Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 143959 | 18 pages | https://doi.org/10.1155/2011/143959

Fixed Point Theorems for Nonlinear Contractions in Menger Spaces

Academic Editor: H. B. Thompson
Received14 May 2011
Accepted24 Jun 2011
Published22 Aug 2011

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].

Δ†irić’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 πΉβˆΆπ‘…β†’π‘…+ is called a distribution if it is nondecreasing left continuous with inf{𝐹(𝑑)βˆΆπ‘‘βˆˆπ‘…}=0 and sup{𝐹(𝑑)βˆΆπ‘‘βˆˆπ‘…}=1. We will denote by 𝐿 the set of all distribution functions. The specific distribution function π»βˆΆπ‘…β†’π‘…+ is defined by 𝐻(𝑑)=0,𝑑≀01,𝑑>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 πΉβˆΆπ‘‹Γ—π‘‹β†’πΏ is defined by (𝑝,π‘ž)→𝐹𝑝,π‘ž, where {𝐹𝑝,π‘žβˆΆπ‘,π‘žβˆˆπ‘‹}βŠ†πΏ, where the functions 𝐹𝑝,π‘ž satisfied the following:(a)𝐹𝑝,π‘ž(π‘₯)=1 for all π‘₯>0 if and only if 𝑝=π‘ž; (b)𝐹𝑝,π‘ž(0)=0; (c)𝐹𝑝,π‘ž=πΉπ‘ž,𝑝; (d)𝐹𝑝,π‘ž(π‘₯)=1 and πΉπ‘ž,π‘Ÿ(𝑦)=1, then 𝐹𝑝,π‘Ÿ(π‘₯+𝑦)=1.

Definition 2.3. A mapping π‘‘βˆΆ[0,1]Γ—[0,1]β†’[0,1] is called a 𝑑-norm if(e)𝑑(0,0)=0 and 𝑑(π‘Ž,1)=π‘Ž for all π‘Žβˆˆ[0,1]; (f)𝑑(π‘Ž,𝑏)=𝑑(𝑏,π‘Ž) for all π‘Ž,π‘βˆˆ[0,1]; (g)𝑑(π‘Ž,𝑏)≀𝑑(𝑐,𝑑) for all π‘Ž,𝑏,𝑐,π‘‘βˆˆ[0,1] with π‘Žβ‰€π‘ and 𝑏≀𝑑; (h)𝑑(𝑑(π‘Ž,𝑏),𝑐)=𝑑(π‘Ž,𝑑(𝑐,𝑑)) for all π‘Ž,𝑏,π‘βˆˆ[0,1].

Definition 2.4. Menger space is a triplet (𝑋,𝐹,𝑑), where (𝑋,𝐹) is PM space and 𝑑 is a 𝑑 norm such that for all 𝑝,π‘ž,π‘Ÿβˆˆπ‘‹ and all π‘₯,𝑦β‰₯0, 𝐹𝑝,π‘Ÿξ€·πΉ(π‘₯+𝑦)β‰₯𝑑𝑝,π‘ž(π‘₯),πΉπ‘ž,π‘Ÿξ€Έ(𝑦).(2.2)

Definition 2.5 (see [17]). Let (𝑋,𝐹,𝑑) 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 πœ€>0 and πœ†>0, there exists a positive integer 𝑀(πœ€,πœ†) such that 𝐹𝑝𝑛,𝑝(πœ€)>1βˆ’πœ† for all 𝑛β‰₯𝑀(πœ€,πœ†).(2)A sequence {𝑝𝑛} in 𝑋 is said to be Cauchy if for each πœ€>0 and πœ†>0, there is a positive integer 𝑀(πœ€,πœ†) such that 𝐹𝑝𝑛,π‘π‘š(πœ€)β‰₯1βˆ’πœ† for all 𝑛,π‘šβˆˆπ‘ with 𝑛,π‘šβ‰₯𝑀(πœ€,πœ†).(3)A Menger space (𝑋,𝐹,𝑑) is said to be complete if every Cauchy sequence in 𝑋 is converged to a point in 𝑋.

