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Abstract and Applied Analysis
Volume 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

Jeong Sheok Ume

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 [3–5].

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 [8–16] 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 𝐹∶𝑅→𝑅+ is called a distribution if it is nondecreasing left continuous with inf{𝐹(𝑡)∶𝑡∈𝑅}=0 and sup{𝐹(𝑡)∶𝑡∈𝑅}=1. We will denote by 𝐿 the set of all distribution functions. The specific distribution function 𝐻∶𝑅→𝑅+ is defined by 𝐻(𝑡)=0,𝑡≤01,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 𝐹∶𝑋×𝑋→𝐿 is defined by (𝑝,𝑞)→𝐹𝑝,𝑞, where {𝐹𝑝,𝑞∶𝑝,𝑞∈𝑋}⊆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 𝑎≤𝑐 and 𝑏≤𝑑; (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,𝑟∈𝑋 and all 𝑥,𝑦≥0, 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 𝑝𝑛→𝑝) if for every 𝜀>0 and 𝜆>0, there exists a positive integer 𝑀(𝜀,𝜆) such that 𝐹𝑝𝑛,𝑝(𝜀)>1−𝜆 for all 𝑛≥𝑀(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)≥1−𝜆 for all 𝑛,𝑚∈𝑁 with 𝑛,𝑚≥𝑀(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>1−𝛿 and 𝑛≥1 imply 𝑡𝑛(𝑎)>1−𝜀. 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 𝑇∶𝑋→𝑋 be a selfmapping. For each 𝑝∈𝑋, 𝑥>0 and 𝑛∈𝑁, let 𝑀𝐹(𝑝,𝑥,𝑛)=min𝑇𝑘𝑝,𝑇𝑙𝑝(𝑥)∶𝑘,𝑙≤𝑛and𝑘,𝑙∈𝑍+,𝑀1𝐹(𝑝,𝑥,𝑛)=min𝑇𝑘𝑝,𝑇𝑙𝑝,𝑀(𝑥)∶𝑘,𝑙≤𝑛and𝑘,𝑙∈𝑁2𝐹(𝑝,𝑥,𝑛)=min𝑝,𝑇𝑙𝑝,𝑇(𝑥)∶𝑙≤𝑛and𝑙∈𝑁𝑂(𝑝,𝑛)=𝑘𝑝∶𝑘≤𝑛and𝑘∈𝑍+,𝑇𝑂(𝑝,∞)=𝑘𝑝∶𝑘∈𝑍+,(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 𝑝∈𝑋 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 𝑇∶𝑋→𝑋 be a selfmapping on 𝑋. Define 𝐹∶𝑋×𝑋→𝐿 by []𝐹(𝑝,𝑞)(𝑥)≡𝐹𝑝,𝑞(𝑥)=𝐻(𝑥−𝑑(𝑝,𝑞))(2.4) for all 𝑝,𝑞∈𝑋 and 𝑥∈𝑅, where {𝐹𝑝,𝑞∶𝑝,𝑞∈𝑋}⊆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 𝑇∶𝑋→𝑋 be a generalized 𝜑-probabilistic contraction, that is, 𝐹𝑇𝑝,𝑇𝑞(𝜑(𝑥))≥𝐹𝑝,𝑞(𝑥)(∗1) for all 𝑝,𝑞∈𝑋 and 𝑥>0, where 𝜑∶[0,∞)→[0,∞) satisfies the following conditions: 𝜑(0)=0, 𝜑(𝑥)<𝑥, and lim𝑟→𝑥+inf𝜑(𝑟)<𝑥 for each 𝑥>0. Then, 𝑇 has a unique fixed point 𝑢∈𝑋 and {𝑇𝑛(𝑝)} converges to 𝑢 for each 𝑝∈𝑋.

