Some Generalizations of Fixed Point Results for Multivalued Contraction Mappings

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, ℕ_𝑒, and ℕ_𝑜 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)=maxsup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑(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 {𝑥_𝑛} in 𝑋 and 𝑥∈𝑋 such that 𝑥_𝑛→𝑥, we have 𝑓(𝑥)≤liminf_𝑛→∞𝑓(𝑥_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 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 limsup_𝑟→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 limsup_𝑟→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 limsup_𝑟→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),limsup_𝑟→𝑡+𝜑(𝑟)<limsup_𝑟→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 limsup_𝑟→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 {𝑥_𝑛} and {𝑦_𝑛} be two sequences in 𝑋, let {𝑎_𝑛} and {𝑏_𝑛} be sequences in [0,∞) converging to zero, and let 𝑧∈𝑋. Then the following hold: (1)if 𝑣(𝑥_𝑛,𝑦_𝑛)≤𝑎_𝑛 and 𝑣(𝑥_𝑛,𝑧)≤𝑏_𝑛 for any 𝑛∈ℕ, then {𝑦_𝑛} converges to 𝑧; (2)if 𝑣(𝑥_𝑛,𝑥_𝑝)≤𝑎_𝑛 for any 𝑛,𝑝∈ℕ with 𝑝>𝑛, then {𝑥_𝑛} 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 {𝑥_𝑛}_𝑛≥0 in 𝑋 is called an orbit of 𝑇 at 𝑥₀∈𝑋 if 𝑥_𝑛∈𝑇(𝑥_𝑛−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,𝜀l∈im(s0u,p_𝑟→𝑡+1),𝑡^𝑘≤𝛼(𝑡),foreach𝑡∈[0,𝜀],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 𝛼𝑓𝑓𝑥₁𝑥₀𝑣𝑥₀,𝑥₁≤𝑓𝑥₀,≤𝛽𝑓𝑥₀𝑣𝑥₀,𝑥₁.(2.3) Then, from (2.3), we have 𝑓𝑥₁≤𝛽𝑓𝑥⁰𝛼𝑓𝑥₀𝑓𝑥⁰.(2.4) Continuing this process, we can define an orbit {𝑥_𝑛} of 𝑇 in 𝑋, such that 𝛼𝑓𝑥_𝑛𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑓𝑥_𝑛,𝑓𝑥_𝑛+1≤𝛽𝑓𝑥_𝑛𝑣𝑥_𝑛,𝑥_𝑛+1,(2.5) for all 𝑛≥0, which imply that 𝑓𝑥_𝑛+1≤𝛽𝑓𝑥^𝑛𝛼𝑓𝑥_𝑛𝑓𝑥^𝑛.(2.6) We can assume that 𝑓(𝑥_𝑛)>0 for all 𝑛≥0, since 𝑓(𝑥_𝑛)=0 for some 𝑛, then from (2.5), 𝑣(𝑥_𝑛,𝑥_𝑛+1)=𝑓(𝑥_𝑛+1)=0, and so for every 𝑚∈ℕ, there exists 𝑦_𝑚∈𝑇(𝑥_𝑛+1) such that 𝑣(𝑥_𝑛+1,𝑦_𝑚)≤1/𝑚; consequently, 𝑣(𝑥_𝑛,𝑦_𝑚)≤𝑣(𝑥_𝑛,𝑥_𝑛+1)+𝑣(𝑥_𝑛+1,𝑦_𝑚)≤1/𝑚; so by Lemma 1.8, 𝑦_𝑚→𝑥_𝑛+1 and thus 𝑥_𝑛+1∈𝑇(𝑥_𝑛+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𝑥_𝑛+1<𝑓𝑥_𝑛.