Definition 2.6 (see [18]). 𝑑-norm 𝑑 is said to be of 𝐻 type if a family of functions {𝑑𝑛(π‘Ž)}βˆžπ‘›=1 is equicontinuous at π‘Ž=1, that is, for any πœ€βˆˆ(0,1), there exists π›Ώβˆˆ(0,1), such that π‘Ž>1βˆ’π›Ώ and 𝑛β‰₯1 imply 𝑑𝑛(π‘Ž)>1βˆ’πœ€. The 𝑑-norm 𝑑=min is a trivial example of 𝑑-norm of 𝐻 type, but there are 𝑑-norms of 𝐻 type with 𝑑-norm β‰ min (see, e.g., HadziΔ‡ [19]).

Definition 2.7. Let (𝑋,𝐹,𝑑) be a Menger space, and let π‘‡βˆΆπ‘‹β†’π‘‹ be a selfmapping. For each π‘βˆˆπ‘‹, π‘₯>0 and π‘›βˆˆπ‘, let 𝑀𝐹(𝑝,π‘₯,𝑛)=minπ‘‡π‘˜π‘,𝑇𝑙𝑝(π‘₯)βˆΆπ‘˜,𝑙≀𝑛andπ‘˜,π‘™βˆˆπ‘+ξ€Ύ,𝑀1𝐹(𝑝,π‘₯,𝑛)=minπ‘‡π‘˜π‘,𝑇𝑙𝑝,𝑀(π‘₯)βˆΆπ‘˜,𝑙≀𝑛andπ‘˜,π‘™βˆˆπ‘2𝐹(𝑝,π‘₯,𝑛)=min𝑝,𝑇𝑙𝑝,𝑇(π‘₯)βˆΆπ‘™β‰€π‘›andπ‘™βˆˆπ‘π‘‚(𝑝,𝑛)=π‘˜π‘βˆΆπ‘˜β‰€π‘›andπ‘˜βˆˆπ‘+ξ€Ύ,𝑇𝑂(𝑝,∞)=π‘˜π‘βˆΆπ‘˜βˆˆπ‘+ξ€Ύ,(2.3) where it is understood that 𝑇0𝑝=𝑝.
A Menger space (𝑋,𝐹,𝑑) is said to be 𝑇 orbitally complete if and only if every Cauchy sequence which is contained in 𝑂(𝑝,∞) for some π‘βˆˆπ‘‹ 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 π‘‡βˆΆπ‘‹β†’π‘‹ be a selfmapping on 𝑋. Define πΉβˆΆπ‘‹Γ—π‘‹β†’πΏ by []𝐹(𝑝,π‘ž)(π‘₯)≑𝐹𝑝,π‘ž(π‘₯)=𝐻(π‘₯βˆ’π‘‘(𝑝,π‘ž))(2.4) for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯βˆˆπ‘…, where {𝐹𝑝,π‘žβˆΆπ‘,π‘žβˆˆπ‘‹}βŠ†πΏ. Suppose that 𝑑-norm π‘‘βˆΆ[0,1]Γ—[0,1]β†’[0,1] is defined by 𝑑(π‘Ž,𝑏)=min{π‘Ž,𝑏} for all π‘Ž,π‘βˆˆ[0,1]. Then, (1)(𝑋,𝐹,𝑑) is a Menger space; (2)If (𝑋,𝑑) is 𝑇 orbitally complete, then (𝑋,𝐹,𝑑) 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,𝑑), if 𝑑(π‘₯,π‘₯)β‰₯π‘₯ for all π‘₯∈[0,1], then 𝑑(π‘Ž,𝑏)=min{π‘Ž,𝑏} for all π‘Ž,π‘βˆˆ[0,1].

3. Ćirić-Type Fixed Point Theorems

In 2010, Ćirić proved the following theorem.

