International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

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Volume 2011 |Article ID 924396 | https://doi.org/10.5402/2011/924396

A. Azizi, H. P. Masiha, "Some Generalizations of Fixed Point Results for Multivalued Contraction Mappings", International Scholarly Research Notices, vol. 2011, Article ID 924396, 13 pages, 2011. https://doi.org/10.5402/2011/924396

Some Generalizations of Fixed Point Results for Multivalued Contraction Mappings

Academic Editor: G. Ólafsson
Received18 Aug 2011
Accepted26 Sep 2011
Published25 Dec 2011

Abstract

We establish some fixed point results for multivalued contraction type mappings in terms of a 𝑤-distance in a complete metric space. Our results generalize very recent results of some authors (Ćirić, 2008, 2009; Feng and Liu 2006; Klim and Wardowski 2007; and Latif and Abdou 2011).

1. Introduction

The Banach contraction principle [1] plays an important role in nonlinear analysis. Following the Banach contraction principle, Nadler [2] first initiated the study of fixed point theorems for multivalued contraction mappings and inspired by his results, the fixed point theory of multi-valued contraction has been further developed in different directions by many authors, in particular, by Reich, Mizoguchi-Takahashi, Feng-Liu, and many others.

The aim of this paper is to present results which are generalizations of the very recent results of Klim and Wardowski [3], Ćirić [4, 5], and Latif and Abdou [6], as well as of the result of Mizoguchi and Takahashi [7] and many others.

Let (𝑋,𝑑) be a metric space. We denote the collection of all nonempty closed bounded subsets of 𝑋 by CB(𝑋) and the collectifon of all nonempty closed subsets and all nonempty compact subsets of 𝑋 by 𝐶(𝑋) and 𝐾(𝑋), respectively. Throughout this paper, we assume that ℝ, ℕ, ℕ𝑒, and ℕ𝑜 denote the sets of all real numbers, positive integers, even positive integers, and odd positive integers, respectively. Let 𝐻 be the Hausdorff metric induced by 𝑑, that is, 𝐻(𝐴,𝐵)=maxsup𝑥∈𝐴𝑑(𝑥,𝐵),sup𝑦∈𝐵𝑑(𝑦,𝐴),(1.1) for all 𝐴,𝐵∈CB(𝑋), where 𝑑(𝑥,𝐵)=inf{𝑑(𝑥,𝑦)∶𝑦∈𝐵}. An element 𝑥∈𝑋 is said to be a fixed point of a multi-valued mapping 𝑇∶𝑋→𝐶(𝑋) if 𝑥∈𝑇(𝑥). A map 𝑓∶𝑋→ℝ is called lower semicontinuous if for any sequence {𝑥𝑛} in 𝑋 and 𝑥∈𝑋 such that 𝑥𝑛→𝑥, we have 𝑓(𝑥)≤liminfğ‘›â†’âˆžğ‘“(𝑥𝑛).

The following theorem is an extension of Banach contraction principle for multi-valued mappings, which was obtained by Nadler in [2].

Theorem 1.1 (see [2]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→CB(𝑋) be a multi-valued mapping. Assume that there exists 𝑟∈[0,1) such that for all𝑥,𝑦∈𝑋, 𝐻(𝑇(𝑥),𝑇(𝑦))≤𝑟𝑑(𝑥,𝑦),(1.2) then there exists 𝑧∈𝑋 such that 𝑧∈𝑇(𝑧).
A generalization of Theorem 1.1 was proved by Mizoguchi and Takahashi that is, in fact, a partial answer of question of Reich [8].

Theorem 1.2 (see [7]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→CB(𝑋) be a multi-valued mapping. If there exists a function 𝜑∶(0,∞)→[0,1) such that limsup𝑟→𝑡+𝜑(𝑟)<1 for each 𝑡∈[0,∞), and if for all 𝑥,𝑦∈𝑋, 𝐻(𝑇(𝑥),𝑇(𝑦))≤𝜑(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦),(1.3) then there exists 𝑧∈𝑋 such that 𝑧∈𝑇(𝑧).
Recently, an interesting result had been obtained by Feng and Liu [9]. They extended Theorem 1.1 in another direction different from that of Mizoguchi-Takahashi's theorem.

Theorem 1.3 (see [9]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝐶(𝑋) be a multi-valued mapping. If there exist constants 𝛼,𝛽∈(0,1), 𝛽<𝛼, such that for any 𝑥∈𝑋 there is 𝑦∈𝑇(𝑥) satisfying the following two conditions: (i)𝛼𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑇(𝑥)),(ii)𝑑(𝑦,𝑇(𝑦))≤𝛽𝑑(𝑥,𝑇(𝑥)), then there exists 𝑧∈𝑋 such that 𝑧∈𝑇(𝑧) provided the function 𝑓(𝑥)=𝑑(𝑥,𝑇(𝑥)) is lower semicontinuous.

By using the ideas of Mizoguchi-Takahashi and Feng-Liu, Klim and Wardowski proved the following two theorems that are different from Theorem 1.2.

