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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 143959, 18 pages
http://dx.doi.org/10.1155/2011/143959
Research Article

Fixed Point Theorems for Nonlinear Contractions in Menger Spaces

Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea

Received 14 May 2011; Accepted 24 June 2011

Academic Editor: H. B. Thompson

Copyright © 2011 Jeong Sheok Ume. 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

The main purpose of this paper is to introduce a new class of Ćirić-type contraction and to present some fixed point theorems for this mapping as well as for Caristi-type contraction. Several examples are given to show that our results are proper extension of many known results.

1. Introduction

Probabilistic metric space has been introduced and studied in 1942 by Menger in USA [1], and since then the theory of probabilistic metric spaces has developed in many directions [2]. The idea of Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to the situation when we do not know exactly the distance between two points, we know only probabilistic of metric spaces to be well adapted for the investigation of physiological thresholds and physical quantities particularly in connections with both string and 𝐸 infinity which were introduced and studied by a well-known scientific hero, El Naschie [35].

Ćirić’s fixed point theorem [6] and Caristi’s fixed point theorem [7] have many applications in nonlinear analysis. These theorems are extended by several authors, see [816] and the references therein.

In this paper, we introduce a new class of Ćirić-type contraction and present some fixed point theorems for this mapping as well as for Caristi-type contraction. Several examples are given to show that our results are proper extension of many known results.

2. Preliminaries

Throughout this paper, we denote by 𝑁 the set of all positive integers, by 𝑍+ the set of all nonnegative integers, by 𝑅 the set of all real numbers, and by 𝑅+ the set of all nonnegative real numbers. We shall recall some definitions and lemmas related to Menger space.

Definition 2.1. A mapping 𝐹𝑅𝑅+ is called a distribution if it is nondecreasing left continuous with inf{𝐹(𝑡)𝑡𝑅}=0 and sup{𝐹(𝑡)𝑡𝑅}=1. We will denote by 𝐿 the set of all distribution functions. The specific distribution function 𝐻𝑅𝑅+ is defined by 𝐻(𝑡)=0,𝑡01,𝑡>0.(2.1)

Definition 2.2 (see [17]). Probabilistic metric space (PM-space) is an ordered pair (𝑋,𝐹), where 𝑋 is an abstract set of elements, and 𝐹𝑋×𝑋𝐿 is defined by (𝑝,𝑞)𝐹𝑝,𝑞, where {𝐹𝑝,𝑞𝑝,𝑞𝑋}𝐿, where the functions 𝐹𝑝,𝑞 satisfied the following:(a)𝐹𝑝,𝑞(𝑥)=1 for all 𝑥>0 if and only if 𝑝=𝑞; (b)𝐹𝑝,𝑞(0)=0; (c)𝐹𝑝,𝑞=𝐹𝑞,𝑝; (d)𝐹𝑝,𝑞(𝑥)=1 and 𝐹𝑞,𝑟(𝑦)=1, then 𝐹𝑝,𝑟(𝑥+𝑦)=1.

Definition 2.3. A mapping 𝑡[0,1]×[0,1][0,1] is called a 𝑡-norm if(e)𝑡(0,0)=0 and 𝑡(𝑎,1)=𝑎 for all 𝑎[0,1]; (f)𝑡(𝑎,𝑏)=𝑡(𝑏,𝑎) for all 𝑎,𝑏[0,1]; (g)𝑡(𝑎,𝑏)𝑡(𝑐,𝑑) for all 𝑎,𝑏,𝑐,𝑑[0,1] with 𝑎𝑐 and 𝑏𝑑; (h)𝑡(𝑡(𝑎,𝑏),𝑐)=𝑡(𝑎,𝑡(𝑐,𝑑)) for all 𝑎,𝑏,𝑐[0,1].

Definition 2.4. Menger space is a triplet (𝑋,𝐹,𝑡), where (𝑋,𝐹) is PM space and 𝑡 is a 𝑡 norm such that for all 𝑝,𝑞,𝑟𝑋 and all 𝑥,𝑦0, 𝐹𝑝,𝑟𝐹(𝑥+𝑦)𝑡𝑝,𝑞(𝑥),𝐹𝑞,𝑟(𝑦).(2.2)

Definition 2.5 (see [17]). Let (𝑋,𝐹,𝑡) be a Menger space.(1)A sequence {𝑝𝑛} in 𝑋 is said to converge to a point 𝑝 in 𝑋 (written as 𝑝𝑛𝑝) if for every 𝜀>0 and 𝜆>0, there exists a positive integer 𝑀(𝜀,𝜆) such that 𝐹𝑝𝑛,𝑝(𝜀)>1𝜆 for all 𝑛𝑀(𝜀,𝜆).(2)A sequence {𝑝𝑛} in 𝑋 is said to be Cauchy if for each 𝜀>0 and 𝜆>0, there is a positive integer 𝑀(𝜀,𝜆) such that 𝐹𝑝𝑛,𝑝𝑚(𝜀)1𝜆 for all 𝑛,𝑚𝑁 with 𝑛,𝑚𝑀(𝜀,𝜆).(3)A Menger space (𝑋,𝐹,𝑡) is said to be complete if every Cauchy sequence in 𝑋 is converged to a point in 𝑋.

Definition 2.6 (see [18]). 𝑡-norm 𝑡 is said to be of 𝐻 type if a family of functions {𝑡𝑛(𝑎)}𝑛=1 is equicontinuous at 𝑎=1, that is, for any 𝜀(0,1), there exists 𝛿(0,1), such that 𝑎>1𝛿 and 𝑛1 imply 𝑡𝑛(𝑎)>1𝜀. The 𝑡-norm 𝑡=min is a trivial example of 𝑡-norm of 𝐻 type, but there are 𝑡-norms of 𝐻 type with 𝑡-norm min (see, e.g., Hadzić [19]).

Definition 2.7. Let (𝑋,𝐹,𝑡) be a Menger space, and let 𝑇𝑋𝑋 be a selfmapping. For each 𝑝𝑋, 𝑥>0 and 𝑛𝑁, let 𝑀𝐹(𝑝,𝑥,𝑛)=min𝑇𝑘𝑝,𝑇𝑙𝑝(𝑥)𝑘,𝑙𝑛and𝑘,𝑙𝑍+,𝑀1𝐹(𝑝,𝑥,𝑛)=min𝑇𝑘𝑝,𝑇𝑙𝑝,𝑀(𝑥)𝑘,𝑙𝑛and𝑘,𝑙𝑁2𝐹(𝑝,𝑥,𝑛)=min𝑝,𝑇𝑙𝑝,𝑇(𝑥)𝑙𝑛and𝑙𝑁𝑂(𝑝,𝑛)=𝑘𝑝𝑘𝑛and𝑘𝑍+,𝑇𝑂(𝑝,)=𝑘𝑝𝑘𝑍+,(2.3) where it is understood that 𝑇0𝑝=𝑝.
A Menger space (𝑋,𝐹,𝑡) is said to be 𝑇 orbitally complete if and only if every Cauchy sequence which is contained in 𝑂(𝑝,) for some 𝑝𝑋 converges in 𝑋.

