Some Endpoint Results for β-Generalized Weak Contractive Multifunctions

Abstract

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_x∈A d(x, B), sup_y∈B d(y, A)} for all A, B∈CB(X), where d(x, B) = inf⁡_y∈B d(x, y). Let T : X → 2^X 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⁡_x∈Xsup⁡_y∈Tx d(x, y) = 0 [1]. Let f : X → X be a mapping. We say that f has the approximate endpoint property whenever inf⁡_x∈X d(x, fx) = 0 [1]. Also, the function g : ℝ → ℝ is called upper semicontinuous whenever limsup⁡_n→∞ g(λ _n) ≤ g(λ) for all sequences {λ _n}_n≥1 with λ _n → 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→∞(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].

Theorem —

Let ψ : [0, ∞)→[0, ∞) be an upper semicontinuous function such that ψ(t) < t and liminf⁡_t→∞(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^X × 2^X → [0, ∞) a mapping. A multifunction T : X → 2^X 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

$$\begin{array}{c}\beta \left(Tx,Ty\right)H\left(Tx,Ty\right)\le \psi \left(N\left(x,y\right)\right)\end{array}$$ (1)

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 _n} in X with x _n → x and β(Tx _n−1, Tx _n) ≥ 1 for all n ≥ 1, we have β(Tx _n, 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 _n} in X with β(Tx _n−1, Tx _n) ≥ 1 for all n ≥ 1, there exists a natural number k such that β(Tx _m, Tx _n) ≥ 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⁡_a∈Tz d(z, a) < ɛ implies that for every x ∈ X there exists y ∈ Tx such that H(Tx, Ty) = sup⁡_b∈Ty d(y, b). A multifunction T : X → 2^X is called lower semicontinuous at x ₀ ∈ X whenever for each sequence {x _n} in X with x _n → x ₀ and every y ∈ Tx ₀, there exists a sequence {y _n} in X with y _n ∈ Tx _n for all n ≥ 1 such that y _n → y [14].

2. Main Results

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

Theorem —

Let (X, d) be a complete metric space, β : 2^X × 2^X → [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⁡_a∈Tz d(z, a) < ɛ. Now by using the condition (H), choose x ₁ ∈ Tx ₀ such that H(Tx ₀, Tx ₁) = sup⁡_a∈Tx1 d(x ₁, a). Also, choose x ₂ ∈ Tx ₁ such that H(Tx ₁, Tx ₂) = sup⁡_a∈Tx2 d(x ₂, a), and by continuing this process, we find a sequence {x _n} in X such that x _n ∈ Tx _n−1 and

$$\begin{array}{c}H\left(T{x}_{n-\mathrm{1}},T{x}_n\right)=\underset{a\in T{x}_n}{\sup }d\left(x_n,a\right),\end{array}$$ (2)

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 _n−1, Tx _n) ≥ 1 for all n ≥ 1. Thus, we obtain

$$\begin{array}{c}d\left(x_n,x_{n+\mathrm{1}}\right)\le \underset{a\in T{x}_n}{\sup }d\left(x_n,a\right)=H\left(T{x}_{n-\mathrm{1}},T{x}_n\right)\\ \le \beta \left(T{x}_{n-\mathrm{1}},T{x}_n\right)H\left(T{x}_{n-\mathrm{1}},T{x}_n\right)\\ \le \psi \left(N\left(x_{n-\mathrm{1}},x_n\right)\right)\end{array}$$ (3)

for all n ≥ 1. If N(x _n−1, x _n) = d(x _n−1, x _n), then

$$\begin{array}{c}d\left(x_n,x_{n+\mathrm{1}}\right)\le \psi \left(d\left(x_{n-\mathrm{1}},x_n\right)\right).\end{array}$$ (4)