Theorem 1.4 (see [3]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝐶(𝑋) be a multi-valued mapping. Assume that the following conditions hold:(i)the map 𝑓∶𝑋→ℝ, defined by 𝑓(𝑥)=𝑑(𝑥,𝑇(𝑥)), 𝑥∈𝑋, is lower semi-continuous; (ii)there exists a constant 𝛼∈(0,1) and a function 𝜑∶[0,∞)→[0,𝛼) satisfying limsup𝑟→𝑡+𝜑(𝑟)<𝛼, for each 𝑡∈[0,∞), and for any 𝑥∈𝑋 there is 𝑦∈𝑇(𝑥) satisfying 𝛼𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑇(𝑥)),𝑑(𝑦,𝑇(𝑦))≤𝜑(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦).(1.4) Then there exists 𝑧∈𝑋 such that 𝑧∈𝑇(𝑧).

Theorem 1.5 (see [3]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝐾(𝑋) be a multi-valued mapping. Assume that the following conditions hold:(i)the map 𝑓∶𝑋→ℝ, defined by 𝑓(𝑥)=𝑑(𝑥,𝑇(𝑥)), 𝑥∈𝑋, is lower semi-continuous;(ii)there exists a function 𝜑∶[0,∞)→[0,1) satisfying limsup𝑟→𝑡+𝜑(𝑟)<1, for each 𝑡∈[0,∞), and for any 𝑥∈𝑋 there is 𝑦∈𝑇(𝑥) satisfying 𝑑(𝑥,𝑦)=𝑑(𝑥,𝑇(𝑥)),𝑑(𝑦,𝑇(𝑦))≤𝜑(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦).(1.5) Then there exists 𝑧∈𝑋 such that 𝑧∈𝑇(𝑧).

In [4, 5], Ćirić proved a few interesting theorems which are generalizations of the above-mentioned theorems, one of which is as follows.

Theorem 1.6 (see [4]). Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝐶(𝑋) be a multi-valued mapping. If there exist a function 𝜑∶[0,∞)→[0,1) and a nondecreasing function 𝜓∶[0,∞)→[𝛼,1), 𝛼>0, such that 𝜑(𝑡)<𝜓(𝑡),limsup𝑟→𝑡+𝜑(𝑟)<limsup𝑟→𝑡+[𝜓(𝑟),∀𝑡∈0,∞),(1.6) and for any 𝑥∈𝑋 there is 𝑦∈𝑇(𝑥) satisfying the following two conditions: 𝜓(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑇(𝑥)),𝑑(𝑦,𝑇(𝑦))≤𝜑(𝑑(𝑥,𝑦))𝑑(𝑥,𝑦),(1.7) then 𝑇 has a fixed point in 𝑋 provided 𝑓(𝑥)=𝑑(𝑥,𝑇(𝑥)) is lower semi-continuous.

Very recently the fixed point theorems of Ćirić were extended by Liu et al. [10] and by Nicolae [11] in a new direction. Also Latif and Abdou [6] improve two theorems of Ćirić with respect to 𝑤-distance. The concept of 𝑤-distance was introduced by Kada et al. [12] on a metric space as follows.

Let (𝑋,𝑑) be a metric space, then a function 𝑣∶𝑋×𝑋→[0,∞) is called a 𝑤-distance on 𝑋 if the following axioms are satisfied: (1)𝑣(𝑥,𝑧)≤𝑣(𝑥,𝑦)+𝑣(𝑦,𝑧), for any 𝑥,𝑦,𝑧∈𝑋; (2)for any 𝑥∈𝑋, 𝑣(𝑥,⋅)∶𝑋→[0,∞) is lower semi-continuous; (3)for any 𝜀>0, there exists 𝛿>0 such that 𝑣(𝑧,𝑥)≤𝛿 and 𝑣(𝑧,𝑦)≤𝛿 imply 𝑑(𝑥,𝑦)≤𝜀.

The metric 𝑑 is a 𝑤-distance on 𝑋. One can see other examples of 𝑤-distances in [12]. One of theorems, which was proved by Latif and Abdou, is as follows.

Theorem 1.7 (see [6]). Let (𝑋,𝑑) be a complete metric space with a 𝑤-distance 𝑣. Let 𝑇∶𝑋→𝐶(𝑋) be a multi-valued map. Assume that the following conditions hold: (i)there exist a constant 𝛼∈(0,1) and a function 𝜑∶[0,∞)→[𝛼,1) such that limsup𝑟→𝑡+𝜑(𝑟)<1, for each 𝑡∈[0,∞); (ii)the map 𝑓∶𝑋→ℝ, defined by 𝑓(𝑥)=𝑣(𝑥,𝑇(𝑥)), 𝑥∈𝑋, is lower semi-continuous; (iii)for any 𝑥∈𝑋, there is 𝑦∈𝑇(𝑥) satisfying √𝜑(𝑓(𝑥))𝑣(𝑥,𝑦)≤𝑓(𝑥),𝑓(𝑦)≤𝜑(𝑓(𝑥))𝑣(𝑥,𝑦).(1.8) Then there exists 𝑧∈𝑋 such that 𝑓(𝑧)=0. Further, if 𝑣(𝑧,𝑧)=0, then 𝑧∈𝑇(𝑧).

