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

Some Fixed Point Theorems for Nonlinear Set-Valued Contractive Mappings

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, [19] 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 𝑇𝑋𝐶𝐵(𝑋) be a set-valued mapping such that 𝐻(𝑇𝑥,𝑇𝑦)𝑟𝑑(𝑥,𝑦),𝑥,𝑦𝑋,(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 𝑇𝑋𝐶(𝑋) satisfy that 𝐻(𝑇𝑥,𝑇𝑦)𝜑(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦),𝑥,𝑦𝑋,(1.2) where [𝜑(0,+)0,1)𝑤𝑖𝑡limsup𝑟𝑡+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 𝑇𝑋𝐶𝐵(𝑋) with (1.2). The affirmative answer under the hypothesis of limsup𝑟𝑡+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 𝑇𝑋𝐶𝐵(𝑋) satisfy (1.2), where [𝜑(0,+)0,1)𝑤𝑖𝑡limsup𝑟𝑡+[𝜑(𝑟)<1,𝑡0,+).(1.4) Then 𝑇 has a fixed point.

Remark 1.4. It is clear that the mappings 𝑇 in Theorems 1.11.3 are continuous on 𝑋.

Remark 1.5. Each of Theorems 1.2 and 1.3 ensures that 𝑇 has a fixed point 𝑎𝑇𝑎𝑋, 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 [68]. Two nontrivial examples are given to show that our results are genuine generalizations or different from these results in [68].
Throughout this paper, we assume that =(,+),+=[0,+), and 0 denote the sets of all positive integers and nonnegative integers, respectively, and Θ=𝜃𝜃++satises(a)(d),(1.5) where(a)𝜃is nondecreasing on +;(b)𝜃(𝑡)>0, for  all 𝑡(0,+);(c)𝜃 is subadditive in (0,+), that is, 𝜃𝑡1+𝑡2𝑡𝜃1𝑡+𝜃2,𝑡1,𝑡2(0,+);(1.6)(d)𝜃(+)=+. Clearly (a)–(d) imply that(e)𝜃 is strictly inverse on +, that is, if there exist 𝑡,𝑠+ 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 𝑥𝑋 and 𝐴,𝐵𝐶𝐿(𝑋), put 𝑑(𝑥,𝐴)=inf{𝑑(𝑥,𝑦)𝑦𝐴} and 𝐻(𝐴,𝐵)=maxsup𝑥𝐴𝑑(𝑥,𝐵),sup𝑦𝐵𝑑(𝑦,𝐴),ifthemaximumexists+,otherwise.(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 𝑇𝑋𝐶𝐿(𝑋) be a set-valued mapping, 𝑥0𝑋 and 𝑓𝑋+ be defined by 𝑓(𝑥)=𝑑(𝑥,𝑇𝑥),𝑥𝑋.(1.8) A sequence {𝑥𝑛}𝑛0 is said to be an orbit o 𝑓𝑇 if it satisfies that {𝑥𝑛}𝑛0𝑋 and 𝑥𝑛𝑇𝑥𝑛1 for each 𝑛0. The function 𝑓𝑋+ is said to be 𝑇𝑜𝑟𝑏𝑖𝑡𝑎𝑙𝑙𝑦𝑙𝑜𝑤𝑒𝑟𝑠𝑒𝑚𝑖𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑎𝑡𝑧𝑋 if for each orbit {𝑥𝑛}𝑛0𝑋 of 𝑇 with lim𝑛𝑥𝑛=𝑧, we have that 𝑓(𝑧)liminf𝑛𝑓(𝑥𝑛).

2. Main Results

The following lemmas play important roles in this paper.

Lemma 2.1. Let (𝑋,𝑑) be a metric space and 𝐵𝐶𝐿(𝑋). Then for each 𝑥𝑋 and 𝜀>0 there exists 𝑏𝐵 satisfying 𝑑(𝑥,𝑏)𝑑(𝑥,𝐵)+𝜀.