Definition 3.1. Let (𝑋,𝐹,𝑡) be a Menger space with 𝑡(𝑥,𝑥)≥𝑥 for all 𝑥∈[0,1], and let 𝑇∶𝑋→𝑋 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,𝑞∈𝑋 and 𝑥>0, where 𝜑∶[0,∞)→[0,∞) is a mapping and for all 𝑝,𝑞∈𝑋 and 𝑥∈𝑅, 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,∞), 𝑇∶𝑋→𝑋 be defined by 𝑇𝑥=𝑥+1, and let 𝜑∶[0,∞)→[0,∞) be defined by ⎧⎪⎨⎪⎩𝑥𝜑(𝑥)=1+𝑥,0≤𝑥≤1,𝑥−1,1<𝑥.(3.1) For each 𝑝,𝑞∈𝑋, let 𝐹𝑝,𝑞∶𝑅→𝑅+ be defined by 𝐹𝑝,𝑞(𝑥)=𝐻(𝑥−𝑑(𝑝,𝑞)) for all 𝑥∈𝑅, where 𝐻 is the same as in Definition 2.1, and 𝑑 is a usual metric on 𝑅×𝑅. Then, since max{|𝑝−𝑞−1|,|𝑞−𝑝−1|}=|𝑝−𝑞|+1 for all 𝑝,𝑞∈𝑋, we have 𝐹𝑇𝑝,𝑇𝑞(𝜑(𝑥))≥min{𝐹𝑝,𝑇𝑞(𝑥),𝐹𝑞,𝑇𝑝(𝑥)} for all 𝑝,𝑞∈𝑋 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,𝑞∈𝑋 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,∞), 𝑇∶𝑋→𝑋 be defined by 𝑇𝑥=2𝑥 and let 𝜑∶[0,∞)→[0,∞) be defined by 𝜑(𝑥)=𝑘𝑥, 0<𝑘<1. For each 𝑝,𝑞∈𝑋, let 𝐹𝑝,𝑞∶𝑅→𝑅+ be defined by 𝐹𝑝,𝑞(𝑥)=𝐻(𝑥−𝑑(𝑝,𝑞)) for all 𝑥∈𝑅, 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 𝑇∶𝑋→𝑋 be a self mapping of 𝑋. We will say that 𝑇 is a mapping of type 𝕌 if there exists 𝑝∈𝑋 such that 𝐹𝑝,𝑇𝑝𝐹((𝐼−𝜑)(𝑥))≤inf𝑇𝑘𝑝,𝑇𝑙𝑝(𝑥)∶𝑘,𝑙∈𝑍+∀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,∞), 𝑇∶𝑋→𝑋 be defined by 𝑇𝑥=𝑥+4, and let 𝜑∶[0,∞)→[0,∞) be defined by ⎧⎪⎨⎪⎩𝑥𝜑(𝑥)=2,0≤𝑥≤4,𝑥−2,4<𝑥.(3.4) For each 𝑝,𝑞∈𝑋, let 𝐹𝑝,𝑞∶𝑅→𝑅+ be defined by 𝐹𝑝,𝑞(𝑥)=𝐻(𝑥−𝑑(𝑝,𝑞)) for all 𝑥∈𝑅, where 𝐻 is the same as in Definition 2.1, and 𝑑 is a usual metric on 𝑅×𝑅. Then, since max{|𝑝−𝑞−4|,|𝑞−𝑝−4|}=|𝑝−𝑞|+4 for all 𝑝,𝑞∈𝑋, we have 𝐹𝑇𝑝,𝑇𝑞(𝜑(𝑥))≥min{𝐹𝑝,𝑇𝑞(𝑥),𝐹𝑞,𝑇𝑝(𝑥)} for all 𝑝,𝑞∈𝑋 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,𝑞∈𝑋 and 𝑥>0, which implies (*2). It is easy to see that there exists 𝑝=1∈𝑋 such that 𝐹𝑝,𝑇𝑝𝐹((𝐼−𝜑)(𝑥))≤inf𝑇𝑘𝑝,𝑇𝑙𝑝(𝑥)∶𝑘,𝑙∈𝑍+∀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 𝜑∶𝑅+→𝑅+ is a mapping such that (i)𝜑(𝑥)<𝑥 for all 𝑥>0 and lim𝑥→∞(𝐼−𝜑)(𝑥)=∞, where 𝐼∶𝑅+→𝑅+ is identity mapping, (ii)𝜑 and 𝐼−𝜑 are strictly increasing and onto mappings, (iii)lim𝑛→∞𝜑−𝑛(𝑥)=∞ for each 𝑥>0, where 𝜑−𝑛 is 𝑛-time repeated composition of 𝜑−1 with itself. Then, (a)𝑀(𝑝,𝑥,𝑛)=min{𝑀1(𝑝,𝑥,𝑛),𝑀2(𝑝,𝑥,𝑛)} for all 𝑝∈𝑋, 𝑥>0, and 𝑛∈𝑁, (b)𝑀(𝑝,𝑥,𝑛)=𝑀2(𝑝,𝑥,𝑛) for all 𝑝∈𝑋, 𝑥>0 and 𝑛∈𝑁, (c){𝑇𝑛𝑝} is Cauchy sequence for each 𝑝∈𝑈, where||𝐹𝑈=𝑝∈𝑋𝑝,𝑇𝑝𝐹((𝐼−𝜑)(𝑥))≤inf𝑇𝑘𝑝,𝑇𝑙𝑝(𝑥)∶𝑘,𝑙∈𝑍+∀𝑥>0,(3.7)(d)𝑇 has a unique fixed point in 𝑋.