For the proof of the main results, we need the following crucial lemma [13].

Lemma 1.8. Let (X,d) be a metric space, and let 𝑣 be a 𝑤-distance on 𝑋. Let {𝑥𝑛} and {𝑦𝑛} be two sequences in 𝑋, let {ğ‘Žğ‘›} and {𝑏𝑛} be sequences in [0,∞) converging to zero, and let 𝑧∈𝑋. Then the following hold: (1)if 𝑣(𝑥𝑛,𝑦𝑛)â‰¤ğ‘Žğ‘› and 𝑣(𝑥𝑛,𝑧)≤𝑏𝑛 for any 𝑛∈ℕ, then {𝑦𝑛} converges to 𝑧; (2)if 𝑣(𝑥𝑛,𝑥𝑝)â‰¤ğ‘Žğ‘› for any 𝑛,𝑝∈ℕ with 𝑝>𝑛, then {𝑥𝑛} is a Cauchy sequence.

In the present paper, using the concept of 𝑤-distance, in the same direction which has been used in [10, 11], we introduce some new contraction conditions for multi-valued mappings in complete metric spaces and three fixed point theorems for such contractions are proved. Our results generalize Theorems 1.6 and 1.7, and many other theorems.

2. Main Results

Recall that a sequence {𝑥𝑛}𝑛≥0 in 𝑋 is called an orbit of 𝑇 at 𝑥0∈𝑋 if 𝑥𝑛∈𝑇(𝑥𝑛−1), for all 𝑛≥1, and 𝑣(𝑥,𝐺)=inf{𝑣(𝑥,𝑦)∶𝑦∈𝐺} for any 𝑥∈𝑋 and 𝐺∈𝐶(𝑋).

Now, we shall prove a theorem which is a generalization of Theorem 1.7.

Theorem 2.1. Let (𝑋,𝑑) be a complete metric space with a 𝑤-distance 𝑣, and let 𝑇 be a multi-valued mapping from 𝑋 into 𝐶(𝑋). Assume that the following conditions hold: (i)the map 𝑓∶𝑋→ℝ, defined by 𝑓(𝑥)=𝑣(𝑥,𝑇(𝑥)), 𝑥∈𝑋, is lower semi-continuous;(ii)there exist the functions 𝛼∶[0,∞)→(0,1], 𝛽∶[0,∞)→[0,1) which satisfy ∃𝑘,𝜀∈(0,1),𝑡𝑘[],≤𝛼(𝑡),foreach𝑡∈0,𝜀limsup𝑟→𝑡+𝛽(𝑟)[𝛼(𝑟)<1,foreach𝑡∈0,∞);(2.1)(iii)for any 𝑥∈𝑋, there is 𝑦∈𝑇(𝑥) satisfying 𝛼(𝑓(𝑥))𝑣(𝑥,𝑦)≤𝑓(𝑥),𝑓(𝑦)≤𝛽(𝑓(𝑥))𝑣(𝑥,𝑦).(2.2) Then, there exists 𝑧∈𝑋 such that 𝑧∈𝑇(𝑧).