From Definition 2.1~  Definition 2.5, we can prove easily the following lemmas.

Lemma 2.8 (see [20]). Let (𝑋,𝑑) be a metric space, and let 𝑇𝑋𝑋 be a selfmapping on 𝑋. Define 𝐹𝑋×𝑋𝐿 by []𝐹(𝑝,𝑞)(𝑥)𝐹𝑝,𝑞(𝑥)=𝐻(𝑥𝑑(𝑝,𝑞))(2.4) for all 𝑝,𝑞𝑋 and 𝑥𝑅, where {𝐹𝑝,𝑞𝑝,𝑞𝑋}𝐿. Suppose that 𝑡-norm 𝑡[0,1]×[0,1][0,1] is defined by 𝑡(𝑎,𝑏)=min{𝑎,𝑏} for all 𝑎,𝑏[0,1]. Then, (1)(𝑋,𝐹,𝑡) is a Menger space; (2)If (𝑋,𝑑) is 𝑇 orbitally complete, then (𝑋,𝐹,𝑡) is 𝑇 orbitally complete.
Menger space generated by a metric is called the induced Menger space.

Lemma 2.9. In a Menger space (𝑋,𝐹,𝑡), if 𝑡(𝑥,𝑥)𝑥 for all 𝑥[0,1], then 𝑡(𝑎,𝑏)=min{𝑎,𝑏} for all 𝑎,𝑏[0,1].

3. Ćirić-Type Fixed Point Theorems

In 2010, Ćirić proved the following theorem.

Theorem A (see Ćirić [9], 2010). Let (𝑋,𝐹,𝑡) be a complete Menger space under a 𝑡-norm 𝑡 of 𝐻 type. Let 𝑇𝑋𝑋 be a generalized 𝜑-probabilistic contraction, that is, 𝐹𝑇𝑝,𝑇𝑞(𝜑(𝑥))𝐹𝑝,𝑞(𝑥)(1) for all 𝑝,𝑞𝑋 and 𝑥>0, where 𝜑[0,)[0,) satisfies the following conditions: 𝜑(0)=0, 𝜑(𝑥)<𝑥, and lim𝑟𝑥+inf𝜑(𝑟)<𝑥 for each 𝑥>0. Then, 𝑇 has a unique fixed point 𝑢𝑋 and {𝑇𝑛(𝑝)} converges to 𝑢 for each 𝑝𝑋.

Definition 3.1. Let (𝑋,𝐹,𝑡) be a Menger space with 𝑡(𝑥,𝑥)𝑥 for all 𝑥[0,1], and let 𝑇𝑋𝑋 be a mapping of 𝑋. We will say that 𝑇 is Ćirić-type-generalized contraction if 𝐹𝑇𝑝,𝑇𝑞𝐹(𝜑(𝑥))min𝑝,𝑞(𝑥),𝐹𝑝,𝑇𝑝(𝑥),𝐹𝑞,𝑇𝑞(𝑥),𝐹𝑝,𝑇𝑞(𝑥),𝐹𝑞,𝑇𝑝(𝑥)(2) for all 𝑝,𝑞𝑋 and 𝑥>0, where 𝜑[0,)[0,) is a mapping and for all 𝑝,𝑞𝑋 and 𝑥𝑅, 𝐹𝑝,𝑞(𝑥) is the same as in Definition 2.2.
It is clear that (*1) implies (*2).

The following example shows that a Ćirić-type-generalized contraction need not be a generalized 𝜑-probabilistic contraction.

Example 3.2. Let 𝑋=[0,), 𝑇𝑋𝑋 be defined by 𝑇𝑥=𝑥+1, and let 𝜑[0,)[0,) be defined by 𝑥𝜑(𝑥)=1+𝑥,0𝑥1,𝑥1,1<𝑥.(3.1) For each 𝑝,𝑞𝑋, let 𝐹𝑝,𝑞𝑅𝑅+ be defined by 𝐹𝑝,𝑞(𝑥)=𝐻(𝑥𝑑(𝑝,𝑞)) for all 𝑥𝑅, where 𝐻 is the same as in Definition 2.1, and 𝑑 is a usual metric on 𝑅×𝑅. Then, since max{|𝑝𝑞1|,|𝑞𝑝1|}=|𝑝𝑞|+1 for all 𝑝,𝑞𝑋, we have 𝐹𝑇𝑝,𝑇𝑞(𝜑(𝑥))min{𝐹𝑝,𝑇𝑞(𝑥),𝐹𝑞,𝑇𝑝(𝑥)} for all 𝑝,𝑞𝑋 and 𝑥>0. Thus, 𝐹𝑇𝑝,𝑇𝑞𝐹(𝜑(𝑥))min𝑝,𝑞(𝑥),𝐹𝑝,𝑇𝑝(𝑥),𝐹𝑞,𝑇𝑞(𝑥),𝐹𝑝,𝑇𝑞(𝑥),𝐹𝑞,𝑇𝑝(𝑥)(3.2) for all 𝑝,𝑞𝑋 and 𝑥>0, which satisfies (*2). If 𝑥=2,𝑝=0 and 𝑞=3/2, then 𝐹𝑇0,𝑇3/2(𝜑(2))=0 and 𝐹0,3/2(2)=1. Thus, 𝐹𝑇0,𝑇3/2(𝜑(2))<𝐹0,3/2(2), which does not satisfy (*1).

In the next example, we shall show that there exists 𝑇 that does not satisfy (*2) with 𝜑(𝑡)=𝑘𝑡,   0<𝑘<1.

