Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011Β (2011), Article IDΒ 924396, 13 pages
http://dx.doi.org/10.5402/2011/924396
Research Article

Some Generalizations of Fixed Point Results for Multivalued Contraction Mappings

Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran

Received 18 August 2011; Accepted 26 September 2011

Academic Editors: A.Β Levy and G.Β Γ“lafsson

Copyright Β© 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.

Abstract

We establish some fixed point results for multivalued contraction type mappings in terms of a 𝑀-distance in a complete metric space. Our results generalize very recent results of some authors (Δ†iriΔ‡, 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 𝐢(𝑋) and 𝐾(𝑋), respectively. Throughout this paper, we assume that ℝ, β„•, ℕ𝑒, and β„•π‘œ 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,𝐡)=maxsupπ‘₯βˆˆπ΄π‘‘(π‘₯,𝐡),supπ‘¦βˆˆπ΅ξƒ°π‘‘(𝑦,𝐴),(1.1) for all 𝐴,𝐡∈CB(𝑋), where 𝑑(π‘₯,𝐡)=inf{𝑑(π‘₯,𝑦)βˆΆπ‘¦βˆˆπ΅}. An element π‘₯βˆˆπ‘‹ is said to be a fixed point of a multi-valued mapping π‘‡βˆΆπ‘‹β†’πΆ(𝑋) if π‘₯βˆˆπ‘‡(π‘₯). A map π‘“βˆΆπ‘‹β†’β„ is called lower semicontinuous if for any sequence {π‘₯𝑛} in 𝑋 and π‘₯βˆˆπ‘‹ such that π‘₯𝑛→π‘₯, we have 𝑓(π‘₯)≀liminfπ‘›β†’βˆžπ‘“(π‘₯𝑛).

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 π‘‡βˆΆπ‘‹β†’CB(𝑋) be a multi-valued mapping. Assume that there exists π‘Ÿβˆˆ[0,1) such that for allπ‘₯,π‘¦βˆˆπ‘‹, 𝐻(𝑇(π‘₯),𝑇(𝑦))β‰€π‘Ÿπ‘‘(π‘₯,𝑦),(1.2) then there exists π‘§βˆˆπ‘‹ such that π‘§βˆˆπ‘‡(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 π‘‡βˆΆπ‘‹β†’CB(𝑋) be a multi-valued mapping. If there exists a function πœ‘βˆΆ(0,∞)β†’[0,1) such that limsupπ‘Ÿβ†’π‘‘+πœ‘(π‘Ÿ)<1 for each π‘‘βˆˆ[0,∞), and if for all π‘₯,π‘¦βˆˆπ‘‹, 𝐻(𝑇(π‘₯),𝑇(𝑦))β‰€πœ‘(𝑑(π‘₯,𝑦))𝑑(π‘₯,𝑦),(1.3) then there exists π‘§βˆˆπ‘‹ such that π‘§βˆˆπ‘‡(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 π‘‡βˆΆπ‘‹β†’πΆ(𝑋) be a multi-valued mapping. If there exist constants 𝛼,π›½βˆˆ(0,1), 𝛽<𝛼, such that for any π‘₯βˆˆπ‘‹ there is π‘¦βˆˆπ‘‡(π‘₯) satisfying the following two conditions: (i)𝛼𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑇(π‘₯)),(ii)𝑑(𝑦,𝑇(𝑦))≀𝛽𝑑(π‘₯,𝑇(π‘₯)), then there exists π‘§βˆˆπ‘‹ such that π‘§βˆˆπ‘‡(𝑧) provided the function 𝑓(π‘₯)=𝑑(π‘₯,𝑇(π‘₯)) 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 π‘‡βˆΆπ‘‹β†’πΆ(𝑋) be a multi-valued mapping. Assume that the following conditions hold:(i)the map π‘“βˆΆπ‘‹β†’β„, defined by 𝑓(π‘₯)=𝑑(π‘₯,𝑇(π‘₯)), π‘₯βˆˆπ‘‹, is lower semi-continuous; (ii)there exists a constant π›Όβˆˆ(0,1) and a function πœ‘βˆΆ[0,∞)β†’[0,𝛼) satisfying limsupπ‘Ÿβ†’π‘‘+πœ‘(π‘Ÿ)<𝛼, for each π‘‘βˆˆ[0,∞), and for any π‘₯βˆˆπ‘‹ there is π‘¦βˆˆπ‘‡(π‘₯) satisfying 𝛼𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑇(π‘₯)),𝑑(𝑦,𝑇(𝑦))β‰€πœ‘(𝑑(π‘₯,𝑦))𝑑(π‘₯,𝑦).(1.4) Then there exists π‘§βˆˆπ‘‹ such that π‘§βˆˆπ‘‡(𝑧).