Proof. Let 𝑥0∈𝑋. Then there exists 𝑥1∈𝑇(𝑥) such that 𝛼𝑓𝑥0𝑣𝑥0,𝑥1𝑥≤𝑓0,𝑓𝑥1𝑓𝑥≤𝛽0𝑣𝑥0,𝑥1.(2.3) Then, from (2.3), we have 𝑓𝑥1≤𝛽𝑓𝑥0𝛼𝑓𝑥0𝑓𝑥0.(2.4) Continuing this process, we can define an orbit {𝑥𝑛} of 𝑇 in 𝑋, such that 𝛼𝑓𝑥𝑛𝑣𝑥𝑛,𝑥𝑛+1𝑥≤𝑓𝑛𝑥,𝑓𝑛+1𝑓𝑥≤𝛽𝑛𝑣𝑥𝑛,𝑥𝑛+1,(2.5) for all 𝑛≥0, which imply that 𝑓𝑥𝑛+1≤𝛽𝑓𝑥𝑛𝛼𝑓𝑥𝑛𝑓𝑥𝑛.(2.6) We can assume that 𝑓(𝑥𝑛)>0 for all 𝑛≥0, since 𝑓(𝑥𝑛)=0 for some 𝑛, then from (2.5), 𝑣(𝑥𝑛,𝑥𝑛+1)=𝑓(𝑥𝑛+1)=0, and so for every 𝑚∈ℕ, there exists 𝑦𝑚∈𝑇(𝑥𝑛+1) such that 𝑣(𝑥𝑛+1,𝑦𝑚)≤1/𝑚; consequently, 𝑣(𝑥𝑛,𝑦𝑚)≤𝑣(𝑥𝑛,𝑥𝑛+1)+𝑣(𝑥𝑛+1,𝑦𝑚)≤1/𝑚; so by Lemma 1.8, 𝑦𝑚→𝑥𝑛+1 and thus 𝑥𝑛+1∈𝑇(𝑥𝑛+1), that is, the assertion of the theorem is proved. Since 𝛽(𝑓(𝑥))/𝛼(𝑓(𝑥))<1 for all 𝑥∈𝑋, then we have 𝑓𝑥𝑛+1𝑥<𝑓𝑛.(2.7) Thus {𝑓(𝑥𝑛)} is a decreasing sequence of positive real numbers and hence converges to a nonnegative number 𝜃, 𝜃≥0. Let 𝜏=limsupğ‘›â†’âˆžğ›½(𝑓(𝑥𝑛))/𝛼(𝑓(𝑥𝑛))<1. Then, for 𝑠=(𝜏+1)/2, we can take ̂𝑛0∈ℕ such that 𝛽𝑓𝑥𝑛𝛼𝑓𝑥𝑛<𝑠,∀𝑛≥̂𝑛0.(2.8) Then, 𝑓𝑥𝑛+1≤𝑠𝑛−̂𝑛0+1𝑓𝑥̂𝑛0,∀𝑛≥̂𝑛0,(2.9) thus, 𝜃=0. Therefore, by (ii) we can take ̃𝑛0∈ℕ such that 𝑓𝑥𝑛∈[]𝑥0,𝜀,𝑓𝑛𝑘𝑓𝑥≤𝛼𝑛,∀𝑛≥̃𝑛0.(2.10) Now, from (2.5), (2.9), and (2.10), we have 𝑣𝑥𝑛,𝑥𝑛+1≤𝑓𝑥𝑛𝛼𝑓𝑥𝑛≤𝑓𝑥𝑛𝑓𝑥𝑛𝑘𝑥=𝑓𝑛1âˆ’ğ‘˜â‰¤ğ‘žğ‘›âˆ’ğ‘›0𝑓𝑥𝑛01−𝑘,(2.11) for all 𝑛≥𝑛0, where 𝑛0=max{̂𝑛0,̃𝑛0} and ğ‘ž=𝑠1−𝑘<1. Then for any 𝑝,𝑛∈ℕ, 𝑝>𝑛≥𝑛0, we have 𝑣𝑥𝑛,𝑥𝑝≤𝑝−1𝑖=𝑛𝑣𝑥𝑖,𝑥𝑖+1≤𝑝−1𝑖=ğ‘›ğ‘žğ‘–âˆ’ğ‘›0𝑓𝑥𝑛01âˆ’ğ‘˜â‰¤ğ‘žğ‘›âˆ’ğ‘›0𝑓𝑥1âˆ’ğ‘žğ‘›01−𝑘.(2.12) Hence, by Lemma 1.8,  {𝑥𝑛} is a Cauchy sequence so there exists 𝑧∈𝑋 such that lim𝑛𝑥𝑛=𝑧. Since 𝑓 is lower semi-continuous, we obtain 0≤𝑓(𝑧)≤liminfğ‘›â†’âˆžğ‘“î€·ğ‘¥ğ‘›î€¸=0,(2.13) and thus, 𝑓(𝑧)=𝑣(𝑧,𝑇(𝑧))=0.(2.14) Now, we clime that 𝑧∈𝑇(𝑧). Notice that the function 𝑣(𝑥,⋅) is lower semi-continuous for all 𝑥∈𝑋. Since 𝑥𝑝→𝑧, then by (2.12), 𝑣𝑥𝑛,𝑧≤liminfğ‘â†’âˆžğ‘£î€·ğ‘¥ğ‘›,ğ‘¥ğ‘î€¸â‰¤ğ‘žğ‘›âˆ’ğ‘›0𝑓𝑥1âˆ’ğ‘žğ‘›01−𝑘,(2.15) for all 𝑛≥𝑛0. On the other hand, from (2.14), for every 𝑛∈ℕ, there exists 𝑦𝑛∈𝑇(𝑧) such that 𝑣(𝑧,𝑦𝑛)≤1/𝑛. Then, for all 𝑛≥𝑛0, we have 𝑣𝑥𝑛,𝑦𝑛𝑥≤𝑣𝑛,𝑧+𝑣𝑧,ğ‘¦ğ‘›î€¸â‰¤ğ‘žğ‘›âˆ’ğ‘›0𝑓𝑥1âˆ’ğ‘žğ‘›01−𝑘+1𝑛.(2.16) Therefore, from (2.15), (2.16), we can find the sequences {ğ‘Žğ‘›} and {𝑏𝑛} in [0,∞) converging to zero, such that 𝑣(𝑥𝑛,𝑦𝑛)â‰¤ğ‘Žğ‘› and 𝑣(𝑥𝑛,𝑧)≤𝑏𝑛 for any 𝑛∈ℕ; then by Lemma 1.8  𝑦𝑛→𝑧. The closedness of 𝑇(𝑧) implies 𝑧∈𝑇(𝑧).

Now, we shall prove a theorem which is different from Theorem 2.1 and is a generalization of Theorem 1.6.

