Some Fixed Point Theorems for Nonlinear Set-Valued Contractive Mappings

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→𝐶𝐵(𝑋) be a set-valued mapping such that 𝐻(𝑇𝑥,𝑇𝑦)≤𝑟𝑑(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(𝑇𝑥,𝑇𝑦)≤𝜑(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦),∀𝑥,𝑦∈𝑋,(1.2) where 𝜑∶(0,+∞)⟶[0,1)𝑤𝑖𝑡ℎlimsup_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→𝐶𝐵(𝑋) with (1.2). The affirmative answer under the hypothesis of limsup_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→𝐶𝐵(𝑋) satisfy (1.2), where 𝜑∶(0,+∞)⟶[0,1)𝑤𝑖𝑡ℎlimsup_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∈𝑇𝑎⊆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 ℕ₀ denote the sets of all positive integers and nonnegative integers, respectively, and Θ=𝜃∶𝜃∶ℝ⁺⟶ℝ⁺satisfies(a)−(d),(1.5) where(a)𝜃is nondecreasing on ℝ⁺;(b)𝜃(𝑡)>0, for  all 𝑡∈(0,+∞);(c)𝜃 is subadditive in (0,+∞), that is, 𝜃𝑡₁+𝑡₂≤𝜃𝑡₁+𝜃𝑡₂,∀𝑡₁,𝑡₂∈(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, 𝐶𝐿(𝑋), 𝐶𝐵(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∈𝐶𝐿(𝑋), put 𝑑(𝑥,𝐴)=inf{𝑑(𝑥,𝑦)∶𝑦∈𝐴} and 𝐻(𝐴,𝐵)=⎧⎪⎨⎪⎩maxsup_𝑥∈𝐴+∞,𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑(𝑦,𝐴),iofththerewmisaex.imumexists(1.7) Such a mapping 𝐻 is called a generalized Hausdorff metric induced b 𝑦𝑑 in 𝐶𝐿(𝑋). It is well known that 𝐻 is a metric on 𝐶𝐵(𝑋). Let 𝑇∶𝑋→𝐶𝐿(𝑋) be a set-valued mapping, 𝑥₀∈𝑋 and 𝑓∶𝑋→ℝ⁺ be defined by 𝑓(𝑥)=𝑑(𝑥,𝑇𝑥),∀𝑥∈𝑋.(1.8) A sequence {𝑥_𝑛}_𝑛∈ℕ0 is said to be an orbit o 𝑓𝑇 if it satisfies that {𝑥_𝑛}_𝑛∈ℕ0⊂𝑋 and 𝑥_𝑛∈𝑇𝑥_𝑛−1 for each 𝑛∈ℕ₀. The function 𝑓∶𝑋→ℝ⁺ is said to be 𝑇−𝑜𝑟𝑏𝑖𝑡𝑎𝑙𝑙𝑦𝑙𝑜𝑤𝑒𝑟𝑠𝑒𝑚𝑖𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡𝑧∈𝑋 if for each orbit {𝑥_𝑛}_𝑛∈ℕ0⊂𝑋 of 𝑇 with lim_𝑛→∞𝑥_𝑛=𝑧, we have that 𝑓(𝑧)≤liminf_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∈𝐶𝐿(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 𝑥₀∈𝑋 and 𝜀₀>0 such that 𝑑𝑥₀,𝑏>𝑑𝑥₀,𝐵+𝜀₀,∀𝑏∈𝐵,(2.1) which yields that 𝑑𝑥₀,𝐵=inf_𝑏∈𝐵𝑑𝑥₀,𝑏≥𝑑𝑥₀,𝐵+𝜀₀>𝑑𝑥₀,𝐵,(2.2) which is a contradiction. This completes the proof.