Proof. Let 𝑝∈𝑋, 𝑥>0 and 𝑛∈𝑁 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,𝑙≤𝑛and𝑘,𝑙∈𝑁=min𝑇𝑇𝑘−1𝑝,𝑇𝑇𝑙−1𝑝𝐹(𝜑(𝑥))∣𝑘,𝑙≤𝑛and𝑘,𝑙∈𝑁≥minmin𝑇𝑘−1𝑝,𝑇𝑙−1𝑝(𝑥),𝐹𝑇𝑘−1𝑝,𝑇𝑘𝑝(𝑥),𝐹𝑇𝑙−1𝑝,𝑇𝑙𝑝𝐹(𝑥),𝑇𝑘−1𝑝,𝑇𝑙𝑝(𝑥),𝐹𝑇𝑙−1𝑝,𝑇𝑘𝑝(𝑥)∶𝑘,𝑙≤𝑛and𝑘,𝑙∈𝑁≥𝑀(𝑝,𝑥,𝑛).(3.8) By virtue of (i), (ii), (3.8), and (a), we obtain 𝑀1𝑀(𝑝,𝑥,𝑛)≥min1𝑝,𝜑−1(𝑥),𝑛,𝑀2𝑝,𝜑−1𝑀(𝑥),𝑛≥min1𝑝,𝜑−1(𝑥),𝑛,𝑀2.(𝑝,𝑥,𝑛)(3.9) By repeating application of (3.9), we have 𝑀1𝑀(𝑝,𝑥,𝑛)≥min1(𝑝,𝜑−𝑚(𝑥),𝑛),𝑀2(𝑝,𝑥,𝑛).(3.10) Since 𝑀1(𝑝,𝜑−𝑚(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 𝑝∈𝑋, 𝑥>0 and 𝑛∈𝑁, which implies (b). To prove (c), let 𝑛 and 𝑚 be two positive integers with 𝑛<𝑚, 𝑝∈𝑈, and let 𝑥 be any positive real number. By (*2) and (b), we have 𝐹𝑇𝑛𝑝,𝑇𝑚𝑝(𝜑(𝑥))=𝐹𝑇𝑇𝑛−1𝑝,𝑇𝑇𝑚−1𝑝𝐹(𝜑(𝑥))≥min𝑇𝑛−1𝑝,𝑇𝑚−1𝑝(𝑥),𝐹𝑇𝑛−1𝑝,𝑇𝑛𝑝(𝑥),𝐹𝑇𝑚−1𝑝,𝑇𝑚𝑝(𝐹𝑥),𝑇𝑛−1𝑝,𝑇𝑚𝑝(𝑥),𝐹𝑇𝑛𝑝,𝑇𝑚−1𝑝(𝐹𝑥)≥min𝑇𝑖𝑇𝑛−1𝑝,𝑇𝑗𝑇𝑛−1𝑝(𝑥)∶𝑖,𝑗≤𝑚−𝑛+1,𝑖,𝑗∈𝑍+𝐹=min𝑇𝑛−1𝑝,𝑇𝑙𝑇𝑛−1𝑝,𝐹(𝑥)∶𝑙≤𝑚−𝑛+1,𝑙∈𝑁(3.12)min𝑇𝑛−1𝑝,𝑇𝑙𝑇𝑛−1𝑝𝐹(𝜑(𝑥))∶𝑙≤𝑚−𝑛+1,𝑙∈𝑁=min𝑇𝑇𝑛−2𝑝,𝑇𝑇𝑙+𝑛−2𝑝𝐹(𝜑(𝑥))∶𝑙≤𝑚−𝑛+1,𝑙∈𝑁≥minmin𝑇𝑛−2𝑝,𝑇𝑙+𝑛−2𝑝(𝑥),𝐹𝑇𝑛−2𝑝,𝑇𝑛−1𝑝(𝑥),𝐹𝑇𝑙+𝑛−2𝑝,𝑇𝑙+𝑛−1𝑝𝐹(𝑥),𝑇𝑛−2𝑝,𝑇𝑙+𝑛−1𝑝(𝑥),𝐹𝑇𝑛−1𝑝,𝑇𝑙+𝑛−2𝑝𝐹(𝑥)∶𝑙≤𝑚−𝑛+1,𝑙∈𝑁≥minmin𝑇𝑖𝑇𝑛−2𝑝,𝑇𝑗𝑇𝑛−2𝑝(𝑥)∶0≤𝑖,𝑗≤𝑙+1,𝑖,𝑗∈𝑍+𝐹∶𝑙≤𝑚−𝑛+1,𝑙∈𝑁≥min𝑇𝑖𝑇𝑛−2𝑝,𝑇𝑗𝑇𝑛−2𝑝(𝑥)∶0≤𝑖,𝑗≤𝑚−𝑛+2,𝑖,𝑗∈𝑍+𝐹=min𝑇𝑛−2𝑝,𝑇𝑙𝑇𝑛−2𝑝(.