If N(x _n−1, x _n) = d(x _n−1, Tx _n−1), then

$$\begin{array}{c}d\left(x_n,x_{n+\mathrm{1}}\right)\le \psi \left(d\left(x_{n-\mathrm{1}},T{x}_{n-\mathrm{1}}\right)\right)\le \psi \left(d\left(x_{n-\mathrm{1}},x_n\right)\right).\end{array}$$ (5)

If N(x _n−1, x _n) = d(x _n, Tx _n), then

$$\begin{array}{c}d\left(x_n,x_{n+\mathrm{1}}\right)\le \psi \left(d\left(x_n,T{x}_n\right)\right)\le \psi \left(d\left(x_n,x_{n+\mathrm{1}}\right)\right),\end{array}$$ (6)

and so d(x _n, x _n+1) = 0. Thus, d(x _n, x _n+1) ≤ ψ(d(x _n−1, x _n)). If

$$\begin{array}{c}N\left(x_{n-\mathrm{1}},x_n\right)=\frac{d\left(x_n,T{x}_{n-\mathrm{1}}\right)+d\left(x_{n-\mathrm{1}},T{x}_n\right)}{\mathrm{2}}\\ =\frac{d\left(x_{n-\mathrm{1}},T{x}_n\right)}{\mathrm{2}}\frac{d\left(x_{n-\mathrm{1}},T{x}_n\right)}{\mathrm{2}}\\ \le \frac{d\left(x_{n-\mathrm{1}},x_{n+\mathrm{1}}\right)}{\mathrm{2}}\\ \le \frac{d\left(x_{n-\mathrm{1}},x_n\right)+d\left(x_n,x_{n+\mathrm{1}}\right)}{\mathrm{2}}\\ \le \max \left\{d\left(x_{n-\mathrm{1}},x_n\right),d\left(x_n,x_{n+\mathrm{1}}\right)\right\},\end{array}$$ (7)

then d(x _n, x _n+1) ≤ ψ(d(x _n−1, x _n)) (other case implies that d(x _n, x _n+1) = 0). Thus,

$$\begin{array}{c}d\left(x_n,x_{n+\mathrm{1}}\right)\le \psi \left(d\left(x_{n-\mathrm{1}},x_n\right)\right)\end{array}$$ (8)

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 _n = d(x _n, x _n+1) for all n. If there exists a natural number n ₀ such that d _n0 = 0, then it is easy to see that d _n = 0 for all n ≥ n ₀, and so lim⁡_n→∞ d _n = 0. Now suppose that d _n ≠ 0 for all n. In this case, we have d _n ≤ ψ(d _n−1) < d _n−1 for all n. Hence, {d _n} is a decreasing sequence, and so there exists d ≥ 0 such that lim⁡_n→∞ d _n = d. If d > 0, then d _n > 0 for all n, and so d _n ≤ ψ(d _n−1) < d _n−1 for all n. Since ψ is upper and semicontinuous, we obtain d = lim⁡_n→∞ d _n ≤ lim⁡_n→∞ ψ(d _n−1) ≤ ψ(lim⁡_n→∞ d _n−1) = ψ(d) < d which is a contradiction. Thus, lim⁡_n→∞ d _n = 0. Now, we prove that {x _n} is a Cauchy sequence. If {x _n} is not a Cauchy sequence, then there exist ɛ > 0 and natural numbers m _k, n _k such that m _k > n _k ≥ k and d(x _mk, x _nk) ≥ ɛ for all k ≥ 1. Also, we choose m _k as small as possible such that

$$\begin{array}{c}d\left(x_{m_k-\mathrm{1}},x_{n_k}\right)<\varepsilon .\end{array}$$ (9)