Theorem 1.5 (see [3]). Let (𝑋,𝑑) be a complete metric space, and let π‘‡βˆΆπ‘‹β†’πΎ(𝑋) be a multi-valued mapping. Assume that the following conditions hold:(i)the map π‘“βˆΆπ‘‹β†’β„, defined by 𝑓(π‘₯)=𝑑(π‘₯,𝑇(π‘₯)), π‘₯βˆˆπ‘‹, is lower semi-continuous;(ii)there exists a function πœ‘βˆΆ[0,∞)β†’[0,1) satisfying limsupπ‘Ÿβ†’π‘‘+πœ‘(π‘Ÿ)<1, for each π‘‘βˆˆ[0,∞), and for any π‘₯βˆˆπ‘‹ there is π‘¦βˆˆπ‘‡(π‘₯) satisfying 𝑑(π‘₯,𝑦)=𝑑(π‘₯,𝑇(π‘₯)),𝑑(𝑦,𝑇(𝑦))β‰€πœ‘(𝑑(π‘₯,𝑦))𝑑(π‘₯,𝑦).(1.5) Then there exists π‘§βˆˆπ‘‹ such that π‘§βˆˆπ‘‡(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 π‘‡βˆΆπ‘‹β†’πΆ(𝑋) be a multi-valued mapping. If there exist a function πœ‘βˆΆ[0,∞)β†’[0,1) and a nondecreasing function πœ“βˆΆ[0,∞)β†’[𝛼,1), 𝛼>0, such that πœ‘(𝑑)<πœ“(𝑑),limsupπ‘Ÿβ†’π‘‘+πœ‘(π‘Ÿ)<limsupπ‘Ÿβ†’π‘‘+[πœ“(π‘Ÿ),βˆ€π‘‘βˆˆ0,∞),(1.6) and for any π‘₯βˆˆπ‘‹ there is π‘¦βˆˆπ‘‡(π‘₯) satisfying the following two conditions: πœ“(𝑑(π‘₯,𝑦))𝑑(π‘₯,𝑦)≀𝑑(π‘₯,𝑇(π‘₯)),𝑑(𝑦,𝑇(𝑦))β‰€πœ‘(𝑑(π‘₯,𝑦))𝑑(π‘₯,𝑦),(1.7) then 𝑇 has a fixed point in 𝑋 provided 𝑓(π‘₯)=𝑑(π‘₯,𝑇(π‘₯)) is lower semi-continuous.

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

Let (𝑋,𝑑) be a metric space, then a function π‘£βˆΆπ‘‹Γ—π‘‹β†’[0,∞) is called a 𝑀-distance on 𝑋 if the following axioms are satisfied: (1)𝑣(π‘₯,𝑧)≀𝑣(π‘₯,𝑦)+𝑣(𝑦,𝑧), for any π‘₯,𝑦,π‘§βˆˆπ‘‹; (2)for any π‘₯βˆˆπ‘‹, 𝑣(π‘₯,β‹…)βˆΆπ‘‹β†’[0,∞) is lower semi-continuous; (3)for any πœ€>0, there exists 𝛿>0 such that 𝑣(𝑧,π‘₯)≀𝛿 and 𝑣(𝑧,𝑦)≀𝛿 imply 𝑑(π‘₯,𝑦)β‰€πœ€.

The metric 𝑑 is a 𝑀-distance on 𝑋. One can see other examples of 𝑀-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 𝑀-distance 𝑣. Let π‘‡βˆΆπ‘‹β†’πΆ(𝑋) be a multi-valued map. Assume that the following conditions hold: (i)there exist a constant π›Όβˆˆ(0,1) and a function πœ‘βˆΆ[0,∞)β†’[𝛼,1) such that limsupπ‘Ÿβ†’π‘‘+πœ‘(π‘Ÿ)<1, for each π‘‘βˆˆ[0,∞); (ii)the map π‘“βˆΆπ‘‹β†’β„, defined by 𝑓(π‘₯)=𝑣(π‘₯,𝑇(π‘₯)), π‘₯βˆˆπ‘‹, is lower semi-continuous; (iii)for any π‘₯βˆˆπ‘‹, there is π‘¦βˆˆπ‘‡(π‘₯) satisfying βˆšπœ‘(𝑓(π‘₯))𝑣(π‘₯,𝑦)≀𝑓(π‘₯),𝑓(𝑦)β‰€πœ‘(𝑓(π‘₯))𝑣(π‘₯,𝑦).(1.8) Then there exists π‘§βˆˆπ‘‹ such that 𝑓(𝑧)=0. Further, if 𝑣(𝑧,𝑧)=0, then π‘§βˆˆπ‘‡(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 𝑀-distance on 𝑋. Let {π‘₯𝑛} and {𝑦𝑛} be two sequences in 𝑋, let {π‘Žπ‘›} and {𝑏𝑛} be sequences in [0,∞) converging to zero, and let π‘§βˆˆπ‘‹. Then the following hold: (1)if 𝑣(π‘₯𝑛,𝑦𝑛)β‰€π‘Žπ‘› and 𝑣(π‘₯𝑛,𝑧)≀𝑏𝑛 for any π‘›βˆˆβ„•, then {𝑦𝑛} converges to 𝑧; (2)if 𝑣(π‘₯𝑛,π‘₯𝑝)β‰€π‘Žπ‘› for any 𝑛,π‘βˆˆβ„• with 𝑝>𝑛, then {π‘₯𝑛} is a Cauchy sequence.