Theorem A (see Δ†iriΔ‡ [9], 2010). Let (𝑋,𝐹,𝑑) be a complete Menger space under a 𝑑-norm 𝑑 of 𝐻 type. Let π‘‡βˆΆπ‘‹β†’π‘‹ be a generalized πœ‘-probabilistic contraction, that is, 𝐹𝑇𝑝,π‘‡π‘ž(πœ‘(π‘₯))β‰₯𝐹𝑝,π‘ž(π‘₯)(βˆ—1) for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯>0, where πœ‘βˆΆ[0,∞)β†’[0,∞) satisfies the following conditions: πœ‘(0)=0, πœ‘(π‘₯)<π‘₯, and limπ‘Ÿβ†’π‘₯+infπœ‘(π‘Ÿ)<π‘₯ for each π‘₯>0. Then, 𝑇 has a unique fixed point π‘’βˆˆπ‘‹ and {𝑇𝑛(𝑝)} converges to 𝑒 for each π‘βˆˆπ‘‹.

Definition 3.1. Let (𝑋,𝐹,𝑑) be a Menger space with 𝑑(π‘₯,π‘₯)β‰₯π‘₯ for all π‘₯∈[0,1], and let π‘‡βˆΆπ‘‹β†’π‘‹ be a mapping of 𝑋. We will say that 𝑇 is Δ†iriΔ‡-type-generalized contraction if 𝐹𝑇𝑝,π‘‡π‘žξ€½πΉ(πœ‘(π‘₯))β‰₯min𝑝,π‘ž(π‘₯),𝐹𝑝,𝑇𝑝(π‘₯),πΉπ‘ž,π‘‡π‘ž(π‘₯),𝐹𝑝,π‘‡π‘ž(π‘₯),πΉπ‘ž,𝑇𝑝(π‘₯)(βˆ—2) for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯>0, where πœ‘βˆΆ[0,∞)β†’[0,∞) is a mapping and for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯βˆˆπ‘…, 𝐹𝑝,π‘ž(π‘₯) is the same as in Definition 2.2.
It is clear that (*1) implies (*2).

The following example shows that a Δ†iriΔ‡-type-generalized contraction need not be a generalized πœ‘-probabilistic contraction.

Example 3.2. Let 𝑋=[0,∞), π‘‡βˆΆπ‘‹β†’π‘‹ be defined by 𝑇π‘₯=π‘₯+1, and let πœ‘βˆΆ[0,∞)β†’[0,∞) be defined by ⎧βŽͺ⎨βŽͺ⎩π‘₯πœ‘(π‘₯)=1+π‘₯,0≀π‘₯≀1,π‘₯βˆ’1,1<π‘₯.(3.1) For each 𝑝,π‘žβˆˆπ‘‹, let 𝐹𝑝,π‘žβˆΆπ‘…β†’π‘…+ be defined by 𝐹𝑝,π‘ž(π‘₯)=𝐻(π‘₯βˆ’π‘‘(𝑝,π‘ž)) for all π‘₯βˆˆπ‘…, where 𝐻 is the same as in Definition 2.1, and 𝑑 is a usual metric on 𝑅×𝑅. Then, since max{|π‘βˆ’π‘žβˆ’1|,|π‘žβˆ’π‘βˆ’1|}=|π‘βˆ’π‘ž|+1 for all 𝑝,π‘žβˆˆπ‘‹, we have 𝐹𝑇𝑝,π‘‡π‘ž(πœ‘(π‘₯))β‰₯min{𝐹𝑝,π‘‡π‘ž(π‘₯),πΉπ‘ž,𝑇𝑝(π‘₯)} for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯>0. Thus, 𝐹𝑇𝑝,π‘‡π‘žξ€½πΉ(πœ‘(π‘₯))β‰₯min𝑝,π‘ž(π‘₯),𝐹𝑝,𝑇𝑝(π‘₯),πΉπ‘ž,π‘‡π‘ž(π‘₯),𝐹𝑝,π‘‡π‘ž(π‘₯),πΉπ‘ž,𝑇𝑝(π‘₯)(3.2) for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯>0, which satisfies (*2). If π‘₯=2,𝑝=0 and π‘ž=3/2, then 𝐹𝑇0,𝑇3/2(πœ‘(2))=0 and 𝐹0,3/2(2)=1. Thus, 𝐹𝑇0,𝑇3/2(πœ‘(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 πœ‘(𝑑)=π‘˜π‘‘,   0<π‘˜<1.