Lemma 2.2. Let (𝑋,𝑑) be a metric space, 𝐵∈𝐶𝐿(𝑋) and 𝜃∈Θ. Then for each 𝑥∈𝑋 and 𝑞>1 there exists 𝑏∈𝐵 such that 𝜃(𝑑(𝑥,𝑏))≤𝑞𝜃(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⁺⧵{0}=𝜃ℝ⁺⧵{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(𝑑(𝑥,𝑏))≤≤𝜃𝜃((𝑑𝑑((𝑥𝑥,,𝐵𝐵)))++𝜀𝜃)≤𝜃𝜃₋₁(𝑑(𝑥,𝐵))+𝜃(𝜀)((𝑞−1)𝜃(𝑑(𝑥,𝐵)))=𝑞𝜃(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→𝐶𝐿(𝑋) satisfy that 𝜃(𝑑(𝑦,𝑇𝑦))≤𝜑(𝑑(𝑥,𝑦))𝜃(𝑑(𝑥,𝑦)),∀(𝑥,𝑦)∈𝑋×𝑇𝑥,(2.6) where 𝜃∈Θ and 𝜑∶ℝ⁺⟶[0,1)𝑤𝑖𝑡ℎlimsup_𝑟→𝑡+𝜑(𝑟)<1,∀𝑡∈ℝ⁺.(2.7) Then for each 𝑥₀∈𝑋, there exists an orbit {𝑥_𝑛}_𝑛∈ℕ0 of 𝑇 and 𝑧∈𝑋 such that lim_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 𝑥₀∈𝑋 be any initial point and choose 𝑥₁∈𝑇𝑥₀. It follows from (2.6), (2.7) and Lemma 2.2 that for 𝑞₁=1/max{√𝜑(𝑑(𝑥₀,𝑥₁)),1/2}>1 there exists 𝑥₂∈𝑇𝑥₁ satisfying 𝜃𝑑𝑥₁,𝑥₂≤𝜃𝑑𝑥¹,𝑇𝑥₁max𝜑𝑑𝑥₀,𝑥₁,1/2≤𝜑𝑑𝑥⁰,𝑥₁𝜃𝑑𝑥₀,𝑥₁≤max𝜑𝑑𝑥₀,𝑥₁,1/2𝜑𝑑𝑥₀,𝑥₁𝜃𝑑𝑥₀,𝑥₁,(2.8) and for 𝑞₂=1/max{√𝜑(𝑑(𝑥₁,𝑥₂)),1/3}>1 there exists 𝑥₃∈𝑇𝑥₂ satisfying 𝜃𝑑𝑥₂,𝑥₃≤𝜃𝑑𝑥²,𝑇𝑥₂max𝜑𝑑𝑥₁,𝑥₂,1/3≤𝜑𝑑𝑥¹,𝑥₂𝜃𝑑𝑥₁,𝑥₂≤max𝜑𝑑𝑥₁,𝑥₂,1/3𝜑𝑑𝑥₁,𝑥₂𝜃𝑑𝑥₁,𝑥₂.(2.9) Repeating the above argument we obtain a sequence {𝑥_𝑛}_𝑛∈ℕ0⊂𝑋 such that 𝑥_𝑘∈𝑇𝑥_𝑘−1 for 1≤𝑘≤𝑛 and for 𝑞_𝑛=1/max{√𝜑(𝑑(𝑥_𝑛−1,𝑥_𝑛)),1/(𝑛+1)}>1, there exists 𝑥_𝑛+1∈𝑇𝑥_𝑛 satisfying 𝜃𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜃𝑑𝑥^𝑛,𝑇𝑥_𝑛≤max𝜑𝑑𝑥𝜑𝑑𝑥_𝑛−1,𝑥_𝑛,1/(𝑛+1)_𝑛−1,𝑥_𝑛𝜃𝑑𝑥_𝑛−1,𝑥_𝑛≤max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.(2.10)
Suppose that there exists some 𝑛₀∈ℕ₀ satisfying 𝑥_𝑛0=𝑥_𝑛0+1∈𝑇𝑥_𝑛0. It follows from (a), (b), and (2.10) that 𝑥_𝑛=𝑥_𝑛0 for all 𝑛≥𝑛₀+1. It is clear the conclusion of Theorem 2.3 holds.