Proof. Suppose that there exist 𝑥0𝑋 and 𝜀0>0 such that 𝑑𝑥0𝑥,𝑏>𝑑0,𝐵+𝜀0,𝑏𝐵,(2.1) which yields that 𝑑𝑥0,𝐵=inf𝑏𝐵𝑑𝑥0𝑥,𝑏𝑑0,𝐵+𝜀0𝑥>𝑑0,,𝐵(2.2) which is a contradiction. This completes the proof.

Lemma 2.2. Let (𝑋,𝑑) be a metric space, 𝐵𝐶𝐿(𝑋) and 𝜃Θ. Then for each 𝑥𝑋 and 𝑞>1 there exists 𝑏𝐵 such that 𝜃(𝑑(𝑥,𝑏))𝑞𝜃(𝑑(𝑥,𝐵)).(2.3)

Proof. Let 𝑥𝑋 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 𝑥𝐵. Put 𝑏=𝑥. Clearly (2.3) holds.
Case  2. Suppose that 𝜃(𝑑(𝑥,𝐵))>0. Note that (b) and (d) mean that (𝑞1)𝜃(𝑑(𝑥,𝐵))+{0}=𝜃+{0}.(2.4) Choose 𝑝𝜃1((𝑞1)𝜃(𝑑(𝑥,𝐵))) and 𝜀=𝑝/2>0. Lemma 2.1 ensures that there exists 𝑏𝐵 satisfying 𝑑(𝑥,𝑏)𝑑(𝑥,𝐵)+𝜀, which together with (a) and (c) gives that 𝜃𝜃(𝑑(𝑥,𝑏))𝜃(𝑑(𝑥,𝐵)+𝜀)𝜃(𝑑(𝑥,𝐵))+𝜃(𝜀)𝜃(𝑑(𝑥,𝐵))+𝜃1((𝑞1)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 𝑇𝑋𝐶𝐿(𝑋) satisfy that 𝜃(𝑑(𝑦,𝑇𝑦))𝜑(𝑑(𝑥,𝑦))𝜃(𝑑(𝑥,𝑦)),(𝑥,𝑦)𝑋×𝑇𝑥,(2.6) where 𝜃Θ and 𝜑+[0,1)𝑤𝑖𝑡limsup𝑟𝑡+𝜑(𝑟)<1,𝑡+.(2.7) Then for each 𝑥0𝑋, there exists an orbit {𝑥𝑛}𝑛0 of 𝑇 and 𝑧𝑋 such that lim𝑛𝑥𝑛=𝑧. Furthermore, 𝑧𝑋 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 𝑥0𝑋 be any initial point and choose 𝑥1𝑇𝑥0. 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 𝑥2𝑇𝑥1 satisfying 𝜃𝑑𝑥1,𝑥2𝜃𝑑𝑥1,𝑇𝑥1max𝜑𝑑𝑥0,𝑥1𝜑𝑑𝑥,1/20,𝑥1𝜃𝑑𝑥0,𝑥1max𝜑𝑑𝑥0,𝑥1,1/2𝜑𝑑𝑥0,𝑥1𝜃𝑑𝑥0,𝑥1,(2.8) and for 𝑞2=1/max{𝜑(𝑑(𝑥1,𝑥2)),1/3}>1 there exists 𝑥3𝑇𝑥2 satisfying 𝜃𝑑𝑥2,𝑥3𝜃𝑑𝑥2,𝑇𝑥2max𝜑𝑑𝑥1,𝑥2𝜑𝑑𝑥,1/31,𝑥2𝜃𝑑𝑥1,𝑥2max𝜑𝑑𝑥1,𝑥2,1/3𝜑𝑑𝑥1,𝑥2𝜃𝑑𝑥1,𝑥2.