Example 3.3. Let 𝑋=[0,∞), π‘‡βˆΆπ‘‹β†’π‘‹ be defined by 𝑇π‘₯=2π‘₯ and let πœ‘βˆΆ[0,∞)β†’[0,∞) be defined by πœ‘(π‘₯)=π‘˜π‘₯, 0<π‘˜<1. For each 𝑝,π‘žβˆˆπ‘‹, let 𝐹𝑝,π‘žβˆΆπ‘…β†’π‘…+ be defined by 𝐹𝑝,π‘ž(π‘₯)=𝐻(π‘₯βˆ’π‘‘(𝑝,π‘ž)) for all π‘₯βˆˆπ‘…, where 𝐻 is the same as in Definition 2.1, and 𝑑 is a usual metric on 𝑅×𝑅. If 𝑝=0, π‘ž=1 and π‘₯=2/π‘˜>0, then for simple calculations, 𝐹𝑇0,𝑇1(πœ‘(2/π‘˜))=0 and 𝐹min0,1ξ‚€2π‘˜ξ‚,𝐹0,𝑇0ξ‚€2π‘˜ξ‚,𝐹1,𝑇1ξ‚€2π‘˜ξ‚,𝐹0,𝑇1ξ‚€2π‘˜ξ‚,𝐹1,𝑇0ξ‚€2π‘˜ξ‚ξ‚‡=1.(3.3) Therefore, for 𝑝=0, π‘ž=1, and π‘₯=2/π‘˜>0, the mapping 𝑇 does not satisfy (*2). Thus, we showed that there exists 𝑇 that does not satisfy (*2) with πœ‘(𝑑)=π‘˜π‘‘,0<π‘˜<1.

Definition 3.4. Let (𝑋,𝐹,𝑑) be a Menger space with 𝑑(π‘₯,π‘₯)β‰₯π‘₯ for all π‘₯∈[0,1] and let π‘‡βˆΆπ‘‹β†’π‘‹ be a self mapping of 𝑋. We will say that 𝑇 is a mapping of type π•Œ if there exists π‘βˆˆπ‘‹ such that 𝐹𝑝,𝑇𝑝𝐹((πΌβˆ’πœ‘)(π‘₯))≀infπ‘‡π‘˜π‘,𝑇𝑙𝑝(π‘₯)βˆΆπ‘˜,π‘™βˆˆπ‘+ξ€Ύβˆ€π‘₯>0,(βˆ—3) where πœ‘βˆΆ[0,∞)β†’[0,∞) is a mapping, and 𝐼∢[0,∞)β†’[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,∞), π‘‡βˆΆπ‘‹β†’π‘‹ be defined by 𝑇π‘₯=π‘₯+4, and let πœ‘βˆΆ[0,∞)β†’[0,∞) be defined by ⎧βŽͺ⎨βŽͺ⎩π‘₯πœ‘(π‘₯)=2,0≀π‘₯≀4,π‘₯βˆ’2,4<π‘₯.(3.4) For each 𝑝,π‘žβˆˆπ‘‹, let 𝐹𝑝,π‘žβˆΆπ‘…β†’π‘…+ be defined by 𝐹𝑝,π‘ž(π‘₯)=𝐻(π‘₯βˆ’π‘‘(𝑝,π‘ž)) for all π‘₯βˆˆπ‘…, where 𝐻 is the same as in Definition 2.1, and 𝑑 is a usual metric on 𝑅×𝑅. Then, since max{|π‘βˆ’π‘žβˆ’4|,|π‘žβˆ’π‘βˆ’4|}=|π‘βˆ’π‘ž|+4 for all 𝑝,π‘žβˆˆπ‘‹, we have 𝐹𝑇𝑝,π‘‡π‘ž(πœ‘(π‘₯))β‰₯min{𝐹𝑝,π‘‡π‘ž(π‘₯),πΉπ‘ž,𝑇𝑝(π‘₯)} for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯>0. Thus, 𝐹𝑇𝑝,π‘‡π‘žξ€½πΉ(πœ‘(π‘₯))β‰₯min𝑝,π‘ž(π‘₯),𝐹𝑝,𝑇𝑝(π‘₯),πΉπ‘ž,π‘‡π‘ž(π‘₯),𝐹𝑝,π‘‡π‘ž(π‘₯),πΉπ‘ž,𝑇𝑝(π‘₯)(3.5) for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯>0, which implies (*2). It is easy to see that there exists 𝑝=1βˆˆπ‘‹ such that 𝐹𝑝,𝑇𝑝𝐹((πΌβˆ’πœ‘)(π‘₯))≀infπ‘‡π‘˜π‘,𝑇𝑙𝑝(π‘₯)βˆΆπ‘˜,π‘™βˆˆπ‘+ξ€Ύβˆ€π‘₯>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 Δ†iriΔ‡-type fixed point theorem which is generalization of Δ†irić’s fixed point theorems [6, 9].