Theorem 2.2. Let (𝑋,𝑑) be a complete metric space with a 𝑤-distance 𝑣, and let 𝑇 be a multi-valued mapping from 𝑋 into 𝐶(𝑋). Assume that the following conditions hold: (i)the map 𝑓∶𝑋→ℝ, defined by 𝑓(𝑥)=𝑣(𝑥,𝑇(𝑥)), 𝑥∈𝑋, is lower semi-continuous;(ii)there exist the functions 𝛼∶[0,∞)→(0,1], 𝛽∶[0,∞)→[0,1) which satisfy ∃𝑘,𝜀∈(0,1),𝑡𝑘[],≤𝛼(𝑡),foreach𝑡∈0,𝜀limsup𝑟→𝑡+𝛽(𝑟)[𝛼(𝑟)<1,foreach𝑡∈0,∞),(2.17) and 𝛼 is nondecreasing; (iii)for any 𝑥∈𝑋, there is 𝑦∈𝑇(𝑥) satisfying 𝛼(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦)≤𝑓(𝑥),𝑓(𝑦)≤𝛽(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦).(2.18) Then, there exists 𝑧∈𝑋 such that 𝑧∈𝑇(𝑧).

Proof. By following the lines in the proof of Theorem 2.1, one can construct an orbit {𝑥𝑛}âˆžğ‘›=0 of 𝑇 in 𝑋 such that 𝛼𝑣𝑥𝑛,𝑥𝑛+1𝑣𝑥𝑛,𝑥𝑛+1𝑥≤𝑓𝑛,(2.19)𝑓𝑥𝑛+1𝑣𝑥≤𝛽𝑛,𝑥𝑛+1𝑣𝑥𝑛,𝑥𝑛+1,(2.20) for all 𝑛≥0, which imply that 𝑓𝑥𝑛+1≤𝛽𝑣𝑥𝑛,𝑥𝑛+1𝛼𝑣𝑥𝑛,𝑥𝑛+1𝑓𝑥𝑛.(2.21) Thus, {𝑓(𝑥𝑛)} is a decreasing sequence, and so there exists 𝜃≥0 such that 𝑓(𝑥𝑛)→𝜃. From (2.19) and (2.20), we have 𝑣𝑥𝑛+1,𝑥𝑛+2≤𝑓𝑥𝑛+1𝛼𝑣𝑥𝑛+1,𝑥𝑛+2≤𝛽𝑣𝑥𝑛,𝑥𝑛+1𝛼𝑣𝑥𝑛+1,𝑥𝑛+2𝑣𝑥𝑛,𝑥𝑛+1,(2.22) for all 𝑛≥0. Now we clime that {𝑣(𝑥𝑛,𝑥𝑛+1)} is a nonincreasing sequence. Suppose not. Then there exists 𝑛0≥0 such that 𝑣(𝑥𝑛0+1,𝑥𝑛0+2)>𝑣(𝑥𝑛0,𝑥𝑛0+1). Since 𝛼 is nondecreasing, then from (2.22), we get that 𝑣𝑥𝑛0,𝑥𝑛0+1𝑥<𝑣𝑛0+1,𝑥𝑛0+2≤𝛽𝑣𝑥𝑛0,𝑥𝑛0+1𝛼𝑣𝑥𝑛0+1,𝑥𝑛0+2𝑣𝑥𝑛0,𝑥𝑛0+1≤𝛽𝑣𝑥𝑛0,𝑥𝑛0+1𝛼𝑣𝑥𝑛0,𝑥𝑛0+1𝑣𝑥𝑛0,𝑥𝑛0+1𝑥<𝑣𝑛0,𝑥𝑛0+1,(2.23) which is a contradiction. Then, {𝑣(𝑥𝑛,𝑥𝑛+1)} is a nonincreasing sequence and so is convergent. Now by using the same argument as in the proof of Theorem 2.1, we obtain the existence of a real number 𝑠∈(0,1) and ̂𝑛0∈ℕ such that 𝑓𝑥𝑛+1≤𝑠𝑛−̂𝑛0+1𝑓𝑥̂𝑛0,∀𝑛≥̂𝑛0,(2.24) thus 𝜃=0. On the other hand, since 𝛼 is nondecreasing, then by (2.19), we have 𝑣𝑥𝑛,𝑥𝑛+1≤𝑓𝑥𝑛𝛼𝑣𝑥𝑛,𝑥𝑛+1≤𝑓𝑥𝑛𝛼𝑓𝑥𝑛.(2.25) For the rest of the proof, we can go on as in the proof of Theorem 2.1.

In the same manner, we can present the following theorem.