(2.9) Repeating the above argument we obtain a sequence {𝑥𝑛}𝑛0𝑋 such that 𝑥𝑘𝑇𝑥𝑘1 for 1𝑘𝑛 and for 𝑞𝑛=1/max{𝜑(𝑑(𝑥𝑛1,𝑥𝑛)),1/(𝑛+1)}>1, there exists 𝑥𝑛+1𝑇𝑥𝑛 satisfying 𝜃𝑑𝑥𝑛,𝑥𝑛+1𝜃𝑑𝑥𝑛,𝑇𝑥𝑛max𝜑𝑑𝑥𝑛1,𝑥𝑛𝜑𝑑𝑥,1/(𝑛+1)𝑛1,𝑥𝑛𝜃𝑑𝑥𝑛1,𝑥𝑛max𝜑𝑑𝑥𝑛1,𝑥𝑛,1/(𝑛+1)𝜑𝑑𝑥𝑛1,𝑥𝑛𝜃𝑑𝑥𝑛1,𝑥𝑛,𝑛1.(2.10)
Suppose that there exists some 𝑛00 satisfying 𝑥𝑛0=𝑥𝑛0+1𝑇𝑥𝑛0. It follows from (a), (b), and (2.10) that 𝑥𝑛=𝑥𝑛0 for all 𝑛𝑛0+1. It is clear the conclusion of Theorem 2.3 holds.
Suppose that 𝑥𝑛+1𝑇𝑥𝑛{𝑥𝑛} for any 𝑛0. It follows that 𝑑(𝑥𝑛,𝑥𝑛+1)>0 for each 𝑛0. Note that (b), (2.7), and (2.10) give that {𝜃(𝑑(𝑥𝑛,𝑥𝑛+1))}𝑛0 is a positive and decreasing sequence. It follows from (e) that {𝑑(𝑥𝑛,𝑥𝑛+1)}𝑛0 is decreasing. Therefore, there exist constants 𝑝 and 𝑞 satisfying lim𝑛𝜃𝑑𝑥𝑛,𝑥𝑛+1=𝑝0,lim𝑛𝑑𝑥𝑛,𝑥𝑛+1=𝑞0.(2.11) Notice that (2.7) implies that there exists a constant 𝑟 satisfying limsup𝑛𝜑𝑑𝑥𝑛,𝑥𝑛1limsup𝑡𝑞+𝜑[(𝑡)=𝑟0,1).(2.12) Taking upper limits in (2.10) and by (2.11) and (2.12) we get that 𝑝limsup𝑛𝜑𝑑𝑥𝑛1,𝑥𝑛limsup𝑛𝜃𝑑𝑥𝑛1,𝑥𝑛𝑟𝑝,(2.13) which implies that 𝑝=0.
Next we assert that 𝑞=0. Since {𝑑(𝑥𝑛,𝑥𝑛+1)}𝑛0 is a decreasing sequence, it follows from (a) and (2.11) that 𝑑𝑥0𝜃(𝑞)<𝜃𝑛,𝑥𝑛+1𝑝=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 satisfying 𝜑(𝑡)<𝑐2𝑥,𝑡(0,𝛿),𝑑𝑛,𝑥𝑛+1<𝛿,𝑛𝑁,(2.15) which implies that 𝜑𝑑𝑥𝑛,𝑥𝑛+1<𝑐2,𝑛𝑁.