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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 786061, 13 pages
doi:10.1155/2012/786061
Research Article

Some Fixed Point Theorems for Nonlinear Set-Valued Contractive Mappings

Zeqing Liu,1 Zhihua Wu,1 Shin Min Kang,2 and Sunhong Lee2

1Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China
2Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 10 April 2012; Accepted 9 May 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Four fixed point theorems for nonlinear set-valued contractive mappings in complete metric spaces are proved. The results presented in this paper are extensions of a few well-known fixed point theorems. Two examples are also provided to illustrate our results.

1. Introduction and Preliminaries

The existence of fixed points for various set-valued contractive mappings had been researched by many authors under different conditions, see, for example, [1–9] and the references cited therein. In 1969, Nadler [7] proved a well-known fixed point theorem for the set-valued contraction mapping  (1.1) below.

Theorem 1.1 (see [7]). Let ( 𝑋 , 𝑑 ) be a complete metric space and 𝑇 ∶ 𝑋 → 𝐶 𝐵 ( 𝑋 ) be a set-valued mapping such that 𝐻 ( 𝑇 𝑥 , 𝑇 𝑦 ) ≤ 𝑟 𝑑 ( 𝑥 , 𝑦 ) , ∀ 𝑥 , 𝑦 ∈ 𝑋 , ( 1 . 1 ) where 𝑟 ∈ ( 0 , 1 ) is a constant. Then 𝑇 has a fixed point.

In 1972, Reich [8] extended Nadler's result and established an interesting fixed point theorem for the set-valued contraction mapping (1.2) below.

Theorem 1.2 (see [8]). Let ( 𝑋 , 𝑑 ) be a complete metric space and 𝑇 ∶ 𝑋 → 𝐶 ( 𝑋 ) satisfy that 𝐻 ( 𝑇 𝑥 , 𝑇 𝑦 ) ≤ 𝜑 ( 𝑑 ( 𝑥 , 𝑦 ) ) 𝑑 ( 𝑥 , 𝑦 ) , ∀ 𝑥 , 𝑦 ∈ 𝑋 , ( 1 . 2 ) where [ 𝜑 ∶ ( 0 , + ∞ ) ⟶ 0 , 1 ) 𝑤 𝑖 𝑡 ℎ l i m s u p 𝑟 → 𝑡 + 𝜑 ( 𝑟 ) < 1 , ∀ 𝑡 ∈ ( 0 , + ∞ ) . ( 1 . 3 ) Then 𝑇 has a fixed point.

In [8] Reich posed the question whether Theorem 1.2 is also true for the set-valued contractive mapping 𝑇 ∶ 𝑋 → 𝐶 𝐵 ( 𝑋 ) with (1.2). The affirmative answer under the hypothesis of l i m s u p 𝑟 → 𝑡 + 𝜑 ( 𝑟 ) < 1 , for  all 𝑡 ∈ [ 0 , + ∞ ) was given by Mizoguchi and Takahashi in [6]. They deduced the following fixed point theorem which is a generalization of the Nadler fixed point theorem.

Theorem 1.3 (see [6]). Let ( 𝑋 , 𝑑 ) be a complete metric space and 𝑇 ∶ 𝑋 → 𝐶 𝐵 ( 𝑋 ) satisfy (1.2), where [ 𝜑 ∶ ( 0 , + ∞ ) ⟶ 0 , 1 ) 𝑤 𝑖 𝑡 ℎ l i m s u p 𝑟 → 𝑡 + [ 𝜑 ( 𝑟 ) < 1 , ∀ 𝑡 ∈ 0 , + ∞ ) . ( 1 . 4 ) Then 𝑇 has a fixed point.

Remark 1.4. It is clear that the mappings 𝑇 in Theorems 1.1–1.3 are continuous on 𝑋 .

