Some endpoint results for β-generalized weak contractive multifunctions.

We introduce β-generalized weak contractive multifunctions and give some results about endpoints of the multifunctions. Also, we give some results about role of a point in the existence of endpoints.

1. Introduction

Let (X, d) be a metric space, CB(X) the collection of all nonempty bounded and closed subsets of X, and H the Hausdorff metric with respect to d; that is, H(A, B) = max⁡{sup d(x, B), sup d(y, A)} for all A, B∈CB(X), where d(x, B) = inf⁡ d(x, y). Let T : X → 2 be a multifunction. An element x ∈ X is said to be a fixed point of T whenever x ∈ Tx. Also, an element x ∈ X is said to be an endpoint of T whenever Tx = {x} [1]. We say that T has the approximate endpoint property whenever inf⁡sup⁡ d(x, y) = 0 [1]. Let f : X → X be a mapping. We say that f has the approximate endpoint property whenever inf⁡ d(x, fx) = 0 [1]. Also, the function g : ℝ → ℝ is called upper semicontinuous whenever limsup⁡ g(λ ) ≤ g(λ) for all sequences {λ } with λ → λ [2]. In 2010, Amini-Harandi defined the concept of approximate endpoint property for multifunctions and proved the following result (see [1]).

Theorem

Let ψ : [0, ∞)→[0, ∞) be an upper semicontinuous function such that ψ(t) < t and liminf⁡(t − ψ(t)) > 0 for all t > 0, (X, d) a complete metric space, and T : X → CB(X) a multifunction satisfing H(Tx, Ty) ≤ ψ(d(x, y)) for all x, y ∈ X. Then T has a unique endpoint if and only if T has the approximate endpoint property.

Then Moradi and Khojasteh introduced the concept of generalized weak contractive multifunctions and improved Theorem 1 by providing the following result [3].

Let ψ : [0, ∞)→[0, ∞) be an upper semicontinuous function such that ψ(t) < t and liminf⁡(t − ψ(t)) > 0 for all t > 0, (X, d) a complete metric space, and T : X → CB(X) a generalized weak contractive multifunction; that is, T satisfies H(Tx, Ty) ≤ ψ(N(x, y)) for all x, y ∈ X, where N(x, y) = max⁡{d(x, y), d(x, Tx), d(y, Ty), (d(x, Ty) + d(y, Tx))/2}. Then T has a unique endpoint if and only if T has the approximate endpoint property.

In this paper, we introduce β-generalized weak contractive multifunctions, and by adding some conditions to assumptions of the results, we give some results about endpoints of β-generalized weak contractive multifunctions. In 2012, the technique of α-ψ-contractive mappings was introduced by Samet et al. [4]. Later, some authors used it for some subjects in fixed point theory (see for example [5-8]) or generalized it by using the method of β-ψ-contractive multifunctions (see e.g., [9-12]).

Let (X, d) be a metric space and β : 2 × 2 → [0, ∞) a mapping. A multifunction T : X → 2 is called β-generalized weak contraction whenever there exists a nondecreasing, upper, semicontinuous function ψ : [0, +∞)→[0, +∞) such that ψ(t) < t for all t > 0 and for all x, y ∈ X. We say that T is β-admissible whenever β(A, B) ≥ 1 implies that β(Tx, Ty) ≥ 1 for all x ∈ A, and y ∈ B, where A and B are subsets of X. We say that T has the property (R) whenever for each convergent sequence {x } in X with x → x and β(Tx , Tx ) ≥ 1 for all n ≥ 1, we have β(Tx , Tx) ≥ 1. One can find idea of the property (R) for mappings in [13]. We say that T has the property (K) whenever for each sequence {x } in X with β(Tx , Tx ) ≥ 1 for all n ≥ 1, there exists a natural number k such that β(Tx , Tx ) ≥ 1 for all m > n ≥ k. Finally, we say that T has the property (H) whenever for each ɛ > 0, there exists z ∈ X such that sup⁡ d(z, a) < ɛ implies that for every x ∈ X there exists y ∈ Tx such that H(Tx, Ty) = sup⁡ d(y, b). A multifunction T : X → 2 is called lower semicontinuous at x 0 ∈ X whenever for each sequence {x } in X with x → x 0 and every y ∈ Tx 0, there exists a sequence {y } in X with y ∈ Tx for all n ≥ 1 such that y → y [14].