(2.16) Note that (2.10) and (2.16) mean that 𝜃𝑑𝑥𝑛,𝑥𝑛+1𝑛1𝑘=𝑁𝜑𝑑𝑥𝑘,𝑥𝑘+1𝜃𝑑𝑥𝑁,𝑥𝑁+1𝑐𝑛𝑁𝜃𝑑𝑥𝑁,𝑥𝑁+1,𝑛𝑁.(2.17) Given 𝜀>0. Since lim𝑛𝑐𝑛𝑁𝜃(𝑑(𝑥𝑁,𝑥𝑁+1))=0, it follows from (b) that there exists 𝑁1>𝑁 satisfying 𝑐𝑛𝑁𝜃𝑑𝑥1𝑐𝑁,𝑥𝑁+1<𝜃(𝜀),𝑛𝑁1,(2.18) which together with (2.17), (a), and (c) gives that 𝜃𝑑𝑥𝑛,𝑥𝑚𝜃𝑚1𝑘=𝑛𝑑𝑥𝑘,𝑥𝑘+1𝑚1𝑘=𝑛𝜃𝑑𝑥𝑘,𝑥𝑘+1𝑚1𝑘=𝑛𝑐𝑘𝑁𝜃𝑑𝑥𝑁,𝑥𝑁+1𝑐𝑛𝑁𝜃𝑑𝑥1𝑐𝑁,𝑥𝑁+1<𝜃(𝜀),𝑚>𝑛𝑁1.(2.19) In view of (e) and (2.19), we deduce that 𝑑(𝑥𝑛,𝑥𝑚)<𝜀, for all 𝑚>𝑛𝑁1, which means that {𝑥𝑛}𝑛0 is a Cauchy sequence. Hence there exists 𝑧𝑋 such that lim𝑛𝑥𝑛=𝑧 by completeness of 𝑋.
Suppose that 𝑓 is 𝑇 orbitally lower semicontinuous at 𝑧. Since {𝑥𝑛}𝑛0 is an orbit of 𝑇 with lim𝑛𝑥𝑛=𝑧, it follows that 𝑓(𝑧)liminf𝑛𝑓𝑥𝑛.(2.20) Using (2.6) and (2.7), we infer that 𝜃𝑑𝑥𝑛,𝑇𝑥𝑛𝑑𝑥𝜑𝑛1,𝑥𝑛𝜃𝑑𝑥𝑛1,𝑥𝑛𝑑𝑥<𝜃𝑛1,𝑥𝑛,𝑛,(2.21) which together with (e), (2.11), and 𝑞=0 implies that 𝑥0<𝑑𝑛,𝑇𝑥𝑛𝑥<𝑑𝑛1,𝑥𝑛0as𝑛,(2.22) that is, lim𝑛𝑑(𝑥𝑛,𝑇𝑥𝑛)=0, which together with (2.20) yields that 0𝑑(𝑧,𝑇𝑧)=𝑓(𝑧)liminf𝑛𝑓𝑥𝑛=lim𝑛𝑑𝑥𝑛,𝑇𝑥𝑛=0,(2.23) which gives that 𝑑(𝑧,𝑇𝑧)=0, that is, 𝑧𝑇𝑧.
Conversely, suppose that 𝑧𝑋 is a fixed point of 𝑇. Let {𝑦𝑛}𝑛0𝑋 be an arbitrarily orbit of 𝑇 with lim𝑛𝑦𝑛=𝑧. It is clear that 𝑓(𝑧)=𝑑(𝑧,𝑇𝑧)=0liminf𝑛𝑓𝑦𝑛,(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 𝑦𝑇𝑥. In light of Theorem 2.3, we have