Remark 1.5. Each of Theorems 1.2 and 1.3 ensures that 𝑇 has a fixed point 𝑎 ∈ 𝑇 𝑎 ⊆ 𝑋 , which together with (1.2) implies that 𝜑 ( 0 ) = 𝜑 ( 𝑑 ( 𝑎 , 𝑎 ) ) , that is, 𝜑 is defined at 0 . Thus the domain of 𝜑 in each of (1.3) and (1.4) should be [ 0 , + ∞ ) but not ( 0 , + ∞ ) .
The aim of this paper is to present four fixed point theorems for some nonlinear set-valued contractive mappings. Our results extend, improve, and unify the corresponding results in [6–8]. Two nontrivial examples are given to show that our results are genuine generalizations or different from these results in [6–8].
Throughout this paper, we assume that ℝ = ( − ∞ , + ∞ ) , ℝ + = [ 0 , + ∞ ) , ℕ and ℕ 0 denote the sets of all positive integers and nonnegative integers, respectively, and Θ = 𝜃 ∶ 𝜃 ∶ ℝ + ⟶ ℝ + s a t i s fi e s ( a ) − ( d ) , ( 1 . 5 ) where(a) 𝜃 is nondecreasing on ℝ + ;(b) 𝜃 ( 𝑡 ) > 0 , for  all 𝑡 ∈ ( 0 , + ∞ ) ;(c) 𝜃 is subadditive in ( 0 , + ∞ ) , that is, 𝜃 𝑡 1 + 𝑡 2 𝑡 ≤ 𝜃 1 𝑡 + 𝜃 2 , ∀ 𝑡 1 , 𝑡 2 ∈ ( 0 , + ∞ ) ; ( 1 . 6 ) (d) 𝜃 ( ℝ + ) = ℝ + . Clearly (a)–(d) imply that(e) 𝜃 is strictly inverse on ℝ + , that is, if there exist 𝑡 , 𝑠 ∈ ℝ + satisfying 𝜃 ( 𝑡 ) < 𝜃 ( 𝑠 ) , then 𝑡 < 𝑠 .
Let ( 𝑋 , 𝑑 ) be a metric space, 𝐶 𝐿 ( 𝑋 ) , 𝐶 𝐵 ( 𝑋 ) , and 𝐶 ( 𝑋 ) denote the families of all nonempty closed, all nonempty bounded closed, and all nonempty compact subsets of 𝑋 . For 𝑥 ∈ 𝑋 and 𝐴 , 𝐵 ∈ 𝐶 𝐿 ( 𝑋 ) , put 𝑑 ( 𝑥 , 𝐴 ) = i n f { 𝑑 ( 𝑥 , 𝑦 ) ∶ 𝑦 ∈ 𝐴 } and ⎧ ⎪ ⎨ ⎪ ⎩ 𝐻 ( 𝐴 , 𝐵 ) = m a x s u p 𝑥 ∈ 𝐴 𝑑 ( 𝑥 , 𝐵 ) , s u p 𝑦 ∈ 𝐵 𝑑 ( 𝑦 , 𝐴 ) , i f t h e m a x i m u m e x i s t s + ∞ , o t h e r w i s e . ( 1 . 7 ) Such a mapping 𝐻 is called a generalized Hausdorff metric induced b 𝑦 𝑑 in 𝐶 𝐿 ( 𝑋 ) . It is well known that 𝐻 is a metric on 𝐶 𝐵 ( 𝑋 ) . Let 𝑇 ∶ 𝑋 → 𝐶 𝐿 ( 𝑋 ) be a set-valued mapping, 𝑥 0 ∈ 𝑋 and 𝑓 ∶ 𝑋 → ℝ + be defined by 𝑓 ( 𝑥 ) = 𝑑 ( 𝑥 , 𝑇 𝑥 ) , ∀ 𝑥 ∈ 𝑋 . ( 1 . 8 ) A sequence { 𝑥 𝑛 } 𝑛 ∈ ℕ 0 is said to be an orbit o 𝑓 𝑇 if it satisfies that { 𝑥 𝑛 } 𝑛 ∈ ℕ 0 ⊂ 𝑋 and 𝑥 𝑛 ∈ 𝑇 𝑥 𝑛 − 1 for each 𝑛 ∈ ℕ 0 . The function 𝑓 ∶ 𝑋 → ℝ + is said to be 𝑇 − 𝑜 𝑟 𝑏 𝑖 𝑡 𝑎 𝑙 𝑙 𝑦 𝑙 𝑜 𝑤 𝑒 𝑟 𝑠 𝑒 𝑚 𝑖 𝑐 𝑜 𝑛 𝑡 𝑖 𝑛 𝑢 𝑜 𝑢 𝑠 𝑎 𝑡 𝑧 ∈ 𝑋 if for each orbit { 𝑥 𝑛 } 𝑛 ∈ ℕ 0 ⊂ 𝑋 of 𝑇 with l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑧 , we have that 𝑓 ( 𝑧 ) ≤ l i m i n f 𝑛 → ∞ 𝑓 ( 𝑥 𝑛 ) .

2. Main Results

The following lemmas play important roles in this paper.

Lemma 2.1. Let ( 𝑋 , 𝑑 ) be a metric space and 𝐵 ∈ 𝐶 𝐿 ( 𝑋 ) . Then for each 𝑥 ∈ 𝑋 and 𝜀 > 0 there exists 𝑏 ∈ 𝐵 satisfying 𝑑 ( 𝑥 , 𝑏 ) ≤ 𝑑 ( 𝑥 , 𝐵 ) + 𝜀 .

Proof. Suppose that there exist 𝑥 0 ∈ 𝑋 and 𝜀 0 > 0 such that 𝑑 𝑥 0 𝑥 , 𝑏 > 𝑑 0 , 𝐵 + 𝜀 0 , ∀ 𝑏 ∈ 𝐵 , ( 2 . 1 ) which yields that 𝑑 𝑥 0 , 𝐵 = i n f 𝑏 ∈ 𝐵 𝑑 𝑥 0 𝑥 , 𝑏 ≥ 𝑑 0 , 𝐵 + 𝜀 0 𝑥 > 𝑑 0 , , 𝐵 ( 2 . 2 ) which is a contradiction. This completes the proof.

Lemma 2.2. Let ( 𝑋 , 𝑑 ) be a metric space, 𝐵 ∈ 𝐶 𝐿 ( 𝑋 ) and 𝜃 ∈ Θ . Then for each 𝑥 ∈ 𝑋 and 𝑞 > 1 there exists 𝑏 ∈ 𝐵 such that 𝜃 ( 𝑑 ( 𝑥 , 𝑏 ) ) ≤ 𝑞 𝜃 ( 𝑑 ( 𝑥 , 𝐵 ) ) . ( 2 . 3 )

Proof. Let 𝑥 ∈ 𝑋 and 𝑞 > 1 . Now we consider two possible cases as follows.
Case  1. Suppose that 𝜃 ( 𝑑 ( 𝑥 , 𝐵 ) ) = 0 . It follows from (b) and (d) that 𝑑 ( 𝑥 , 𝐵 ) = 0 . Since 𝐵 is a closed subset of 𝑋 , it follows that 𝑥 ∈ 𝐵 . Put 𝑏 = 𝑥 . Clearly (2.3) holds.
Case  2. Suppose that 𝜃 ( 𝑑 ( 𝑥 , 𝐵 ) ) > 0 . Note that (b) and (d) mean that ( 𝑞 − 1 ) 𝜃 ( 𝑑 ( 𝑥 , 𝐵 ) ) ∈ ℝ + ⧵ ℝ { 0 } = 𝜃 + ⧵ { 0 } . ( 2 . 4 ) Choose 𝑝 ∈ 𝜃 − 1 ( ( 𝑞 − 1 ) 𝜃 ( 𝑑 ( 𝑥 , 𝐵 ) ) ) and 𝜀 = 𝑝 / 2 > 0 . Lemma 2.1 ensures that there exists 𝑏 ∈ 𝐵 satisfying 𝑑 ( 𝑥 , 𝑏 ) ≤ 𝑑 ( 𝑥 , 𝐵 ) + 𝜀 , which together with (a) and (c) gives that 𝜃 𝜃 ( 𝑑 ( 𝑥 , 𝑏 ) ) ≤ 𝜃 ( 𝑑 ( 𝑥 , 𝐵 ) + 𝜀 ) ≤ 𝜃 ( 𝑑 ( 𝑥 , 𝐵 ) ) + 𝜃 ( 𝜀 ) ≤ 𝜃 ( 𝑑 ( 𝑥 , 𝐵 ) ) + 𝜃 − 1 ( ( 𝑞 − 1 ) 𝜃 ( 𝑑 ( 𝑥 , 𝐵 ) ) ) = 𝑞 𝜃 ( 𝑑 ( 𝑥 , 𝐵 ) ) . ( 2 . 5 ) That is, (2.3) holds. This completes the proof.