Thus, ɛ ≤ d(x _mk, x _nk) ≤ d(x _mk, x _mk−1) + d(x _mk−1, x _nk) ≤ d _mk−1 + ɛ for all k. Hence, lim⁡_k→∞ d(x _mk, x _nk) = ɛ. Since T has the property (K), we obtain

$$\begin{array}{c}d\left(x_{m_k},x_{n_k}\right)\le d(x_{m_k},x_{m_k+\mathrm{1}})+d(x_{m_k+\mathrm{1}},x_{n_k+\mathrm{1}})\\ \hspace{1em}+d(x_{n_k+\mathrm{1}},x_{n_k})\\ \le d_{m_k}+H(T{x}_{m_k},T{x}_{n_k})+d_{n_k}\\ \le d_{m_k}+\beta (T{x}_{m_k},T{x}_{n_k})\\ \hspace{1em}\times H(T{x}_{m_k},T{x}_{n_k})+d_{n_k}\\ \le d_{m_k}+\psi \left(N(x_{m_k},x_{n_k})\right)+d_{n_k}\end{array}$$ (∗)

for all k. Since lim⁡_k→∞ d(x _mk, x _nk) = ɛ, lim⁡_k→∞ N(x _mk, x _nk) = ɛ. In fact,

$$\begin{array}{c}d\left(x_{m_k},x_{n_k}\right)\\ \hspace{1em}\le N\left(x_{m_k},x_{n_k}\right)\\ \hspace{1em}\hspace{1em}=\max \{d\left(x_{m_k},x_{n_k}\right),d\left(x_{m_k},T{x}_{m_k}\right),d\left(x_{n_k},T{x}_{n_k}\right),\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{d\left(x_{m_k},T{x}_{n_k}\right)+d\left(x_{n_k},T{x}_{m_k}\right)}{\mathrm{2}}\}\\ \hspace{1em}\le \max \{d\left(x_{m_k},x_{n_k}\right),d\left(x_{m_k},x_{m_{k+\mathrm{1}}}\right),d\left(x_{n_k},x_{n_{k+\mathrm{1}}}\right),\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{d\left(x_{m_k},x_{n_{k+\mathrm{1}}}\right)+d\left(x_{n_k},x_{m_{k+\mathrm{1}}}\right)}{\mathrm{2}}\}\\ \hspace{1em}\le \max \{d\left(x_{m_k},x_{n_k}\right),d\left(x_{m_k},x_{m_{k+\mathrm{1}}}\right),d\left(x_{n_k},x_{n_{k+\mathrm{1}}}\right),\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{0.5em}(d\left(x_{m_k},x_{n_k}\right)+d\left(x_{n_k},x_{n_{k+\mathrm{1}}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{0.5em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+d\left(x_{n_k},x_{m_k}\right)+d\left(x_{m_k},x_{m_{\text{k}+\mathrm{1}}}\right))\\ {\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{0.5em}\times \left(\mathrm{2}\right)}^{-1}\},\end{array}$$ (10)

and so ɛ = lim⁡_k→+∞ d(x _mk, x _nk) ≤ lim⁡_k→∞ N(x _mk, x _nk) ≤ ɛ. Since ψ is upper semicontinuous, by using ((∗)) we obtain

$$\begin{array}{c}\varepsilon =\underset{k\to \infty }{\lim }d\left(x_{m_k},x_{n_k}\right)\\ \le \underset{k\to \infty }{\lim }\psi \left(N\left(x_{m_k},x_{n_k}\right)\right)\le \psi \left(\varepsilon \right)<\varepsilon ,\end{array}$$ (11)

which is a contradiction, and so {x _n} is a Cauchy sequence. Choose x* ∈ X such that x _n → x*. Now, note that

$$\begin{array}{c}H\left(\left\{x_n\right\},T{x}_n\right)=\max \left\{d\left(x_n,T{x}_n\right),\underset{y\in T{x}_n}{\sup }d\left(x_n,y\right)\right\}\\ =H\left(T{x}_{n-\mathrm{1}},T{x}_n\right)\end{array}$$ (12)