2. Main Results

Now, we are ready to state and prove our main results.

Let (X, d) be a complete metric space, β : 2 × 2 → [0, ∞) a mapping, and T : X → CB(X) a β-admissible, β-generalized weak contractive multifunction which has the properties (R), (K), and (H). Suppose that there exist a subset A of X and x 0 ∈ A such that β(A, Tx 0) ≥ 1. Then T has an endpoint if and only if T has the approximate endpoint property.

Proof

It is clear that if T has an endpoint, then T has the approximate endpoint property. Conversely, suppose that T has the approximate endpoint property. Choose A ⊂ X and x 0 ∈ A such that β(A, Tx 0) ≥ 1. Since T has the approximate endpoint property, for each ɛ > 0, there exists z ∈ X such that sup⁡ d(z, a) < ɛ. Now by using the condition (H), choose x 1 ∈ Tx 0 such that H(Tx 0, Tx 1) = sup⁡ d(x 1, a). Also, choose x 2 ∈ Tx 1 such that H(Tx 1, Tx 2) = sup⁡ d(x 2, a), and by continuing this process, we find a sequence {x } in X such that x ∈ Tx and for all n ≥ 1. Since β(A, Tx 0) ≥ 1 and T is β-admissible, β(Tx 0, Tx 1) ≥ 1. By using induction, it is easy to see that β(Tx , Tx ) ≥ 1 for all n ≥ 1. Thus, we obtain for all n ≥ 1. If N(x , x ) = d(x , x ), then If N(x , x ) = d(x , Tx ), then If N(x , x ) = d(x , Tx ), then and so d(x , x ) = 0. Thus, d(x , x ) ≤ ψ(d(x , x )). If then d(x , x ) ≤ ψ(d(x , x )) (other case implies that d(x , x ) = 0). Thus, for all n ≥ 1. We claim that ψ(0) = 0. If ψ(0) > 0, then ψ 2(0) ≥ ψ(0) > 0 because ψ is nondecreasing. On the other hand, since ψ(t) < t for all t > 0, we have ψ 2(0) < ψ(0) which is a contradiction. Hence, ψ(0) = 0. Let d = d(x , x ) for all n. If there exists a natural number n 0 such that d = 0, then it is easy to see that d = 0 for all n ≥ n 0, and so lim⁡ d = 0. Now suppose that d ≠ 0 for all n. In this case, we have d ≤ ψ(d ) < d for all n. Hence, {d } is a decreasing sequence, and so there exists d ≥ 0 such that lim⁡ d = d. If d > 0, then d > 0 for all n, and so d ≤ ψ(d ) < d for all n. Since ψ is upper and semicontinuous, we obtain d = lim⁡ d ≤ lim⁡ ψ(d ) ≤ ψ(lim⁡ d ) = ψ(d) < d which is a contradiction. Thus, lim⁡ d = 0. Now, we prove that {x } is a Cauchy sequence. If {x } is not a Cauchy sequence, then there exist ɛ > 0 and natural numbers m , n such that m > n ≥ k and d(x , x ) ≥ ɛ for all k ≥ 1. Also, we choose m as small as possible such that Thus, ɛ ≤ d(x , x ) ≤ d(x , x ) + d(x , x ) ≤ d + ɛ for all k. Hence, lim⁡ d(x , x ) = ɛ. Since T has the property (K), we obtain for all k. Since lim⁡ d(x , x ) = ɛ, lim⁡ N(x , x ) = ɛ. In fact, and so ɛ = lim⁡ d(x , x ) ≤ lim⁡ N(x , x ) ≤ ɛ. Since ψ is upper semicontinuous, by using ((∗)) we obtain which is a contradiction, and so {x } is a Cauchy sequence. Choose x* ∈ X such that x → x*. Now, note that for all n, and so for all n, and so lim⁡ H({x }, Tx ) = 0. Since T has the property (R), we obtain for all n. If N(x , x*) = d(x*, Tx*), then we have for all n. This implies that H({x*}, Tx*) ≤ ψ(H({x*}, Tx*)), and so If N(x , x*) = d(x , x*) or N(x , x*) ≤ d(x , x ), then it is easy to see that H({x*}, Tx*) = 0. Thus, x* is an endpoint of T.