Now we prove four fixed point theorems for the nonlinear set-valued contractive mappings (2.6), (2.25), (2.26), and (2.36) below in complete metric spaces.

Theorem 2.3. Let ( 𝑋 , 𝑑 ) be a complete metric space and 𝑇 ∶ 𝑋 → 𝐶 𝐿 ( 𝑋 ) satisfy that 𝜃 ( 𝑑 ( 𝑦 , 𝑇 𝑦 ) ) ≤ 𝜑 ( 𝑑 ( 𝑥 , 𝑦 ) ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) , ∀ ( 𝑥 , 𝑦 ) ∈ 𝑋 × 𝑇 𝑥 , ( 2 . 6 ) where 𝜃 ∈ Θ and 𝜑 ∶ ℝ + ⟶ [ 0 , 1 ) 𝑤 𝑖 𝑡 ℎ l i m s u p 𝑟 → 𝑡 + 𝜑 ( 𝑟 ) < 1 , ∀ 𝑡 ∈ ℝ + . ( 2 . 7 ) Then for each 𝑥 0 ∈ 𝑋 , there exists an orbit { 𝑥 𝑛 } 𝑛 ∈ ℕ 0 of 𝑇 and 𝑧 ∈ 𝑋 such that l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑧 . Furthermore, 𝑧 ∈ 𝑋 is fixed point of 𝑇 if and only if the function 𝑓 defined by (1.8) is 𝑇 orbitally lower semicontinuous at 𝑧 .