In the present paper, using the concept of 𝑀-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 {π‘₯𝑛}𝑛β‰₯0 in 𝑋 is called an orbit of 𝑇 at π‘₯0βˆˆπ‘‹ if π‘₯π‘›βˆˆπ‘‡(π‘₯π‘›βˆ’1), for all 𝑛β‰₯1, and 𝑣(π‘₯,𝐺)=inf{𝑣(π‘₯,𝑦)βˆΆπ‘¦βˆˆπΊ} for any π‘₯βˆˆπ‘‹ and 𝐺∈𝐢(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 𝑀-distance 𝑣, and let 𝑇 be a multi-valued mapping from 𝑋 into 𝐢(𝑋). Assume that the following conditions hold: (i)the map π‘“βˆΆπ‘‹β†’β„, defined by 𝑓(π‘₯)=𝑣(π‘₯,𝑇(π‘₯)), π‘₯βˆˆπ‘‹, is lower semi-continuous;(ii)there exist the functions π›ΌβˆΆ[0,∞)β†’(0,1], π›½βˆΆ[0,∞)β†’[0,1) which satisfy βˆƒπ‘˜,πœ€βˆˆ(0,1),π‘‘π‘˜[],≀𝛼(𝑑),foreachπ‘‘βˆˆ0,πœ€limsupπ‘Ÿβ†’π‘‘+𝛽(π‘Ÿ)[𝛼(π‘Ÿ)<1,foreachπ‘‘βˆˆ0,∞);(2.1)(iii)for any π‘₯βˆˆπ‘‹, there is π‘¦βˆˆπ‘‡(π‘₯) satisfying 𝛼(𝑓(π‘₯))𝑣(π‘₯,𝑦)≀𝑓(π‘₯),𝑓(𝑦)≀𝛽(𝑓(π‘₯))𝑣(π‘₯,𝑦).(2.2) Then, there exists π‘§βˆˆπ‘‹ such that π‘§βˆˆπ‘‡(𝑧).