Theorem 2.4. Let (𝑋,𝑑) be a complete metric space and 𝑇𝑋𝐶𝐿(𝑋) satisfy that 𝜃(𝐻(𝑇𝑥,𝑇𝑦))𝜑(𝑑(𝑥,𝑦))𝜃(𝑑(𝑥,𝑦)),(𝑥,𝑦)𝑋×𝑇𝑥,(2.25) where 𝜃Θ and 𝜑 satisfies (2.7). Then for each 𝑥0𝑋, there exists an orbit {𝑥𝑛}𝑛0 of𝑇 and 𝑧𝑋 such that lim𝑛𝑥𝑛=𝑧. Furthermore, 𝑧𝑋 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 𝑇𝑋𝐶𝐿(𝑋) satisfy that 𝜃(𝑑(𝑦,𝑇𝑦))𝜑(𝑑(𝑥,𝑇𝑥))𝜃(𝑑(𝑥,𝑦)),(𝑥,𝑦)𝑋×𝑇𝑥,(2.26) where 𝜃Θ and 𝜑 satisfies (2.7). Then for each 𝑥0𝑋, there exists an orbit {𝑥𝑛}𝑛0 of 𝑇 and 𝑧𝑋 such that lim𝑛𝑥𝑛=𝑧. Furthermore, 𝑧𝑋 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 𝑥0𝑋 be any initial point and choose 𝑥1𝑇𝑥0. 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 𝑥2𝑇𝑥1 such that 𝜃𝑑𝑥1,𝑥2𝜃𝑑𝑥1,𝑇𝑥1max𝜑𝑑𝑥0,𝑇𝑥0,𝜑𝑑𝑥1,𝑇𝑥1𝜑𝑑𝑥,1/20,𝑇𝑥0𝜃𝑑𝑥0,𝑥1max𝜑𝑑𝑥0,𝑇𝑥0,𝜑𝑑𝑥1,𝑇𝑥1,1/2𝜑𝑑𝑥0,𝑇𝑥0𝜃𝑑𝑥0,𝑥1,𝜃𝑑𝑥2,𝑇𝑥2𝑑𝑥𝜑1,𝑇𝑥1𝜃𝑑𝑥1,𝑥2𝜑𝑑𝑥1,𝑇𝑥1𝜃𝑑𝑥1,𝑇𝑥1max𝜑𝑑𝑥0,𝑇𝑥0,𝜑𝑑𝑥1,𝑇𝑥1,1/2𝜑𝑑𝑥1,𝑇𝑥1𝜃𝑑𝑥1,𝑇𝑥1.(2.27) Repeating the above argument we obtain a sequence {𝑥𝑛}𝑛0𝑋 satisfying 𝑥𝑛+1𝑇𝑥𝑛 for each 𝑛0, 𝜃𝑑𝑥𝑛,𝑥𝑛+1𝜃𝑑𝑥𝑛,𝑇𝑥𝑛max𝜑𝑑𝑥𝑛1,𝑇𝑥𝑛1,𝜑𝑑𝑥𝑛,𝑇𝑥𝑛𝜑𝑑𝑥,1/(𝑛+1)𝑛1,𝑇𝑥𝑛1𝜃𝑑𝑥𝑛1,𝑥𝑛max𝜑𝑑𝑥𝑛1,𝑇𝑥𝑛1,𝜑𝑑𝑥𝑛,𝑇𝑥𝑛,1/(𝑛+1)𝜑𝑑𝑥𝑛1,𝑇𝑥𝑛1𝜃𝑑𝑥𝑛1,𝑥𝑛𝜃𝑑𝑥,𝑛,(2.28)𝑛+1,𝑇𝑥𝑛+1𝑑𝑥𝜑𝑛,𝑇𝑥𝑛𝜃𝑑𝑥𝑛,𝑥𝑛+1𝜑𝑑𝑥𝑛,𝑇𝑥𝑛𝜃𝑑𝑥𝑛,𝑇𝑥𝑛max𝜑𝑑𝑥𝑛1,𝑇𝑥𝑛1,𝜑𝑑𝑥𝑛,𝑇𝑥𝑛,1/(𝑛+1)𝜑𝑑𝑥𝑛,𝑇𝑥𝑛𝜃𝑑𝑥𝑛,𝑇𝑥𝑛,𝑛.(2.29)
Suppose that 𝑥𝑛0𝑇𝑥𝑛0 for some 𝑛00. It is easy to verify that 𝑥𝑛=𝑥𝑛0 for all 𝑛𝑛0 and the conclusion of Theorem 2.5 holds.