Example 3.3. Let 𝑋=[0,), 𝑇𝑋𝑋 be defined by 𝑇𝑥=2𝑥 and let 𝜑[0,)[0,) be defined by 𝜑(𝑥)=𝑘𝑥, 0<𝑘<1. For each 𝑝,𝑞𝑋, let 𝐹𝑝,𝑞𝑅𝑅+ be defined by 𝐹𝑝,𝑞(𝑥)=𝐻(𝑥𝑑(𝑝,𝑞)) for all 𝑥𝑅, where 𝐻 is the same as in Definition 2.1, and 𝑑 is a usual metric on 𝑅×𝑅. If 𝑝=0, 𝑞=1 and 𝑥=2/𝑘>0, then for simple calculations, 𝐹𝑇0,𝑇1(𝜑(2/𝑘))=0 and 𝐹min0,12𝑘,𝐹0,𝑇02𝑘,𝐹1,𝑇12𝑘,𝐹0,𝑇12𝑘,𝐹1,𝑇02𝑘=1.(3.3) Therefore, for 𝑝=0, 𝑞=1, and 𝑥=2/𝑘>0, the mapping 𝑇 does not satisfy (*2). Thus, we showed that there exists 𝑇 that does not satisfy (*2) with 𝜑(𝑡)=𝑘𝑡,0<𝑘<1.

Definition 3.4. Let (𝑋,𝐹,𝑡) be a Menger space with 𝑡(𝑥,𝑥)𝑥 for all 𝑥[0,1] and let 𝑇𝑋𝑋 be a self mapping of 𝑋. We will say that 𝑇 is a mapping of type 𝕌 if there exists 𝑝𝑋 such that 𝐹𝑝,𝑇𝑝𝐹((𝐼𝜑)(𝑥))inf𝑇𝑘𝑝,𝑇𝑙𝑝(𝑥)𝑘,𝑙𝑍+𝑥>0,(3) where 𝜑[0,)[0,) is a mapping, and 𝐼[0,)[0,) is identity mapping.

The following example shows that 𝑇 has no fixed point, even though 𝑇 satisfies (*2) and (*3).

Example 3.5. Let 𝑋=[0,), 𝑇𝑋𝑋 be defined by 𝑇𝑥=𝑥+4, and let 𝜑[0,)[0,) be defined by 𝑥𝜑(𝑥)=2,0𝑥4,𝑥2,4<𝑥.(3.4) For each 𝑝,𝑞𝑋, let 𝐹𝑝,𝑞𝑅𝑅+ be defined by 𝐹𝑝,𝑞(𝑥)=𝐻(𝑥𝑑(𝑝,𝑞)) for all 𝑥𝑅, where 𝐻 is the same as in Definition 2.1, and 𝑑 is a usual metric on 𝑅×𝑅. Then, since max{|𝑝𝑞4|,|𝑞𝑝4|}=|𝑝𝑞|+4 for all 𝑝,𝑞𝑋, we have 𝐹𝑇𝑝,𝑇𝑞(𝜑(𝑥))min{𝐹𝑝,𝑇𝑞(𝑥),𝐹𝑞,𝑇𝑝(𝑥)} for all 𝑝,𝑞𝑋 and 𝑥>0. Thus, 𝐹𝑇𝑝,𝑇𝑞𝐹(𝜑(𝑥))min𝑝,𝑞(𝑥),𝐹𝑝,𝑇𝑝(𝑥),𝐹𝑞,𝑇𝑞(𝑥),𝐹𝑝,𝑇𝑞(𝑥),𝐹𝑞,𝑇𝑝(𝑥)(3.5) for all 𝑝,𝑞𝑋 and 𝑥>0, which implies (*2). It is easy to see that there exists 𝑝=1𝑋 such that 𝐹𝑝,𝑇𝑝𝐹((𝐼𝜑)(𝑥))inf𝑇𝑘𝑝,𝑇𝑙𝑝(𝑥)𝑘,𝑙𝑍+𝑥>0,(3.6) which implies (*3). But 𝑇 has no fixed point.

Remark 3.6. It follows from Example 3.5 that 𝑇 must satisfy (*2), (*3), and other conditions in order to have fixed point of 𝑇.
The following is Ćirić-type fixed point theorem which is generalization of Ćirić’s fixed point theorems [6, 9].

Theorem 3.7. Let (𝑋,𝐹,𝑡) be a Menger space with continuous 𝑡 norm and 𝑡(𝑥,𝑥)𝑥 for all 𝑥[0,1], let 𝑇 be a self-mapping on 𝑋 satisfying (*2) and (*3). Let (𝑋,𝐹,𝑡) be 𝑇 orbitally complete. Suppose that 𝜑𝑅+𝑅+ is a mapping such that (i)𝜑(𝑥)<𝑥 for all 𝑥>0 and lim𝑥(𝐼𝜑)(𝑥)=, where 𝐼𝑅+𝑅+ is identity mapping, (ii)𝜑 and 𝐼𝜑 are strictly increasing and onto mappings, (iii)lim𝑛𝜑𝑛(𝑥)= for each 𝑥>0, where 𝜑𝑛 is 𝑛-time repeated composition of 𝜑1 with itself. Then, (a)𝑀(𝑝,𝑥,𝑛)=min{𝑀1(𝑝,𝑥,𝑛),𝑀2(𝑝,𝑥,𝑛)} for all 𝑝𝑋, 𝑥>0, and 𝑛𝑁, (b)𝑀(𝑝,𝑥,𝑛)=𝑀2(𝑝,𝑥,𝑛) for all 𝑝𝑋, 𝑥>0 and 𝑛𝑁, (c){𝑇𝑛𝑝} is Cauchy sequence for each 𝑝𝑈, where||𝐹𝑈=𝑝𝑋𝑝,𝑇𝑝𝐹((𝐼𝜑)(𝑥))inf𝑇𝑘𝑝,𝑇𝑙𝑝(𝑥)𝑘,𝑙𝑍+𝑥>0,(3.7)(d)𝑇 has a unique fixed point in 𝑋.

