from X × X × ℝ₊ × ℝ₊ into ℝ₊ such that η is nondecreasing in its third and fourth variable, respectively satisfying (u2)η ~ (u5)η, where (u2)η ~ (u5)η stand for substituting η for θ in (u2) ~ (u5), respectively

(b) In the light of (a), we may assume that θ is nondecreasing in its third and fourth variables, respectively, for a function θ from X × X × ℝ₊ × ℝ₊ into ℝ₊ satisfying (u2) ~ (u5).

(c) Each τ-distance p on a metric space (X, d) is also a u-distance on X.

We present some examples of u-distance which are not τ-distance. (For the detail, see [20]).

Example 1.1. Let X = ℝ₊ with the usual metric. Define p: X × X → ℝ₊ by . Then p is a u-distance on X but not a τ-distance on X.

Example 1.2. Let X be a normed space with norm ||.||. Then a function p: X × X → ℝ₊ defined by p(x, y) = ||x|| for every x, y ∈ X is a u-distance on X but not a τ-distance.

It follows from the above examples and Remark 1.1(c) that u-distance is a proper extension of τ-distance. Other useful examples are also given in [20]).

Let X be a metric space with a metric d and let p be a u-distance on X. Then a sequence {x_n} in X is called p-Cauchy [20] if there exists a function θ from X × X × ℝ₊ × ℝ₊ into ℝ₊ satisfying (u2) ~ (u5) and a sequence {z_n} of X such that

or

The following lemmas concerning u-distance are crucial for the proofs of our results.

Lemma 1.1. Let X be a metric space with a metric d and let p be a u-distance on X. If {x_n} is a p-Cauchy sequence, then {x_n} is a Cauchy sequence.

Lemma 1.2. Let X be a metric space with a metric d and let p be a u-distance on X.

(1) If sequences {x_n} and {y_n} of X satisfy lim_n→∞ p(z, x_n) = 0, and lim_n→∞ p(z, y_n) = 0 for some z ∈ X, then lim_n→∞ d(x_n, y_n) = 0.

(2) If p(z, x) = 0 and p(z, y) = 0, then x = y.

(3) Suppose that sequences {x_n} and {y_n} of X satisfy lim_n→∞ p(x_n, z) = 0, and lim_n→∞ p(y_n, z) = 0 for some z ∈ X. Then lim_n→∞ d(x_n, y_n) = 0.

(4) If p(x, z) = 0 and p(y, z) = 0, then x = y.

Lemma 1.3. Let X be a metric space with a metric d and let p be a u-distance on X. Suppose that a sequence {x_n} of X satisfies

or

Then {x_n} is a p-Cauchy sequence.

Using u-distance, Ume [20] generalized Caristi's fixed point theorem as follows:

Theorem 1.2. Let X be a complete metric space with metric d, let ϕ : X → (-∞, ∞] be a proper lower semicontinuous function which is bounded from below. Let p be a u-distance on X. Suppose that f is a single-valued selfmap of X such that

for all x ∈ X. Then there exists x₀ ∈ X such that fx₀ = x₀, and p(x₀, x₀) = 0.

We say a multivalued map T : X → 2^X is contractive with respect to u-distance p on X (in short, p-contractive) if there exist a u-distance p on X and a constant r ∈ (0, 1) such that for any x, y ∈ X and u ∈ T(x), there is υ ∈ T(y) satisfying

In particular, a single-valued map g : X → X is p-contractive if there exist a u-distance p on X and a constant r ∈ (0, 1) such that for each x, y ∈ X

In this article, using the concept of u-distance, first we prove a useful lemma for multivalued mappings in metric spaces. Then using our lemma we prove a fixed point result for closed valued p-contraction mappings. Also, we prove multivalued version of the Caristi's fixed point theorem and then applying this result we establish common fixed point theorem. Consequently, several known fixed point results are either improved or generalized.

Using Lemma 1.3, we prove the following key lemma in the setting of metric spaces.

Lemma 2.1. Let X be a metric space with metric d. Let T : X → Cl(X) be a p-contractive map. Then, there exists an orbit {u_n} of T at u₀ such that {u_n} is a Cauchy sequence.

Proof. Let u₀ be an arbitrary but fixed element of X and let u₁ ∈ Tu₀ be fixed. Since T is p-contractive, there exists u₂ ∈ Tu₁ such that

where r ∈ (0, 1). Continuing this process, we get a sequence {u_n} in X such that u_n+1 ∈ Tu_n and

for all n ∈ ℕ. Thus for any n ∈ ℕ, we have

Now, for any n, m ∈ ℕ with m > n,

and hence

By Lemma 1.3, {u_n} is a p-Cauchy sequence and hence by Lemma 1.1, {u_n} is a Cauchy sequence.

Now, applying Lemma 2.1 we prove the following fixed point result for multivalued p-contractive maps.

Theorem 2.2. Let X be a complete metric space with metric d and let T : X → Cl(X) be p-contractive map. Then there exists x₀ ∈ X such that x₀ ∈ Tx₀ and p(x₀, x₀) = 0.

Proof. By Lemma 2.1, there exists a Cauchy sequence {u_n} in X such that u_n ∈ Tu_n-1 for each n ∈ ℕ. Since X is complete, {u_n} converges to some υ₀ ∈ X. For n ∈ ℕ, from (u₃) and the proof of Lemma 2.1, we have

Since u_n ∈ Tu_n-1 and T is p-contractive, there exist w_n ∈ Tv₀ such that

Thus for any n ∈ ℕ

and so . Now, since it follows from Lemma 1.2 that

Since the sequence {w_n} ⊂ Tυ₀ and Tυ₀ is closed, we get υ₀ ∈ Tυ₀. Since T is p-contractive map so for such υ₀ there is υ₁ ∈ Tυ₀ such that

Thus, we also have a sequence {υ_n} in X such that υ_n+1 ∈ Tυ_n and

for all n ∈ ℕ. Now, as in the proof of Lemma 2.1 we get υ_n is a p-Cauchy sequence in X and thus it converges to some x₀ ∈ X. Moreover, we have , which implies p(v₀, x₀) = 0. So for any n ∈ ℕ we have

Now, since and , so by Lemma 1.2, it follows that d(x₀, υ₀) = 0. Hence we get x₀ = υ₀ and p(υ₀, v₀) = 0.