for all n, and so

$$\begin{array}{c}H\left(\left\{x_n\right\},T{x}_n\right)=H\left(T{x}_{n-\mathrm{1}},T{x}_n\right)\\ \le \beta \left(T{x}_{n-\mathrm{1}},T{x}_n\right)H\left(T{x}_{n-\mathrm{1}},T{x}_n\right)\\ \le \psi \left(N\left(x_{n-\mathrm{1}},x_n\right)\right)\\ \le \psi \left(d\left(x_{n-\mathrm{1}},x_n\right)\right)\le d\left(x_{n-\mathrm{1}},x_n\right)\end{array}$$ (13)

for all n, and so lim⁡_n→∞ H({x _n}, Tx _n) = 0. Since T has the property (R), we obtain

$$\begin{array}{c}H\left(\left\{x^{\ast }\right\},T{x}^{\ast }\right)\le d\left(x^{\ast },x_n\right)\\ \hspace{1em}+H\left(\left\{x_n\right\},T{x}_n\right)+H\left(T{x}_n,T{x}^{\ast }\right)\\ \le d\left(x^{\ast },x_n\right)+H\left(\left\{x_n\right\},T{x}_n\right)\\ \hspace{1em}+\beta \left(T{x}_n,T{x}^{\ast }\right)H\left(T{x}_n,T{x}^{\ast }\right)\\ \le d\left(x^{\ast },x_n\right)+H\left(\left\{x_n\right\},T{x}_n\right)\\ \hspace{1em}+\psi \left(N\left(x_n,x^{\ast }\right)\right)\end{array}$$ (14)

for all n. If N(x _n, x*) = d(x*, Tx*), then we have

$$\begin{array}{c}H\left(\left\{x^{\ast }\right\},T{x}^{\ast }\right)\\ \hspace{1em}\hspace{1em}\le d\left(x^{\ast },x_n\right)+H\left(\left\{x_n\right\},T{x}_n\right)+\psi \left(H\left(\left\{x^{\ast }\right\},T{x}^{\ast }\right)\right)\end{array}$$ (15)

for all n. This implies that H({x*}, Tx*) ≤ ψ(H({x*}, Tx*)), and so

$$\begin{array}{c}H\left(\left\{x^{\ast }\right\},T{x}^{\ast }\right)=\mathrm{0}.\end{array}$$ (16)

If N(x _n, x*) = d(x _n, x*) or N(x _n, x*) ≤ d(x _n, x _n+1), 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,

$$\begin{array}{c}H\left(Tx,Ty\right)=\mathrm{2}\gneqq \psi \left(\mathrm{2}\right)=\psi \left(N\left(x,y\right)\right).\end{array}$$ (17)

Thus, T is not a generalized weak contractive multifunction. Now, suppose that ψ(t) = t/2 for all t ≥ 0 and define β : 2^X × 2^X → [0, ∞) by β(A, B) = 1/2 for all subsets A and B of X. Then, we have

$$\begin{array}{c}\beta \left(Tx,Ty\right)H\left(Tx,Ty\right)=\frac{\mathrm{1}}{\mathrm{2}}d\left(x,y\right)\\ =\psi \left(d\left(x,y\right)\right)=\psi \left(N\left(x,y\right)\right)\end{array}$$ (18)

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.

Example —

Let X = [0, 9/2] and let d(x, y) = |x − y|. Define T : X → CB(X) by

$$\begin{array}{c}Tx=\{\begin{array}{cc}\left\{\frac{x}{\mathrm{2}}\right\}\hfill & \mathrm{0}\le x\le \mathrm{1}\hfill \\ \left\{\mathrm{4}x-\frac{\mathrm{3}}{\mathrm{2}}\right\}\hfill & \mathrm{1}<x\le \frac{\mathrm{3}}{\mathrm{2}}\hfill \\ \left\{\mathrm{0}\right\}\hfill & \frac{\mathrm{3}}{\mathrm{2}}<x\le \frac{\mathrm{9}}{\mathrm{2}}.\hfill \end{array}\end{array}$$ (19)