Proof. Let 𝑝𝑋, 𝑥>0 and 𝑛𝑁 be arbitrary. By Definition 2.2 and Definition 2.7, clearly, we have 𝑀(𝑝,𝑥,𝑛)=min{𝑀1(𝑝,𝑥,𝑛),𝑀2(𝑝,𝑥,𝑛)} which implies (a). From (i), (ii), and (*2), we have 𝑀1𝐹(𝑝,𝜑(𝑥),𝑛)=min𝑇𝑘𝑝,𝑇𝑙𝑝𝐹(𝜑(𝑥))𝑘,𝑙𝑛and𝑘,𝑙𝑁=min𝑇𝑇𝑘1𝑝,𝑇𝑇𝑙1𝑝𝐹(𝜑(𝑥))𝑘,𝑙𝑛and𝑘,𝑙𝑁minmin𝑇𝑘1𝑝,𝑇𝑙1𝑝(𝑥),𝐹𝑇𝑘1𝑝,𝑇𝑘𝑝(𝑥),𝐹𝑇𝑙1𝑝,𝑇𝑙𝑝𝐹(𝑥),𝑇𝑘1𝑝,𝑇𝑙𝑝(𝑥),𝐹𝑇𝑙1𝑝,𝑇𝑘𝑝(𝑥)𝑘,𝑙𝑛and𝑘,𝑙𝑁𝑀(𝑝,𝑥,𝑛).(3.8) By virtue of (i), (ii), (3.8), and (a), we obtain 𝑀1𝑀(𝑝,𝑥,𝑛)min1𝑝,𝜑1(𝑥),𝑛,𝑀2𝑝,𝜑1𝑀(𝑥),𝑛min1𝑝,𝜑1(𝑥),𝑛,𝑀2.(𝑝,𝑥,𝑛)(3.9) By repeating application of (3.9), we have 𝑀1𝑀(𝑝,𝑥,𝑛)min1(𝑝,𝜑𝑚(𝑥),𝑛),𝑀2(𝑝,𝑥,𝑛).(3.10) Since 𝑀1(𝑝,𝜑𝑚(𝑥),𝑛) converges to 1 when 𝑚 tends to infinity, it follows that 𝑀1(𝑝,𝑥,𝑛)𝑀2(𝑝,𝑥,𝑛).(3.11) On account of (a) and (3.11), we have 𝑀(𝑝,𝑥,𝑛)=𝑀2(𝑝,𝑥,𝑛) for 𝑝𝑋, 𝑥>0 and 𝑛𝑁, which implies (b). To prove (c), let 𝑛 and 𝑚 be two positive integers with 𝑛<𝑚, 𝑝𝑈, and let 𝑥 be any positive real number. By (*2) and (b), we have 𝐹𝑇𝑛𝑝,𝑇𝑚𝑝(𝜑(𝑥))=𝐹𝑇𝑇𝑛1𝑝,𝑇𝑇𝑚1𝑝𝐹(𝜑(𝑥))min𝑇𝑛1𝑝,𝑇𝑚1𝑝(𝑥),𝐹𝑇𝑛1𝑝,𝑇𝑛𝑝(𝑥),𝐹𝑇𝑚1𝑝,𝑇𝑚𝑝(𝐹𝑥),𝑇𝑛1𝑝,𝑇𝑚𝑝(𝑥),𝐹𝑇𝑛𝑝,𝑇𝑚1𝑝(𝐹𝑥)min𝑇𝑖𝑇𝑛1𝑝,𝑇𝑗𝑇𝑛1𝑝(𝑥)𝑖,𝑗𝑚𝑛+1,𝑖,𝑗𝑍+𝐹=min𝑇𝑛1𝑝,𝑇𝑙𝑇𝑛1𝑝,𝐹(𝑥)𝑙𝑚𝑛+1,𝑙𝑁(3.12)min𝑇𝑛1𝑝,𝑇𝑙𝑇𝑛1𝑝𝐹(𝜑(𝑥))𝑙𝑚𝑛+1,𝑙𝑁=min𝑇𝑇𝑛2𝑝,𝑇𝑇𝑙+𝑛2𝑝𝐹(𝜑(𝑥))𝑙𝑚𝑛+1,𝑙𝑁minmin𝑇𝑛2𝑝,𝑇𝑙+𝑛2𝑝(𝑥),𝐹𝑇𝑛2𝑝,𝑇𝑛1𝑝(𝑥),𝐹𝑇𝑙+𝑛2𝑝,𝑇𝑙+𝑛1𝑝𝐹(𝑥),𝑇𝑛2𝑝,𝑇𝑙+𝑛1𝑝(𝑥),𝐹𝑇𝑛1𝑝,𝑇𝑙+𝑛2𝑝𝐹(𝑥)𝑙𝑚𝑛+1,𝑙𝑁minmin𝑇𝑖𝑇𝑛2𝑝,𝑇𝑗𝑇𝑛2𝑝(𝑥)0𝑖,𝑗𝑙+1,𝑖,𝑗𝑍+𝐹𝑙𝑚𝑛+1,𝑙𝑁min𝑇𝑖𝑇𝑛2𝑝,𝑇𝑗𝑇𝑛2𝑝(𝑥)0𝑖,𝑗𝑚𝑛+2,𝑖,𝑗𝑍+𝐹=min𝑇𝑛2𝑝,𝑇𝑙𝑇𝑛2𝑝(.𝑥)𝑙𝑚𝑛+2,𝑙𝑁(3.13) In terms of (i), (ii), and (3.13), we get 𝐹𝑇𝑛𝑝,𝑇𝑚𝑝𝐹(𝑥)min𝑇𝑛1𝑝,𝑇𝑙𝑇𝑛1𝑝𝜑1𝐹(𝑥)𝑙𝑚𝑛+1,𝑙𝑁min𝑇𝑛2𝑝,𝑇𝑙𝑇𝑛2𝑝𝜑2.(𝑥)𝑙𝑚𝑛+2,𝑙𝑁(3.14) By repeating the same method as in (3.13) and (3.14), we have 𝐹𝑇𝑛𝑝,𝑇𝑚𝑝𝐹(𝑥)min𝑝,𝑇𝑙𝑝(𝜑𝑛𝐹(𝑥))𝑙𝑚,𝑙𝑁inf𝑝,𝑇𝑙𝑝(𝜑𝑛.(𝑥))𝑙𝑁(3.15) On account of (iii), (*3), (b), (3.15), and Definition 2.2, we have lim𝑛𝐹inf𝑝,𝑇𝑙𝑝(𝜑𝑛(𝑥))𝑙𝑁lim𝑛𝐹inf𝑇𝑘𝑝,𝑇𝑙𝑝(𝜑𝑛(𝑥))𝑘,𝑙𝑍+lim𝑛𝐹𝑝,𝑇𝑝((1𝜑)(𝜑𝑛(𝑥)))=1for𝑥>0.(3.16) It follows from (3.15) and (3.16) that lim𝑛𝐹𝑇𝑛𝑝,𝑇𝑚𝑝(𝑥)=1for𝑝𝑈,𝑥>0.(3.17) This implies that {𝑇𝑛𝑝} is a Cauchy sequence for 𝑝𝑈. This is the proof of (c). Since 𝑋 is 𝑇 orbitally complete, and {𝑇𝑛𝑝} is a Cauchy sequence for 𝑝𝑈, {𝑇𝑛𝑝} has a limit 𝑢 in 𝑋. To prove (d), let us consider the following inequality; 𝐹𝑇𝑢,𝑇𝑛+1𝑝𝐹(𝜑(𝑥))min𝑢,𝑇𝑛𝑝(𝑥),𝐹𝑢,𝑇𝑢(𝑥),𝐹𝑇𝑛𝑝,𝑇𝑛+1𝑝(𝑥),𝐹𝑢,𝑇𝑛+1𝑝(𝑥),𝐹𝑇𝑛𝑝,𝑇𝑢(𝑥)𝑥>0.(3.18) Since lim𝑛𝑇𝑛𝑝=𝑢, from (3.18), we get 𝐹𝑇𝑢,𝑢(𝜑(𝑥))𝐹𝑇𝑢,𝑢(𝑥)𝑥>0.(3.19) In terms of (3.19), (i), (ii), (iii), and Definition 2.2, we deduce that 𝑇𝑢=𝑢, that is, 𝑢 is a fixed point of 𝑇. To prove uniqueness of a fixed point of 𝑇, let 𝑤 be another fixed point of 𝑇. Then 𝑇𝑤=𝑤. Putting 𝑝=𝑢 and 𝑞=𝑤 in (*2), we get 𝐹𝑇𝑢,𝑇𝑤(𝜑(𝑥))=𝐹𝑢,𝑤𝐹(𝜑(𝑥))min𝑢,𝑤(𝑥),𝐹𝑢,𝑇𝑢(𝑥),𝐹𝑤,𝑇𝑤(𝑥),𝐹𝑢,𝑇𝑤(𝑥),𝐹𝑤,𝑇𝑢(𝑥)=𝐹𝑢,𝑤(𝑥)𝑥>0,(3.20) which gives 𝑢=𝑤. Thus, 𝑢 is a unique fixed point of 𝑇, which implies (d).