Next example shows that a β-generalized weak contractive multifunction is not necessarily a generalized weak contractive multifunction.

Example

Let X = ℝ. Define T : X → CB(X) by Tx = [x, x + 2] for all x ∈ X. Suppose that ψ : [0, +∞)→[0, +∞) is an arbitrary upper semicontinuous function such that ψ(t) < t for all t > 0. If x = 0 and y = 2, then H(Tx, Ty) = H([0,2], [2,4]) = 2 and N(x, y) = 2. Hence, Thus, T is not a generalized weak contractive multifunction. Now, suppose that ψ(t) = t/2 for all t ≥ 0 and define β : 2 × 2 → [0, ∞) by β(A, B) = 1/2 for all subsets A and B of X. Then, we have for all x, y ∈ ℝ. Thus, T is a β-generalized weak contractive multifunction.

Next example shows that there are multifunctions which satisfy the conditions of Theorem 3, while they are not generalized weak contractive multifunctions.

Let X = [0, 9/2] and let d(x, y) = |x − y|. Define T : X → CB(X) by If x = 1 and y = 3/2, then where ψ : [0, +∞)→[0, +∞) is an arbitrary upper semicontinuous function such that ψ(t) < t for all t > 0. Thus, T is not a generalized weak contractive multifunction. Now, we show that T satisfies all conditions of Theorem 3. For this aim, define ψ(t) = t/2 and β(A, B) = 1 whenever A and B are subsets of [0,1] and β(A, B) = 0 otherwise. First suppose that x ∉ [0,1] or that y ∉ [0,1]. If x, y ∈ (3/2, 9/2], then Tx, Ty ⊂ [0,1] and β(Tx, Ty) = 1. But, H(Tx, Ty) = 0, and so β(Tx, Ty)H(Tx, Ty) ≤ ψ(N(x, y)). If x ∈ (1, 3/2] or y ∈ (1, 3/2], then Tx⊈[0,1], or Ty⊈[0,1] and so β(Tx, Ty) = 0. Hence, β(Tx, Ty)H(Tx, Ty) ≤ ψ(N(x, y)). Now, suppose that x, y ∈ [0,1]. In this case, we have β(Tx, Ty) ≥ 1, H(Tx, Ty) = H({x/2}, {y/2}) = (1/2)d(x, y), and N(x, y) = max⁡{d(x, y), x/2, y/2, (d(x, y/2) + d(y, x/2))/2}. Thus, d(x, y) ≤ N(x, y), and so Therefore, T is a β-generalized weak contractive multifunction. Now, we show that T is β-admissible. If β(A, B) ≥ 1, then A, B ⊂ [0,1], and so Tx = {x/2}∈[0,1] and Ty = {y/2}∈[0,1] for all x ∈ A and y ∈ B. Thus, β(Tx, Ty) ≥ 1 for all x ∈ A and y ∈ B. Now, suppose A = [0, 1/2] and x 0 = 1/4. Then, Tx 0 = {1/8}∈[0,1] and [0, 1/2]⊂[0,1]. Hence, β(A, Tx 0) ≥ 1. Now, we show that T satisfies the condition (H). First note that, for each ɛ > 0, there exists z ∈ X such that sup⁡ d(z, a) < ɛ. Now, we show that for each x ∈ X there exists y ∈ Tx such that H(Tx, Ty) = sup⁡ d(y, b). If 0 ≤ x ≤ 1, then Tx = {x/2}, T(x/2) = {x/4}, and Since for 1 < x ≤ 3/2 we have 5/2 < 4x − (3/2) ≤ 9/2, T(4x − (3/2)) = {0}. Thus, If 3/2 < x ≤ 9/2, then Tx = {0} and T(0) = {0}. Hence, It is easy to check that T satisfies the conditions (R) and (K). Note that, 0 is the endpoint of T.