Proof. Let 𝑥 0 ∈ 𝑋 be any initial point and choose 𝑥 1 ∈ 𝑇 𝑥 0 . It follows from (2.6), (2.7) and Lemma 2.2 that for 𝑞 1 √ = 1 / m a x { 𝜑 ( 𝑑 ( 𝑥 0 , 𝑥 1 ) ) , 1 / 2 } > 1 there exists 𝑥 2 ∈ 𝑇 𝑥 1 satisfying 𝜃 𝑑 𝑥 1 , 𝑥 2 ≤ 𝜃 𝑑 𝑥 1 , 𝑇 𝑥 1 m a x 𝜑 𝑑 𝑥 0 , 𝑥 1 ≤ 𝜑 𝑑 𝑥 , 1 / 2 0 , 𝑥 1 𝜃 𝑑 𝑥 0 , 𝑥 1 m a x 𝜑 𝑑 𝑥 0 , 𝑥 1 ≤ , 1 / 2 𝜑 𝑑 𝑥 0 , 𝑥 1 𝜃 𝑑 𝑥 0 , 𝑥 1 , ( 2 . 8 ) and for 𝑞 2 √ = 1 / m a x { 𝜑 ( 𝑑 ( 𝑥 1 , 𝑥 2 ) ) , 1 / 3 } > 1 there exists 𝑥 3 ∈ 𝑇 𝑥 2 satisfying 𝜃 𝑑 𝑥 2 , 𝑥 3 ≤ 𝜃 𝑑 𝑥 2 , 𝑇 𝑥 2 m a x 𝜑 𝑑 𝑥 1 , 𝑥 2 ≤ 𝜑 𝑑 𝑥 , 1 / 3 1 , 𝑥 2 𝜃 𝑑 𝑥 1 , 𝑥 2 m a x 𝜑 𝑑 𝑥 1 , 𝑥 2 ≤ , 1 / 3 𝜑 𝑑 𝑥 1 , 𝑥 2 𝜃 𝑑 𝑥 1 , 𝑥 2 . ( 2 . 9 ) Repeating the above argument we obtain a sequence { 𝑥 𝑛 } 𝑛 ∈ ℕ 0 ⊂ 𝑋 such that 𝑥 𝑘 ∈ 𝑇 𝑥 𝑘 − 1 for 1 ≤ 𝑘 ≤ 𝑛 and for 𝑞 𝑛 √ = 1 / m a x { 𝜑 ( 𝑑 ( 𝑥 𝑛 − 1 , 𝑥 𝑛 ) ) , 1 / ( 𝑛 + 1 ) } > 1 , there exists 𝑥 𝑛 + 1 ∈ 𝑇 𝑥 𝑛 satisfying 𝜃 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 1 ≤ 𝜃 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 m a x 𝜑 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 ≤ 𝜑 𝑑 𝑥 , 1 / ( 𝑛 + 1 ) 𝑛 − 1 , 𝑥 𝑛 𝜃 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 m a x 𝜑 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 ≤ , 1 / ( 𝑛 + 1 ) 𝜑 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 𝜃 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 , ∀ 𝑛 ≥ 1 . ( 2 . 1 0 )
Suppose that there exists some 𝑛 0 ∈ ℕ 0 satisfying 𝑥 𝑛 0 = 𝑥 𝑛 0 + 1 ∈ 𝑇 𝑥 𝑛 0 . It follows from (a), (b), and (2.10) that 𝑥 𝑛 = 𝑥 𝑛 0 for all 𝑛 ≥ 𝑛 0 + 1 . It is clear the conclusion of Theorem 2.3 holds.
Suppose that 𝑥 𝑛 + 1 ∈ 𝑇 𝑥 𝑛 ⧵ { 𝑥 𝑛 } for any 𝑛 ∈ ℕ 0 . It follows that 𝑑 ( 𝑥 𝑛 , 𝑥 𝑛 + 1 ) > 0 for each 𝑛 ∈ ℕ 0 . Note that (b), (2.7), and (2.10) give that { 𝜃 ( 𝑑 ( 𝑥 𝑛 , 𝑥 𝑛 + 1 ) ) } 𝑛 ∈ ℕ 0 is a positive and decreasing sequence. It follows from (e) that { 𝑑 ( 𝑥 𝑛 , 𝑥 𝑛 + 1 ) } 𝑛 ∈ ℕ 0 is decreasing. Therefore, there exist constants 𝑝 and 𝑞 satisfying l i m 𝑛 → ∞ 𝜃 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 1 = 𝑝 ≥ 0 , l i m 𝑛 → ∞ 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 1 = 𝑞 ≥ 0 . ( 2 . 1 1 ) Notice that (2.7) implies that there exists a constant 𝑟 satisfying l i m s u p 𝑛 → ∞ 𝜑 𝑑 𝑥 𝑛 , 𝑥 𝑛 − 1 ≤ l i m s u p 𝑡 → 𝑞 + 𝜑 [ ( 𝑡 ) = 𝑟 ∈ 0 , 1 ) . ( 2 . 1 2 ) Taking upper limits in (2.10) and by (2.11) and (2.12) we get that 𝑝 ≤ l i m s u p 𝑛 → ∞ 𝜑 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 l i m s u p 𝑛 → ∞ 𝜃 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 ≤ √ 𝑟 𝑝 , ( 2 . 1 3 ) which implies that 𝑝 = 0 .
Next we assert that 𝑞 = 0 . Since { 𝑑 ( 𝑥 𝑛 , 𝑥 𝑛 + 1 ) } 𝑛 ∈ ℕ 0 is a decreasing sequence, it follows from (a) and (2.11) that 𝑑 𝑥 0 ≤ 𝜃 ( 𝑞 ) < 𝜃 𝑛 , 𝑥 𝑛 + 1 ⟶ 𝑝 = 0 a s 𝑛 ⟶ ∞ , ( 2 . 1 4 ) that is, 𝜃 ( 𝑞 ) = 0 , which together with (b) and (d) yields that 𝑞 = 0 .
Put 𝑐 = ( 1 + 𝑟 ) / 2 . It follows from (2.12) that 𝑐 ∈ ( 𝑟 , 1 ) ⊂ [ 0 , 1 ) , which gives that 𝑐 2 ∈ ( 𝑟 , 1 ) . Notice that (2.11), (2.12), and 𝑞 = 0 ensure that there exist 𝛿 > 0 and 𝑁 ∈ ℕ satisfying 𝜑 ( 𝑡 ) < 𝑐 2 𝑥 , ∀ 𝑡 ∈ ( 0 , 𝛿 ) , 𝑑 𝑛 , 𝑥 𝑛 + 1 < 𝛿 , ∀ 𝑛 ≥ 𝑁 , ( 2 . 1 5 ) which implies that 𝜑 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 1 < 𝑐 2 , ∀ 𝑛 ≥ 𝑁 . ( 2 . 1 6 ) Note that (2.10) and (2.16) mean that 𝜃 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 1 ≤ 𝑛 − 1 𝑘 = 𝑁 𝜑 𝑑 𝑥 𝑘 , 𝑥 𝑘 + 1 𝜃 𝑑 𝑥 𝑁 , 𝑥 𝑁 + 1 ≤ 𝑐 𝑛 − 𝑁 𝜃 𝑑 𝑥 𝑁 , 𝑥 𝑁 + 1 , ∀ 𝑛 ≥ 𝑁 . ( 2 . 1 7 ) Given 𝜀 > 0 . Since l i m 𝑛 → ∞ 𝑐 𝑛 − 𝑁 𝜃 ( 𝑑 ( 𝑥 𝑁 , 𝑥 𝑁 + 1 ) ) = 0 , it follows from (b) that there exists 𝑁 1 > 𝑁 satisfying 𝑐 𝑛 − 𝑁 𝜃 𝑑 𝑥 1 − 𝑐 𝑁 , 𝑥 𝑁 + 1 < 𝜃 ( 𝜀 ) , ∀ 𝑛 ≥ 𝑁 1 , ( 2 . 1 8 ) which together with (2.17), (a), and (c) gives that 𝜃 𝑑 𝑥 𝑛 , 𝑥 𝑚 ≤ 𝜃 𝑚 − 1 𝑘 = 𝑛 𝑑 𝑥 𝑘 , 𝑥 𝑘 + 1 ≤ 𝑚 − 1 𝑘 = 𝑛 𝜃 𝑑 𝑥 𝑘 , 𝑥 𝑘 + 1 ≤ 𝑚 − 1 𝑘 = 𝑛 𝑐 𝑘 − 𝑁 𝜃 𝑑 𝑥 𝑁 , 𝑥 𝑁 + 1 ≤ 𝑐 𝑛 − 𝑁 𝜃 𝑑 𝑥 1 − 𝑐 𝑁 , 𝑥 𝑁 + 1 < 𝜃 ( 𝜀 ) , ∀ 𝑚 > 𝑛 ≥ 𝑁 1 . ( 2 . 1 9 ) In view of (e) and (2.19), we deduce that 𝑑 ( 𝑥 𝑛 , 𝑥 𝑚 ) < 𝜀 , for all 𝑚 > 𝑛 ≥ 𝑁 1 , which means that { 𝑥 𝑛 } 𝑛 ∈ ℕ 0 is a Cauchy sequence. Hence there exists 𝑧 ∈ 𝑋 such that l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑧 by completeness of 𝑋 .
Suppose that 𝑓 is 𝑇 orbitally lower semicontinuous at 𝑧 . Since { 𝑥 𝑛 } 𝑛 ≥ 0 is an orbit of 𝑇 with l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑧 , it follows that 𝑓 ( 𝑧 ) ≤ l i m i n f 𝑛 → ∞ 𝑓 𝑥 𝑛 . ( 2 . 2 0 ) Using (2.6) and (2.7), we infer that 𝜃 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 𝑑 𝑥 ≤ 𝜑 𝑛 − 1 , 𝑥 𝑛 𝜃 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 𝑑 𝑥 < 𝜃 𝑛 − 1 , 𝑥 𝑛 , ∀ 𝑛 ∈ ℕ , ( 2 . 2 1 ) which together with (e), (2.11), and 𝑞 = 0 implies that 𝑥 0 < 𝑑 𝑛 , 𝑇 𝑥 𝑛 𝑥 < 𝑑 𝑛 − 1 , 𝑥 𝑛 ⟶ 0 a s 𝑛 ⟶ ∞ , ( 2 . 2 2 ) that is, l i m 𝑛 → ∞ 𝑑 ( 𝑥 𝑛 , 𝑇 𝑥 𝑛 ) = 0 , which together with (2.20) yields that 0 ≤ 𝑑 ( 𝑧 , 𝑇 𝑧 ) = 𝑓 ( 𝑧 ) ≤ l i m i n f 𝑛 → ∞ 𝑓 𝑥 𝑛 = l i m 𝑛 → ∞ 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 = 0 , ( 2 . 2 3 ) which gives that 𝑑 ( 𝑧 , 𝑇 𝑧 ) = 0 , that is, 𝑧 ∈ 𝑇 𝑧 .
Conversely, suppose that 𝑧 ∈ 𝑋 is a fixed point of 𝑇 . Let { 𝑦 𝑛 } 𝑛 ∈ ℕ 0 ⊂ 𝑋 be an arbitrarily orbit of 𝑇 with l i m 𝑛 → ∞ 𝑦 𝑛 = 𝑧 . It is clear that 𝑓 ( 𝑧 ) = 𝑑 ( 𝑧 , 𝑇 𝑧 ) = 0 ≤ l i m i n f 𝑛 → ∞ 𝑓 𝑦 𝑛 , ( 2 . 2 4 ) which implies that 𝑓 is 𝑇 orbitally lower semicontinuous at 𝑧 . This completes the proof.