Suppose that 𝑥_𝑛+1∈𝑇𝑥_𝑛⧵{𝑥_𝑛} for any 𝑛∈ℕ₀. It follows that 𝑑(𝑥_𝑛,𝑥_𝑛+1)>0 for each 𝑛∈ℕ₀. Note that (b), (2.7), and (2.10) give that {𝜃(𝑑(𝑥_𝑛,𝑥_𝑛+1))}_𝑛∈ℕ0 is a positive and decreasing sequence. It follows from (e) that {𝑑(𝑥_𝑛,𝑥_𝑛+1)}_𝑛∈ℕ0 is decreasing. Therefore, there exist constants 𝑝 and 𝑞 satisfying lim_𝑛→∞𝜃𝑑𝑥_𝑛,𝑥_𝑛+1=𝑝≥0,lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=𝑞≥0.(2.11) Notice that (2.7) implies that there exists a constant 𝑟 satisfying limsup_𝑛→∞𝜑𝑑𝑥_𝑛,𝑥_𝑛−1≤limsup_𝑡→𝑞+𝜑(𝑡)=𝑟∈[0,1).(2.12) Taking upper limits in (2.10) and by (2.11) and (2.12) we get that 𝑝≤limsup_𝑛→∞𝜑𝑑𝑥_𝑛−1,𝑥_𝑛limsup_𝑛→∞𝜃𝑑𝑥_𝑛−1,𝑥_𝑛≤√𝑟𝑝,(2.13) which implies that 𝑝=0.
Next we assert that 𝑞=0. Since {𝑑(𝑥_𝑛,𝑥_𝑛+1)}_𝑛∈ℕ0 is a decreasing sequence, it follows from (a) and (2.11) that 0≤𝜃(𝑞)<𝜃𝑑𝑥_𝑛,𝑥_𝑛+1⟶𝑝=0as𝑛⟶∞,(2.14) 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 𝑐²∈(𝑟,1). Notice that (2.11), (2.12), and 𝑞=0 ensure that there exist 𝛿>0 and 𝑁∈ℕ satisfying 𝜑(𝑡)<𝑐²,∀𝑡∈(0,𝛿),𝑑𝑥_𝑛,𝑥_𝑛+1<𝛿,∀𝑛≥𝑁,(2.15) which implies that 𝜑𝑑𝑥_𝑛,𝑥_𝑛+1<𝑐²,∀𝑛≥𝑁.(2.16) Note that (2.10) and (2.16) mean that 𝜃𝑑𝑥_𝑛,𝑥_𝑛+1≤^𝑛−1_𝑘=𝑁≤𝑐_𝑛−𝑁𝜑𝑑𝑥_𝑘,𝑥_𝑘+1𝜃𝑑𝑥_𝑁,𝑥_𝑁+1𝜃𝑑𝑥_𝑁,𝑥_𝑁+1,∀𝑛≥𝑁.(2.17) Given 𝜀>0. Since lim_𝑛→∞𝑐^𝑛−𝑁𝜃(𝑑(𝑥_𝑁,𝑥_𝑁+1))=0, it follows from (b) that there exists 𝑁₁>𝑁 satisfying 𝑐^𝑛−𝑁1−𝑐𝜃𝑑𝑥^𝑁,𝑥_𝑁+1<𝜃(𝜀),∀𝑛≥𝑁₁,(2.18) which together with (2.17), (a), and (c) gives that 𝜃𝑑𝑥_𝑛,𝑥_𝑚≤𝜃^𝑚−1_𝑘=𝑛𝑑𝑥_𝑘,𝑥_𝑘+1≤^𝑚−1_𝑘=𝑛≤𝜃𝑑𝑥_𝑘,𝑥_𝑘+1_𝑚−1_𝑘=𝑛≤𝑐𝑐^𝑘−𝑁𝜃𝑑𝑥_𝑁,𝑥_𝑁+1_𝑛−𝑁1−𝑐𝜃𝑑𝑥^𝑁,𝑥_𝑁+1<𝜃(𝜀),∀𝑚>𝑛≥𝑁₁.(2.19) In view of (e) and (2.19), we deduce that 𝑑(𝑥_𝑛,𝑥_𝑚)<𝜀, for all 𝑚>𝑛≥𝑁₁, which means that {𝑥_𝑛}_𝑛∈ℕ0 is a Cauchy sequence. Hence there exists 𝑧∈𝑋 such that lim_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 lim_𝑛→∞𝑥_𝑛=𝑧, it follows that 𝑓(𝑧)≤liminf_𝑛→∞𝑓𝑥_𝑛.(2.20) Using (2.6) and (2.7), we infer that 𝜃𝑑𝑥_𝑛,𝑇𝑥_𝑛≤𝜑𝑑𝑥_𝑛−1,𝑥_𝑛𝜃𝑑𝑥_𝑛−1,𝑥_𝑛<𝜃𝑑𝑥_𝑛−1,𝑥_𝑛,∀𝑛∈ℕ,(2.21) which together with (e), (2.11), and 𝑞=0 implies that 0<𝑑𝑥_𝑛,𝑇𝑥_𝑛<𝑑𝑥_𝑛−1,𝑥_𝑛⟶0as𝑛⟶∞,(2.22) that is, lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0, which together with (2.20) yields that 0≤𝑑(𝑧,𝑇𝑧)=𝑓(𝑧)≤liminf_𝑛→∞𝑓𝑥_𝑛=lim_𝑛→∞𝑑𝑥_𝑛,𝑇𝑥_𝑛=0,(2.23) which gives that 𝑑(𝑧,𝑇𝑧)=0, that is, 𝑧∈𝑇𝑧.