Corollary 3.8 (see [6]). let 𝑇 be a quasicontraction on a metric space (𝑋,𝑑), that is, there exists 𝑘(0,1) such that 𝑑(𝑇𝑝,𝑇𝑞)𝑘max{𝑑(𝑝,𝑞),𝑑(𝑝,𝑇𝑝),𝑑(𝑞,𝑇𝑞),𝑑(𝑝,𝑇𝑞),𝑑(𝑞,𝑇𝑝)},𝑝,𝑞𝑋.(3.21) Suppose that 𝑋 is 𝑇 orbitally complete. Then, 𝑇 has a unique fixed point in 𝑋.

Proof. Define 𝐹𝑋×𝑋𝐿 by 𝐹(𝑝,𝑞)=𝐹𝑝,𝑞 for all 𝑝,𝑞𝑋 and 𝐹𝑝,𝑞(𝑥)=𝐻(𝑥𝑑(𝑝,𝑞)) for all 𝑝,𝑞𝑋 and 𝑥𝑅, where 𝐻 and 𝐿 are the same as in Definition 2.1. Let 𝑡[0,1]×[0,1][0,1] be defined by 𝑡(𝑎,𝑏)=min{𝑎,𝑏} for all 𝑎,𝑏[0,1]. Let 𝜑[0,)[0,) be defined by 𝜑(𝑥)=𝑘𝑥,0<𝑘<1.(3.22) Then from Lemma 2.8, (𝑋,𝐹,𝑡) is a 𝑇 orbitally complete Menger space. It follows from (3.21), Lemma 2.8, Lemma 2.9, and [6, lemma 2] that all conditions of Theorem 3.7 are satisfied. Therefore, 𝑇 has a unique fixed point in 𝑋.

Now we shall present an example to show that all conditions of Theorem 3.7 are satisfied but condition (3.21) in Corollary 3.8 and condition (*1) in Theorem A are not satisfied.

Example 3.9. Let 𝑋=[1,1] be the closed interval with the usual metric and 𝑇𝑋𝑋 and 𝜑𝑅+𝑅+ be mappings defined as follows: 𝑝𝑇𝑝=0,1𝑝<0,41+𝑝,0𝑝<57or81<𝑝1,416𝑝,57𝑝8,𝑡(3.23)𝜑(𝑡)=𝑡287,0𝑡1,8𝑡,1<𝑡.(3.24) Define 𝐹𝑝,𝑞𝑅𝑅+ by 𝐹𝑝,𝑞||||(𝑥)=𝐻𝑥𝑝𝑞(3.25) for all 𝑝,𝑞𝑋 and 𝑥𝑅, where 𝐹𝑝,𝑞 and 𝐻 are the same as in Definition 2.1 and Definition 2.2. Let 𝑡[0,1]×[0,1][0,1] be defined by 𝑡(𝑎,𝑏)=min{𝑎,𝑏} for all 𝑎,𝑏[0,1]. Then from Lemma 2.8, (𝑋,𝐹,𝑡) is a 𝑇 orbitally complete Menger space, and 𝜑 is continuous function on 𝑅+ which satisfy (i), (ii), and (iii). Clearly 𝑝=0𝑋 satisfies (*3). To show that condition (*2) is satisfied, we need to consider several possible cases.

Case 1. Let 𝑝,𝑞[1,0). Then ||||𝑑(𝑇𝑝,𝑇𝑞)=𝑇𝑝𝑇𝑞=0𝜑(𝑑(𝑝,𝑞)).(3.26)

Case 2. Let 𝑝[1,0) and 𝑞[0,4/5)(7/8,1]. Then ||||=||||=𝑞𝑑(𝑇𝑝,𝑇𝑞)=𝑇𝑝𝑇𝑞𝑇𝑞𝑞1+𝑞𝑞28||||=𝜑(𝑞)=𝜑𝑞𝑇𝑝=𝜑(𝑑(𝑞,𝑇𝑝)).(3.27)

Case 3. Let 𝑝[1,0) and 𝑞[4/5,7/8]. Then ||||=||||=1𝑑(𝑇𝑝,𝑇𝑞)=𝑇𝑝𝑇𝑞𝑇𝑞𝑞16𝑞𝑞28||||=𝜑(𝑞)=𝜑𝑞𝑇𝑝=𝜑(𝑑(𝑞,𝑇𝑝)).(3.28)

Case 4. Let 𝑝,𝑞[0,4/5)(7/8,1]. Then, by simple calculation, 𝑑||||=||||𝑝(𝑇𝑝,𝑇𝑞)=𝑇𝑝𝑇𝑞𝑞1+𝑝||||||||||||1+𝑞𝑝𝑞𝑝𝑞28||||=𝜑𝑝𝑞=𝜑(𝑑(𝑝,𝑞)).(3.29)