𝑥)∶𝑙≤𝑚−𝑛+2,𝑙∈𝑁(3.13) In terms of (i), (ii), and (3.13), we get 𝐹𝑇𝑛𝑝,𝑇𝑚𝑝𝐹(𝑥)≥min𝑇𝑛−1𝑝,𝑇𝑙𝑇𝑛−1𝑝𝜑−1𝐹(𝑥)∶𝑙≤𝑚−𝑛+1,𝑙∈𝑁≥min𝑇𝑛−2𝑝,𝑇𝑙𝑇𝑛−2𝑝𝜑−2.(𝑥)∶𝑙≤𝑚−𝑛+2,𝑙∈𝑁(3.14) By repeating the same method as in (3.13) and (3.14), we have 𝐹𝑇𝑛𝑝,𝑇𝑚𝑝𝐹(𝑥)≥min𝑝,𝑇𝑙𝑝(𝜑−𝑛𝐹(𝑥))∶𝑙≤𝑚,𝑙∈𝑁≥inf𝑝,𝑇𝑙𝑝(𝜑−𝑛.(𝑥))∶𝑙∈𝑁(3.15) On account of (iii), (*3), (b), (3.15), and Definition 2.2, we have lim𝑛→∞𝐹inf𝑝,𝑇𝑙𝑝(𝜑−𝑛(𝑥))∶𝑙∈𝑁≥lim𝑛→∞𝐹inf𝑇𝑘𝑝,𝑇𝑙𝑝(𝜑−𝑛(𝑥))∶𝑘,𝑙∈𝑍+≥lim𝑛→∞𝐹𝑝,𝑇𝑝((1−𝜑)(𝜑−𝑛(𝑥)))=1for𝑥>0.(3.16) It follows from (3.15) and (3.16) that lim𝑛→∞𝐹𝑇𝑛𝑝,𝑇𝑚𝑝(𝑥)=1for𝑝∈𝑈,𝑥>0.(3.17) This implies that {𝑇𝑛𝑝} is a Cauchy sequence for 𝑝∈𝑈. This is the proof of (c). Since 𝑋 is 𝑇 orbitally complete, and {𝑇𝑛𝑝} is a Cauchy sequence for 𝑝∈𝑈, {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 lim𝑛→∞𝑇𝑛𝑝=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)≤𝑘⋅max{𝑑(𝑝,𝑞),𝑑(𝑝,𝑇𝑝),𝑑(𝑞,𝑇𝑞),𝑑(𝑝,𝑇𝑞),𝑑(𝑞,𝑇𝑝)},∀𝑝,𝑞∈𝑋.(3.21) Suppose that 𝑋 is 𝑇 orbitally complete. Then, 𝑇 has a unique fixed point in 𝑋.