Proof. Let π‘₯0βˆˆπ‘‹. Then there exists π‘₯1βˆˆπ‘‡(π‘₯) such that 𝛼𝑓π‘₯0𝑣π‘₯ξ€Έξ€Έ0,π‘₯1ξ€Έξ€·π‘₯≀𝑓0ξ€Έ,𝑓π‘₯1𝑓π‘₯≀𝛽0𝑣π‘₯ξ€Έξ€Έ0,π‘₯1ξ€Έ.(2.3) Then, from (2.3), we have 𝑓π‘₯1≀𝛽𝑓π‘₯0𝛼𝑓π‘₯0𝑓π‘₯ξ€Έξ€Έ0ξ€Έ.(2.4) Continuing this process, we can define an orbit {π‘₯𝑛} of 𝑇 in 𝑋, such that 𝛼𝑓π‘₯𝑛𝑣π‘₯𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯≀𝑓𝑛π‘₯,𝑓𝑛+1𝑓π‘₯≀𝛽𝑛𝑣π‘₯𝑛,π‘₯𝑛+1ξ€Έ,(2.5) for all 𝑛β‰₯0, which imply that 𝑓π‘₯𝑛+1≀𝛽𝑓π‘₯𝑛𝛼𝑓π‘₯𝑛𝑓π‘₯𝑛.(2.6) We can assume that 𝑓(π‘₯𝑛)>0 for all 𝑛β‰₯0, since 𝑓(π‘₯𝑛)=0 for some 𝑛, then from (2.5), 𝑣(π‘₯𝑛,π‘₯𝑛+1)=𝑓(π‘₯𝑛+1)=0, and so for every π‘šβˆˆβ„•, there exists π‘¦π‘šβˆˆπ‘‡(π‘₯𝑛+1) such that 𝑣(π‘₯𝑛+1,π‘¦π‘š)≀1/π‘š; consequently, 𝑣(π‘₯𝑛,π‘¦π‘š)≀𝑣(π‘₯𝑛,π‘₯𝑛+1)+𝑣(π‘₯𝑛+1,π‘¦π‘š)≀1/π‘š; so by Lemma 1.8, π‘¦π‘šβ†’π‘₯𝑛+1 and thus π‘₯𝑛+1βˆˆπ‘‡(π‘₯𝑛+1), that is, the assertion of the theorem is proved. Since 𝛽(𝑓(π‘₯))/𝛼(𝑓(π‘₯))<1 for all π‘₯βˆˆπ‘‹, then we have 𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯<𝑓𝑛.(2.7) Thus {𝑓(π‘₯𝑛)} is a decreasing sequence of positive real numbers and hence converges to a nonnegative number πœƒ, πœƒβ‰₯0. Let 𝜏=limsupπ‘›β†’βˆžπ›½(𝑓(π‘₯𝑛))/𝛼(𝑓(π‘₯𝑛))<1. Then, for 𝑠=(𝜏+1)/2, we can take ̂𝑛0βˆˆβ„• such that 𝛽𝑓π‘₯𝑛𝛼𝑓π‘₯𝑛<𝑠,βˆ€π‘›β‰₯̂𝑛0.(2.8) Then, 𝑓π‘₯𝑛+1ξ€Έβ‰€π‘ π‘›βˆ’Μ‚π‘›0+1𝑓π‘₯̂𝑛0,βˆ€π‘›β‰₯̂𝑛0,(2.9) thus, πœƒ=0. Therefore, by (ii) we can take ̃𝑛0βˆˆβ„• such that 𝑓π‘₯π‘›ξ€Έβˆˆ[]ξ€·π‘₯0,πœ€,π‘“π‘›ξ€Έπ‘˜ξ€·π‘“ξ€·π‘₯≀𝛼𝑛,βˆ€π‘›β‰₯̃𝑛0.(2.10) Now, from (2.5), (2.9), and (2.10), we have 𝑣π‘₯𝑛,π‘₯𝑛+1≀𝑓π‘₯𝑛𝛼𝑓π‘₯𝑛≀𝑓π‘₯𝑛𝑓π‘₯π‘›ξ€Έπ‘˜ξ€·π‘₯=𝑓𝑛1βˆ’π‘˜β‰€π‘žπ‘›βˆ’π‘›0𝑓π‘₯𝑛0ξ€Έ1βˆ’π‘˜,(2.11) for all 𝑛β‰₯𝑛0, where 𝑛0=max{̂𝑛0,̃𝑛0} and π‘ž=𝑠1βˆ’π‘˜<1. Then for any 𝑝,π‘›βˆˆβ„•, 𝑝>𝑛β‰₯𝑛0, we have 𝑣π‘₯𝑛,π‘₯π‘ξ€Έβ‰€π‘βˆ’1𝑖=𝑛𝑣π‘₯𝑖,π‘₯𝑖+1ξ€Έβ‰€π‘βˆ’1𝑖=π‘›π‘žπ‘–βˆ’π‘›0𝑓π‘₯𝑛0ξ€Έ1βˆ’π‘˜β‰€π‘žπ‘›βˆ’π‘›0𝑓π‘₯1βˆ’π‘žπ‘›0ξ€Έ1βˆ’π‘˜.(2.12) Hence, by Lemma 1.8,  {π‘₯𝑛} is a Cauchy sequence so there exists π‘§βˆˆπ‘‹ such that lim𝑛π‘₯𝑛=𝑧. Since 𝑓 is lower semi-continuous, we obtain 0≀𝑓(𝑧)≀liminfπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛=0,(2.13) and thus, 𝑓(𝑧)=𝑣(𝑧,𝑇(𝑧))=0.(2.14) Now, we clime that π‘§βˆˆπ‘‡(𝑧). Notice that the function 𝑣(π‘₯,β‹…) is lower semi-continuous for all π‘₯βˆˆπ‘‹. Since π‘₯𝑝→𝑧, then by (2.12), 𝑣π‘₯𝑛,𝑧≀liminfπ‘β†’βˆžπ‘£ξ€·π‘₯𝑛,π‘₯π‘ξ€Έβ‰€π‘žπ‘›βˆ’π‘›0𝑓π‘₯1βˆ’π‘žπ‘›0ξ€Έ1βˆ’π‘˜,(2.15) for all 𝑛β‰₯𝑛0. On the other hand, from (2.14), for every π‘›βˆˆβ„•, there exists π‘¦π‘›βˆˆπ‘‡(𝑧) such that 𝑣(𝑧,𝑦𝑛)≀1/𝑛. Then, for all 𝑛β‰₯𝑛0, we have 𝑣π‘₯𝑛,𝑦𝑛π‘₯≀𝑣𝑛,𝑧+𝑣𝑧,π‘¦π‘›ξ€Έβ‰€π‘žπ‘›βˆ’π‘›0𝑓π‘₯1βˆ’π‘žπ‘›0ξ€Έ1βˆ’π‘˜+1𝑛.(2.16) Therefore, from (2.15), (2.16), we can find the sequences {π‘Žπ‘›} and {𝑏𝑛} in [0,∞) converging to zero, such that 𝑣(π‘₯𝑛,𝑦𝑛)β‰€π‘Žπ‘› and 𝑣(π‘₯𝑛,𝑧)≀𝑏𝑛 for any π‘›βˆˆβ„•; then by Lemma 1.8  𝑦𝑛→𝑧. The closedness of 𝑇(𝑧) implies π‘§βˆˆπ‘‡(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 𝑀-distance 𝑣, and let 𝑇 be a multi-valued mapping from 𝑋 into 𝐢(𝑋). Assume that the following conditions hold: (i)the map π‘“βˆΆπ‘‹β†’β„, defined by 𝑓(π‘₯)=𝑣(π‘₯,𝑇(π‘₯)), π‘₯βˆˆπ‘‹, is lower semi-continuous;(ii)there exist the functions π›ΌβˆΆ[0,∞)β†’(0,1], π›½βˆΆ[0,∞)β†’[0,1) which satisfy βˆƒπ‘˜,πœ€βˆˆ(0,1),π‘‘π‘˜[],≀𝛼(𝑑),foreachπ‘‘βˆˆ0,πœ€limsupπ‘Ÿβ†’π‘‘+𝛽(π‘Ÿ)[𝛼(π‘Ÿ)<1,foreachπ‘‘βˆˆ0,∞),(2.17) and 𝛼 is nondecreasing; (iii)for any π‘₯βˆˆπ‘‹, there is π‘¦βˆˆπ‘‡(π‘₯) satisfying 𝛼(𝑣(π‘₯,𝑦))𝑣(π‘₯,𝑦)≀𝑓(π‘₯),𝑓(𝑦)≀𝛽(𝑣(π‘₯,𝑦))𝑣(π‘₯,𝑦).(2.18) Then, there exists π‘§βˆˆπ‘‹ such that π‘§βˆˆπ‘‡(𝑧).