(2.7) Thus {𝑓(𝑥_𝑛)} is a decreasing sequence of positive real numbers and hence converges to a nonnegative number 𝜃, 𝜃≥0. Let 𝜏=limsup_𝑛→∞𝛽(𝑓(𝑥_𝑛))/𝛼(𝑓(𝑥_𝑛))<1. Then, for 𝑠=(𝜏+1)/2, we can take 𝑛₀̂∈ℕ such that 𝛽𝑓𝑥_𝑛𝛼𝑓𝑥_𝑛<𝑠,∀𝑛≥𝑛⁰̂.(2.8) Then, 𝑓𝑥_𝑛+1≤𝑠^𝑛−𝑛̂₀₊₁𝑓𝑥𝑛₀̂,∀𝑛≥𝑛⁰̂,(2.9) thus, 𝜃=0. Therefore, by (ii) we can take 𝑛₀̃∈ℕ such that 𝑓𝑥_𝑛∈[0,𝜀],𝑓𝑥_𝑛^𝑘≤𝛼𝑓𝑥_𝑛,∀𝑛≥𝑛₀̃.(2.10) Now, from (2.5), (2.9), and (2.10), we have 𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑓𝑥^𝑛𝛼𝑓𝑥_𝑛≤𝑓𝑥^𝑛𝑓𝑥_𝑛^𝑘=𝑓𝑥_𝑛^1−𝑘≤𝑞^𝑛−𝑛0𝑓𝑥_𝑛0^1−𝑘,(2.11) for all 𝑛≥𝑛₀, where 𝑛₀=max{𝑛₀̂,𝑛₀̃} and 𝑞=𝑠^1−𝑘<1. Then for any 𝑝,𝑛∈ℕ, 𝑝>𝑛≥𝑛₀, we have 𝑣𝑥_𝑛,𝑥_𝑝≤^𝑝−1_𝑖=𝑛𝑣𝑥_𝑖,𝑥_𝑖+1≤^𝑝−1_𝑖=𝑛𝑞^𝑖−𝑛0𝑓𝑥_𝑛0^1−𝑘≤𝑞^𝑛−𝑛01−𝑞𝑓𝑥^𝑛0^1−𝑘.(2.12) Hence, by Lemma 1.8,  {𝑥_𝑛} is a Cauchy sequence so there exists 𝑧∈𝑋 such that lim_𝑛𝑥_𝑛=𝑧. Since 𝑓 is lower semi-continuous, we obtain 0≤𝑓(𝑧)≤liminf_𝑛→∞𝑓𝑥_𝑛=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 𝑥_𝑝→𝑧, then by (2.12), 𝑣𝑥_𝑛,𝑧≤liminf_𝑝→∞𝑣𝑥_𝑛,𝑥_𝑝≤𝑞^𝑛−𝑛01−𝑞𝑓𝑥^𝑛0^1−𝑘,(2.15) for all 𝑛≥𝑛₀. On the other hand, from (2.14), for every 𝑛∈ℕ, there exists 𝑦_𝑛∈𝑇(𝑧) such that 𝑣(𝑧,𝑦_𝑛)≤1/𝑛. Then, for all 𝑛≥𝑛₀, we have 𝑣𝑥_𝑛,𝑦_𝑛≤𝑣𝑥_𝑛,𝑧+𝑣𝑧,𝑦_𝑛≤𝑞^𝑛−𝑛01−𝑞𝑓𝑥^𝑛0^1−𝑘+1𝑛.(2.16) Therefore, from (2.15), (2.16), we can find the sequences {𝑎_𝑛} and {𝑏_𝑛} in [0,∞) converging to zero, such that 𝑣(𝑥_𝑛,𝑦_𝑛)≤𝑎_𝑛 and 𝑣(𝑥_𝑛,𝑧)≤𝑏_𝑛 for any 𝑛∈ℕ; then by Lemma 1.8  𝑦_𝑛→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,𝜀l∈im(s0u,p_𝑟→𝑡+1),𝑡^𝑘≤𝛼(𝑡),foreach𝑡∈[0,𝜀],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 {𝑥_𝑛}^∞𝑛=0 of 𝑇 in 𝑋 such that 𝛼𝑣𝑥_𝑛,𝑥_𝑛+1𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑓𝑥_𝑛,(2.19)𝑓𝑥_𝑛+1≤𝛽𝑣𝑥_𝑛,𝑥_𝑛+1𝑣𝑥_𝑛,𝑥_𝑛+1,(2.20) for all 𝑛≥0, which imply that 𝑓𝑥_𝑛+1≤𝛽𝑣𝑥^𝑛,𝑥_𝑛+1𝛼𝑣𝑥_𝑛,𝑥_𝑛+1𝑓𝑥^𝑛.(2.21) Thus, {𝑓(𝑥_𝑛)} is a decreasing sequence, and so there exists 𝜃≥0 such that 𝑓(𝑥_𝑛)→𝜃. From (2.19) and (2.20), we have 𝑣𝑥_𝑛+1,𝑥_𝑛+2≤𝑓𝑥^𝑛+1𝛼𝑣𝑥_𝑛+1,𝑥_𝑛+2≤𝛽𝑣𝑥^𝑛,𝑥_𝑛+1𝛼𝑣𝑥_𝑛+1,𝑥_𝑛+2𝑣𝑥^𝑛,𝑥_𝑛+1,(2.22) for all 𝑛≥0. Now we clime that {𝑣(𝑥_𝑛,𝑥_𝑛+1)} is a nonincreasing sequence. Suppose not. Then there exists 