Conversely, suppose that 𝑧∈𝑋 is a fixed point of 𝑇. Let {𝑦_𝑛}_𝑛∈ℕ0⊂𝑋 be an arbitrarily orbit of 𝑇 with lim_𝑛→∞𝑦_𝑛=𝑧. It is clear that 𝑓(𝑧)=𝑑(𝑧,𝑇𝑧)=0≤liminf_𝑛→∞𝑓𝑦_𝑛,(2.24) which implies that 𝑓 is 𝑇 orbitally lower semicontinuous at 𝑧. This completes the proof.

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

Theorem 2.4. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝐶𝐿(𝑋) satisfy that 𝜃(𝐻(𝑇𝑥,𝑇𝑦))≤𝜑(𝑑(𝑥,𝑦))𝜃(𝑑(𝑥,𝑦)),∀(𝑥,𝑦)∈𝑋×𝑇𝑥,(2.25) where 𝜃∈Θ and 𝜑 satisfies (2.7). Then for each 𝑥₀∈𝑋, there exists an orbit {𝑥_𝑛}_𝑛∈ℕ0 of𝑇 and 𝑧∈𝑋 such that lim_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,𝑇𝑥)), one has

Theorem 2.5. Let (𝑋,𝑑) be a complete metric space and 𝑇∶𝑋→𝐶𝐿(𝑋) satisfy that 𝜃(𝑑(𝑦,𝑇𝑦))≤𝜑(𝑑(𝑥,𝑇𝑥))𝜃(𝑑(𝑥,𝑦)),∀(𝑥,𝑦)∈𝑋×𝑇𝑥,(2.26) where 𝜃∈Θ and 𝜑 satisfies (2.7). Then for each 𝑥₀∈𝑋, there exists an orbit {𝑥_𝑛}_𝑛∈ℕ0 of 𝑇 and 𝑧∈𝑋 such that lim_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 𝑥₀∈𝑋 be any initial point and choose 𝑥₁∈𝑇𝑥₀. It follows from (2.7), (2.26), and Lemma 2.2 that for 𝑞=1/max{√𝜑(𝑑(𝑥₀,𝑇𝑥₀)),√𝜑(𝑑(𝑥₁,𝑇𝑥₁)),1/2}>1 there exists 𝑥₂∈𝑇𝑥₁ such that 𝜃𝑑𝑥₁,𝑥₂≤𝜃𝑑𝑥¹,𝑇𝑥₁≤max𝜑𝜑𝑑𝑑𝑥𝑥₀,𝑇𝑥₀,𝜑𝑑𝑥₁,𝑇𝑥₁,1/2₀,𝑇𝑥₀𝜃𝑑𝑥₀,𝑥₁≤max𝜑𝑑𝑥₀,𝑇𝑥₀,𝜑𝑑𝑥₁,𝑇𝑥₁,1/2𝜃𝑑𝑥𝜑𝑑𝑥₀,𝑇𝑥₀𝜃𝑑𝑥₀,𝑥₁₂,,𝑇𝑥₂≤≤𝜑𝑑𝑥₁,𝑇𝜑𝑥₁𝑑𝜃𝑥𝑑𝑥₁,𝑥₂₁,𝑇𝑥₁𝜃𝑑𝑥₁,𝑇𝑥₁≤max𝜑𝑑𝑥₀,𝑇𝑥₀,𝜑𝑑𝑥₁,𝑇𝑥₁,1/2𝜑𝑑𝑥₁,𝑇𝑥₁𝜃𝑑𝑥₁,𝑇𝑥₁.