Suppose that 𝑥𝑛𝑇𝑥𝑛 for each 𝑛0. It follows that {𝑑(𝑥𝑛,𝑇𝑥𝑛)}𝑛0 and {𝑑(𝑥𝑛,𝑥𝑛+1)}𝑛0 are positive sequences. Combining (2.7), (2.28), (2.29), (b) and (e), we infer that {𝜃(𝑑(𝑥𝑛,𝑥𝑛+1))}𝑛0 and {𝜃(𝑑(𝑥𝑛,𝑇𝑥𝑛))}𝑛0 are both positive and decreasing, so do {𝑑(𝑥𝑛,𝑥𝑛+1)}𝑛0 and {𝑑(𝑥𝑛,𝑇𝑥𝑛)}𝑛0. It follows that there exist constants 𝛼,𝛽,𝑠 and 𝑡 satisfying lim𝑛𝜃𝑑𝑥𝑛,𝑥𝑛+1=𝛼0,lim𝑛𝑑𝑥𝑛,𝑥𝑛+1=𝛽0,lim𝑛𝜃𝑑𝑥𝑛,𝑇𝑥𝑛=𝑠0,lim𝑛𝑑𝑥𝑛,𝑇𝑥𝑛=𝑡0.(2.30) Notice that (2.7) implies that there exists a constant 𝑟 such that limsup𝑛𝜑𝑑𝑥𝑛,𝑇𝑥𝑛limsup𝑙𝑡+𝜑[(𝑙)=𝑟0,1).(2.31) Taking upper limits in (2.29) and by (2.30) and (2.31) we get that 𝑠limsup𝑛𝜑𝑑𝑥𝑛,𝑇𝑥𝑛limsup𝑛𝜃𝑑𝑥𝑛,𝑇𝑥𝑛𝑟𝑠,(2.32) which implies that 𝑠=0, which together with (2.30) and (a) ensures that 𝑑𝑥0𝜃(𝑡)<𝜃𝑛,𝑇𝑥𝑛0,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 𝛼limsup𝑛𝜑𝑑𝑥𝑛,𝑇𝑥𝑛limsup𝑛𝜃𝑑𝑥𝑛1,𝑥𝑛𝑟𝛼,(2.34) which yields that 𝛼=0. Notice that (2.30) and (a) guarantee that 𝑑𝑥0𝜃(𝛽)<𝜃𝑛,𝑥𝑛+10,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 𝑇𝑋𝐶𝐿(𝑋) satisfy that 𝜃(𝐻(𝑇𝑥,𝑇𝑦))𝜑(𝑑(𝑥,𝑇𝑥))𝜃(𝑑(𝑥,𝑦)),(𝑥,𝑦)𝑋×𝑇𝑥,(2.36) where 𝜃Θ and 𝜑 satisfies (2.7). Then for each 𝑥0𝑋, there exists an orbit {𝑥𝑛}𝑛0 of𝑇 and 𝑧𝑋 such that lim𝑛𝑥𝑛=𝑧. Furthermore, 𝑧𝑋 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 [68].