Proof. Define 𝐹∶𝑋×𝑋→𝐿 by 𝐹(𝑝,𝑞)=𝐹𝑝,𝑞 for all 𝑝,𝑞∈𝑋 and 𝐹𝑝,𝑞(𝑥)=𝐻(𝑥−𝑑(𝑝,𝑞)) for all 𝑝,𝑞∈𝑋 and 𝑥∈𝑅, 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 𝑇∶𝑋→𝑋 and 𝜑∶𝑅+→𝑅+ be mappings defined as follows: ⎧⎪⎪⎨⎪⎪⎩𝑝𝑇𝑝=0,−1≤𝑝<0,41+𝑝,0≤𝑝<57or8−1<𝑝≤1,416𝑝,57≤𝑝≤8,⎧⎪⎨⎪⎩𝑡(3.23)𝜑(𝑡)=𝑡−287,0≤𝑡≤1,8𝑡,1<𝑡.(3.24) Define 𝐹𝑝,𝑞∶𝑅→𝑅+ by 𝐹𝑝,𝑞||||(𝑥)=𝐻𝑥−𝑝−𝑞(3.25) for all 𝑝,𝑞∈𝑋 and 𝑥∈𝑅, 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=0∈𝑋 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)=𝑇𝑝−𝑇𝑞=0≤𝜑(𝑑(𝑝,𝑞)).(3.26)

Case 2. Let 𝑝∈[−1,0) and 𝑞∈[0,4/5)∪(7/8,1]. Then ||||=||||=𝑞𝑑(𝑇𝑝,𝑇𝑞)=𝑇𝑝−𝑇𝑞𝑇𝑞𝑞1+𝑞≤𝑞−28||||=𝜑(𝑞)=𝜑𝑞−𝑇𝑝=𝜑(𝑑(𝑞,𝑇𝑝)).(3.27)

Case 3. Let 𝑝∈[−1,0) and 𝑞∈[4/5,7/8]. Then ||||=||||=1𝑑(𝑇𝑝,𝑇𝑞)=𝑇𝑝−𝑇𝑞𝑇𝑞𝑞16𝑞≤𝑞−28||||=𝜑(𝑞)=𝜑𝑞−𝑇𝑝=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)=𝑇𝑝−𝑇𝑞−𝑞1+𝑝||||≤||||−||||1+𝑞𝑝−𝑞𝑝−𝑞28||||=𝜑𝑝−𝑞=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)=𝑇𝑝−𝑇𝑞−−11+𝑝𝑞||||=𝑝16+11+𝑝𝑞≤1162+1×7168=71𝜑|||||||−1128=𝜑(𝑑(𝑝,𝑞)),(3.30)(𝑑(𝑞,𝑇𝑞))=𝜑𝑞−𝑇𝑞=𝜑𝑞−𝑞|||16=𝜑17𝑞416≥𝜑5×17=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)=𝑇𝑝−𝑇𝑞𝑝−−116𝑞|||=116||||||||16𝑝−𝑞≤𝜑𝑝−𝑞=𝜑(𝑑(𝑝,𝑞)).(3.33) Hence, we obtain [],𝑑(𝑇𝑝,𝑇𝑞)≤𝜑(𝑀(𝑝,𝑞))∀𝑝,𝑞∈−1,1(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,𝑞∈𝑋 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=0∈𝑋 and 𝑞∈𝑋 with 0<𝑞<min{4/5,(1/𝑘)−1}, we have 1𝑘⋅max{𝑑(𝑝,𝑞),𝑑(𝑝,𝑇𝑝),𝑑(𝑞,𝑇𝑞),𝑑(𝑝,𝑇𝑞),𝑑(𝑞,𝑇𝑝)}=𝑘⋅𝑑(𝑝,𝑞)<𝑞1+𝑞𝑑(𝑝,𝑞)=1+𝑞=𝑑(0,𝑇𝑞)=𝑑(𝑇𝑝,𝑇𝑞).(3.37) Thus, 𝑑(𝑇𝑝,𝑇𝑞)>𝑘⋅max{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,𝑞∈𝑋 and 𝑥>0 such that 𝐹𝑇𝑝,𝑇𝑞(𝜑(𝑥))<𝐹𝑝,𝑞(𝑥). Let 𝑝=4/5, 𝑞=(4/5)−(1/100), and 𝑥=4−116−8×+2079179.