Case 5. Let 𝑝[0,4/5)(7/8,1] and 𝑞[4/5,7/8]. Then 𝑑||||=||||𝑝(𝑇𝑝,𝑇𝑞)=𝑇𝑝𝑇𝑞11+𝑝𝑞||||=𝑝16+11+𝑝𝑞1162+1×7168=71𝜑|||||||1128=𝜑(𝑑(𝑝,𝑞)),(3.30)(𝑑(𝑞,𝑇𝑞))=𝜑𝑞𝑇𝑞=𝜑𝑞𝑞|||16=𝜑17𝑞416𝜑5×17=1617×20143>16071.128(3.31) Thus, 𝑑(𝑇𝑝,𝑇𝑞)71<12817×20143160<𝜑(𝑑(𝑞,𝑇𝑞)).(3.32)

Case 6. Let 𝑝,𝑞[4/5,7/8]. Then, ||||=|||1𝑑(𝑇𝑝,𝑇𝑞)=𝑇𝑝𝑇𝑞𝑝116𝑞|||=116||||||||16𝑝𝑞𝜑𝑝𝑞=𝜑(𝑑(𝑝,𝑞)).(3.33) Hence, we obtain [],𝑑(𝑇𝑝,𝑇𝑞)𝜑(𝑀(𝑝,𝑞))𝑝,𝑞1,1(3.34) where 𝑀(𝑝,𝑞)=max{𝑑(𝑝,𝑞),𝑑(𝑝,𝑇𝑝),𝑑(𝑞,𝑇𝑞),𝑑(𝑝,𝑇𝑞),𝑑(𝑞,𝑇𝑝)}.(3.35) From (3.25) and (3.34), we have 𝐹𝑇𝑝,𝑇𝑞𝐹(𝜑(𝑥))min𝑝,𝑞(𝑥),𝐹𝑝,𝑇𝑝(𝑥),𝐹𝑞,𝑇𝑞(𝑥),𝐹𝑝,𝑇𝑞(𝑥),𝐹𝑞,𝑇𝑝(𝑥)(3.36) for all 𝑝,𝑞𝑋 and 𝑥>0, which implies (*2). Therefore, all hypotheses of Example 3.9 satisfy that of Theorem 3.7. Hence, 𝑇 has a unique fixed point 0 in 𝑋. On the other hand, let 𝑘(0,1) be any fixed number. Then, for 𝑝=0𝑋 and 𝑞𝑋 with 0<𝑞<min{4/5,(1/𝑘)1}, we have 1𝑘max{𝑑(𝑝,𝑞),𝑑(𝑝,𝑇𝑝),𝑑(𝑞,𝑇𝑞),𝑑(𝑝,𝑇𝑞),𝑑(𝑞,𝑇𝑝)}=𝑘𝑑(𝑝,𝑞)<𝑞1+𝑞𝑑(𝑝,𝑞)=1+𝑞=𝑑(0,𝑇𝑞)=𝑑(𝑇𝑝,𝑇𝑞).(3.37) Thus, 𝑑(𝑇𝑝,𝑇𝑞)>𝑘max{𝑑(𝑝,𝑞),𝑑(𝑝,𝑇𝑝),𝑑(𝑞,𝑇𝑞),𝑑(𝑝,𝑇𝑞),𝑑(𝑞,𝑇𝑝)},(3.38) which shows that 𝑇 does not satisfy (3.21).

Finally, in above Example 3.9, we shall show that 𝑇 does not satisfy (*1). In fact, we need to show that there are 𝑝,𝑞𝑋 and 𝑥>0 such that 𝐹𝑇𝑝,𝑇𝑞(𝜑(𝑥))<𝐹𝑝,𝑞(𝑥). Let 𝑝=4/5, 𝑞=(4/5)(1/100), and 𝑥=41168×+2079179.(3.39) Then, 1/100<𝑥<1,  𝜑(𝑥)|𝑇𝑝𝑇𝑞|=0, and 𝑥|𝑝𝑞|>0. Hence, 𝐻(𝜑(𝑥)|𝑇𝑝𝑇𝑞|)=0 and 𝐻(𝑥|𝑝𝑞|)=1. Thus, 𝐹𝑇𝑝,𝑇𝑞(𝜑(𝑥))<𝐹𝑝,𝑞(𝑥). Therefore, Theorem 3.7 is a proper extension of Theorem A and Corollary 3.8.

4. Caristi-Type Fixed Point Theorems

The following Lemma plays an important role to prove Caristi-type fixed point theorem which is generalization of Caristi’s fixed point theorem [7].

Lemma 4.1. Let (𝑋,𝐹,𝑡) be a Menger space with continuous 𝑡 norm and 𝑡(𝑥,𝑥)𝑥 for all 𝑥[0,1], and let 𝐻𝑅𝑅+ be the same as in Definition 2.1. Suppose that 𝑔𝑋×𝑋𝑅+ and 𝑓𝑋(,] are mappings satisfying the following conditions: (1)𝑔(𝑢,𝑤)𝑔(𝑢,𝑣)+𝑔(𝑣,𝑤) for all 𝑢,𝑣,𝑤𝑋, (2)𝑓 is a proper function which is bounded from below, (3)for any sequence {𝑢𝑛}𝑛=1 in 𝑋 satisfying lim𝑛𝑔𝑢sup𝑛,𝑢𝑚𝑚>𝑛=0,(4.1)there exists 𝑢0𝑋 such that lim𝑛𝑢𝑛=𝑢0,𝑔𝑢𝑛,𝑢0liminf𝑚𝑔𝑢𝑛,𝑢𝑚,𝑓𝑢0liminf𝑛𝑓𝑢𝑛,(4.2)(4)for any 𝑢𝑋 with inf𝑣𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤𝑋{𝑢} such that 𝐹𝑔(𝑢,𝑤)𝑓(𝑢)𝑓(𝑤),(4.3)𝑢,𝑤1(𝑘𝑥)𝐻𝑥𝑘[]𝑓(𝑢)𝑓(𝑤),𝑥>0andsome𝑘(0,1),(4.4)where 𝐿 is the set of all distribution functions,𝐹𝑝,𝑞𝑝,𝑞𝑋𝐿,𝐹𝑋×𝑋𝐿(4.5)is defined by 𝐹(𝑝,𝑞)=𝐹𝑝,𝑞 for all 𝑝,𝑞𝑋. Then, there exists 𝑤0𝑋 such that inf𝑣𝑋𝑓(𝑣)=𝑓(𝑤0).