If x = 1 and y = 3/2, then

$$\begin{array}{c}H\left(Tx,Ty\right)=H\left(\left\{\frac{\mathrm{1}}{\mathrm{2}}\right\},\left\{\frac{\mathrm{9}}{\mathrm{2}}\right\}\right)=\mathrm{4}>\mathrm{3}\\ =N\left(x,y\right)>\psi \left(N\left(x,y\right)\right),\end{array}$$ (20)

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

$$\begin{array}{c}\beta \left(Tx,Ty\right)H\left(Tx,Ty\right)=\frac{\mathrm{1}}{\mathrm{2}}d\left(x,y\right)\\ \le \psi \left(d\left(x,y\right)\right)\le \psi \left(N\left(x,y\right)\right).\end{array}$$ (21)

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⁡_a∈Tz d(z, a) < ɛ. Now, we show that for each x ∈ X there exists y ∈ Tx such that H(Tx, Ty) = sup⁡_b∈Ty d(y, b). If 0 ≤ x ≤ 1, then Tx = {x/2}, T(x/2) = {x/4}, and

$$\begin{array}{c}H\left(Tx,T\left(\frac{x}{\mathrm{2}}\right)\right)=H\left(\left\{\frac{x}{\mathrm{2}}\right\},\left\{\frac{x}{\mathrm{4}}\right\}\right)\\ \begin{array}{c}=\frac{x}{\mathrm{4}}=\underset{b\in T\left(x/\mathrm{2}\right)}{\sup }d\left(\frac{x}{\mathrm{2}},b\right).\end{array}\end{array}$$ (22)

Since for 1 < x ≤ 3/2 we have 5/2 < 4x − (3/2) ≤ 9/2, T(4x − (3/2)) = {0}. Thus,

$$\begin{array}{c}H\left(Tx,T\left(\mathrm{4}x-\frac{\mathrm{3}}{\mathrm{2}}\right)\right)=H\left(\left\{\mathrm{4}x-\frac{\mathrm{3}}{\mathrm{2}}\right\},\left\{\mathrm{0}\right\}\right)\\ =\mathrm{4}x-\frac{\mathrm{3}}{\mathrm{2}}=\underset{b\in T\left(\mathrm{4}x-\mathrm{3}/\mathrm{2}\right)}{\sup }d\left(\mathrm{4}x-\frac{\mathrm{3}}{\mathrm{2}},b\right).\end{array}$$ (23)

If 3/2 < x ≤ 9/2, then Tx = {0} and T(0) = {0}. Hence,

$$\begin{array}{c}H\left(Tx,T\left(\mathrm{0}\right)\right)=H\left(\left\{\mathrm{0}\right\},\left\{\mathrm{0}\right\}\right)\\ \begin{array}{c}=\mathrm{0}=\underset{b\in T\left(\mathrm{0}\right)}{\sup }d\left(\mathrm{0},b\right).\end{array}\end{array}$$ (24)

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^X × 2^X → [0, ∞) a map. We say that the set X has the property (G _β) 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^X × 2^X → [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 _β), then T has a unique endpoint.

Proof —

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 _β). Hence,

$$\begin{array}{c}d\left(x^{\ast },y^{\ast }\right)\le H\left(T{x}^{\ast },T{y}^{\ast }\right)\\ \le \beta \left(T{x}^{\ast },T{y}^{\ast }\right)H\left(T{x}^{\ast },T{y}^{\ast }\right)\\ \le \psi \left(N\left(x^{\ast },y^{\ast }\right)\right)<N\left(x\ast ,y\ast \right)\\ =d\left(x^{\ast },y^{\ast }\right),\end{array}$$ (25)

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 _β). Also, T has the property (R), while T is not lower semicontinuous. To see this, consider the sequence {x _n} defined by

$$\begin{array}{c}{x}_n=\{\begin{array}{cc}\mathrm{1}-\frac{\mathrm{1}}{n}\hfill & n=\mathrm{2}k\hfill \\ \mathrm{1}+\frac{\mathrm{1}}{n}\hfill & n=\mathrm{2}k-\mathrm{1}\hfill \end{array}\end{array}$$ (26)

for k ≥ 1 and put y = 1/2 and x ₀ = 1. Then x _n → 1 and y ∈ Tx ₀ = {1/2}. Let {y _n} be an arbitrary sequence in X such that y _n ∈ Tx _n for all n ≥ 1. Then, y _2k−1 ∈ Tx _2k−1 and y _2k ∈ Tx _2k for all k. But, y _2k−1 = 4x _2k−1 − (3/2) for sufficiently large k and y _2k = x _2k/2 for all k since y _2k−1 → 5/2, y _n↛1/2. This implies that T is not lower semicontinuous.