(2.27) Repeating the above argument we obtain a sequence {𝑥_𝑛}_𝑛∈ℕ0⊂𝑋 satisfying 𝑥_𝑛+1∈𝑇𝑥_𝑛 for each 𝑛∈ℕ₀, 𝜃𝑑𝑥_𝑛,𝑥_𝑛+1≤𝜃𝑑𝑥^𝑛,𝑇𝑥_𝑛≤max𝜑𝜑𝑑𝑑𝑥𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝜑𝑑𝑥_𝑛,𝑇𝑥_𝑛,1/(𝑛+1)_𝑛−1,𝑇𝑥_𝑛−1𝜃𝑑𝑥_𝑛−1,𝑥_𝑛≤max𝜑𝑑𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝜑𝑑𝑥_𝑛,𝑇𝑥_𝑛,1/(𝑛+1)𝜃𝑑𝑥𝜑𝑑𝑥_𝑛−1,𝑇𝑥_𝑛−1𝜃𝑑𝑥_𝑛−1,𝑥_𝑛,∀𝑛∈ℕ,(2.28)_𝑛+1,𝑇𝑥_𝑛+1≤≤𝜑𝑑𝑥_𝑛,𝑇𝑥_𝑛𝜑𝜃𝑑𝑑𝑥𝑥_𝑛,𝑥_𝑛+1_𝑛,𝑇𝑥_𝑛𝜃𝑑𝑥_𝑛,𝑇𝑥_𝑛≤max𝜑𝑑𝑥_𝑛−1,𝑇𝑥_𝑛−1,𝜑𝑑𝑥_𝑛,𝑇𝑥_𝑛,1/(𝑛+1)𝜑𝑑𝑥_𝑛,𝑇𝑥_𝑛𝜃𝑑𝑥_𝑛,𝑇𝑥_𝑛,∀𝑛∈ℕ.(2.29)
Suppose that 𝑥_𝑛0∈𝑇𝑥_𝑛0 for some 𝑛₀∈ℕ₀. It is easy to verify that 𝑥_𝑛=𝑥_𝑛0 for all 𝑛≥𝑛₀ and the conclusion of Theorem 2.5 holds.
Suppose that 𝑥_𝑛∉𝑇𝑥_𝑛 for each 𝑛∈ℕ₀. It follows that {𝑑(𝑥_𝑛,𝑇𝑥_𝑛)}_𝑛∈ℕ0 and {𝑑(𝑥_𝑛,𝑥_𝑛+1)}_𝑛∈ℕ0 are positive sequences. Combining (2.7), (2.28), (2.29), (b) and (e), we infer that {𝜃(𝑑(𝑥_𝑛,𝑥_𝑛+1))}_𝑛∈ℕ0 and {𝜃(𝑑(𝑥_𝑛,𝑇𝑥_𝑛))}_𝑛∈ℕ0 are both positive and decreasing, so do {𝑑(𝑥_𝑛,𝑥_𝑛+1)}_𝑛∈ℕ0 and {𝑑(𝑥_𝑛,𝑇𝑥_𝑛)}_𝑛∈ℕ0. It follows that there exist constants 𝛼,𝛽,𝑠 and 𝑡 satisfying lilmim_{𝑛→∞𝑛→∞}𝜃𝑑𝑥_𝑛,𝑥_𝑛+1=𝛼≥0,lim_𝑛→∞𝑑𝑥_𝑛,𝑥_𝑛+1=𝛽≥0,𝜃𝑑𝑥_𝑛,𝑇𝑥_𝑛=𝑠≥0,lim_𝑛→∞𝑑𝑥_𝑛,𝑇𝑥_𝑛=𝑡≥0.(2.30) Notice that (2.7) implies that there exists a constant 𝑟 such that limsup_𝑛→∞𝜑𝑑𝑥_𝑛,𝑇𝑥_𝑛≤limsup_𝑙→𝑡+𝜑(𝑙)=𝑟∈[0,1).(2.31) Taking upper limits in (2.29) and by (2.30) and (2.31) we get that 𝑠≤limsup_𝑛→∞𝜑𝑑𝑥_𝑛,𝑇𝑥_𝑛limsup_𝑛→∞𝜃𝑑𝑥_𝑛,𝑇𝑥_𝑛≤√𝑟𝑠,(2.32) which implies that 𝑠=0, which together with (2.30) and (a) ensures that 0≤𝜃(𝑡)<𝜃𝑑𝑥_𝑛,𝑇𝑥_𝑛⟶0,𝑛⟶∞,(2.33) 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≤limsup_𝑛→∞𝜑𝑑𝑥_𝑛,𝑇𝑥_𝑛limsup_𝑛→∞𝜃𝑑𝑥_𝑛−1,𝑥_𝑛≤√𝑟𝛼,(2.34) which