Fixed Point Theorems for Nonlinear Contractions in Menger Spaces

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)=01,,𝑡𝑡≤>00.(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)→𝐹_𝑝,𝑞, where {𝐹_𝑝,𝑞∶𝑝,𝑞∈𝑋}⊆𝐿, where the functions 𝐹_𝑝,𝑞 satisfied the following:(a)𝐹_𝑝,𝑞(𝑥)=1 for all 𝑥>0 if and only if 𝑝=𝑞; (b)𝐹_𝑝,𝑞(0)=0; (c)𝐹_𝑝,𝑞=𝐹_𝑞,𝑝; (d)𝐹_𝑝,𝑞(𝑥)=1 and 𝐹_𝑞,𝑟(𝑦)=1, then 𝐹_𝑝,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, 𝐹_𝑝,𝑟(𝑥+𝑦)≥𝑡𝐹_𝑝,𝑞(𝑥),𝐹_𝑞,𝑟(𝑦).(2.2)

Definition 2.5 (see [17]). Let (𝑋,𝐹,𝑡) be a Menger space.(1)A sequence {𝑝_𝑛} in 𝑋 is said to converge to a point 𝑝 in 𝑋 (written as 𝑝_𝑛→𝑝) if for every 𝜀>0 and 𝜆>0, there exists a positive integer 𝑀(𝜀,𝜆) such that 𝐹_{𝑝𝑛,𝑝}(𝜀)>1−𝜆 for all 𝑛≥𝑀(𝜀,𝜆).(2)A sequence {𝑝_𝑛} 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 𝐹_{𝑝𝑛,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 {𝑡^𝑛(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 𝑡^𝑛(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 𝑀𝑀₁(𝑝,𝑥,𝑛)=min𝐹_𝑇^𝑘_𝑝,𝑇^𝑙_𝑝(𝑥)∶𝑘,𝑙≤𝑛and𝑘,𝑙∈𝑍⁺,𝑀(𝑝,𝑥,𝑛)=min𝐹_𝑇^𝑘_𝑝,𝑇^𝑙_𝑝(𝑥)∶𝑘,𝑙≤𝑛and𝑘,𝑙∈𝑁₂,(𝑝,𝑂𝑥,(𝑝𝑛,)𝑛=)=min𝑇_𝑘𝐹_𝑝,𝑇^𝑙_𝑝(𝑥)∶𝑙≤𝑛and𝑙∈𝑁,𝑂(𝑝,∞)𝑝=∶𝑘𝑇_𝑘≤𝑛and𝑘∈𝑍⁺,𝑝∶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)≡𝐹_𝑝,𝑞(𝑥)=𝐻(𝑥−𝑑(𝑝,𝑞))(2.4) for all 𝑝,𝑞∈𝑋 and 𝑥∈𝑅, where {𝐹_𝑝,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, 𝐹_{𝑇𝑝,𝑇𝑞}(𝜑(𝑥))≥𝐹_𝑝,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 lim_𝑟→𝑥+inf𝜑(𝑟)<𝑥 for each 𝑥>0. Then, 𝑇 has a unique fixed point 𝑢∈𝑋 and {𝑇^𝑛(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 𝐹_{𝑇𝑝,𝑇𝑞}(𝜑(𝑥))≥min𝐹_𝑝,𝑞(𝑥),𝐹_{𝑝,𝑇𝑝}(𝑥),𝐹_{𝑞,𝑇𝑞}(𝑥),𝐹_{𝑝,𝑇𝑞}(𝑥),𝐹_{𝑞,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, 𝐹_𝑝,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 𝜑(𝑥)=⎧⎪⎨⎪⎩𝑥1−+𝑥1𝑥,,10<≤𝑥𝑥.≤1,(3.1) For each 𝑝,𝑞∈𝑋, let 𝐹_𝑝,𝑞∶𝑅→𝑅⁺ be defined by 𝐹_𝑝,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 𝐹_{𝑇𝑝,𝑇𝑞}(𝜑(𝑥))≥min{𝐹_{𝑝,𝑇𝑞}(𝑥),𝐹_{𝑞,𝑇𝑝}(𝑥)} for all 𝑝,𝑞∈𝑋 and 𝑥>0. Thus, 𝐹_{𝑇𝑝,𝑇𝑞}(𝜑(𝑥))≥min𝐹_𝑝,𝑞(𝑥),𝐹_{𝑝,𝑇𝑝}(𝑥),𝐹_{𝑞,𝑇𝑞}(𝑥),𝐹_{𝑝,𝑇𝑞}(𝑥),𝐹_{𝑞,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 𝐹_{𝑇0,𝑇3/2}(𝜑(2))=0 and 𝐹_0,3/2(2)=1. Thus, 𝐹_{𝑇0,𝑇3/2}(𝜑(2))<𝐹_0,3/2(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 𝐹_𝑝,𝑞∶𝑅→𝑅⁺ be defined by 𝐹_𝑝,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, 𝐹_𝑇0,𝑇1(𝜑(2/𝑘))=0 and min𝐹_0,12𝑘,𝐹^0,𝑇0𝑘2,𝐹^1,𝑇1𝑘2,𝐹^0,𝑇1𝑘2,𝐹^1,𝑇0𝑘2=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 𝐹_{𝑝,𝑇𝑝}((𝐼−𝜑)(𝑥))≤inf𝐹_𝑇^𝑘_𝑝,𝑇^𝑙_𝑝(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 𝜑(𝑥)=⎧⎪⎨⎪⎩𝑥𝑥2,−2,04≤<𝑥𝑥.≤4,(3.4) For each 𝑝,𝑞∈𝑋, let 𝐹_𝑝,𝑞∶𝑅→𝑅⁺ be defined by 𝐹_𝑝,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 𝐹_{𝑇𝑝,𝑇𝑞}(𝜑(𝑥))≥min{𝐹_{𝑝,𝑇𝑞}(𝑥),𝐹_{𝑞,𝑇𝑝}(𝑥)} for all 𝑝,𝑞∈𝑋 and 𝑥>0. Thus, 𝐹_{𝑇𝑝,𝑇𝑞}(𝜑(𝑥))≥min𝐹_𝑝,𝑞(𝑥),𝐹_{𝑝,𝑇𝑝}(𝑥),𝐹_{𝑞,𝑇𝑞}(𝑥),𝐹_{𝑝,𝑇𝑞}(𝑥),𝐹_{𝑞,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 𝐹_{𝑝,𝑇𝑝}((𝐼−𝜑)(𝑥))≤inf𝐹_𝑇^𝑘_𝑝,𝑇^𝑙_𝑝(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 lim_𝑥→∞(𝐼−𝜑)(𝑥)=∞, where 𝐼∶𝑅⁺→𝑅⁺ is identity mapping, (ii)𝜑 and 𝐼−𝜑 are strictly