Proof. By following the lines in the proof of Theorem 2.1, one can construct an orbit {π‘₯𝑛}βˆžπ‘›=0 of 𝑇 in 𝑋 such that 𝛼𝑣π‘₯𝑛,π‘₯𝑛+1𝑣π‘₯𝑛,π‘₯𝑛+1ξ€Έξ€·π‘₯≀𝑓𝑛,(2.19)𝑓π‘₯𝑛+1𝑣π‘₯≀𝛽𝑛,π‘₯𝑛+1𝑣π‘₯𝑛,π‘₯𝑛+1ξ€Έ,(2.20) for all 𝑛β‰₯0, which imply that 𝑓π‘₯𝑛+1≀𝛽𝑣π‘₯𝑛,π‘₯𝑛+1𝛼𝑣π‘₯𝑛,π‘₯𝑛+1𝑓π‘₯𝑛.(2.21) Thus, {𝑓(π‘₯𝑛)} is a decreasing sequence, and so there exists πœƒβ‰₯0 such that 𝑓(π‘₯𝑛)β†’πœƒ. From (2.19) and (2.20), we have 𝑣π‘₯𝑛+1,π‘₯𝑛+2≀𝑓π‘₯𝑛+1𝛼𝑣π‘₯𝑛+1,π‘₯𝑛+2≀𝛽𝑣π‘₯𝑛,π‘₯𝑛+1𝛼𝑣π‘₯𝑛+1,π‘₯𝑛+2𝑣π‘₯𝑛,π‘₯𝑛+1ξ€Έ,(2.22) for all 𝑛β‰₯0. Now we clime that {𝑣(π‘₯𝑛,π‘₯𝑛+1)} is a nonincreasing sequence. Suppose not. Then there exists 𝑛0β‰₯0 such that 𝑣(π‘₯𝑛0+1,π‘₯𝑛0+2)>𝑣(π‘₯𝑛0,π‘₯𝑛0+1). Since 𝛼 is nondecreasing, then from (2.22), we get that 𝑣π‘₯𝑛0,π‘₯𝑛0+1ξ€Έξ€·π‘₯<𝑣𝑛0+1,π‘₯𝑛0+2≀𝛽𝑣π‘₯𝑛0,π‘₯𝑛0+1𝛼𝑣π‘₯𝑛0+1,π‘₯𝑛0+2𝑣π‘₯𝑛0,π‘₯𝑛0+1≀𝛽𝑣π‘₯𝑛0,π‘₯𝑛0+1𝛼𝑣π‘₯𝑛0,π‘₯𝑛0+1𝑣π‘₯𝑛0,π‘₯𝑛0+1ξ€Έξ€·π‘₯<𝑣𝑛0,π‘₯𝑛0+1ξ€Έ,(2.23) which is a contradiction. Then, {𝑣(π‘₯𝑛,π‘₯𝑛+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 π‘ βˆˆ(0,1) and ̂𝑛0βˆˆβ„• such that 𝑓π‘₯𝑛+1ξ€Έβ‰€π‘ π‘›βˆ’Μ‚π‘›0+1𝑓π‘₯̂𝑛0,βˆ€π‘›β‰₯̂𝑛0,(2.24) thus πœƒ=0. On the other hand, since 𝛼 is nondecreasing, then by (2.19), we have 𝑣π‘₯𝑛,π‘₯𝑛+1≀𝑓π‘₯𝑛𝛼𝑣π‘₯𝑛,π‘₯𝑛+1≀𝑓π‘₯𝑛𝛼𝑓π‘₯𝑛.(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 𝑀-distance 𝑣, and let 𝑇 be a multi-valued mapping from 𝑋 into 𝐢(𝑋). Assume that the following conditions hold: (i)the map π‘“βˆΆπ‘‹β†’β„, defined by 𝑓(π‘₯)=𝑣(π‘₯,𝑇(π‘₯)), π‘₯βˆˆπ‘‹, is lower semi-continuous,(ii)there exist the functions π›ΌβˆΆ[0,∞)β†’(0,1], π›½βˆΆ[0,∞)β†’[0,1) which satisfy βˆƒπ‘˜,πœ€βˆˆ(0,1),π‘‘π‘˜[],𝛽≀𝛽(𝑑),foreachπ‘‘βˆˆ0,πœ€(𝑑)<𝛼(𝑑),limsupπ‘Ÿβ†’π‘‘+𝛽(π‘Ÿ)𝛼(π‘Ÿ)<1,foreacht∈[0,∞),(2.26) and also one of 𝛼 and 𝛽 is nondecreasing; (iii)for any π‘₯βˆˆπ‘‹, there is π‘¦βˆˆπ‘‡(π‘₯) satisfying𝛼(𝑣(π‘₯,𝑦))𝑣(π‘₯,𝑦)≀𝑓(π‘₯),𝑓(𝑦)≀𝛽(𝑣(π‘₯,𝑦))𝑣(π‘₯,𝑦).(2.27) Then, there exists π‘§βˆˆπ‘‹ such that π‘§βˆˆπ‘‡(z).