Theorem 2.3. Let (𝑋,𝑑) be a complete metric space with a 𝑤-distance 𝑣, and let 𝑇 be a multi-valued mapping from 𝑋 into 𝐶(𝑋). Assume that the following conditions hold: (i)the map 𝑓∶𝑋→ℝ, defined by 𝑓(𝑥)=𝑣(𝑥,𝑇(𝑥)), 𝑥∈𝑋, is lower semi-continuous,(ii)there exist the functions 𝛼∶[0,∞)→(0,1], 𝛽∶[0,∞)→[0,1) which satisfy ∃𝑘,𝜀∈(0,1),𝑡𝑘[],𝛽≤𝛽(𝑡),foreach𝑡∈0,𝜀(𝑡)<𝛼(𝑡),limsup𝑟→𝑡+𝛽(𝑟)𝛼(𝑟)<1,foreacht∈[0,∞),(2.26) and also one of 𝛼 and 𝛽 is nondecreasing; (iii)for any 𝑥∈𝑋, there is 𝑦∈𝑇(𝑥) satisfying𝛼(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦)≤𝑓(𝑥),𝑓(𝑦)≤𝛽(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦).(2.27) Then, there exists 𝑧∈𝑋 such that 𝑧∈𝑇(z).

Proof. As in the proof of Theorem 2.1, one can construct an orbit {𝑥𝑛}âˆžğ‘›=0 of 𝑇 in 𝑋 such that (2.19), (2.20), (2.21), and (2.22) hold. Then, {𝑓(𝑥𝑛} is a decreasing sequence and so there exists 𝜃≥0 such that 𝑓(𝑥𝑛)→𝜃. Now we clime that {𝑣(𝑥𝑛,𝑥𝑛+1)} is a nonincreasing sequence. Suppose not. Then there exists 𝑛0≥0 such that 𝑣(𝑥𝑛0+1,𝑥𝑛0+2)>𝑣(𝑥𝑛0,𝑥𝑛0+1). Since one of 𝛼 and 𝛽 is nondecreasing, it follows from (2.22) that 𝑣𝑥𝑛0,𝑥𝑛0+1𝑥<𝑣𝑛0+1,𝑥𝑛0+2≤𝛽𝑣𝑥𝑛0,𝑥𝑛0+1𝛼𝑣𝑥𝑛0+1,𝑥𝑛0+2𝑣𝑥𝑛0,𝑥𝑛0+1𝛽𝑣𝑥≤max𝑛0,𝑥𝑛0+1𝛼𝑣𝑥𝑛0,𝑥𝑛0+1,𝛽𝑣𝑥𝑛0+1,𝑥𝑛0+2𝛼𝑣𝑥𝑛0+1,𝑥𝑛0+2𝑣𝑥𝑛0,𝑥𝑛0+1𝑥<𝑣𝑛0,𝑥𝑛0+1,(2.28) which is a contradiction. Thus {𝑣(𝑥𝑛,𝑥𝑛+1)} is a nonincreasing sequence and so is convergent. Now as in the proof of Theorem 2.1, we obtain the existence of a real number 𝑠∈(0,1) and ̂𝑛0∈ℕ such that 𝑓𝑥𝑛+1≤𝑠𝑛−̂𝑛0+1𝑓𝑥̂𝑛0,∀𝑛≥̂𝑛0,(2.29) thus 𝜃=0. If 𝛼 is nondecreasing, then from assumption (ii) and (2.19), we have 𝑣𝑥𝑛,𝑥𝑛+1≤𝑓𝑥𝑛𝛼𝑣𝑥𝑛,𝑥𝑛+1≤𝑓𝑥𝑛𝛼𝑓𝑥𝑛≤𝑓𝑥𝑛𝛽𝑓𝑥𝑛,(2.30) and if 𝛽 is nondecreasing then from (2.19) and (2.20), we have 𝑣𝑥𝑛,𝑥𝑛+1≤𝑓𝑥𝑛𝛽𝑣𝑥𝑛,𝑥𝑛+1≤𝑓𝑥𝑛𝛽𝑓𝑥𝑛,(2.31) for all 𝑛≥0. Therefore, in all cases we have shown 𝑣𝑥𝑛,𝑥𝑛+1≤𝑓𝑥𝑛𝛽𝑓𝑥𝑛,∀𝑛≥0,(2.32) and so, by following as in the proof of Theorem 2.1, we can take 𝑛0∈ℕ such that 𝑣𝑥𝑛,𝑥𝑛+1î€¸â‰¤ğ‘žğ‘›âˆ’ğ‘›0𝑓𝑥𝑛01−𝑘,(2.33) for all 𝑛≥𝑛0, where ğ‘ž=𝑠1−𝑘<1. The rest of the proof is similar to that of Theorem 2.1.

Remark 2.4. Theorem 2.1 is a generalization of Theorem 1.7. In fact, if we consider 𝛽(𝑡)=𝜑(𝑡) and √𝛼(𝑡)=𝜑(𝑡), then the assumptions of Theorem 2.1 are satisfied. Also, one can see that Theorem 2.1 generalizes Theorem 2.2 of Nicolae [11].

Remark 2.5. Theorem 2.2 essentially generalizes Theorem 1.6. Indeed, if we consider 𝛽(𝑡)=𝜑(𝑡) and 𝛼(𝑡)=𝜓(𝑡), then all assumptions of Theorem 2.2 are satisfied.

The following example shows that there are mappings which satisfy the assumptions of Theorem 2.1 but do not satisfy the assumptions of Theorem 1.7.