increasing and onto mappings, (iii)lim_𝑛→∞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){𝑇^𝑛𝑝} is Cauchy sequence for each 𝑝∈𝑈, where𝑈=𝑝∈𝑋||𝐹_{𝑝,𝑇𝑝}((𝐼−𝜑)(𝑥))≤inf𝐹_𝑇^𝑘_𝑝,𝑇^𝑙_𝑝(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 𝑀₁(𝑝,𝜑(𝑥),𝑛)==mmiinn𝐹𝐹_𝑇^𝑘_𝑝,𝑇^𝑙_𝑝(𝜑(_{𝑇𝑇𝑘−1𝑝,𝑇𝑇𝑙−1𝑝}𝑥))∣𝑘,𝑙≤𝑛and𝑘,𝑙∈𝑁≥minmin𝐹(𝜑_{𝑇𝑘−1𝑝,𝑇𝑙−1𝑝}(𝑥))∣𝑘,𝑙≤𝑛and𝑘,𝑙∈𝑁𝐹(𝑥),𝐹_𝑇^𝑘−1_𝑝,𝑇^𝑘_𝑝(𝑥),𝐹_𝑇^𝑙−1_𝑝,𝑇^𝑙_𝑝(𝑥),_{𝑇𝑘−1𝑝,𝑇𝑙𝑝}≥𝑀(𝑝,𝑥,𝑛).(𝑥),𝐹_𝑇^𝑙−1_𝑝,𝑇^𝑘_𝑝(𝑥)∶𝑘,𝑙≤𝑛and𝑘,𝑙∈𝑁(3.8) By virtue of (i), (ii), (3.8), and (a), we obtain 𝑀₁(𝑝,𝑥,𝑛)≥≥mmiinn𝑀𝑀₁₁𝑝,𝜑⁻¹(𝑥),𝑛,𝑀₂𝑝,𝜑⁻¹(𝑥),𝑛𝑝,𝜑⁻¹(𝑥),𝑛,𝑀₂(𝑝,𝑥,𝑛).(3.9) By repeating application of (3.9), we have 𝑀₁(𝑝,𝑥,𝑛)≥min𝑀₁(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 𝐹≥_𝑇^𝑛_𝑝,𝑇^𝑚_𝑝(𝜑m(in𝑥))𝐹=𝐹_𝑇𝑇^𝑛−1_{𝑝,𝑇𝑇}^𝑚−1_{𝑝𝑇𝑛−1𝑝,𝑇𝑚−1𝑝}(𝜑(𝑥))𝐹(𝑥),𝐹_𝑇^𝑛−1_𝑝,𝑇^𝑛_𝑝(𝑥),𝐹_𝑇^𝑚−1_𝑝,𝑇^𝑚_𝑝(𝑥),_{𝑇𝑛−1𝑝,𝑇𝑚𝑝}≥min𝐹(𝑥),𝐹_𝑇^𝑛_𝑝,𝑇^𝑚−1_{𝑝𝑇𝑖𝑇𝑛−1𝑝,𝑇𝑗𝑇𝑛−1𝑝}(𝑥)=min𝐹_{𝑇𝑛−1𝑝,𝑇𝑙𝑇𝑛−1𝑝}(𝑥)∶𝑖,𝑗≤𝑚−𝑛+1,𝑖,𝑗∈𝑍⁺min𝐹_{𝑇𝑛−1𝑝,𝑇𝑙𝑇𝑛−1𝑝}(𝑥)∶𝑙≤𝑚−𝑛+1,𝑙∈𝑁,(3.12)=min𝐹(𝜑(𝑥))∶_{𝑇𝑇𝑛−2𝑝,𝑇𝑇𝑙+𝑛−2𝑝}𝑙≤𝑚−𝑛+1,𝑙∈𝑁≥minmin𝐹(𝜑_{𝑇𝑛−2𝑝,𝑇𝑙+𝑛−2𝑝}(𝑥))∶𝑙≤𝑚−𝑛+1,𝑙∈𝑁𝐹(𝑥),𝐹_𝑇^𝑛−2_𝑝,𝑇^𝑛−1_𝑝(𝑥),𝐹_𝑇^{𝑙+𝑛−2}_𝑝,𝑇^{𝑙+𝑛−1}_𝑝(𝑥),_{𝑇𝑛−2𝑝,𝑇𝑙+𝑛−1𝑝}≥minmin𝐹(𝑥),_{𝑇𝑖𝑇𝑛−2𝑝,𝑇𝑗𝑇𝑛−2𝑝}𝐹_𝑇^𝑛−1_𝑝,𝑇^{𝑙+𝑛−2}_𝑝(𝑥)∶𝑙≤𝑚−𝑛+1,𝑙∈𝑁≥min∶𝐹𝑙≤𝑚−𝑛+1_{𝑇𝑖𝑇𝑛−2𝑝,𝑇𝑗𝑇𝑛−2𝑝},𝑙∈𝑁(𝑥)∶0≤𝑖,𝑗≤𝑙+1,𝑖,𝑗∈𝑍⁺=min𝐹_{𝑇𝑛−2𝑝,𝑇𝑙𝑇𝑛−2𝑝}(𝑥)∶0≤𝑖,𝑗≤𝑚−𝑛+2,𝑖,𝑗∈𝑍⁺(𝑥)∶𝑙≤𝑚−𝑛+2,𝑙∈𝑁.(3.13) In terms of (i), (ii), and (3.13), we get 𝐹_𝑇^𝑛_𝑝,𝑇^𝑚_𝑝(𝑥)≥≥mmiinn𝐹𝐹_𝑇^𝑛−1_𝑝,𝑇^𝑙_𝑇^𝑛−1_{𝑝𝑇𝑛−2𝑝,𝑇𝑙𝑇𝑛−2𝑝}𝜑⁻¹(𝑥)∶𝑙≤𝑚−𝑛+1,𝑙∈𝑁𝜑⁻²(𝑥)∶𝑙≤𝑚−𝑛+2,𝑙∈𝑁.