Proof. As in the proof of Theorem 2.1, one can construct an orbit {π‘₯𝑛}βˆžπ‘›=0 of 𝑇 in 𝑋 such that (2.19), (2.20), (2.21), and (2.22) hold. Then, {𝑓(π‘₯𝑛} is a decreasing sequence and so there exists πœƒβ‰₯0 such that 𝑓(π‘₯𝑛)β†’πœƒ. Now we clime that {𝑣(π‘₯𝑛,π‘₯𝑛+1)} is a nonincreasing sequence. Suppose not. Then there exists 𝑛0β‰₯0 such that 𝑣(π‘₯𝑛0+1,π‘₯𝑛0+2)>𝑣(π‘₯𝑛0,π‘₯𝑛0+1). Since one of 𝛼 and 𝛽 is nondecreasing, it follows from (2.22) that 𝑣π‘₯𝑛0,π‘₯𝑛0+1ξ€Έξ€·π‘₯<𝑣𝑛0+1,π‘₯𝑛0+2≀𝛽𝑣π‘₯𝑛0,π‘₯𝑛0+1𝛼𝑣π‘₯𝑛0+1,π‘₯𝑛0+2𝑣π‘₯𝑛0,π‘₯𝑛0+1𝛽𝑣π‘₯≀max𝑛0,π‘₯𝑛0+1𝛼𝑣π‘₯𝑛0,π‘₯𝑛0+1,𝛽𝑣π‘₯𝑛0+1,π‘₯𝑛0+2𝛼𝑣π‘₯𝑛0+1,π‘₯𝑛0+2𝑣π‘₯𝑛0,π‘₯𝑛0+1ξ€Έξ€·π‘₯<𝑣𝑛0,π‘₯𝑛0+1ξ€Έ,(2.28) which is a contradiction. Thus {𝑣(π‘₯𝑛,π‘₯𝑛+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 π‘ βˆˆ(0,1) and ̂𝑛0βˆˆβ„• such that 𝑓π‘₯𝑛+1ξ€Έβ‰€π‘ π‘›βˆ’Μ‚π‘›0+1𝑓π‘₯̂𝑛0,βˆ€π‘›β‰₯̂𝑛0,(2.29) thus πœƒ=0. If 𝛼 is nondecreasing, then from assumption (ii) and (2.19), we have 𝑣π‘₯𝑛,π‘₯𝑛+1≀𝑓π‘₯𝑛𝛼𝑣π‘₯𝑛,π‘₯𝑛+1≀𝑓π‘₯𝑛𝛼𝑓π‘₯𝑛≀𝑓π‘₯𝑛𝛽𝑓π‘₯𝑛,(2.30) and if 𝛽 is nondecreasing then from (2.19) and (2.20), we have 𝑣π‘₯𝑛,π‘₯𝑛+1≀𝑓π‘₯𝑛𝛽𝑣π‘₯𝑛,π‘₯𝑛+1≀𝑓π‘₯𝑛𝛽𝑓π‘₯𝑛,(2.31) for all 𝑛β‰₯0. Therefore, in all cases we have shown 𝑣π‘₯𝑛,π‘₯𝑛+1≀𝑓π‘₯𝑛𝛽𝑓π‘₯𝑛,βˆ€π‘›β‰₯0,(2.32) and so, by following as in the proof of Theorem 2.1, we can take 𝑛0βˆˆβ„• such that 𝑣π‘₯𝑛,π‘₯𝑛+1ξ€Έβ‰€π‘žπ‘›βˆ’π‘›0𝑓π‘₯𝑛0ξ€Έ1βˆ’π‘˜,(2.33) for all 𝑛β‰₯𝑛0, where π‘ž=𝑠1βˆ’π‘˜<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(𝑑)=πœ‘(𝑑) and βˆšπ›Ό(𝑑)=πœ‘(𝑑), 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(𝑑)=πœ‘(𝑑) and 𝛼(𝑑)=πœ“(𝑑), 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 π‘₯𝑛=1/𝑛, for π‘›βˆˆβ„•, and π‘₯0=0. Then 𝑋={π‘₯0,π‘₯1,π‘₯2,…} is a bounded complete subset of ℝ. Let 𝑣(π‘₯,𝑦)=𝑑(π‘₯,𝑦), for all π‘₯,π‘¦βˆˆπ‘‹. Define a mapping 𝑇 from 𝑋 into 𝐢(𝑋) by 𝑇π‘₯𝑛=⎧βŽͺ⎨βŽͺ⎩π‘₯0π‘₯if𝑛=0,1π‘₯if𝑛=1,1ξ€½ifπ‘›βˆˆπ‘›βˆˆβ„•π‘’ξ€Ύ,ξ€½π‘₯βˆΆπ‘›β‰₯2𝑛+1,π‘₯𝑛2ξ€Ύξ€½ifπ‘›βˆˆπ‘›βˆˆβ„•π‘œξ€Ύ.βˆΆπ‘›>2(2.34) It is easy to verify that 𝑓π‘₯𝑛π‘₯=𝑣𝑛π‘₯,𝑇𝑛=⎧βŽͺ⎨βŽͺ⎩1ξ€Έξ€Έ0if𝑛=0,1,1βˆ’π‘›ξ€½ifπ‘›βˆˆπ‘›βˆˆβ„•π‘’ξ€Ύ,1βˆΆπ‘›β‰₯2π‘›βˆ’1𝑛+1ifπ‘›βˆˆπ‘›βˆˆβ„•π‘œξ€Ύ,βˆΆπ‘›>2(2.35) is lower semi-continuous in 𝑋. Define 𝛼(𝑑)∢[0,∞)β†’(0,1] and 𝛽(𝑑)∢[0,∞)β†’[0,1) by ξ‚»[βˆšπ›Ό(𝑑)=1ifπ‘‘βˆˆ{0}βˆͺ1,∞),ξ‚»[[𝑑ifπ‘‘βˆˆ(0,1),𝛽(𝑑)=𝑑ifπ‘‘βˆˆ0,1),0ifπ‘‘βˆˆ1,∞).(2.36) Since 𝛽(𝑑)=ξ‚»βˆšπ›Ό(𝑑)[[𝑑ifπ‘‘βˆˆ0,1),0ifπ‘‘βˆˆ1,∞),(2.37) then, we have limsupπ‘Ÿβ†’π‘‘+𝛽(π‘Ÿ)[𝛼(π‘Ÿ)<1,foreachπ‘‘βˆˆ0,∞).