(3.39) Then, 1/100<𝑥<1,  𝜑(𝑥)−|𝑇𝑝−𝑇𝑞|=0, and 𝑥−|𝑝−𝑞|>0. Hence, 𝐻(𝜑(𝑥)−|𝑇𝑝−𝑇𝑞|)=0 and 𝐻(𝑥−|𝑝−𝑞|)=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 𝐻∶𝑅→𝑅+ be the same as in Definition 2.1. Suppose that 𝑔∶𝑋×𝑋→𝑅+ and 𝑓∶𝑋→(−∞,∞] are mappings satisfying the following conditions: (1)𝑔(𝑢,𝑤)≤𝑔(𝑢,𝑣)+𝑔(𝑣,𝑤) for all 𝑢,𝑣,𝑤∈𝑋, (2)𝑓 is a proper function which is bounded from below, (3)for any sequence {𝑢𝑛}∞𝑛=1 in 𝑋 satisfying lim𝑛→∞𝑔𝑢sup𝑛,𝑢𝑚∶𝑚>𝑛=0,(4.1)there exists 𝑢0∈𝑋 such that lim𝑛→∞𝑢𝑛=𝑢0,𝑔𝑢𝑛,𝑢0≤liminf𝑚→∞𝑔𝑢𝑛,𝑢𝑚,𝑓𝑢0≤liminf𝑛→∞𝑓𝑢𝑛,(4.2)(4)for any 𝑢∈𝑋 with inf𝑣∈𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤∈𝑋−{𝑢} such that 𝐹𝑔(𝑢,𝑤)≤𝑓(𝑢)−𝑓(𝑤),(4.3)𝑢,𝑤1(𝑘𝑥)≥𝐻𝑥−𝑘[]𝑓(𝑢)−𝑓(𝑤),∀𝑥>0andsome𝑘∈(0,1),(4.4)where 𝐿 is the set of all distribution functions,𝐹𝑝,𝑞∶𝑝,𝑞∈𝑋⊆𝐿,𝐹∶𝑋×𝑋→𝐿(4.5)is defined by 𝐹(𝑝,𝑞)=𝐹𝑝,𝑞 for all 𝑝,𝑞∈𝑋. Then, there exists 𝑤0∈𝑋 such that inf𝑣∈𝑋𝑓(𝑣)=𝑓(𝑤0).

Proof. Suppose that inf𝑣∈𝑋𝑓(𝑣)<𝑓(𝑢)∀𝑢∈𝑋.(4.6) For each 𝑢∈𝑋, let 𝑆(𝑢)={𝑤∈𝑋∣𝑔(𝑢,𝑤)≤𝑓(𝑢)−𝑓(𝑤)}.(4.7) Then, by (4), (4.6), and (4.7) 𝑆(𝑢) is nonempty for each 𝑢∈𝑋. From (1) and (4.7), we obtain 𝑆(𝑤)⊆𝑆(𝑢),foreach𝑤∈𝑆(𝑢).(4.8) For each 𝑢∈𝑋, let 𝑐(𝑢)=inf{𝑓(𝑤)∣𝑤∈𝑆(𝑢)}.(4.9) Choose 𝑢∈𝑋 with 𝑓(𝑢)<∞. Then from (4.8) and (4.9), there exists a sequence {𝑢𝑛}∞𝑛=1 in 𝑋 such that for all 𝑛∈𝑁𝑢1=𝑢,𝑢𝑛+1𝑢∈𝑆𝑛𝑢,𝑆𝑛𝑢⊆𝑆(𝑢),𝑓𝑛+1𝑢<𝑐𝑛+1𝑛.(4.10) In virtue of (4.7), (4.9), and (4.10), we have 𝑔𝑢𝑛,𝑢𝑛+1𝑢≤𝑓𝑛𝑢−𝑓𝑛+1𝑓𝑢,(4.11)𝑛+1−1𝑛𝑢<𝑐𝑛𝑢≤𝑓𝑛+1(4.12) for all 𝑛∈𝑁. In view of (4.11), {𝑓(𝑢𝑛)}∞𝑛=1 is a nonincreasing sequence of real numbers, and so it converges to some 𝛽∈𝑅. Therefore, due to (4.12), 𝛽=lim𝑛→∞𝑐𝑢𝑛=lim𝑛→∞𝑓𝑢𝑛.(4.13) Combining (1) and (4.11), we get 𝑔𝑢𝑛,𝑢𝑚𝑢≤𝑓𝑛𝑢−𝑓𝑚∀𝑛,𝑚∈𝑁with𝑛<𝑚.(4.14) On account of (4.13) and (4.14), we have lim𝑛→∞𝑔𝑢sup𝑛,𝑢𝑚∶𝑚>𝑛=0.