(3.14) By repeating the same method as in (3.13) and (3.14), we have 𝐹_𝑇^𝑛_𝑝,𝑇^𝑚_𝑝(𝑥)≥≥minifn𝐹𝐹_𝑝,𝑇^𝑙_{𝑝𝑝,𝑇𝑙𝑝}(𝜑^−𝑛(𝑥))∶𝑙≤𝑚,𝑙∈𝑁(𝜑^−𝑛(𝑥))∶𝑙∈𝑁.(3.15) On account of (iii), (*3), (b), (3.15), and Definition 2.2, we have lim_𝑛→∞inf𝐹_𝑝,𝑇^𝑙_𝑝(𝜑^−𝑛(𝑥))∶𝑙∈𝑁≥≥lliimm_{𝑛→∞𝑛→∞}inf𝐹_𝑇^𝑘_𝑝,𝑇^𝑙_𝑝(𝜑^−𝑛(𝑥))∶𝑘,𝑙∈𝑍⁺𝐹_{𝑝,𝑇𝑝}((1−𝜑)(𝜑^−𝑛(𝑥)))=1for𝑥>0.(3.16) It follows from (3.15) and (3.16) that lim_𝑛→∞𝐹_𝑇^𝑛_𝑝,𝑇^𝑚_𝑝(𝑥)=1for𝑝∈𝑈,𝑥>0.(3.17) This implies that {𝑇^𝑛𝑝} is a Cauchy sequence for 𝑝∈𝑈. This is the proof of (c). Since 𝑋 is 𝑇 orbitally complete, and {𝑇^𝑛𝑝} is a Cauchy sequence for 𝑝∈𝑈, {𝑇^𝑛𝑝} has a limit 𝑢 in 𝑋. To prove (d), let us consider the following inequality; 𝐹_{𝑇𝑢,𝑇}^𝑛+1_𝑝(𝜑(𝑥))≥min𝐹_𝑢,𝑇^𝑛_𝑝(𝑥),𝐹_{𝑢,𝑇𝑢}(𝑥),𝐹_𝑇^𝑛_𝑝,𝑇^𝑛+1_𝑝(𝑥),𝐹_𝑢,𝑇^𝑛+1_𝑝(𝑥),𝐹_𝑇^𝑛_{𝑝,𝑇𝑢}(𝑥)∀𝑥>0.(3.18) Since lim_𝑛→∞𝑇^𝑛𝑝=𝑢, from (3.18), we get 𝐹_{𝑇𝑢,𝑢}(𝜑(𝑥))≥𝐹_{𝑇𝑢,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 𝐹_{𝑇𝑢,𝑇𝑤}(𝜑(𝑥))=≥𝐹mi_𝑢,𝑤n(𝜑𝐹(𝑥_𝑢,𝑤))=𝐹_𝑢,𝑤(𝑥),𝐹_{𝑢,𝑇𝑢}(𝑥),𝐹_{𝑤,𝑇𝑤}(𝑥),𝐹_{𝑢,𝑇𝑤}(𝑥),𝐹_{𝑤,𝑇𝑢}(𝑥)(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)=𝐹_𝑝,𝑞 for all 𝑝,𝑞∈𝑋 and 𝐹_𝑝,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: 𝑇𝑝=⎧⎪⎪⎨⎪⎪⎩0,𝑝−1≤𝑝<0,−1+1𝑝,0≤𝑝<45or87<𝑝≤1,𝜑1(6𝑡)𝑝=,⎧⎪⎨⎪⎩45𝑡−≤𝑡𝑝₂≤87,(3.23)78,0≤𝑡≤1,8𝑡,1<𝑡.(3.24) Define 𝐹_𝑝,𝑞∶𝑅→𝑅⁺ by 𝐹_𝑝,𝑞(𝑥)=𝐻𝑥−||𝑝−𝑞||(3.25) for all 𝑝,𝑞∈𝑋 and 𝑥∈𝑅, where 𝐹_𝑝,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 𝑑(𝑇𝑝,𝑇𝑞)=||𝑇𝑝−𝑇𝑞||=||𝑇𝑞||=𝑞=𝜑(𝑞)=𝜑||𝑞−𝑇𝑝||1=+𝑞𝜑(≤𝑑𝑞(𝑞−,𝑇𝑞𝑝8²)).(3.27)

Case 3. Let 𝑝∈[−1,0) and 𝑞∈[4/5,7/8]. Then 𝑑(𝑇𝑝,𝑇𝑞)=||𝑇𝑝−𝑇𝑞||=||𝑇𝑞||=1=𝜑||𝑞−𝑇𝑝||=𝜑(𝑑1(6𝑞𝑞,𝑇≤𝑝)𝑞).−𝑞8²=𝜑(𝑞)(3.28)