(2.38) For π‘₯=π‘₯0,π‘₯1, there exists 𝑦=π‘₯βˆˆπ‘‡(π‘₯) such that 𝛼(𝑓(π‘₯))𝑣(π‘₯,𝑦)=0=𝑓(π‘₯),𝑓(𝑦)=0=𝛽(𝑓(π‘₯))𝑣(π‘₯,𝑦),(2.39) and for π‘₯=π‘₯𝑛, 𝑛β‰₯2, if π‘›βˆˆβ„•π‘’, there exists 𝑦=π‘₯1βˆˆπ‘‡(π‘₯) satisfying ξ‚€1𝛼(𝑓(π‘₯))𝑣(π‘₯,𝑦)=𝛼1βˆ’π‘›11βˆ’π‘›ξ‚<ξ‚€11βˆ’π‘›ξ‚ξ‚€1=𝑓(π‘₯),𝑓(𝑦)=0<1βˆ’π‘›11βˆ’π‘›ξ‚=𝛽(𝑓(π‘₯))𝑣(π‘₯,𝑦),(2.40) and, if π‘›βˆˆβ„•π‘œ, there exists 𝑦=π‘₯𝑛2βˆˆπ‘‡(π‘₯) satisfying 1𝛼(𝑓(π‘₯))𝑣(π‘₯,𝑦)=√ξƒͺ𝑛(𝑛+1)π‘›βˆ’1𝑛2<11𝑛(𝑛+1)=𝑓(π‘₯),𝑓(𝑦)=𝑛2𝑛2ξ€Έ<ξ‚΅1+1𝑛(𝑛+1)π‘›βˆ’1𝑛2=𝛽(𝑓(π‘₯))𝑣(π‘₯,𝑦).(2.41)
Therefore, all assumptions of Theorem 2.1 are satisfied and π‘₯0, π‘₯1 are two fixed points of 𝑇. Let us observe that 𝑇 does not satisfy the assumptions of Theorem 1.7 provided that 𝑣(π‘₯,𝑦)=𝑑(π‘₯,𝑦), for all π‘₯,π‘¦βˆˆπ‘‹. Indeed, for any function πœ‘βˆΆ[0,∞)β†’[𝛼,1), π›Όβˆˆ(0,1), there exists 𝑛>2, π‘›βˆˆβ„•π‘œ, such that for π‘₯=π‘₯𝑛, if 𝑦=π‘₯𝑛2βˆˆπ‘‡(π‘₯), we have 1𝑓(π‘₯)=𝑛<√(𝑛+1)π›Όξ‚€π‘›βˆ’1𝑛2ξ‚β‰€βˆšπœ‘(𝑓(π‘₯))𝑣(π‘₯,𝑦),(2.42) and if 𝑦=π‘₯𝑛+1βˆˆπ‘‡(π‘₯), we have 1𝑣(𝑦,𝑇(𝑦))=1βˆ’>1𝑛+1𝑛(𝑛+1)β‰₯πœ‘(𝑓(π‘₯))𝑣(π‘₯,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(π‘₯,𝑦)=𝑦, for all π‘₯,π‘¦βˆˆπ‘‹. Note that 𝑣 is a 𝑀-distance on 𝑋. Define a mapping 𝑇 from 𝑋 into 𝐢(𝑋) by 𝑇π‘₯𝑛=⎧βŽͺ⎨βŽͺ⎩π‘₯0ξ€½π‘₯if𝑛=0,0,π‘₯1ξ€Ύπ‘₯if𝑛=1,1ξ€½ifπ‘›βˆˆπ‘›βˆˆβ„•π‘’ξ€Ύ,ξ€½π‘₯βˆΆπ‘›β‰₯2𝑛4βˆ’1,π‘₯𝑛3ξ€Ύξ€½ifπ‘›βˆˆπ‘›βˆˆβ„•π‘œξ€Ύ.βˆΆπ‘›>2(2.44) Clearly, 𝑓π‘₯𝑛π‘₯=𝑣𝑛π‘₯,𝑇𝑛=⎧βŽͺ⎨βŽͺβŽ©ξ€½ξ€Έξ€Έ0if𝑛=0,1,1ifπ‘›βˆˆπ‘›βˆˆβ„•π‘’ξ€Ύ,1βˆΆπ‘›β‰₯2𝑛4ξ€½βˆ’1ifπ‘›βˆˆπ‘›βˆˆβ„•π‘œξ€ΎβˆΆπ‘›>2(2.45) is lower semi-continuous in 𝑋. Define 𝛼(𝑑)∢[0,∞)β†’(0,1] and 𝛽(𝑑)∢[0,∞)β†’[0,1) by ξ‚»[𝑑𝛼(𝑑)=1ifπ‘‘βˆˆ{0}βˆͺ1,∞),1/3π›½βŽ§βŽͺ⎨βŽͺβŽ©π‘‘ifπ‘‘βˆˆ(0,1),(𝑑)=5/61ifπ‘‘βˆˆ0,2ξ‚„,ξ‚€125/6ξ‚€1ifπ‘‘βˆˆ2.,∞(2.46) Note that 𝛽(𝑑) is nondecreasing and 𝑑5/6≀𝛽(𝑑), for each π‘‘βˆˆ[0,(1/2)]. Since 𝛽(𝑑)=⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π‘‘π›Ό(𝑑)1/21ifπ‘‘βˆˆ0,2ξ‚„,ξ‚€125/6π‘‘βˆ’1/3ξ‚€1ifπ‘‘βˆˆ2,ξ‚€1,125/6[ifπ‘‘βˆˆ1,∞),(2.47) then, limsupπ‘Ÿβ†’π‘‘+𝛽(π‘Ÿ)𝛼(π‘Ÿ)<1,foreach[π‘‘βˆˆ0,∞).(2.48) For π‘₯=π‘₯0,π‘₯1, there exists 𝑦=π‘₯0βˆˆπ‘‡(π‘₯) such that 𝛼(𝑣(π‘₯,𝑦))𝑣(π‘₯,𝑦)=0=𝑓(π‘₯),𝑓(𝑦)=0=𝛽(𝑣(π‘₯,𝑦))𝑣(π‘₯,𝑦),(2.49) and for π‘₯=π‘₯𝑛, 𝑛β‰₯2, if π‘›βˆˆβ„•π‘’, there exists 𝑦=π‘₯1βˆˆπ‘‡(π‘₯) satisfying 𝑓𝛼(𝑣(π‘₯,𝑦))𝑣(π‘₯,𝑦)=1=𝑓(π‘₯),(𝑦)=0<𝛽(𝑣(π‘₯,𝑦))𝑣(π‘₯,𝑦),(2.50) and, if π‘›βˆˆβ„•π‘œ, there exists 𝑦=π‘₯𝑛3βˆˆπ‘‡(π‘₯) satisfying ξ‚€1𝛼(𝑣(π‘₯,𝑦))𝑣(π‘₯,𝑦)=𝑛1𝑛3<1𝑛41βˆ’1=𝑓(π‘₯),𝑓(𝑦)=𝑛12<ξ‚€1βˆ’1𝑛35/6ξ‚€1𝑛3=𝛽(𝑣(π‘₯,𝑦))𝑣(π‘₯,𝑦).(2.51) Then, all assumptions of Theorem 2.3 are satisfied and π‘₯0, π‘₯1 are two fixed points of 𝑇. Note that 𝑣(π‘₯1,π‘₯1)β‰ 0.