Now, we add an assumption to obtain uniqueness of endpoint. In this way, we introduce a new notion. Let X be a set and β : 2 × 2 → [0, ∞) a map. We say that the set X has the property (G ) whenever β(A, B) ≥ 1 for all subsets A and B of X with A⊈B or B⊈A.

Corollary

Let (X, d) be a complete metric space, β : 2 × 2 → [0, ∞) a mapping, and T : X → CB(X) a β-admissible, β-generalized weak contractive multifunction which has the properties (R), (K), and (H). Suppose that there exist a subset A of X and x 0 ∈ A such that β(A, Tx 0) ≥ 1. If T has the approximate endpoint property and X has the property (G ), then T has a unique endpoint.

By using Theorem 3, T has a endpoint. If T has two distinct endpoints x* and y*, then β(Tx*, Ty*) = β({x*}, {y*}) ≥ 1 because X has the property (G ). Hence, which is a contradiction. Thus, T has a unique endpoint.

In Example 5, T has a unique endpoint, while X does has not the property (G ). Also, T has the property (R), while T is not lower semicontinuous. To see this, consider the sequence {x } defined by for k ≥ 1 and put y = 1/2 and x 0 = 1. Then x → 1 and y ∈ Tx 0 = {1/2}. Let {y } be an arbitrary sequence in X such that y ∈ Tx for all n ≥ 1. Then, y 2 ∈ Tx 2 and y 2 ∈ Tx 2 for all k. But, y 2 = 4x 2 − (3/2) for sufficiently large k and y 2 = x 2/2 for all k since y 2 → 5/2, y ↛1/2. This implies that T is not lower semicontinuous.

Let (X, d) be a complete metric space, β : 2 × 2 → [0, ∞) a mapping, and T : X → CB(X) a β-admissible multifunction which has the properties (R), (K), and (H). Suppose that X has the property (G ), and there exist a subset A of X, x 0 ∈ A and k ∈ [0,1) such that β(A, Tx 0) ≥ 1 and β(Tx, Ty)H(Tx, Ty) ≤ kN(x, y) for all x, y ∈ X. Then T has a unique endpoint if and only if T has the approximate endpoint property.

It is sufficient that we define ψ(t) = kt for all t ≥ 0. Then, Theorem 3 and Corollary 6 guarantee the result.

It has been proved that lower semicontinuity of the multifunction T and the property (R) are independent conditions [9]. We can replace lower semicontinuity of the multifunction instead of the property (R) to obtain the next result. Its proof is similar to the proof of Theorem 3.

Let (X, d) be a complete metric space, β : 2 × 2 → [0, ∞) a mapping, and T : X → CB(X) a lower semicontinuous, β-admissible, β-generalized weak contractive multifunction which has the properties (K) and (H). Suppose that there exist a subset A of X and x 0 ∈ A such that β(A, Tx 0) ≥ 1. Then T has the approximate endpoint property if and only if T has an endpoint.

Let (X, d) be a complete metric space, β : 2 × 2 → [0, ∞) a mapping, and T : X → CB(X) a lower semicontinuous, β-admissible, β-generalized weak contractive multifunction which has the properties (K) and (H). Suppose that there exist a subset A of X and x 0 ∈ A such that β(A, Tx 0) ≥ 1. If T has the approximate endpoint property and X has the property (G ), then T has a unique endpoint.

Let (X, d) be a complete metric space, β : 2 × 2 → [0, ∞) a mapping, and T : X → CB(X) a β-admissible multifunction which has the properties (R), (K), and (H). Suppose that X has the property (G ), and there exist a subset A of X, x 0 ∈ A and k ∈ [0,1) such that β(A, Tx 0) ≥ 1 and β(Tx, Ty)H(Tx, Ty) ≤ kN(x, y) for all x, y ∈ X. If T has the approximate endpoint property, then Fix⁡(T) = End(T) = {x}.