Case 4. Let 𝑝,𝑞∈[0,4/5)∪(7/8,1]. Then, by simple calculation, 𝑑(𝑇𝑝,𝑇𝑞)=||𝑇𝑝−𝑇𝑞||=||||𝑝≤||𝑝−𝑞||−||𝑝−1𝑞+||₂𝑝−1+𝑞𝑞||||=𝜑(𝑑(𝑝,𝑞)).8=𝜑||𝑝−𝑞||(3.29)

Case 5. Let 𝑝∈[0,4/5)∪(7/8,1] and 𝑞∈[4/5,7/8]. Then 𝑑(𝑇𝑝,𝑇𝑞)=||𝑇𝑝−𝑇𝑞||=||||𝑝≤11+𝑝−−116𝑞||||=1+𝑝𝑝+116𝑞𝜑(𝑑(𝑞,𝑇𝑞))==𝜑𝜑2(+𝑑||𝑞1(1−𝑝6,𝑇𝑞×𝑞))||87,==𝜑17218|||𝑞−−1(3.30)≥𝜑416𝑞|||=𝜑1167𝑞5×1176=1207×114630>17218.(3.31) Thus, 𝑑(𝑇𝑝,𝑇𝑞)≤71128<1270×114630<𝜑(𝑑(𝑞,𝑇𝑞)).(3.32)

Case 6. Let 𝑝,𝑞∈[4/5,7/8]. Then, 𝑑(𝑇𝑝,𝑇𝑞)=||𝑇𝑝−𝑇𝑞||=|||−116𝑝−−116𝑞|||=116||𝑝−𝑞||≤𝜑||𝑝−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 𝐹_{𝑇𝑝,𝑇𝑞}(𝜑(𝑥))≥min𝐹_𝑝,𝑞(𝑥),𝐹_{𝑝,𝑇𝑝}(𝑥),𝐹_{𝑞,𝑇𝑞}(𝑥),𝐹_{𝑝,𝑇𝑞}(𝑥),𝐹_{𝑞,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 𝑘⋅m=ax𝑘{⋅𝑑𝑑((𝑝𝑝,,𝑞𝑞)),<𝑑(𝑝,1𝑇𝑝),𝑑(𝑞,𝑇𝑞),𝑑(𝑝,𝑇𝑞),𝑑(𝑞,𝑇𝑝)}1+𝑞𝑑(𝑝,𝑞)=1+𝑞𝑞=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 𝐹_{𝑇𝑝,𝑇𝑞}(𝜑(𝑥))<𝐹_𝑝,𝑞(𝑥). Let 𝑝=4/5, 𝑞=(4/5)−(1/100), and 𝑥=4−16−8×120+17799.(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, 𝐹_{𝑇𝑝,𝑇𝑞}(𝜑(𝑥))<𝐹_𝑝,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 {𝑢_𝑛}^∞𝑛=1 in 𝑋 satisfying lim_𝑛→∞sup𝑔𝑢_𝑛,𝑢_𝑚∶𝑚>𝑛=0,(4.1)there exists 𝑢₀∈𝑋 such that 𝑔𝑢_𝑛lim_𝑛→∞𝑢_𝑛=𝑢₀,𝑓,𝑢𝑢₀₀≤liminf_𝑚→∞𝑔𝑢_𝑛,𝑢_𝑚,≤liminf_𝑛→∞𝑓𝑢_𝑛,(4.2)(4)for any 𝑢∈𝑋 with inf_𝑣∈𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤∈𝑋−{𝑢} such that 𝐹𝑔(𝑢,𝑤)≤𝑓(𝑢)−𝑓(𝑤),(4.3)_𝑢,𝑤(𝑘𝑥)≥𝐻𝑥−1𝑘[𝑓(𝑢)−𝑓(𝑤)],∀𝑥>0andsome𝑘∈(0,1),(4.4)where 𝐿 is the set of all distribution functions,𝐹_𝑝,𝑞∶𝑝,𝑞∈𝑋⊆𝐿,𝐹∶𝑋×𝑋→𝐿(4.5)is defined by 𝐹(𝑝,𝑞)=𝐹_𝑝,𝑞 for all 𝑝,𝑞∈𝑋. Then, there exists 𝑤₀∈𝑋 such that inf_𝑣∈𝑋𝑓(𝑣)=𝑓(𝑤₀).