Example 2.6. Consider 𝑥𝑛=1/𝑛, for 𝑛∈ℕ, and 𝑥0=0. Then 𝑋={𝑥0,𝑥1,𝑥2,…} is a bounded complete subset of ℝ. Let 𝑣(𝑥,𝑦)=𝑑(𝑥,𝑦), for all 𝑥,𝑦∈𝑋. Define a mapping 𝑇 from 𝑋 into 𝐶(𝑋) by 𝑇𝑥𝑛=âŽ§âŽªâŽ¨âŽªâŽ©ğ‘¥0𝑥if𝑛=0,1𝑥if𝑛=1,1if𝑛∈𝑛∈ℕ𝑒,𝑥∶𝑛≥2𝑛+1,𝑥𝑛2if𝑛∈𝑛∈ℕ𝑜.∶𝑛>2(2.34) It is easy to verify that 𝑓𝑥𝑛𝑥=𝑣𝑛𝑥,𝑇𝑛=⎧⎪⎨⎪⎩10if𝑛=0,1,1−𝑛if𝑛∈𝑛∈ℕ𝑒,1∶𝑛≥2𝑛−1𝑛+1if𝑛∈𝑛∈ℕ𝑜,∶𝑛>2(2.35) is lower semi-continuous in 𝑋. Define 𝛼(𝑡)∶[0,∞)→(0,1] and 𝛽(𝑡)∶[0,∞)→[0,1) by [√𝛼(𝑡)=1if𝑡∈{0}∪1,∞),[[𝑡if𝑡∈(0,1),𝛽(𝑡)=𝑡if𝑡∈0,1),0if𝑡∈1,∞).(2.36) Since 𝛽(𝑡)=√𝛼(𝑡)[[𝑡if𝑡∈0,1),0if𝑡∈1,∞),(2.37) then, we have limsup𝑟→𝑡+𝛽(𝑟)[𝛼(𝑟)<1,foreach𝑡∈0,∞).(2.38) For 𝑥=𝑥0,𝑥1, there exists 𝑦=𝑥∈𝑇(𝑥) such that 𝛼(𝑓(𝑥))𝑣(𝑥,𝑦)=0=𝑓(𝑥),𝑓(𝑦)=0=𝛽(𝑓(𝑥))𝑣(𝑥,𝑦),(2.39) and for 𝑥=𝑥𝑛, 𝑛≥2, if 𝑛∈ℕ𝑒, there exists 𝑦=𝑥1∈𝑇(𝑥) satisfying 1𝛼(𝑓(𝑥))𝑣(𝑥,𝑦)=𝛼1−𝑛11−𝑛<11−𝑛1=𝑓(𝑥),𝑓(𝑦)=0<1−𝑛11−𝑛=𝛽(𝑓(𝑥))𝑣(𝑥,𝑦),(2.40) and, if 𝑛∈ℕ𝑜, there exists 𝑦=𝑥𝑛2∈𝑇(𝑥) satisfying 1𝛼(𝑓(𝑥))𝑣(𝑥,𝑦)=√𝑛(𝑛+1)𝑛−1𝑛2<11𝑛(𝑛+1)=𝑓(𝑥),𝑓(𝑦)=𝑛2𝑛2<1+1𝑛(𝑛+1)𝑛−1𝑛2=𝛽(𝑓(𝑥))𝑣(𝑥,𝑦).(2.41)
Therefore, all assumptions of Theorem 2.1 are satisfied and 𝑥0, 𝑥1 are two fixed points of 𝑇. Let us observe that 𝑇 does not satisfy the assumptions of Theorem 1.7 provided that 𝑣(𝑥,𝑦)=𝑑(𝑥,𝑦), for all 𝑥,𝑦∈𝑋. Indeed, for any function 𝜑∶[0,∞)→[𝛼,1), 𝛼∈(0,1), there exists 𝑛>2, 𝑛∈ℕ𝑜, such that for 𝑥=𝑥𝑛, if 𝑦=𝑥𝑛2∈𝑇(𝑥), we have 1𝑓(𝑥)=𝑛<√(𝑛+1)𝛼𝑛−1𝑛2≤√𝜑(𝑓(𝑥))𝑣(𝑥,𝑦),(2.42) and if 𝑦=𝑥𝑛+1∈𝑇(𝑥), we have 1𝑣(𝑦,𝑇(𝑦))=1−>1𝑛+1𝑛(𝑛+1)≥𝜑(𝑓(𝑥))𝑣(𝑥,𝑦),(2.43) that is, the assumptions of Theorem 1.7 are not satisfied. The next example is an application of Theorem 2.3.