Theorem 3.7. Let (𝑋,𝐹,𝑑) be a Menger space with continuous 𝑑 norm and 𝑑(π‘₯,π‘₯)β‰₯π‘₯ for all π‘₯∈[0,1], let 𝑇 be a self-mapping on 𝑋 satisfying (*2) and (*3). Let (𝑋,𝐹,𝑑) be 𝑇 orbitally complete. Suppose that πœ‘βˆΆπ‘…+→𝑅+ is a mapping such that (i)πœ‘(π‘₯)<π‘₯ for all π‘₯>0 and limπ‘₯β†’βˆž(πΌβˆ’πœ‘)(π‘₯)=∞, where πΌβˆΆπ‘…+→𝑅+ is identity mapping, (ii)πœ‘ and πΌβˆ’πœ‘ are strictly increasing and onto mappings, (iii)limπ‘›β†’βˆžπœ‘βˆ’π‘›(π‘₯)=∞ for each π‘₯>0, where πœ‘βˆ’π‘› is 𝑛-time repeated composition of πœ‘βˆ’1 with itself. Then, (a)𝑀(𝑝,π‘₯,𝑛)=min{𝑀1(𝑝,π‘₯,𝑛),𝑀2(𝑝,π‘₯,𝑛)} for all π‘βˆˆπ‘‹, π‘₯>0, and π‘›βˆˆπ‘, (b)𝑀(𝑝,π‘₯,𝑛)=𝑀2(𝑝,π‘₯,𝑛) for all π‘βˆˆπ‘‹, π‘₯>0 and π‘›βˆˆπ‘, (c){𝑇𝑛𝑝} is Cauchy sequence for each π‘βˆˆπ‘ˆ, whereξ€½||πΉπ‘ˆ=π‘βˆˆπ‘‹π‘,𝑇𝑝𝐹((πΌβˆ’πœ‘)(π‘₯))≀infπ‘‡π‘˜π‘,𝑇𝑙𝑝(π‘₯)βˆΆπ‘˜,π‘™βˆˆπ‘+ξ€»ξ€Ύβˆ€π‘₯>0,(3.7)(d)𝑇 has a unique fixed point in 𝑋.