Acknowledgment

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

References

  1. S. Banach, β€œSur les opΓ©rations dans les ensembles abstraits et leur applications aux Γ©quations intΓ©grales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922. View at Google Scholar
  2. S. B. Nadler, β€œMulti-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969. View at Google Scholar
  3. D. Klim and D. Wardowski, β€œFixed 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 Β· View at Google Scholar Β· View at Scopus
  4. L. B. Δ†iriΔ‡, β€œFixed 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. Δ†iriΔ‡, β€œMulti-valued nonlinear contraction mappings,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 7-8, pp. 2716–2723, 2009. View at Publisher Β· View at Google Scholar Β· View at Scopus
  6. A. Latif and A. A. N. Abdou, β€œMultivalued 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, β€œFixed 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 Β· View at Scopus
  8. S. Reich, β€œSome 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, β€œFixed 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 Β· View at Google Scholar Β· View at Scopus
  10. Z. Liu, W. Sun, S. M. Kang, and J. S. Ume, β€œOn fixed point theorems for multi-valued contractions,” Fixed Point Theory and Applications, vol. 2010, Article ID 870980, 18 pages, 2010. View at Publisher Β· View at Google Scholar
  11. A. Nicolae, β€œFixed 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, β€œNonconvex 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, β€œFixed point theorems and characterizations of metric completeness,” Topological Methods in Nonlinear Analysis, vol. 8, pp. 371–382, 1996. View at Google Scholar