(4.15) Thus, by virtue of (3), (4.13), (4.14), and (4.15), there exists 𝑢0∈𝑋 such that lim𝑛→∞𝑢𝑛=𝑢0,𝑓𝑢(4.16)0≤lim𝑛→∞𝑓𝑢𝑛𝑔𝑢=𝛽,(4.17)𝑛,𝑢0≤liminf𝑚→∞𝑔𝑢𝑛,𝑢𝑚.(4.18) Using (4.14), (4.17), and (4.18), we obtain 𝑓𝑢0≤𝛽=limsup𝑚→∞𝑓𝑢𝑚≤limsup𝑚→∞𝑓𝑢𝑛𝑢−𝑔𝑛,𝑢𝑚𝑢=𝑓𝑛+limsup𝑚→∞𝑢−𝑔𝑛,𝑢𝑚𝑢=𝑓𝑛−liminf𝑚→∞𝑔𝑢𝑛,𝑢𝑚𝑢≤𝑓𝑛𝑢−𝑔𝑛,𝑢0.(4.19) Combining (4.7), (4.9), and (4.19), it follows that 𝑢0∈𝑆(𝑢𝑛) and, hence, 𝑐𝑢𝑛𝑢≤𝑓0,∀𝑛∈𝑁.(4.20) Taking the limit in inequality (4.20) when 𝑛 tends to infinity, we have lim𝑛→∞𝑐𝑢𝑛𝑢≤𝑓0.(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: thereexists𝑤1𝑢∈𝑋−0satisfying𝑤1𝑢∈𝑆0.(4.23) In terms of (4.7), (4.8), (4.9), (4.20), and (4.23), we deduce that 𝑤1𝑢∈𝑆𝑛𝑐𝑢,∀𝑛∈𝑁,(4.24)𝑛𝑤≤𝑓1.(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)≥𝐻𝑥−𝑘𝑓𝑢0𝑤−𝑓1=𝐻(𝑥)≥𝐹𝑢0,𝑤1(𝑥)∀𝑥>0.(4.27) By virtue of (4.27), we obtain 𝐹𝑢0,𝑤1(𝑥)≥𝐹𝑢0,𝑤1𝑘−1𝑥,∀𝑥>0.(4.28) By repeating the application of inequality (4.28), we get 𝐹𝑢0,𝑤1(𝑥)≥𝐹𝑢0,𝑤1(𝑘−𝑚𝑥),∀𝑥>0,𝑚∈𝑁.(4.29) In terms of (4.29), we deduce that 𝐹𝑢0,𝑤1(𝑘−𝑚𝑥) 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 𝑤0∈𝑋 such that inf𝑣∈𝑋𝑓𝑤(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 𝑢∈𝑋 with inf𝑣∈𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤∈𝑋−{𝑢} such that 1𝑔(𝑢,𝑤)≤𝑓(𝑢)−𝑓(𝑤),(4.32)𝐻(𝑘𝑥−𝑑(𝑢,𝑤))≥𝐻𝑥−𝑘[]𝑓(𝑢)−𝑓(𝑤)(4.33) for all 𝑥>0 and some 𝑘∈(0,1). Thus, there exists 𝑤0∈𝑋 such that inf𝑣∈𝑋𝑓𝑤(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(𝑥−𝑑(𝑝,𝑞)),∀𝑝,𝑞∈𝑋,𝑥∈𝑅.(4.35)

Corollary 4.3 (see [12]). Let (𝑋,𝑑) be a complete metric space, and let 𝑓∶𝑋→(−∞,∞] be a proper lower semicontinuous function, bounded from below. Assume that for any 𝑢∈𝑋 with inf𝑣∈𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤∈𝑋 with 𝑤≠𝑢 and 𝑓(𝑤)+𝑑(𝑢,𝑤)≤𝑓(𝑢). Then there exists 𝑤𝑜∈𝑋 such that inf𝑣∈𝑋𝑓(𝑣)=𝑓(𝑤0).