Corollary —

Let (X, d) be a complete metric space, β : 2^X × 2^X → [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 _β), 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.

Proof —

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.

Theorem —

Let (X, d) be a complete metric space, β : 2^X × 2^X → [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.

Corollary —

Let (X, d) be a complete metric space, β : 2^X × 2^X → [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 _β), then T has a unique endpoint.

Corollary —

Let (X, d) be a complete metric space, β : 2^X × 2^X → [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 _β), 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}.

Proof —

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 _β). Also, we have

$$\begin{array}{c}d\left(x,y\right)\le H\left(\left\{x\right\},Ty\right)=H\left(Tx,Ty\right)\\ \le \beta \left(Tx,Ty\right)H\left(Tx,Ty\right)\\ \le kN\left(x,y\right).\end{array}$$ (27)

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.

Corollary —

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 _n} in X with x _n → x and x* ∈ Tx _n−1∩Tx _n, for all n ≥ 1, one has x* ∈ Tx _n∩Tx. Also, for each sequence {x _n} in X with x* ∈ Tx _n−1∩Tx _n for all n ≥ 1, there exists a natural number k such that x* ∈ Tx _m∩Tx _n for all m > n ≥ k. Then T has an endpoint if and only if T has the approximate endpoint property.

Proof —

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

Corollary —

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

$$\begin{array}{c}H\left(Tx,Ty\right)\le \psi \left(N\left(x,y\right)\right)\end{array}$$ (28)

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 _n} in X with x _n → x and x* ∈ Tx _n−1∩Tx _n for all n ≥ 1, we have x* ∈ Tx _n∩Tx. Then T has an endpoint if and only if T has the approximate endpoint property.

Proof —

It is sufficient to define β : 2^X × 2^X → [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.

Theorem —

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 _n} in X with x _n → x and Tx _n−1⪯Tx _n, for all n ≥ 1, one has Tx _n⪯Tx. Also, for each sequence {x _n} in X with Tx _n−1⪯Tx _n for all n ≥ 1, there exists a natural number k such that Tx _m⪯Tx _n for all m > n ≥ k. Then T has an endpoint if and only if T has the approximate endpoint property.

Proof —

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

Corollary —

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 _n} in X with x _n → x and Tx _n−1⪯Tx _n, for all n ≥ 1, one has Tx _n⪯Tx. Also, for each sequence {x _n} in X with Tx _n−1⪯Tx _n for all n ≥ 1, there exists a natural number k such that Tx _m⪯Tx _n 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.

Proof —

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^X 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⁡_b∈Ty 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.

Theorem —

Let (X, d) be a complete metric space, β : 2^X × 2^X → [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.

Theorem —

Let (X, d) be a complete metric space, β : 2^X × 2^X → [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.

Corollary —

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 _n} in X with x _n → x and x* ∈ Tx _n−1∩Tx _n for all n ≥ 1 one has x* ∈ Tx _n∩Tx. Also, for each sequence {x _n} in X with x* ∈ Tx _n−1∩Tx _n for all n ≥ 1, there exists a natural number k such that x* ∈ Tx _m∩Tx _n 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.

Corollary —

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 _n} in X with Tx _n−1⪯Tx _n for all n ≥ 1, there exists a natural number k such that Tx _m⪯Tx _n for all m > n ≥ k. Then T has an endpoint, and so T has the approximate endpoint property.