Example 2.7. Let 𝑋 be as in the Example 2.6, and let 𝑣(𝑥,𝑦)=𝑦, for all 𝑥,𝑦∈𝑋. Note that 𝑣 is a 𝑤-distance on 𝑋. Define a mapping 𝑇 from 𝑋 into 𝐶(𝑋) by 𝑇𝑥𝑛=âŽ§âŽªâŽ¨âŽªâŽ©ğ‘¥0𝑥if𝑛=0,0,𝑥1𝑥if𝑛=1,1if𝑛∈𝑛∈ℕ𝑒,𝑥∶𝑛≥2𝑛4−1,𝑥𝑛3if𝑛∈𝑛∈ℕ𝑜.∶𝑛>2(2.44) Clearly, 𝑓𝑥𝑛𝑥=𝑣𝑛𝑥,𝑇𝑛=⎧⎪⎨⎪⎩0if𝑛=0,1,1if𝑛∈𝑛∈ℕ𝑒,1∶𝑛≥2𝑛4−1if𝑛∈𝑛∈ℕ𝑜∶𝑛>2(2.45) is lower semi-continuous in 𝑋. Define 𝛼(𝑡)∶[0,∞)→(0,1] and 𝛽(𝑡)∶[0,∞)→[0,1) by [𝑡𝛼(𝑡)=1if𝑡∈{0}∪1,∞),1/3ğ›½âŽ§âŽªâŽ¨âŽªâŽ©ğ‘¡if𝑡∈(0,1),(𝑡)=5/61if𝑡∈0,2,125/61if𝑡∈2.,∞(2.46) Note that 𝛽(𝑡) is nondecreasing and 𝑡5/6≤𝛽(𝑡), for each 𝑡∈[0,(1/2)]. Since 𝛽(𝑡)=âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©ğ‘¡ğ›¼(𝑡)1/21if𝑡∈0,2,125/6𝑡−1/31if𝑡∈2,1,125/6[if𝑡∈1,∞),(2.47) then, limsup𝑟→𝑡+𝛽(𝑟)𝛼(𝑟)<1,foreach[𝑡∈0,∞).(2.48) For 𝑥=𝑥0,𝑥1, there exists 𝑦=𝑥0∈𝑇(𝑥) such that 𝛼(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦)=0=𝑓(𝑥),𝑓(𝑦)=0=𝛽(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦),(2.49) and for 𝑥=𝑥𝑛, 𝑛≥2, if 𝑛∈ℕ𝑒, there exists 𝑦=𝑥1∈𝑇(𝑥) satisfying 𝑓𝛼(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦)=1=𝑓(𝑥),(𝑦)=0<𝛽(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦),(2.50) and, if 𝑛∈ℕ𝑜, there exists 𝑦=𝑥𝑛3∈𝑇(𝑥) satisfying 1𝛼(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦)=𝑛1𝑛3<1𝑛41−1=𝑓(𝑥),𝑓(𝑦)=𝑛12<1−1𝑛35/61𝑛3=𝛽(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦).(2.51) Then, all assumptions of Theorem 2.3 are satisfied and 𝑥0, 𝑥1 are two fixed points of 𝑇. Note that 𝑣(𝑥1,𝑥1)≠0.

Acknowledgment

The authors would like to thank the referees for their valuable and useful comments.

References

  1. S. Banach, “Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922. View at: Google Scholar
  2. S. B. Nadler, “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969. View at: Google Scholar
  3. D. Klim and D. Wardowski, “Fixed point theorems for set-valued contractions in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 132–139, 2007. View at: Publisher Site | Google Scholar
  4. L. B. Ćirić, “Fixed point theorems for multi-valued contractions in metric spaces,” Journal of Mathematical Analysis and Applications, vol. 348, pp. 499–507, 2008. View at: Google Scholar
  5. L. Ćirić, “Multi-valued nonlinear contraction mappings,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 7-8, pp. 2716–2723, 2009. View at: Publisher Site | Google Scholar
  6. A. Latif and A. A. N. Abdou, “Multivalued generalised nonlinear contractive maps and fixed points,” Nonlinear Analysis, vol. 74, pp. 1436–1444, 2011. View at: Google Scholar
  7. N. Mizoguchi and W. Takahashi, “Fixed point theorems for multivalued mappings on complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 141, no. 1, pp. 172–188, 1989. View at: Google Scholar
  8. S. Reich, “Some problems and results in fixed point theory,” Contemporary Mathematics, vol. 21, pp. 179–187, 1983. View at: Google Scholar
  9. Y. Feng and S. Liu, “Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings,” Journal of Mathematical Analysis and Applications, vol. 317, no. 1, pp. 103–112, 2006. View at: Publisher Site | Google Scholar
  10. Z. Liu, W. Sun, S. M. Kang, and J. S. Ume, “On fixed point theorems for multi-valued contractions,” Fixed Point Theory and Applications, vol. 2010, Article ID 870980, 18 pages, 2010. View at: Publisher Site | Google Scholar
  11. A. Nicolae, “Fixed point theorems for multi-valued mappings of Feng-Liu type,” Fixed Point Theory, vol. 12, pp. 145–154, 2011. View at: Google Scholar
  12. O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimization theorems and fixed point theorems in complete metric spaces,” Mathematica Japonica, vol. 44, pp. 381–391, 1996. View at: Google Scholar
  13. T. Suzuki and W. Takahashi, “Fixed point theorems and characterizations of metric completeness,” Topological Methods in Nonlinear Analysis, vol. 8, pp. 371–382, 1996. View at: Google Scholar

Copyright © 2011 A. Azizi and H. P. Masiha. 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.


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