Proof. Suppose that inf𝑣𝑋𝑓(𝑣)<𝑓(𝑢)𝑢𝑋.(4.6) For each 𝑢𝑋, let 𝑆(𝑢)={𝑤𝑋𝑔(𝑢,𝑤)𝑓(𝑢)𝑓(𝑤)}.(4.7) Then, by (4), (4.6), and (4.7) 𝑆(𝑢) is nonempty for each 𝑢𝑋. From (1) and (4.7), we obtain 𝑆(𝑤)𝑆(𝑢),foreach𝑤𝑆(𝑢).(4.8) For each 𝑢𝑋, let 𝑐(𝑢)=inf{𝑓(𝑤)𝑤𝑆(𝑢)}.(4.9) Choose 𝑢𝑋 with 𝑓(𝑢)<. Then from (4.8) and (4.9), there exists a sequence {𝑢𝑛}𝑛=1 in 𝑋 such that for all 𝑛𝑁𝑢1=𝑢,𝑢𝑛+1𝑢𝑆𝑛𝑢,𝑆𝑛𝑢𝑆(𝑢),𝑓𝑛+1𝑢<𝑐𝑛+1𝑛.(4.10) In virtue of (4.7), (4.9), and (4.10), we have 𝑔𝑢𝑛,𝑢𝑛+1𝑢𝑓𝑛𝑢𝑓𝑛+1𝑓𝑢,(4.11)𝑛+11𝑛𝑢<𝑐𝑛𝑢𝑓𝑛+1(4.12) for all 𝑛𝑁. In view of (4.11), {𝑓(𝑢𝑛)}𝑛=1 is a nonincreasing sequence of real numbers, and so it converges to some 𝛽𝑅. Therefore, due to (4.12), 𝛽=lim𝑛𝑐𝑢𝑛=lim𝑛𝑓𝑢𝑛.(4.13) Combining (1) and (4.11), we get 𝑔𝑢𝑛,𝑢𝑚𝑢𝑓𝑛𝑢𝑓𝑚𝑛,𝑚𝑁with𝑛<𝑚.(4.14) On account of (4.13) and (4.14), we have lim𝑛𝑔𝑢sup𝑛,𝑢𝑚𝑚>𝑛=0.(4.15) Thus, by virtue of (3), (4.13), (4.14), and (4.15), there exists 𝑢0𝑋 such that lim𝑛𝑢𝑛=𝑢0,𝑓𝑢(4.16)0lim𝑛𝑓𝑢𝑛𝑔𝑢=𝛽,(4.17)𝑛,𝑢0liminf𝑚𝑔𝑢𝑛,𝑢𝑚.(4.18) Using (4.14), (4.17), and (4.18), we obtain 𝑓𝑢0𝛽=limsup𝑚𝑓𝑢𝑚limsup𝑚𝑓𝑢𝑛𝑢𝑔𝑛,𝑢𝑚𝑢=𝑓𝑛+limsup𝑚𝑢𝑔𝑛,𝑢𝑚𝑢=𝑓𝑛liminf𝑚𝑔𝑢𝑛,𝑢𝑚𝑢𝑓𝑛𝑢𝑔𝑛,𝑢0.(4.19) Combining (4.7), (4.9), and (4.19), it follows that 𝑢0𝑆(𝑢𝑛) and, hence, 𝑐𝑢𝑛𝑢𝑓0,𝑛𝑁.(4.20) Taking the limit in inequality (4.20) when 𝑛 tends to infinity, we have lim𝑛𝑐𝑢𝑛𝑢𝑓0.(4.21) In terms of (4.13), (4.17), and (4.21), we deduce that 𝑢𝛽=𝑓0.(4.22) On the other hand, from (4), (4.6), (4.7), and (4.16), we have the following property: thereexists𝑤1𝑢𝑋0satisfying𝑤1𝑢𝑆0.(4.23) In terms of (4.7), (4.8), (4.9), (4.20), and (4.23), we deduce that 𝑤1𝑢𝑆𝑛𝑐𝑢,𝑛𝑁,(4.24)𝑛𝑤𝑓1.(4.25) In view of (4.7), (4.13), (4.22), (4.23), and (4.25), we have 𝑤𝛽=𝑓1.(4.26) Due to (4), (4.22), (4.23), and (4.26), we have the following: 𝐹𝑢0,𝑤11(𝑘𝑥)𝐻𝑥𝑘𝑓𝑢0𝑤𝑓1=𝐻(𝑥)𝐹𝑢0,𝑤1(𝑥)𝑥>0.(4.27) By virtue of (4.27), we obtain 𝐹𝑢0,𝑤1(𝑥)𝐹𝑢0,𝑤1𝑘1𝑥,𝑥>0.(4.28) By repeating the application of inequality (4.28), we get 𝐹𝑢0,𝑤1(𝑥)𝐹𝑢0,𝑤1(𝑘𝑚𝑥),𝑥>0,𝑚𝑁.(4.29) In terms of (4.29), we deduce that 𝐹𝑢0,𝑤1(𝑘𝑚𝑥) converges to 1 as 𝑚 and, hence, 𝐹𝑢0,𝑤1(𝑥)=1,𝑥>0.(4.30) From (4.30) and Definition 2.2, we have 𝑢0=𝑤1. This is a contradiction from (4.23). Therefore, there exists 𝑤0𝑋 such that inf𝑣𝑋𝑓𝑤(𝑣)=𝑓0.(4.31)

Theorem 4.2. Let (𝑋,𝑑) be a metric space and let 𝐻,𝑔,𝐿, and 𝑓 satisfy conditions (1), (2), and (3) in Lemma 4.1. Suppose that for any 𝑢𝑋 with inf𝑣𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤𝑋{𝑢} such that 1𝑔(𝑢,𝑤)𝑓(𝑢)𝑓(𝑤),(4.32)𝐻(𝑘𝑥𝑑(𝑢,𝑤))𝐻𝑥𝑘[]𝑓(𝑢)𝑓(𝑤)(4.33) for all 𝑥>0 and some 𝑘(0,1). Thus, there exists 𝑤0𝑋 such that inf𝑣𝑋𝑓𝑤(𝑣)=𝑓0.(4.34)

Proof. The proof follows from Lemma 4.1 by considering the induced Menger space (𝑋,𝐹,𝑡), where 𝑡(𝑎,𝑏)=min{𝑎,𝑏} and 𝐹𝑝,𝑞(𝑥)=𝐻(𝑥𝑑(𝑝,𝑞)),𝑝,𝑞𝑋,𝑥𝑅.(4.35)

Corollary 4.3 (see [12]). Let (𝑋,𝑑) be a complete metric space, and let 𝑓𝑋(,] be a proper lower semicontinuous function, bounded from below. Assume that for any 𝑢𝑋 with inf𝑣𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤𝑋 with 𝑤𝑢 and 𝑓(𝑤)+𝑑(𝑢,𝑤)𝑓(𝑢). Then there exists 𝑤𝑜𝑋 such that inf𝑣𝑋𝑓(𝑣)=𝑓(𝑤0).