Notice that 𝑑 ( 𝑦 , 𝑇 𝑦 ) ≤ 𝐻 ( 𝑇 𝑥 , 𝑇 𝑦 ) for each 𝑦 ∈ 𝑇 𝑥 . In light of Theorem 2.3, we have

Theorem 2.4. Let ( 𝑋 , 𝑑 ) be a complete metric space and 𝑇 ∶ 𝑋 → 𝐶 𝐿 ( 𝑋 ) satisfy that 𝜃 ( 𝐻 ( 𝑇 𝑥 , 𝑇 𝑦 ) ) ≤ 𝜑 ( 𝑑 ( 𝑥 , 𝑦 ) ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) , ∀ ( 𝑥 , 𝑦 ) ∈ 𝑋 × 𝑇 𝑥 , ( 2 . 2 5 ) where 𝜃 ∈ Θ and 𝜑 satisfies (2.7). Then for each 𝑥 0 ∈ 𝑋 , there exists an orbit { 𝑥 𝑛 } 𝑛 ∈ ℕ 0 of 𝑇 and 𝑧 ∈ 𝑋 such that l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑧 . Furthermore, 𝑧 ∈ 𝑋 is fixed point of 𝑇 if and only if the function 𝑓 defined by (1.8) is 𝑇 orbitally lower semicontinuous at 𝑧 .

If 𝜑 ( 𝑑 ( 𝑥 , 𝑦 ) ) in (2.6) is replaced by 𝜑 ( 𝑑 ( 𝑥 , 𝑇 𝑥 ) ) , one has

Theorem 2.5. Let ( 𝑋 , 𝑑 ) be a complete metric space and 𝑇 ∶ 𝑋 → 𝐶 𝐿 ( 𝑋 ) satisfy that 𝜃 ( 𝑑 ( 𝑦 , 𝑇 𝑦 ) ) ≤ 𝜑 ( 𝑑 ( 𝑥 , 𝑇 𝑥 ) ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) , ∀ ( 𝑥 , 𝑦 ) ∈ 𝑋 × 𝑇 𝑥 , ( 2 . 2 6 ) where 𝜃 ∈ Θ and 𝜑 satisfies (2.7). Then for each 𝑥 0 ∈ 𝑋 , there exists an orbit { 𝑥 𝑛 } 𝑛 ∈ ℕ 0 of 𝑇 and 𝑧 ∈ 𝑋 such that l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑧 . Furthermore, 𝑧 ∈ 𝑋 is fixed point of 𝑇 if and only if the function 𝑓 defined by (1.8) is 𝑇 orbitally lower semicontinuous at 𝑧 .