𝑛₀≥0 such that 𝑣(𝑥_𝑛0+1,𝑥_𝑛0+2)>𝑣(𝑥_𝑛0,𝑥_𝑛0+1). Since 𝛼 is nondecreasing, then from (2.22), we get that 𝑣𝑥_𝑛0,𝑥_𝑛0+1<𝑣𝑥_𝑛0+1,𝑥_𝑛0+2≤𝛽𝑣𝑥^𝑛0,𝑥_𝑛0+1𝛼𝑣𝑥_𝑛0+1,𝑥_𝑛0+2𝑣𝑥^𝑛0≤𝛽𝑣𝑥_𝑛0,𝑥_𝑛0+1,𝑥_𝑛0+1𝛼𝑣𝑥_𝑛0,𝑥_𝑛0+1𝑣𝑥^𝑛0<𝑣𝑥_𝑛0,𝑥_𝑛0+1,𝑥_𝑛0+1,(2.23) which is a contradiction. Then, {𝑣(𝑥_𝑛,𝑥_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 𝑠∈(0,1) and 𝑛₀̂∈ℕ such that 𝑓𝑥_𝑛+1≤𝑠^𝑛−𝑛̂₀₊₁𝑓𝑥𝑛₀̂,∀𝑛≥𝑛⁰̂,(2.24) thus 𝜃=0. On the other hand, since 𝛼 is nondecreasing, then by (2.19), we have 𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑓𝑥^𝑛𝛼𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑓𝑥^𝑛𝛼𝑓𝑥_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 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 𝛽(𝑡∃)𝑘<,𝜀𝛼∈(𝑡)(,0,1),limsu𝑡^𝑘p_𝑟→𝑡+≤𝛽(𝑡),foreach𝑡∈[0,𝜀],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 {𝑥_𝑛}^∞𝑛=0 of 𝑇 in 𝑋 such that (2.19), (2.20), (2.21), and (2.22) hold. Then, {𝑓(𝑥_𝑛} is a decreasing sequence and so there exists 𝜃≥0 such that 𝑓(𝑥_𝑛)→𝜃. Now we clime that {𝑣(𝑥_𝑛,𝑥_𝑛+1)} is a nonincreasing sequence. Suppose not. Then there exists 𝑛₀≥0 such that 𝑣(𝑥_𝑛0+1,𝑥_𝑛0+2)>𝑣(𝑥_𝑛0,𝑥_𝑛0+1). Since one of 𝛼 and 𝛽 is nondecreasing, it follows from (2.22) that 𝑣𝑥_𝑛0,𝑥_𝑛0+1<𝑣𝑥_𝑛0+1,𝑥_𝑛0+2≤𝛽𝑣𝑥^𝑛0,𝑥_𝑛0+1𝛼𝑣𝑥_𝑛0+1,𝑥_𝑛0+2𝑣𝑥^𝑛0≤max𝛽𝑣𝑥_𝑛0,𝑥_𝑛0+1,𝑥_𝑛0+1𝛼𝑣𝑥_𝑛0,𝑥_𝑛0+1,𝛽𝑣𝑥^𝑛0+1,𝑥_𝑛0+2𝛼𝑣𝑥_𝑛0+1,𝑥_𝑛0+2𝑣𝑥^𝑛0<𝑣𝑥_𝑛0,𝑥_𝑛0+1,𝑥_𝑛0+1,(2.28) which is a contradiction. Thus {𝑣(𝑥_𝑛,𝑥_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 𝑠∈(0,1) and 𝑛₀̂∈ℕ such that 𝑓𝑥_𝑛+1≤𝑠^𝑛−𝑛̂₀₊₁𝑓𝑥𝑛₀̂,∀𝑛≥𝑛⁰̂,(2.29) thus 𝜃=0. If 𝛼 is nondecreasing, then from assumption (ii) and (2.19), we have 𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑓𝑥^𝑛𝛼𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑓𝑥^𝑛𝛼𝑓𝑥_𝑛≤𝑓𝑥^𝑛𝛽𝑓𝑥_𝑛,(2.30) and if 𝛽 is nondecreasing then from (2.19) and (2.20), we have 𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑓𝑥^𝑛𝛽𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑓𝑥^𝑛𝛽𝑓𝑥_𝑛,(2.31) for all 𝑛≥0. Therefore, in all cases we have shown 𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑓𝑥^𝑛𝛽𝑓𝑥_𝑛,∀𝑛≥0,(2.32) and