A direct consequence of Theorem 2.2 is the following generalization of the Banach contraction principle.

Corollary 2.3. Let X be a complete metric space with metric d. If a single-valued map T : X → X is p-contractive, then T has a unique fixed point x₀ ∈ X. Further, such x₀ satisfies p(x₀, x₀) = 0.

Proof. By Theorem 2.2, it follows that there exists x₀ ∈ X with Tx₀ = x₀ and p(x₀, x₀) = 0 For the uniqueness of x₀ we let y₀ = Ty₀. Then by the definition of T there exist r ∈ (0, 1) such that p(x₀, y₀) = p(Tx₀, Ty₀) ≤ rp(x₀, y₀), and p(y₀, y₀) = p(Ty₀, Ty₀) ≤ rp(y₀, y₀). Thus

and hence by Lemma 1.2, we have x₀ = y₀.

Remark 2.4. Since w-distance and τ-distance are u-distance, Theorem 2.2 is a generalization of [[9], Theorem 1], while Corollary 2.3 contains [[9], Theorem 2] and [[14], Theorem 2].

We now prove a multivalued version of the Caristi's fixed point theorem with respect to u-distance.

Theorem 2.5. Let X be a complete metric space and let ϕ : X → (-∞, ∞] be proper, lower semicontinuous bounded below function. Let T : X → 2^X. Assume that there exists a u-distance p on X such that for every x ∈ X, there exists y ∈ Tx satisfying

Then T has a fixed point x₀ ∈ X such that p(x₀, x₀) = 0.

Proof. For each x ∈ X, we put f(x) = y, where y ∈ T(x) ⊂ X and ϕ(y) + p(x, y) ≤ ϕ(x). Note that f is a selfmap of X satisfying

for every x ∈ X. Since the map ϕ is proper, there exists u ∈ X with ϕ(u) < ∞ and so we get p(u, u) = 0. Put

and assume that for a sequence {x_n} in M either or . Note that M is nonempty because u ∈ M. Now, we show that the set M is closed. Let {x_n} be a sequence in M which converges to some x ∈ X. Then {x_n} is a p-Cauchy sequence and thus it follows from (u₃) that

Using the lower semicontinuity ϕ it is easy to show that the set M is closed in X. Thus M is a complete metric space. Now, we show that the set M is invariant under f. Note that for each x ∈ M, we have

and thus

It follows that f(x) ∈ M and hence f is a selfmap of M. Applying Theorem 1.2, there exists x₀ ∈ M such that f(x₀) = x₀ ∈ T(x₀) and p(x₀, x₀) = 0.

Remark 2.6. Theorem 2.5 is a multivalued version of Theorem 1.2 due to Ume [20] and generalizes a fixed point result due to Mizoguchi and Takahashi [[5], Theorem 1]. Further, Theorem 2.5 contains [[7], Theorem 2] and [[14], Theorem 3] which are single-valued generalizations of the Caristi's fixed point theorem.

Theorem 2.7. Let X be a complete metric space, f be a single-valued selfmap of X with f(X) = M complete and let T : X → 2^X be such that T(X) ⊂ M. Assume that there exists a u-distance p on X such that for every x ∈ X, there exists y ∈ Tx satisfying

where ϕ : M → (-∞, ∞] is proper, lower semicontinuous, and bounded from below. Then, there exits a point x₀ ∈ M such that x₀ ∈ fT(x₀).

Proof. For each y ∈ M, define

Clearly, J carries M into 2^M. Now, for each s ∈ J(y), there exists some t ∈ T(y) with s = f(t) and p(y, f(t)) ≤ ϕ (y) - ϕ(f(t)), that is; p(y, s) ≤ ϕ(y) - ϕ(s). Since ϕ is proper, there exists z ∈ M with ϕ(z) < + ∞.

Let

and assume that for a sequence {x_n} in Y either or . Note that Y is nonempty closed subset of a complete space M. Thus Y is a complete metric space. Now we show that Y is invariant under the map J. Now, let s ∈ J(y), y ∈ Y. By definition of J, there exists t ∈ T(y) such that s = f(t), and

and hence

proving that s ∈ Y and hence J(y) ⊂ Y for all y ∈ Y. Now, Theorem 2.5 guarantees that there exits x₀ ∈ M such that x₀ ∈ J(x₀) = fT(x₀).

Finally, we obtain a common fixed point result.

Theorem 2.8. Suppose that X, M, f, and T satisfy the assumptions of Theorem 2.7. Moreover, the following conditions hold:

(a) f and T commute weakly.

(b) x ∉ Fix(f) implies x ∉ fT(x).

Then T and f have a common fixed point in M.

Proof. As in the proof of Theorem 2.7, there exits x₀ ∈ M such that x₀ ∈ fT(x₀). Using conditions (a) and (b), we obtain

Thus, x₀ must be a common fixed point of f and T.

The authors declare that they have no competing interests.

All authors participated in the design of this work and performed equally. All authors read and approved the final manuscript.

The authors were grateful to the referees for their valuable comments and suggestions. Also, the authors thank the Deanship of Scientific Research (DSR), King Abdulaziz University for the research grant # 85/363/1431.

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