Proof. Suppose that inf_𝑣∈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 {𝑢_𝑛}^∞𝑛=1 in 𝑋 such that for all 𝑛∈𝑁𝑢₁=𝑢,𝑢_𝑛+1∈𝑆𝑢_𝑛,𝑆𝑢_𝑛⊆𝑆(𝑢),𝑓𝑢_𝑛+1<𝑐𝑢_𝑛+1𝑛.(4.10) In virtue of (4.7), (4.9), and (4.10), we have 𝑓𝑔𝑢𝑢_𝑛,𝑢_𝑛+1≤𝑓𝑢_𝑛−𝑓𝑢_𝑛+1,(4.11)_𝑛+1−1𝑛<𝑐𝑢^𝑛≤𝑓𝑢_𝑛+1(4.12) for all 𝑛∈𝑁. In view of (4.11), {𝑓(𝑢_𝑛)}^∞𝑛=1 is a nonincreasing sequence of real numbers, and so it converges to some 𝛽∈𝑅. Therefore, due to (4.12), 𝛽=lim_𝑛→∞𝑐𝑢_𝑛=lim_𝑛→∞𝑓𝑢_𝑛.(4.13) Combining (1) and (4.11), we get 𝑔𝑢_𝑛,𝑢_𝑚≤𝑓𝑢_𝑛−𝑓𝑢_𝑚∀𝑛,𝑚∈𝑁with𝑛<𝑚.(4.14) On account of (4.13) and (4.14), we have lim_𝑛→∞sup𝑔𝑢_𝑛,𝑢_𝑚∶𝑚>𝑛=0.(4.15) Thus, by virtue of (3), (4.13), (4.14), and (4.15), there exists 𝑢₀∈𝑋 such that 𝑓𝑢lim_𝑛→∞𝑢_𝑛=𝑢₀,(4.16)₀𝑔𝑢≤lim_𝑛→∞𝑓𝑢_𝑛=𝛽,(4.17)_𝑛,𝑢₀≤liminf_𝑚→∞𝑔𝑢_𝑛,𝑢_𝑚.(4.18) Using (4.14), (4.17), and (4.18), we obtain 𝑓𝑢₀≤𝛽=≤lliimmssuup_𝑚→∞p_𝑚→∞𝑓𝑢_𝑚=𝑓𝑢_𝑛𝑓𝑢_𝑛−𝑔𝑢_𝑛,𝑢_𝑚=𝑓𝑢_𝑛+limsup_𝑚→∞−𝑔𝑢_𝑛,𝑢_𝑚≤𝑓𝑢_𝑛−liminf_𝑚→∞𝑔𝑢_𝑛,𝑢_𝑚−𝑔𝑢_𝑛,𝑢₀.(4.19) Combining (4.7), (4.9), and (4.19), it follows that 𝑢₀∈𝑆(𝑢_𝑛) and, hence, 𝑐𝑢_𝑛≤𝑓𝑢₀,∀𝑛∈𝑁.(4.20) Taking the limit in inequality (4.20) when 𝑛 tends to infinity, we have lim_𝑛→∞𝑐𝑢_𝑛≤𝑓𝑢₀.(4.21) In terms of (4.13), (4.17), and (4.21), we deduce that 𝛽=𝑓𝑢₀.(4.22) On the other hand, from (4), (4.6), (4.7), and (4.16), we have the following property: thereexists𝑤₁∈𝑋−𝑢₀satisfying𝑤₁∈𝑆𝑢₀.(4.23) In terms of (4.7), (4.8), (4.9), (4.20), and (4.23), we deduce that 𝑤₁∈𝑐𝑆𝑢𝑢_𝑛,∀𝑛∈𝑁,(4.24)_𝑛≤𝑓𝑤₁.(4.25) In view of (4.7), (4.13), (4.22), (4.23), and (4.25), we have 𝛽=𝑓𝑤₁.(4.26) Due to (4), (4.22), (4.23), and (4.26), we have the following: 𝐹𝑢₀,𝑤₁(𝑘𝑥)≥𝐻𝑥−1=𝐻(𝑥)≥𝑘𝐹𝑢₀𝑓𝑢⁰−𝑓𝑤₁,𝑤₁(𝑥)∀𝑥>0.(4.27) By virtue of (4.27), we obtain 𝐹𝑢₀,𝑤₁(𝑥)≥𝐹𝑢₀,𝑤₁𝑘⁻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 inf_𝑣∈𝑋𝑓(𝑣)=𝑓𝑤₀.(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 inf_𝑣∈𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤∈𝑋−{𝑢} such that 𝐻(𝑘𝑥−𝑑(𝑔𝑢,(𝑤𝑢,)𝑤)≥)≤𝐻𝑓(𝑥𝑢)−−1𝑓(𝑤),(4.32)𝑘[𝑓(𝑢)−𝑓(𝑤)](4.33) for all 𝑥>0 and some 𝑘∈(0,1). Thus, there exists 𝑤₀∈𝑋 such that inf_𝑣∈𝑋𝑓(𝑣)=𝑓𝑤₀.(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 𝐹_𝑝,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 inf_𝑣∈𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤∈𝑋 with 𝑤≠𝑢 and 𝑓(𝑤)+𝑑(𝑢,𝑤)≤𝑓(𝑢). Then there exists 𝑤_𝑜∈𝑋 such that inf_𝑣∈𝑋𝑓(𝑣)=𝑓(𝑤₀).