If we put ψ(t) = kt, then, by using Theorem 2.10 in [9], T has a fixed point. Since T has the approximate endpoint property, by using Corollary 7, T has a unique endpoint such x. Let y ∈ Fix⁡(T). If Tx = Ty, then y = x. If Tx ≠ Ty, then β(Tx, Ty) ≥ 1 because X has the property (G ). Also, we have But, N(x, y) = max⁡{d(x, y), d(x, Tx), d(y, Ty), (d(x, Ty) + d(y, Tx))/2} = d(x, y). Thus, d(x, y) = 0, and so Fix⁡(T) = End(T) = {x}.

Next corollary shows us the role of a point in the existence of endpoints.

Let (X, d) be a complete metric space, x* ∈ X a fixed element, and T : X → CB(X) a multifunction such that T has the property (H) and x* ∈ Tx∩Ty for all subsets A and B of X with x* ∈ A∩B and all x ∈ A and y ∈ B. Assume that H(Tx, Ty) ≤ ψ(N(x, y)) for all x, y ∈ X with x* ∈ Tx∩Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A 0 of X and x 0 ∈ A 0 such that x* ∈ A 0∩Tx 0. Assume that for each convergent sequence {x } in X with x → x and x* ∈ Tx ∩Tx , for all n ≥ 1, one has x* ∈ Tx ∩Tx. Also, for each sequence {x } in X with x* ∈ Tx ∩Tx for all n ≥ 1, there exists a natural number k such that x* ∈ Tx ∩Tx for all m > n ≥ k. Then T has an endpoint if and only if T has the approximate endpoint property.

It is sufficient we define β : 2 × 2 → [0, ∞) by β(A, B) = 1 whenever x* ∈ A∩B and β(A, B) = 0 otherwise, and then we use Theorem 3.

Let (X, d) be a complete metric space, x* ∈ X a fixed element and T : X → CB(X) a lower semicontinuous multifunction such that T has the property (H) and x* ∈ Tx∩Ty for all subsets A and B of X with x* ∈ A∩B and all x ∈ A and y ∈ B. Assume that for all x, y ∈ X with x* ∈ Tx∩Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A 0 of X and x 0 ∈ A 0 such that x* ∈ A 0∩Tx 0. Assume that for each convergent sequence {x } in X with x → x and x* ∈ Tx ∩Tx for all n ≥ 1, we have x* ∈ Tx ∩Tx. Then T has an endpoint if and only if T has the approximate endpoint property.

It is sufficient to define β : 2 × 2 → [0, ∞) by β(A, B) = 1 whenever x* ∈ A∩B and β(A, B) = 0 otherwise, and then we use Theorem 8.

Let (X, d, ≤) be an ordered metric space. Define the order ⪯ on arbitrary subsets A and B of X by A⪯B if and only if for each a ∈ A there exists b ∈ B such that a ≤ b. It is easy to check that (CB(X), ⪯) is a partially ordered set.

Let (X, d, ≤) be a complete ordered metric space and T a closed and bounded valued multifunction on X such that T has the property (H) and Tx⪯Ty for all subsets A and B of X with A⪯B and all x ∈ A and y ∈ B. Assume that H(Tx, Ty) ≤ ψ(N(x, y)) for all x, y ∈ X with Tx⪯Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A 0 of X and x 0 ∈ A 0 such that A 0⪯Tx 0. Assume that for each convergent sequence {x } in X with x → x and Tx ⪯Tx , for all n ≥ 1, one has Tx ⪯Tx. Also, for each sequence {x } in X with Tx ⪯Tx for all n ≥ 1, there exists a natural number k such that Tx ⪯Tx for all m > n ≥ k. Then T has an endpoint if and only if T has the approximate endpoint property.

Define β(A, B) = 1 whenever A⪯B and β(A, B) = 0 otherwise, and then we use Theorem 3.