so, by following as in the proof of Theorem 2.1, we can take 𝑛₀∈ℕ such that 𝑣𝑥_𝑛,𝑥_𝑛+1≤𝑞^𝑛−𝑛0𝑓𝑥_𝑛0^1−𝑘,(2.33) for all 𝑛≥𝑛₀, where 𝑞=𝑠^1−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 𝑥_𝑛=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𝑥_𝑛=⎧⎪⎨⎪⎩𝑥⁰𝑥if𝑛=0,₁𝑥if𝑛=1,₁𝑥if𝑛∈𝑛∈ℕ_𝑒∶𝑛≥2_𝑛+1,,𝑥_𝑛2if𝑛∈𝑛∈ℕ_𝑜∶𝑛>2.(2.34) It is easy to verify that 𝑓𝑥_𝑛=𝑣𝑥_𝑛,𝑇𝑥_𝑛=⎧⎪⎨⎪⎩011−1𝑛iiff𝑛𝑛∈=0,𝑛1,∈ℕ^𝑒∶𝑛≥2,𝑛−𝑛+11if𝑛∈𝑛∈ℕ^𝑜∶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,∞),𝛽(𝑡)=𝑡𝑡0if𝑡iiff∈𝑡𝑡∈∈(0,[[101),,1∞,)),.(2.36) Since 𝛽(𝑡)𝛼(𝑡)=√0𝑡iiff𝑡𝑡∈∈[[01,,1∞)),,(2.37) then, we have limsup_𝑟→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=𝑥_𝑛, 𝑛≥2, if 𝑛∈ℕ_𝑒, there exists 𝑦=𝑥₁∈𝑇(𝑥) satisfying 𝛼(𝑓(𝑥))𝑣(𝑥,𝑦)=𝛼1−1𝑓(𝑦)=0<1−1𝑛1−1𝑛<1−1𝑛=𝑓(𝑥),𝑛1−1𝑛=𝛽(𝑓(𝑥))𝑣(𝑥,𝑦),(2.40) and, if 𝑛∈ℕ_𝑜, there exists 𝑦=𝑥_𝑛2∈𝑇(𝑥) satisfying 𝛼(𝑓(𝑥))𝑣(𝑥,𝑦)=1√𝑛(𝑛+1)𝑛𝑛−²1𝑓(𝑦)=1<𝑛(𝑛1+1)=𝑓(𝑥),𝑛²𝑛²+1<𝑛(𝑛1+1)𝑛𝑛−²1=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∈ℕ_𝑜, such that for 𝑥=𝑥_𝑛, if 𝑦=𝑥_𝑛2∈𝑇(𝑥), we have 𝑓(𝑥)=1𝑛(𝑛+1)<√𝛼𝑛−1𝑛²≤√𝜑(𝑓(𝑥))𝑣(𝑥,𝑦),(2.42) and if 𝑦=𝑥_𝑛+1∈𝑇(𝑥), we have 𝑣(𝑦,𝑇(𝑦))=1−1𝑛+1>𝑛(𝑛1+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𝑥_𝑛=⎧⎪⎨⎪⎩𝑥⁰𝑥if𝑛=0,₀𝑥,𝑥₁if𝑛=1,₁𝑥if𝑛∈𝑛∈ℕ_𝑒∶𝑛≥2_𝑛4−1,,𝑥_𝑛3if𝑛∈𝑛∈ℕ_𝑜∶𝑛>2.(2.44) Clearly, 𝑓𝑥_𝑛=𝑣𝑥_𝑛,𝑇𝑥_𝑛=⎧⎪⎨⎪⎩011iiff𝑛𝑛=∈0,𝑛1,∈ℕ^𝑒∶𝑛≥2,𝑛⁴−1if𝑛∈𝑛∈ℕ^𝑜∶𝑛>2(2.45) is lower semi-continuous in 𝑋. Define 𝛼(𝑡)∶[0,∞)→(0,1] and 𝛽(𝑡)∶[0,∞)→[0,1) by 𝛼(𝑡)=𝑡1if𝑡∈{0}∪[1,∞),_1/3𝛽(𝑡)=⎧⎪⎨⎪⎩𝑡_5/6if𝑡∈(0,1),if𝑡∈0,112,2^5/6if𝑡∈12,∞.(2.46) Note that 𝛽(𝑡) is nondecreasing and 𝑡^5/6≤𝛽(𝑡), for each 𝑡∈[0,(1/2)]. Since 𝛽(𝑡)𝛼(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝑡^1/2if𝑡∈0,112,2^5/6𝑡^−1/3if𝑡∈112,1,2^5/6if𝑡∈[1,∞),(2.47) then, limsup_𝑟→𝑡+𝛽(𝑟)𝛼(𝑟)<1,foreach𝑡∈[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=𝑥_𝑛, 𝑛≥2, if 𝑛∈ℕ_𝑒, there exists 𝑦=𝑥₁∈𝑇(𝑥) satisfying 𝛼𝑓((𝑣𝑦)(𝑥=,𝑦0))<𝑣𝛽(𝑥(,𝑣𝑦()𝑥=,𝑦1))=𝑣(𝑓𝑥,(𝑥𝑦)),,(2.50) and, if 𝑛∈ℕ_𝑜, there exists 𝑦=𝑥_𝑛3∈𝑇(𝑥) satisfying 𝛼(𝑣(𝑥,𝑦))𝑣(𝑥,𝑦)=1𝑓(𝑦)=1𝑛𝑛1³<𝑛⁴1−1=𝑓(𝑥),𝑛¹²−1<𝑛1³^5/6𝑛1₃=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.