yields that 𝛼=0. Notice that (2.30) and (a) guarantee that 0≤𝜃(𝛽)<𝜃𝑑𝑥_𝑛,𝑥_𝑛+1⟶0,𝑛⟶∞,(2.35) 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→𝐶𝐿(𝑋) satisfy that 𝜃(𝐻(𝑇𝑥,𝑇𝑦))≤𝜑(𝑑(𝑥,𝑇𝑥))𝜃(𝑑(𝑥,𝑦)),∀(𝑥,𝑦)∈𝑋×𝑇𝑥,(2.36) where 𝜃∈Θ and 𝜑 satisfies (2.7). Then for each 𝑥₀∈𝑋, there exists an orbit {𝑥_𝑛}_𝑛∈ℕ0 of𝑇 and 𝑧∈𝑋 such that lim_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 𝐶𝐿(𝑋) of the nonlinear set-valued contractive mappings 𝑇 in Theorems 2.3–2.6 are more general than the ranges 𝐶(𝑋) and 𝐶𝐵(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,𝑇𝑥) 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,𝑇𝑦)≤𝐻(𝑇𝑥,𝑇𝑦),∀(𝑥,𝑦)∈𝑋×𝑇𝑥.(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=(−∞,3/10] and 𝑑 be the standard metric in 𝑋. Let 𝜃∶ℝ⁺→ℝ⁺, 𝜑∶ℝ⁺→[0,1) and 𝑇∶𝑋→𝐶𝐿(𝑋) be defined by 𝜃(𝑡)=𝑡^1/2,𝜑(𝑡)=2√65,∀𝑡∈ℝ⁺,𝑇𝑥=⎧⎪⎨⎪⎩0−,∞2𝑥,₂14𝑥,∀𝑥∈(−∞,0),,∀𝑥∈0,130,(3.2) respectively. It is clear that 𝜃∈Θ, 𝜑 satisfies (2.7) and 𝑓(𝑥)=𝑑(𝑥,𝑇𝑥)=0𝑥,−2𝑥₂∀𝑥∈(−∞,0),∀𝑥∈0,310(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∈𝑇𝑥=(−∞,(1/4)𝑥]. It is clear that𝜃(𝑑(𝑦,𝑇𝑦))≤𝜃(𝐻(𝑇𝑥,𝑇𝑦))=12𝜃(𝑑(𝑥,𝑦))≤𝜑(𝑑(𝑥,𝑦))𝜃(𝑑(𝑥,𝑦)).(3.4)
Case  2. Let 𝑥∈[0,3/10] and 𝑦∈𝑇𝑥=[0,2𝑥²]. It follows that 𝜃(𝑑(𝑦,𝑇𝑦))≤≤𝜃√(𝐻(𝑇𝑥,𝑇𝑦))=√2||𝑥+𝑦||^1/2𝜃(𝑑(𝑥,𝑦))2310+590^1/2𝜃(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 𝑥₀=3/10 and 𝑦₀=1/5, we get that 𝐻𝑇𝑥₀,𝑇𝑦₀=𝑑2310²,215²=110≰1𝑟0=𝑟𝑑𝑥⁰,𝑦₀,(3.6) for