Proof. Let π‘βˆˆπ‘‹, π‘₯>0 and π‘›βˆˆπ‘ be arbitrary. By Definition 2.2 and Definition 2.7, clearly, we have 𝑀(𝑝,π‘₯,𝑛)=min{𝑀1(𝑝,π‘₯,𝑛),𝑀2(𝑝,π‘₯,𝑛)} which implies (a). From (i), (ii), and (*2), we have 𝑀1𝐹(𝑝,πœ‘(π‘₯),𝑛)=minπ‘‡π‘˜π‘,𝑇𝑙𝑝𝐹(πœ‘(π‘₯))βˆ£π‘˜,𝑙≀𝑛andπ‘˜,π‘™βˆˆπ‘=minπ‘‡π‘‡π‘˜βˆ’1𝑝,π‘‡π‘‡π‘™βˆ’1𝑝𝐹(πœ‘(π‘₯))βˆ£π‘˜,𝑙≀𝑛andπ‘˜,π‘™βˆˆπ‘β‰₯minminπ‘‡π‘˜βˆ’1𝑝,π‘‡π‘™βˆ’1𝑝(π‘₯),πΉπ‘‡π‘˜βˆ’1𝑝,π‘‡π‘˜π‘(π‘₯),πΉπ‘‡π‘™βˆ’1𝑝,𝑇𝑙𝑝𝐹(π‘₯),π‘‡π‘˜βˆ’1𝑝,𝑇𝑙𝑝(π‘₯),πΉπ‘‡π‘™βˆ’1𝑝,π‘‡π‘˜π‘ξ€»ξ€Ύ(π‘₯)βˆΆπ‘˜,𝑙≀𝑛andπ‘˜,π‘™βˆˆπ‘β‰₯𝑀(𝑝,π‘₯,𝑛).(3.8) By virtue of (i), (ii), (3.8), and (a), we obtain 𝑀1𝑀(𝑝,π‘₯,𝑛)β‰₯min1𝑝,πœ‘βˆ’1ξ€Έ(π‘₯),𝑛,𝑀2𝑝,πœ‘βˆ’1𝑀(π‘₯),𝑛β‰₯min1𝑝,πœ‘βˆ’1ξ€Έ(π‘₯),𝑛,𝑀2ξ€Ύ.(𝑝,π‘₯,𝑛)(3.9) By repeating application of (3.9), we have 𝑀1𝑀(𝑝,π‘₯,𝑛)β‰₯min1(𝑝,πœ‘βˆ’π‘š(π‘₯),𝑛),𝑀2ξ€Ύ(𝑝,π‘₯,𝑛).(3.10) Since 𝑀1(𝑝,πœ‘βˆ’π‘š(π‘₯),𝑛) converges to 1 when π‘š tends to infinity, it follows that 𝑀1(𝑝,π‘₯,𝑛)β‰₯𝑀2(𝑝,π‘₯,𝑛).(3.11) On account of (a) and (3.11), we have 𝑀(𝑝,π‘₯,𝑛)=𝑀2(𝑝,π‘₯,𝑛) for π‘βˆˆπ‘‹, π‘₯>0 and π‘›βˆˆπ‘, which implies (b). To prove (c), let 𝑛 and π‘š be two positive integers with 𝑛<π‘š, π‘βˆˆπ‘ˆ, and let π‘₯ be any positive real number. By (*2) and (b), we have 𝐹𝑇𝑛𝑝,π‘‡π‘šπ‘(πœ‘(π‘₯))=πΉπ‘‡π‘‡π‘›βˆ’1𝑝,π‘‡π‘‡π‘šβˆ’1𝑝𝐹(πœ‘(π‘₯))β‰₯minπ‘‡π‘›βˆ’1𝑝,π‘‡π‘šβˆ’1𝑝(π‘₯),πΉπ‘‡π‘›βˆ’1𝑝,𝑇𝑛𝑝(π‘₯),πΉπ‘‡π‘šβˆ’1𝑝,π‘‡π‘šπ‘(𝐹π‘₯),π‘‡π‘›βˆ’1𝑝,π‘‡π‘šπ‘(π‘₯),𝐹𝑇𝑛𝑝,π‘‡π‘šβˆ’1𝑝(𝐹π‘₯)β‰₯minπ‘‡π‘–π‘‡π‘›βˆ’1𝑝,π‘‡π‘—