Proof. Let 𝑥 0 ∈ 𝑋 be any initial point and choose 𝑥 1 ∈ 𝑇 𝑥 0 . It follows from (2.7), (2.26), and Lemma 2.2 that for √ 𝑞 = 1 / m a x { 𝜑 ( 𝑑 ( 𝑥 0 , 𝑇 𝑥 0 √ ) ) , 𝜑 ( 𝑑 ( 𝑥 1 , 𝑇 𝑥 1 ) ) , 1 / 2 } > 1 there exists 𝑥 2 ∈ 𝑇 𝑥 1 such that 𝜃 𝑑 𝑥 1 , 𝑥 2 ≤ 𝜃 𝑑 𝑥 1 , 𝑇 𝑥 1 m a x 𝜑 𝑑 𝑥 0 , 𝑇 𝑥 0 , 𝜑 𝑑 𝑥 1 , 𝑇 𝑥 1 ≤ 𝜑 𝑑 𝑥 , 1 / 2 0 , 𝑇 𝑥 0 𝜃 𝑑 𝑥 0 , 𝑥 1 m a x 𝜑 𝑑 𝑥 0 , 𝑇 𝑥 0 , 𝜑 𝑑 𝑥 1 , 𝑇 𝑥 1 ≤ , 1 / 2 𝜑 𝑑 𝑥 0 , 𝑇 𝑥 0 𝜃 𝑑 𝑥 0 , 𝑥 1 , 𝜃 𝑑 𝑥 2 , 𝑇 𝑥 2 𝑑 𝑥 ≤ 𝜑 1 , 𝑇 𝑥 1 𝜃 𝑑 𝑥 1 , 𝑥 2 ≤ 𝜑 𝑑 𝑥 1 , 𝑇 𝑥 1 𝜃 𝑑 𝑥 1 , 𝑇 𝑥 1 m a x 𝜑 𝑑 𝑥 0 , 𝑇 𝑥 0 , 𝜑 𝑑 𝑥 1 , 𝑇 𝑥 1 ≤ , 1 / 2 𝜑 𝑑 𝑥 1 , 𝑇 𝑥 1 𝜃 𝑑 𝑥 1 , 𝑇 𝑥 1 . ( 2 . 2 7 ) Repeating the above argument we obtain a sequence { 𝑥 𝑛 } 𝑛 ∈ ℕ 0 ⊂ 𝑋 satisfying 𝑥 𝑛 + 1 ∈ 𝑇 𝑥 𝑛 for each 𝑛 ∈ ℕ 0 , 𝜃 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 1 ≤ 𝜃 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 m a x 𝜑 𝑑 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 − 1 , 𝜑 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 ≤ 𝜑 𝑑 𝑥 , 1 / ( 𝑛 + 1 ) 𝑛 − 1 , 𝑇 𝑥 𝑛 − 1 𝜃 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 m a x 𝜑 𝑑 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 − 1 , 𝜑 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 ≤ , 1 / ( 𝑛 + 1 ) 𝜑 𝑑 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 − 1 𝜃 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 𝜃 𝑑 𝑥 , ∀ 𝑛 ∈ ℕ , ( 2 . 2 8 ) 𝑛 + 1 , 𝑇 𝑥 𝑛 + 1 𝑑 𝑥 ≤ 𝜑 𝑛 , 𝑇 𝑥 𝑛 𝜃 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 1 ≤ 𝜑 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 𝜃 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 m a x 𝜑 𝑑 𝑥 𝑛 − 1 , 𝑇 𝑥 𝑛 − 1 , 𝜑 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 ≤ , 1 / ( 𝑛 + 1 ) 𝜑 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 𝜃 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 , ∀ 𝑛 ∈ ℕ . ( 2 . 2 9 )
Suppose that 𝑥 𝑛 0 ∈ 𝑇 𝑥 𝑛 0 for some 𝑛 0 ∈ ℕ 0 . It is easy to verify that 𝑥 𝑛 = 𝑥 𝑛 0 for all 𝑛 ≥ 𝑛 0 and the conclusion of Theorem 2.5 holds.
Suppose that 𝑥 𝑛 ∉ 𝑇 𝑥 𝑛 for each 𝑛 ∈ ℕ 0 . It follows that { 𝑑 ( 𝑥 𝑛 , 𝑇 𝑥 𝑛 ) } 𝑛 ∈ ℕ 0 and { 𝑑 ( 𝑥 𝑛 , 𝑥 𝑛 + 1 ) } 𝑛 ∈ ℕ 0 are positive sequences. Combining (2.7), (2.28), (2.29), (b) and (e), we infer that { 𝜃 ( 𝑑 ( 𝑥 𝑛 , 𝑥 𝑛 + 1 ) ) } 𝑛 ∈ ℕ 0 and { 𝜃 ( 𝑑 ( 𝑥 𝑛 , 𝑇 𝑥 𝑛 ) ) } 𝑛 ∈ ℕ 0 are both positive and decreasing, so do { 𝑑 ( 𝑥 𝑛 , 𝑥 𝑛 + 1 ) } 𝑛 ∈ ℕ 0 and { 𝑑 ( 𝑥 𝑛 , 𝑇 𝑥 𝑛 ) } 𝑛 ∈ ℕ 0 . It follows that there exist constants 𝛼 , 𝛽 , 𝑠 and 𝑡 satisfying l i m 𝑛 → ∞ 𝜃 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 1 = 𝛼 ≥ 0 , l i m 𝑛 → ∞ 𝑑 𝑥 𝑛 , 𝑥 𝑛 + 1 = 𝛽 ≥ 0 , l i m 𝑛 → ∞ 𝜃 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 = 𝑠 ≥ 0 , l i m 𝑛 → ∞ 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 = 𝑡 ≥ 0 . ( 2 . 3 0 ) Notice that (2.7) implies that there exists a constant 𝑟 such that l i m s u p 𝑛 → ∞ 𝜑 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 ≤ l i m s u p 𝑙 → 𝑡 + 𝜑 [ ( 𝑙 ) = 𝑟 ∈ 0 , 1 ) . ( 2 . 3 1 ) Taking upper limits in (2.29) and by (2.30) and (2.31) we get that 𝑠 ≤ l i m s u p 𝑛 → ∞ 𝜑 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 l i m s u p 𝑛 → ∞ 𝜃 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 ≤ √ 𝑟 𝑠 , ( 2 . 3 2 ) which implies that 𝑠 = 0 , which together with (2.30) and (a) ensures that 𝑑 𝑥 0 ≤ 𝜃 ( 𝑡 ) < 𝜃 𝑛 , 𝑇 𝑥 𝑛 ⟶ 0 , 𝑛 ⟶ ∞ , ( 2 . 3 3 ) that is, 𝜃 ( 𝑡 ) = 0 , which gives that 𝑡 = 0 by (b) and (d). It follows from (2.28), (2.30), and (2.31) that 𝛼 ≤ l i m s u p 𝑛 → ∞ 𝜑 𝑑 𝑥 𝑛 , 𝑇 𝑥 𝑛 l i m s u p 𝑛 → ∞ 𝜃 𝑑 𝑥 𝑛 − 1 , 𝑥 𝑛 ≤ √ 𝑟 𝛼 , ( 2 . 3 4 ) which yields that 𝛼 = 0 . Notice that (2.30) and (a) guarantee that 𝑑 𝑥 0 ≤ 𝜃 ( 𝛽 ) < 𝜃 𝑛 , 𝑥 𝑛 + 1 ⟶ 0 , 𝑛 ⟶ ∞ , ( 2 . 3 5 ) which together with (b) and (d) yields that 𝛽 = 0 . The rest of the proof is similar to that of Theorem 2.3 and is omitted. This completes the proof.

The result below follows from Theorem 2.5.