any 𝑟∈(0,1) and 𝐻𝑇𝑥₀,𝑇𝑦₀=𝑑2310²,215²=110≰110𝜑110=𝜑𝑑𝑥⁰,𝑦₀𝑑𝑥₀,𝑦₀,(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=[−3/10,+∞) and 𝑑 be the standard metric in 𝑋. Define 𝜃∶ℝ⁺→ℝ⁺, 𝜑∶ℝ⁺→[0,1) and 𝑇∶𝑋→𝐶𝐿(𝑋) by 𝜃(𝑡)=𝑡^1/2,∀𝑡∈ℝ⁺,𝜑(𝑡)=⎧⎪⎨⎪⎩2√2𝑡^1/2,∀𝑡∈0,12√8,6𝑇𝑥=⎧⎪⎨⎪⎩𝑥5,∀𝑡∈{0}∪18,+∞,−42(𝑥1₂+𝑥),+∞,∀𝑥∈(0,+∞),,0,∀𝑥∈−310,0,(3.8) respectively. It is easy to see that (2.7) holds and 𝑓(𝑥)=𝑑(𝑥,𝑇𝑥)=0−,2𝑥₂∀𝑥∈(0,+∞),−𝑥,∀𝑥∈−310,0(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∈𝑇𝑥=[𝑥/4(1+𝑥),+∞). It is clear that 𝜃(𝑑(𝑦,𝑇𝑦))=0≤𝜃(𝐻(𝑇𝑥,𝑇𝑦))=||||𝑥4(1+𝑥)−4(1𝑦+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))(5𝑥+4)^1/2≤𝜃(𝑑(𝑥,𝑦))2≤2√6=𝜑(0)𝜃(𝑑(𝑥,𝑦))=𝜑(𝑑(𝑥,𝑇𝑥)5)𝜃(𝜃𝑑(𝑑(𝑥(,𝑥𝑦,)𝑦)).)(3.10)
Case  2. Let 𝑥∈[−3/10,0] and 𝑦∈𝑇𝑥=[−2𝑥²,0]. It follows that 𝜃(𝑑(𝑦,𝑇𝑦))≤𝜃(𝐻(𝑇𝑥,𝑇𝑦))=√2||𝑥+𝑦||^1/2𝜃(𝑑(𝑥,𝑦))≤√2||𝑥−2𝑥²||^1/2𝜃(𝑑(𝑥,𝑦)).(3.11) For 𝑥=0, we have √2||𝑥−2𝑥²||^1/2𝜃(𝑑(𝑥,𝑦))=0≤𝜑(𝑑(𝑥,𝑇𝑥))𝜃(𝑑(𝑥,𝑦)).(3.12) For 𝑥∈[−3/10,−1/4)∪(−1/4,0), we infer that √2||𝑥−2𝑥²||^1/2𝜃(𝑑(𝑥,𝑦))≤2√2−2𝑥²−𝑥^1/2𝜃(𝑑(𝑥,𝑦))=𝜑(𝑑(𝑥,𝑇𝑥))𝜃(𝑑(𝑥,𝑦)).(3.13) For 𝑥=−1/4, we get that √2||𝑥−2𝑥²||^1/2𝜃(𝑑(𝑥,𝑦))=√32𝜃(𝑑(𝑥,𝑦))≤𝜑81𝜃(𝑑(𝑥,𝑦))=𝜑(𝑑(𝑥,𝑇𝑥))𝜃(𝑑(𝑥,𝑦)).(3.14) 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 𝑥₀=1 and 𝑦₀=−3/10, we deduce that 𝐻𝑇𝑥₀,𝑇𝑦₀=𝐻18,+∞,−590,0=+∞≰1130𝑟=𝑟𝑑𝑥⁰,𝑦₀,(3.15) for any 𝑟∈(0,1), and 𝐻𝑇𝑥₀,𝑇𝑦₀=+∞≰2√65⋅1130=𝜑𝑑𝑥⁰,𝑦₀𝑑𝑥₀,𝑦₀,(3.16) 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.