Proof. Let (𝑋,𝑑) be a complete metric space, and let 𝑔∶𝑋×𝑋→𝑅+,𝐻∶𝑅→𝑅+, 𝐹𝑝,𝑞∶𝑅→[0.1](𝑝,𝑞∈𝑋), and 𝑡∶[0,1]×[0,1]→[0,1] be mappings such that 𝐹𝑔(𝑝,𝑞)=𝑑(𝑝,𝑞)∀𝑝,𝑞∈𝑋,𝐻(𝑥)=0if𝑥≤0,𝐻(𝑥)=1if𝑥>0,𝑝,𝑞[].(𝑥)=𝐻(𝑥−𝑑(𝑝,𝑞))∀𝑝,𝑞∈𝑋,𝑥>0,𝑡(𝑎,𝑏)=min{𝑎,𝑏}∀𝑎,𝑏∈0,1(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 𝑔∶𝑋×𝑋→𝑅+,  𝐹∶𝑋×𝑋→𝐿, and 𝑓∶𝑋→(−∞,∞] be mappings defined as follows: 𝑔(𝑢,𝑤)=𝑤∀𝑢,𝑤∈𝑋,(4.37)[]𝐹(𝑢,𝑤)(𝑥)≡𝐹𝑢,𝑤(𝑥)=𝐻(𝑥−|𝑢−𝑤|)∀𝑢,𝑤∈𝑋,𝑥>0,(4.38)⎧⎪⎪⎨⎪⎪⎩𝑓(𝑢)=0,0≤𝑢≤1−3𝑢+7,1<𝑢<22𝑢−2,2≤𝑢≤3.(4.39) Then, for any 𝑢∈𝑋 with inf𝑣∈𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤=1∈𝑋−{𝑢} such that 1𝑔(𝑢,𝑤)≤𝑓(𝑢)−𝑓(𝑤),(4.40)𝐻(𝑘𝑥−|𝑢−𝑤|)≥𝐻𝑥−𝑘[]𝑓(𝑢)−𝑓(𝑤)∀𝑥>0.(4.41) Let {𝑢𝑛} be a sequence of 𝑋 such that lim𝑛→∞𝑢𝑛=0. Then, clearly conditions (1), (2), (3), and (4) in Lemma 4.1 are satisfied. Thus there exists 0∈𝑋 such that inf𝑣∈𝑋𝑓(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)∀𝑢∈𝑋,𝑢,𝑇𝑢1(𝑘𝑥)≥𝐻𝑥−𝑘[]𝑓(𝑢)−𝑓(𝑇𝑢)∀𝑢∈𝑋,𝑥>0andsome𝑘∈(0,1).(4.43) Then, 𝑇 has a fixed point in 𝑋.

Proof. Suppose 𝑢≠𝑇𝑢 for all 𝑢∈𝑋. Then by Lemma 4.1, there exists 𝑤0∈𝑋 such that 𝑓𝑤0=inf𝑣∈𝑋𝑓(𝑣).(4.44) Since 𝑔(𝑤0,𝑇𝑤0)+𝑓(𝑇𝑤0)≤𝑓(𝑤0), we have 𝑓𝑇𝑤0𝑤=𝑓0=inf𝑣∈𝑋𝑓𝐹(𝑣),(4.45)𝑤0,𝑇𝑤01(𝑘𝑥)≥𝐻𝑥−𝑘𝑓𝑤0−𝑓𝑇𝑤0=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 𝑢≠𝑇𝑢 for all 𝑢∈𝑋. The proof of Theorem 4.5 is complete.

Theorem 4.6. Let (𝑋,𝑑) be a metric space, 𝑇∶𝑋→𝑋, 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)∀𝑢∈𝑋,(4.47)𝐻(𝑘𝑥−𝑑(𝑢,𝑇𝑢))≥𝐻𝑥−𝑘[]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 𝑓∶𝑋→(−∞,∞] 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)∀𝑢∈𝑋.(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 𝑓∶𝑋→(−∞,∞] and 𝑇∶𝑋→𝑋 are mappings defined as follows: ⎧⎪⎨⎪⎩3𝑓(𝑥)=21𝑥,if0≤𝑥<2,2𝑥,if2≤𝑥≤3,𝑇𝑥=2𝑥∀𝑥∈𝑋.(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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