Remark 3.1. Theorems 2.3 and 2.4 extend Theorems 1.11.3, and Theorems 2.5 and 2.6 are different from Theorems 1.11.3, respectively, in the following ways:(1)the ranges 𝐶𝐿(𝑋) of the nonlinear set-valued contractive mappings 𝑇 in Theorems 2.32.6 are more general than the ranges 𝐶(𝑋) and 𝐶𝐵(𝑋) of the set-valued contraction mappings 𝑇 in Theorems 1.11.3, respectively;(2)the 𝑇 orbit lower semicontinuity at some 𝑧𝑋 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 1.11.3, 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 𝜃1 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 1.11.3, respectively.

Example 3.2. Let 𝑋=(,3/10] and 𝑑 be the standard metric in 𝑋. Let 𝜃++, 𝜑+[0,1) and 𝑇𝑋𝐶𝐿(𝑋) be defined by 𝜃(𝑡)=𝑡1/22,𝜑(𝑡)=65,𝑡+1,𝑇𝑥=,4𝑥,𝑥(,0),0,2𝑥23,𝑥0,,10(3.2) respectively. It is clear that 𝜃Θ, 𝜑 satisfies (2.7) and 𝑓(𝑥)=𝑑(𝑥,𝑇𝑥)=0,𝑥(,0)𝑥2𝑥23,𝑥0,10(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 𝑦𝑇𝑥=(,(1/4)𝑥]. It is clear that1𝜃(𝑑(𝑦,𝑇𝑦))𝜃(𝐻(𝑇𝑥,𝑇𝑦))=2𝜃(𝑑(𝑥,𝑦))𝜑(𝑑(𝑥,𝑦))𝜃(𝑑(𝑥,𝑦)).(3.4)
Case  2. Let 𝑥[0,3/10] and 𝑦𝑇𝑥=[0,2𝑥2]. It follows that 𝜃(𝑑(𝑦,𝑇𝑦))𝜃(𝐻(𝑇𝑥,𝑇𝑦))=2||||𝑥+𝑦1/2𝜃(𝑑(𝑥,𝑦))23+910501/2𝜃(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 1.11.3 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 𝐻𝑇𝑥0,𝑇𝑦023=𝑑1021,252=1𝑟10𝑥10=𝑟𝑑0,𝑦0,(3.6) for any 𝑟(0,1) and 𝐻𝑇𝑥0,𝑇𝑦023=𝑑1021,252=1110𝜑110𝑑𝑥10=𝜑0,𝑦0𝑑𝑥0,𝑦0,(3.7) for any mapping 𝜑+[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++, 𝜑+[0,1) and 𝑇𝑋𝐶𝐿(𝑋) by 𝜃(𝑡)=𝑡1/2,𝑡+2,𝜑(𝑡)=2𝑡1/21,𝑡0,8,2651,𝑡{0}8,𝑥,+𝑇𝑥=4(1+𝑥),+,𝑥(0,+),2𝑥23,0,𝑥,10,0(3.8) respectively. It is easy to see that (2.7) holds and 𝑓(𝑥)=𝑑(𝑥,𝑇𝑥)=0,𝑥(0,+),2𝑥23𝑥,𝑥10,0(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 𝑦𝑇𝑥=[𝑥/4(1+𝑥),+). It is clear that ||||𝑥𝜃(𝑑(𝑦,𝑇𝑦))=0𝜃(𝐻(𝑇𝑥,𝑇𝑦))=𝑦4(1+𝑥)||||4(1+𝑦)1/2=𝜃(𝑑(𝑥,𝑦))2(1+𝑥)1/2(1+𝑦)1/2𝜃(𝑑(𝑥,𝑦))2(1+𝑥)1/2(1+𝑥/4(1+𝑥))1/2=𝜃(𝑑(𝑥,𝑦))(5𝑥+4)1/2𝜃(𝑑(𝑥,𝑦))2265𝜃(𝑑(𝑥,𝑦))=𝜑(0)𝜃(𝑑(𝑥,𝑦))=𝜑(𝑑(𝑥,𝑇𝑥))𝜃(𝑑(𝑥,𝑦)).(3.10)
Case  2. Let 𝑥[3/10,0] and 𝑦𝑇𝑥=[2𝑥2,0]. It follows that 𝜃(𝑑(𝑦,𝑇𝑦))𝜃(𝐻(𝑇𝑥,𝑇𝑦))=2||||𝑥+𝑦1/2𝜃(𝑑(𝑥,𝑦))2||𝑥2𝑥2||1/2𝜃(𝑑(𝑥,𝑦)).(3.11) For 𝑥=0, we have 2||𝑥2𝑥2||1/2𝜃(𝑑(𝑥,𝑦))=0𝜑(𝑑(𝑥,𝑇𝑥))𝜃(𝑑(𝑥,𝑦)).(3.12) For 𝑥[3/10,1/4)(1/4,0), we infer that 2||𝑥2𝑥2||1/2𝜃(𝑑(𝑥,𝑦))222𝑥2𝑥1/2𝜃(𝑑(𝑥,𝑦))=𝜑(𝑑(𝑥,𝑇𝑥))𝜃(𝑑(𝑥,𝑦)).(3.13) For 𝑥=1/4, we get that 2||𝑥2𝑥2||1/2𝜃(𝑑(𝑥,𝑦))=321𝜃(𝑑(𝑥,𝑦))𝜑8𝜃(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 𝐻𝑇𝑥0,𝑇𝑦01=𝐻8,9,+50,0=+13𝑟𝑥10=𝑟𝑑0,𝑦0,(3.15) for any 𝑟(0,1), and 𝐻𝑇𝑥0,𝑇𝑦02=+6513𝑑𝑥10=𝜑0,𝑦0𝑑𝑥0,𝑦0,(3.16) for any mapping 𝜑+[0,1) with each of (1.3) and (1.4). That is, Theorems 1.11.3 are inapplicable in proving the existence of fixed points for the nonlinear set-valued contractive mapping 𝑇.

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