Proof. Let (𝑋,𝑑) be a complete metric space, and let 𝑔𝑋×𝑋𝑅+,𝐻𝑅𝑅+, 𝐹𝑝,𝑞𝑅[0.1](𝑝,𝑞𝑋), and 𝑡[0,1]×[0,1][0,1] be mappings such that 𝐹𝑔(𝑝,𝑞)=𝑑(𝑝,𝑞)𝑝,𝑞𝑋,𝐻(𝑥)=0if𝑥0,𝐻(𝑥)=1if𝑥>0,𝑝,𝑞[].(𝑥)=𝐻(𝑥𝑑(𝑝,𝑞))𝑝,𝑞𝑋,𝑥>0,𝑡(𝑎,𝑏)=min{𝑎,𝑏}𝑎,𝑏0,1(4.36) Then, all conditions of Corollary 4.3 satisfy all conditions of Lemma 4.1. Therefore, result of Corollary 4.3 follows from Lemma 4.1.

The following example shows that Theorem 4.2 is more general than Corollary 4.3.

Example 4.4. Let 𝐻,𝐹, and 𝐿 be the same as in Theorem 4.2. Let 𝑋=[0,3] be the closed interval with the usual metric,  𝑘=1/2, and let 𝑔𝑋×𝑋𝑅+,  𝐹𝑋×𝑋𝐿, and 𝑓𝑋(,] be mappings defined as follows: 𝑔(𝑢,𝑤)=𝑤𝑢,𝑤𝑋,(4.37)[]𝐹(𝑢,𝑤)(𝑥)𝐹𝑢,𝑤(𝑥)=𝐻(𝑥|𝑢𝑤|)𝑢,𝑤𝑋,𝑥>0,(4.38)𝑓(𝑢)=0,0𝑢13𝑢+7,1<𝑢<22𝑢2,2𝑢3.(4.39) Then, for any 𝑢𝑋 with inf𝑣𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤=1𝑋{𝑢} such that 1𝑔(𝑢,𝑤)𝑓(𝑢)𝑓(𝑤),(4.40)𝐻(𝑘𝑥|𝑢𝑤|)𝐻𝑥𝑘[]𝑓(𝑢)𝑓(𝑤)𝑥>0.(4.41) Let {𝑢𝑛} be a sequence of 𝑋 such that lim𝑛𝑢𝑛=0. Then, clearly conditions (1), (2), (3), and (4) in Lemma 4.1 are satisfied. Thus there exists 0𝑋 such that inf𝑣𝑋𝑓(𝑣)=𝑓(0).(4.42) Therefore, all conditions of Theorem 4.2 are satisfied. Since 𝑓 is not lower semicontinuous at 𝑢=2, and 𝑔 is not metric, Corollary 4.3 cannot be applicable.

Theorem 4.5. Suppose that condition (4) in Lemma 4.1 is replaced with the following conditions.
For self-mapping 𝑇 on 𝑋, 𝐹𝑔(𝑢,𝑇𝑢)𝑓(𝑢)𝑓(𝑇𝑢)𝑢𝑋,𝑢,𝑇𝑢1(𝑘𝑥)𝐻𝑥𝑘[]𝑓(𝑢)𝑓(𝑇𝑢)𝑢𝑋,𝑥>0andsome𝑘(0,1).(4.43) Then, 𝑇 has a fixed point in 𝑋.

Proof. Suppose 𝑢𝑇𝑢 for all 𝑢𝑋. Then by Lemma 4.1, there exists 𝑤0𝑋 such that 𝑓𝑤0=inf𝑣𝑋𝑓(𝑣).(4.44) Since 𝑔(𝑤0,𝑇𝑤0)+𝑓(𝑇𝑤0)𝑓(𝑤0), we have 𝑓𝑇𝑤0𝑤=𝑓0=inf𝑣𝑋𝑓𝐹(𝑣),(4.45)𝑤0,𝑇𝑤01(𝑘𝑥)𝐻𝑥𝑘𝑓𝑤0𝑓𝑇𝑤0=𝐻(𝑥)𝐹𝑤0,𝑇𝑤0(𝑥),𝑥>0andsome𝑘(0,1).(4.46) By the same method as in proof of Lemma 4.1, it follows that 𝑤0=𝑇𝑤0. But this contradicts our assumption that 𝑢𝑇𝑢 for all 𝑢𝑋. The proof of Theorem 4.5 is complete.

Theorem 4.6. Let (𝑋,𝑑) be a metric space, 𝑇𝑋𝑋, and let 𝐻,𝑔, and 𝑓 be satisfied conditions (1), (2), and (3) in Lemma 4.1. Suppose that 1𝑔(𝑢,𝑇𝑢)𝑓(𝑢)𝑓(𝑇𝑢)𝑢𝑋,(4.47)𝐻(𝑘𝑥𝑑(𝑢,𝑇𝑢))𝐻𝑥𝑘[]𝑓(𝑢)𝑓(𝑇𝑢)𝑥>0andsome𝑘(0,1).(4.48) Then, 𝑇 has a fixed point in 𝑋.

Proof. By method similar to Theorem 4.2, the result of Theorem 4.6 follows.

Corollary 4.7 (see [7]). Let (𝑋,𝑑) be a complete metric space, and let 𝑓𝑋(,] is a proper lower semicontinuous function bounded from below. Let 𝑇 be a mapping from 𝑋 into itself such that 𝑑(𝑢,𝑇𝑢)𝑓(𝑢)𝑓(𝑇𝑢)𝑢𝑋.(4.49) Then, 𝑇 has a fixed point in 𝑋.

Proof. By the same method as in Corollary 4.3, the result of Corollary 4.7 follows.

The following example shows that all conditions of Theorem 4.6 are satisfied but not that of Corollary 4.7.

Example 4.8. Let 𝑋,𝐹,𝐻,𝑔, and 𝑘 be the same as in Example 4.4. Suppose that 𝑓𝑋(,] and 𝑇𝑋𝑋 are mappings defined as follows: 3𝑓(𝑥)=21𝑥,if0𝑥<2,2𝑥,if2𝑥3,𝑇𝑥=2𝑥𝑥𝑋.(4.50) Then clearly, all conditions of Theorem 4.6 are satisfied but not that of Corollary 4.7, since 𝑓 is not lower semicontinuous at 𝑥=2.

Natural question arises from Example 3.5.

Question 1. Whether Theorem 3.7 would remain true if (i), (ii), and (iii) in Theorem 3.7 are substituted by some suitable conditions?

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