Theorem 2.6. Let ( 𝑋 , 𝑑 ) be a complete metric space and 𝑇 ∶ 𝑋 → 𝐶 𝐿 ( 𝑋 ) satisfy that 𝜃 ( 𝐻 ( 𝑇 𝑥 , 𝑇 𝑦 ) ) ≤ 𝜑 ( 𝑑 ( 𝑥 , 𝑇 𝑥 ) ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) , ∀ ( 𝑥 , 𝑦 ) ∈ 𝑋 × 𝑇 𝑥 , ( 2 . 3 6 ) where 𝜃 ∈ Θ and 𝜑 satisfies (2.7). Then for each 𝑥 0 ∈ 𝑋 , there exists an orbit { 𝑥 𝑛 } 𝑛 ∈ ℕ 0 of 𝑇 and 𝑧 ∈ 𝑋 such that l i m 𝑛 → ∞ 𝑥 𝑛 = 𝑧 . Furthermore, 𝑧 ∈ 𝑋 is fixed point of   𝑇 if and only if the function 𝑓 defined by (1.8) is 𝑇 orbitally lower semicontinuous at 𝑧 .

3. Comparisons and Examples

Now we construct two examples to compare the results in Section 2 with the corresponding results in [6–8].

Remark 3.1. Theorems 2.3 and 2.4 extend Theorems 1.1–1.3, and Theorems 2.5 and 2.6 are different from Theorems 1.1–1.3, respectively, in the following ways:(1)the ranges 𝐶 𝐿 ( 𝑋 ) of the nonlinear set-valued contractive mappings 𝑇 in Theorems 2.3–2.6 are more general than the ranges 𝐶 ( 𝑋 ) and 𝐶 𝐵 ( 𝑋 ) of the set-valued contraction mappings 𝑇 in Theorems 1.1–1.3, respectively;(2)the 𝑇 orbit lower semicontinuity at some 𝑧 ∈ 𝑋 of the functions 𝑓 ( 𝑥 ) = 𝑑 ( 𝑥 , 𝑇 𝑥 ) in Theorems 2.3 and 2.4 is weaker than the continuity of the set-valued contraction mappings 𝑇 in 𝑋 in Theorems 1.1–1.3, respectively;(3)the set-valued contraction mappings (1.1) and (1.2) are special cases of the nonlinear set-valued contractive mapping (2.6) with 𝜃 ≡ 1 because 𝑑 ( 𝑦 , 𝑇 𝑦 ) ≤ 𝐻 ( 𝑇 𝑥 , 𝑇 𝑦 ) , ∀ ( 𝑥 , 𝑦 ) ∈ 𝑋 × 𝑇 𝑥 . ( 3 . 1 )
Example 3.2 below shows that Theorems 2.3 and 2.4 extend substantively Theorems 1.1–1.3, respectively.

Example 3.2. Let 𝑋 = ( − ∞ , 3 / 1 0 ] and 𝑑 be the standard metric in 𝑋 . Let 𝜃 ∶ ℝ + → ℝ + , 𝜑 ∶ ℝ + → [ 0 , 1 ) and 𝑇 ∶ 𝑋 → 𝐶 𝐿 ( 𝑋 ) be defined by 𝜃 ( 𝑡 ) = 𝑡 1 / 2 2 √ , 𝜑 ( 𝑡 ) = 6 5 , ∀ 𝑡 ∈ ℝ + ⎧ ⎪ ⎨ ⎪ ⎩ 1 , 𝑇 𝑥 = − ∞ , 4 𝑥 , ∀ 𝑥 ∈ ( − ∞ , 0 ) , 0 , 2 𝑥 2 3 , ∀ 𝑥 ∈ 0 , , 1 0 ( 3 . 2 ) respectively. It is clear that 𝜃 ∈ Θ , 𝜑 satisfies (2.7) and 𝑓 ( 𝑥 ) = 𝑑 ( 𝑥 , 𝑇 𝑥 ) = 0 , ∀ 𝑥 ∈ ( − ∞ , 0 ) 𝑥 − 2 𝑥 2 3 , ∀ 𝑥 ∈ 0 , 1 0 ( 3 . 3 ) is 𝑇 orbitally lower semicontinuous in 𝑋 . In order to prove (2.6) holds, we consider two possible cases.
Case  1. Let 𝑥 ∈ ( − ∞ , 0 ) and 𝑦 ∈ 𝑇 𝑥 = ( − ∞ , ( 1 / 4 ) 𝑥 ] . It is clear that 1 𝜃 ( 𝑑 ( 𝑦 , 𝑇 𝑦 ) ) ≤ 𝜃 ( 𝐻 ( 𝑇 𝑥 , 𝑇 𝑦 ) ) = 2 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) ≤ 𝜑 ( 𝑑 ( 𝑥 , 𝑦 ) ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) . ( 3 . 4 )
Case  2. Let 𝑥 ∈ [ 0 , 3 / 1 0 ] and 𝑦 ∈ 𝑇 𝑥 = [ 0 , 2 𝑥 2 ] . It follows that √ 𝜃 ( 𝑑 ( 𝑦 , 𝑇 𝑦 ) ) ≤ 𝜃 ( 𝐻 ( 𝑇 𝑥 , 𝑇 𝑦 ) ) = 2 | | | | 𝑥 + 𝑦 1 / 2 ≤ √ 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) 2 3 + 9 1 0 5 0 1 / 2 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) = 𝜑 ( 𝑑 ( 𝑥 , 𝑦 ) ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) , ( 3 . 5 ) that is, (2.6) holds. Therefore all assumptions of Theorems 2.3 and 2.4 are satisfied. It follows from each of Theorems 2.3 and 2.4 that 𝑇 has a fixed point in 𝑋 . However, we cannot invoke any one of Theorems 1.1–1.3 to show the existence of fixed points for the mapping 𝑇 in 𝑋 . Indeed, taking 𝑥 0 = 3 / 1 0 and 𝑦 0 = 1 / 5 , we get that 𝐻 𝑇 𝑥 0 , 𝑇 𝑦 0 2 3 = 𝑑 1 0 2 1 , 2 5 2 = 1 ≰ 𝑟 1 0 𝑥 1 0 = 𝑟 𝑑 0 , 𝑦 0 , ( 3 . 6 ) for any 𝑟 ∈ ( 0 , 1 ) and 𝐻 𝑇 𝑥 0 , 𝑇 𝑦 0 2 3 = 𝑑 1 0 2 1 , 2 5 2 = 1 ≰ 1 1 0 𝜑 1 1 0 𝑑 𝑥 1 0 = 𝜑 0 , 𝑦 0 𝑑 𝑥 0 , 𝑦 0 , ( 3 . 7 ) for any mapping 𝜑 ∶ ℝ + → [ 0 , 1 ) with each of (1.3) and (1.4).
Next we construct an example to explain Theorems 2.5 and 2.6.