Proof. Let (𝑋,𝑑) be a complete metric space, and let 𝑔∶𝑋×𝑋→𝑅⁺,𝐻∶𝑅→𝑅⁺, 𝐹_𝑝,𝑞∶𝑅→[0.1](𝑝,𝑞∈𝑋), and 𝑡∶[0,1]×[0,1]→[0,1] be mappings such that 𝐻𝐹(𝑥)=𝑔0(𝑝,if𝑞)𝑥=≤𝑑0(,𝑝,𝑞)𝐻∀(𝑥𝑝,)𝑞=∈1𝑋,if𝑥>0,_𝑝,𝑞(𝑡𝑥()𝑎,=𝑏)𝐻=(m𝑥−in𝑑{𝑎(𝑝,,𝑏𝑞}))∀𝑎∀,𝑝𝑏,𝑞∈∈[0𝑋,,1]𝑥.>0,(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)≡𝐹_𝑢,𝑤(𝑥)=𝐻(𝑥−|𝑢−𝑤|)∀𝑢,𝑤∈𝑋,𝑥>0,(4.38)𝑓(𝑢)=⎧⎪⎪⎨⎪⎪⎩0−2,𝑢3𝑢−+2,7,120≤<≤𝑢𝑢𝑢<≤≤123.(4.39) Then, for any 𝑢∈𝑋 with inf_𝑣∈𝑋𝑓(𝑣)<𝑓(𝑢), there exists 𝑤=1∈𝑋−{𝑢} such that 𝐻(𝑘𝑥−|𝑢−𝑤|)𝑔≥(𝑢𝐻,𝑤)𝑥≤−𝑓1(𝑢)−𝑓(𝑤),(4.40)𝑘[𝑓(𝑢)−𝑓(𝑤)]∀𝑥>0.(4.41) Let {𝑢_𝑛} be a sequence of 𝑋 such that lim_𝑛→∞𝑢_𝑛=0. Then, clearly conditions (1), (2), (3), and (4) in Lemma 4.1 are satisfied. Thus there exists 0∈𝑋 such that inf_𝑣∈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,𝑇𝑢}(𝑘𝑥)≥𝐻𝑥−1𝑘[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 𝑓𝑤₀=inf_𝑣∈𝑋𝑓(𝑣).(4.44) Since 𝑔(𝑤₀,𝑇𝑤₀)+𝑓(𝑇𝑤₀)≤𝑓(𝑤₀), we have 𝐹𝑓𝑇𝑤₀=𝑓𝑤₀=inf_𝑣∈𝑋𝑓(𝑣),(4.45)_{𝑤0,𝑇𝑤0}(𝑘𝑥)≥𝐻𝑥−1=𝐻(𝑥)≥𝑘𝐹𝑓𝑤⁰_{𝑤0,𝑇𝑤0}−𝑓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 𝐻(𝑘𝑥−𝑑(𝑢,𝑇𝑢))≥𝐻𝑔(𝑢𝑥,𝑇−𝑢)1≤𝑓(𝑢)−𝑓(𝑇𝑢)∀𝑢∈𝑋,(4.47)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: 𝑓(𝑥)=⎧⎪⎨⎪⎩232𝑥𝑥,,iiff02≤≤𝑥𝑥<≤32,,𝑇𝑥=12𝑥∀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?