Let (X, d, ≤) be a complete ordered metric space and T a closed and bounded valued multifunction on X such that T has the property (H) and Tx⪯Ty for all subsets A and B of X with A⪯B, all x ∈ A, and y ∈ B. Assume that H(Tx, Ty) ≤ ψ(N(x, y)) for all x, y ∈ X with Tx⪯Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A 0 of X and x 0 ∈ A 0 such that A 0⪯Tx 0. Assume that for each convergent sequence {x } in X with x → x and Tx ⪯Tx , for all n ≥ 1, one has Tx ⪯Tx. Also, for each sequence {x } in X with Tx ⪯Tx for all n ≥ 1, there exists a natural number k such that Tx ⪯Tx for all m > n ≥ k. If T has the approximate endpoint property and A⪯B for all subsets A and B of X with A⊈B or B⊈A, then T has a unique endpoint.

Define β(A, B) = 1 whenever A⪯B and β(A, B) = 0 otherwise, and then we use Corollary 6.

Let (X, d) be a metric space and T : X → 2 a multifunction. We say that T is an HS-multifunction whenever for each x ∈ X there exists y ∈ Tx such that H(Tx, Ty) = sup⁡ d(y, b). It is obvious that each HS-multifunction is an multifunction which has the property (H). Thus, one can conclude similar results to above ones for HS-multifunctions. Here, we provide some ones. Although by considering HS-multifunction we restrict ourselves, we obtain strange results with respect to above ones. One can prove the following by reading exactly the proofs of similar above results.

Let (X, d) be a complete metric space, β : 2 × 2 → [0, ∞) a mapping, and T : X → CB(X) a β-admissible, β-generalized weak contractive HS-multifunction which has the properties (R) and (K). Suppose that there exist a subset A of X and x 0 ∈ A such that β(A, Tx 0) ≥ 1. Then T has an endpoint, and so T has the approximate endpoint property.

Let (X, d) be a complete metric space, β : 2 × 2 → [0, ∞) a mapping, and T : X → CB(X) a lower semicontinuous, β-admissible, and β-generalized weak contractive HS-multifunction which has the property (K). Suppose that there exist a subset A of X and x 0 ∈ A such that β(A, Tx 0) ≥ 1. Then T has an endpoint, and so T has the approximate endpoint property.

The next result is a consequence of Theorem 15.

Let (X, d) be a complete metric space, x* ∈ X a fixed element, and T : X → CB(X) an HS-multifunction such that x* ∈ Tx∩Ty for all subsets A and B of X with x* ∈ A∩B, all x ∈ A, and y ∈ B. Assume that H(Tx, Ty) ≤ ψ(N(x, y)) for all x, y ∈ X with x* ∈ Tx∩Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A 0 of X and x 0 ∈ A 0 such that x* ∈ A 0∩Tx 0. Assume that for each convergent sequence {x } in X with x → x and x* ∈ Tx ∩Tx for all n ≥ 1 one has x* ∈ Tx ∩Tx. Also, for each sequence {x } in X with x* ∈ Tx ∩Tx for all n ≥ 1, there exists a natural number k such that x* ∈ Tx ∩Tx for all m > n ≥ k. Then T has an endpoint, and so T has the approximate endpoint property.

The next result is a consequence of Theorem 16.

Let (X, d, ≤) be a complete ordered metric space and T a closed and bounded valued lower semicontinuous HS-multifunction on X such that Tx⪯Ty for all subsets A and B of X with A⪯B, all x ∈ A, and y ∈ B. Assume that H(Tx, Ty) ≤ ψ(N(x, y)) for all x, y ∈ X with Tx⪯Ty, where ψ : [0, +∞)→[0, +∞) is a nondecreasing upper semicontinuous function such that ψ(t) < t for all t > 0. Suppose that there exist a subset A 0 of X and x 0 ∈ A 0 such that A 0⪯Tx 0. Assume that for each sequence {x } in X with Tx ⪯Tx for all n ≥ 1, there exists a natural number k such that Tx ⪯Tx for all m > n ≥ k. Then T has an endpoint, and so T has the approximate endpoint property.