Example 3.3. Let 𝑋 = [ − 3 / 1 0 , + ∞ ) and 𝑑 be the standard metric in 𝑋 . Define 𝜃 ∶ ℝ + → ℝ + , 𝜑 ∶ ℝ + → [ 0 , 1 ) and 𝑇 ∶ 𝑋 → 𝐶 𝐿 ( 𝑋 ) by 𝜃 ( 𝑡 ) = 𝑡 1 / 2 , ∀ 𝑡 ∈ ℝ + ⎧ ⎪ ⎨ ⎪ ⎩ 2 √ , 𝜑 ( 𝑡 ) = 2 𝑡 1 / 2 1 , ∀ 𝑡 ∈ 0 , 8 , 2 √ 6 5 1 , ∀ 𝑡 ∈ { 0 } ∪ 8 , ⎧ ⎪ ⎨ ⎪ ⎩ 𝑥 , + ∞ 𝑇 𝑥 = 4 ( 1 + 𝑥 ) , + ∞ , ∀ 𝑥 ∈ ( 0 , + ∞ ) , − 2 𝑥 2 − 3 , 0 , ∀ 𝑥 ∈ , 1 0 , 0 ( 3 . 8 ) respectively. It is easy to see that (2.7) holds and 𝑓 ( 𝑥 ) = 𝑑 ( 𝑥 , 𝑇 𝑥 ) = 0 , ∀ 𝑥 ∈ ( 0 , + ∞ ) , − 2 𝑥 2 − 3 − 𝑥 , ∀ 𝑥 ∈ 1 0 , 0 ( 3 . 9 ) is 𝑇 orbitally lower semicontinuous in 𝑋 . In order to check (2.26), we have to consider two cases as follows.
Case  1. Let 𝑥 ∈ ( 0 , + ∞ ) and 𝑦 ∈ 𝑇 𝑥 = [ 𝑥 / 4 ( 1 + 𝑥 ) , + ∞ ) . It is clear that | | | | 𝑥 𝜃 ( 𝑑 ( 𝑦 , 𝑇 𝑦 ) ) = 0 ≤ 𝜃 ( 𝐻 ( 𝑇 𝑥 , 𝑇 𝑦 ) ) = − 𝑦 4 ( 1 + 𝑥 ) | | | | 4 ( 1 + 𝑦 ) 1 / 2 = 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) 2 ( 1 + 𝑥 ) 1 / 2 ( 1 + 𝑦 ) 1 / 2 ≤ 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) 2 ( 1 + 𝑥 ) 1 / 2 ( 1 + 𝑥 / 4 ( 1 + 𝑥 ) ) 1 / 2 = 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) ( 5 𝑥 + 4 ) 1 / 2 ≤ 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) 2 ≤ 2 √ 6 5 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) = 𝜑 ( 0 ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) = 𝜑 ( 𝑑 ( 𝑥 , 𝑇 𝑥 ) ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) . ( 3 . 1 0 )
Case  2. Let 𝑥 ∈ [ − 3 / 1 0 , 0 ] and 𝑦 ∈ 𝑇 𝑥 = [ − 2 𝑥 2 , 0 ] . It follows that √ 𝜃 ( 𝑑 ( 𝑦 , 𝑇 𝑦 ) ) ≤ 𝜃 ( 𝐻 ( 𝑇 𝑥 , 𝑇 𝑦 ) ) = 2 | | | | 𝑥 + 𝑦 1 / 2 √ 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) ≤ 2 | | 𝑥 − 2 𝑥 2 | | 1 / 2 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) . ( 3 . 1 1 ) For 𝑥 = 0 , we have √ 2 | | 𝑥 − 2 𝑥 2 | | 1 / 2 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) = 0 ≤ 𝜑 ( 𝑑 ( 𝑥 , 𝑇 𝑥 ) ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) . ( 3 . 1 2 ) For 𝑥 ∈ [ − 3 / 1 0 , − 1 / 4 ) ∪ ( − 1 / 4 , 0 ) , we infer that √ 2 | | 𝑥 − 2 𝑥 2 | | 1 / 2 √ 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) ≤ 2 2 − 2 𝑥 2 − 𝑥 1 / 2 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) = 𝜑 ( 𝑑 ( 𝑥 , 𝑇 𝑥 ) ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) . ( 3 . 1 3 ) For 𝑥 = − 1 / 4 , we get that √ 2 | | 𝑥 − 2 𝑥 2 | | 1 / 2 √ 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) = 3 2 1 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) ≤ 𝜑 8 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) = 𝜑 ( 𝑑 ( 𝑥 , 𝑇 𝑥 ) ) 𝜃 ( 𝑑 ( 𝑥 , 𝑦 ) ) . ( 3 . 1 4 ) Hence (2.26) holds. Thus all assumptions of Theorems 2.5 and 2.6 are satisfied. It follows from each of Theorems 2.5 and 2.6 that 𝑇 has a fixed point in 𝑋 .
Taking 𝑥 0 = 1 and 𝑦 0 = − 3 / 1 0 , we deduce that 𝐻 𝑇 𝑥 0 , 𝑇 𝑦 0 1 = 𝐻 8 , − 9 , + ∞ 5 0 , 0 = + ∞ ≰ 1 3 𝑟 𝑥 1 0 = 𝑟 𝑑 0 , 𝑦 0 , ( 3 . 1 5 ) for any 𝑟 ∈ ( 0 , 1 ) , and 𝐻 𝑇 𝑥 0 , 𝑇 𝑦 0 2 √ = + ∞ ≰ 6 5 ⋅ 1 3 𝑑 𝑥 1 0 = 𝜑 0 , 𝑦 0 𝑑 𝑥 0 , 𝑦 0 , ( 3 . 1 6 ) for any mapping 𝜑 ∶ ℝ + → [ 0 , 1 ) with each of (1.3) and (1.4). That is, Theorems 1.1–1.3 are inapplicable in proving the existence of fixed points for the nonlinear set-valued contractive mapping 𝑇 .

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