π‘‡π‘›βˆ’1𝑝(π‘₯)βˆΆπ‘–,π‘—β‰€π‘šβˆ’π‘›+1,𝑖,π‘—βˆˆπ‘+𝐹=minπ‘‡π‘›βˆ’1𝑝,π‘‡π‘™π‘‡π‘›βˆ’1𝑝,𝐹(π‘₯)βˆΆπ‘™β‰€π‘šβˆ’π‘›+1,π‘™βˆˆπ‘(3.12)minπ‘‡π‘›βˆ’1𝑝,π‘‡π‘™π‘‡π‘›βˆ’1𝑝𝐹(πœ‘(π‘₯))βˆΆπ‘™β‰€π‘šβˆ’π‘›+1,π‘™βˆˆπ‘=minπ‘‡π‘‡π‘›βˆ’2𝑝,𝑇𝑇𝑙+π‘›βˆ’2𝑝𝐹(πœ‘(π‘₯))βˆΆπ‘™β‰€π‘šβˆ’π‘›+1,π‘™βˆˆπ‘β‰₯minminπ‘‡π‘›βˆ’2𝑝,𝑇𝑙+π‘›βˆ’2𝑝(π‘₯),πΉπ‘‡π‘›βˆ’2𝑝,π‘‡π‘›βˆ’1𝑝(π‘₯),𝐹𝑇𝑙+π‘›βˆ’2𝑝,𝑇𝑙+π‘›βˆ’1𝑝𝐹(π‘₯),π‘‡π‘›βˆ’2𝑝,𝑇𝑙+π‘›βˆ’1𝑝(π‘₯),πΉπ‘‡π‘›βˆ’1𝑝,𝑇𝑙+π‘›βˆ’2𝑝𝐹(π‘₯)βˆΆπ‘™β‰€π‘šβˆ’π‘›+1,π‘™βˆˆπ‘β‰₯minminπ‘‡π‘–π‘‡π‘›βˆ’2𝑝,π‘‡π‘—π‘‡π‘›βˆ’2𝑝(π‘₯)∢0≀𝑖,𝑗≀𝑙+1,𝑖,π‘—βˆˆπ‘+ξ€»ξ€Ύξ€½πΉβˆΆπ‘™β‰€π‘šβˆ’π‘›+1,π‘™βˆˆπ‘β‰₯minπ‘‡π‘–π‘‡π‘›βˆ’2𝑝,π‘‡π‘—π‘‡π‘›βˆ’2𝑝(π‘₯)∢0≀𝑖,π‘—β‰€π‘šβˆ’π‘›+2,𝑖,π‘—βˆˆπ‘+𝐹=minπ‘‡π‘›βˆ’2𝑝,π‘‡π‘™π‘‡π‘›βˆ’2𝑝(ξ€Ύ.π‘₯)βˆΆπ‘™β‰€π‘šβˆ’π‘›+2,π‘™βˆˆπ‘(3.13) In terms of (i), (ii), and (3.13), we get 𝐹𝑇𝑛𝑝,π‘‡π‘šπ‘ξ€½πΉ(π‘₯)β‰₯minπ‘‡π‘›βˆ’1𝑝,π‘‡π‘™π‘‡π‘›βˆ’1π‘ξ€·πœ‘βˆ’1𝐹(π‘₯)βˆΆπ‘™β‰€π‘šβˆ’π‘›+1,π‘™βˆˆπ‘β‰₯minπ‘‡π‘›βˆ’2𝑝,π‘‡π‘™π‘‡π‘›βˆ’2π‘ξ€·πœ‘βˆ’2ξ€Έξ€Ύ.(π‘₯)βˆΆπ‘™β‰€π‘šβˆ’π‘›+2,π‘™βˆˆπ‘(3.14) By repeating the same method as in (3.13) and (3.14), we have 𝐹𝑇𝑛𝑝,π‘‡π‘šπ‘ξ€½πΉ(π‘₯)β‰₯min𝑝,𝑇𝑙𝑝(πœ‘βˆ’π‘›ξ€Ύξ€½πΉ(π‘₯))βˆΆπ‘™β‰€π‘š,π‘™βˆˆπ‘β‰₯inf𝑝,𝑇𝑙𝑝(πœ‘βˆ’π‘›ξ€Ύ.(π‘₯))βˆΆπ‘™βˆˆπ‘(3.15) On account of (iii), (*3), (b), (3.15), and Definition 2.2, we have limπ‘›β†’βˆžξ€½πΉinf𝑝,𝑇𝑙𝑝(πœ‘βˆ’π‘›ξ€Ύ(π‘₯))βˆΆπ‘™βˆˆπ‘β‰₯limπ‘›β†’βˆžξ€½πΉinfπ‘‡π‘˜π‘,𝑇𝑙𝑝(πœ‘βˆ’π‘›(π‘₯))βˆΆπ‘˜,π‘™βˆˆπ‘+ξ€Ύβ‰₯limπ‘›β†’βˆžπΉπ‘,𝑇𝑝((1βˆ’πœ‘)(πœ‘βˆ’π‘›(π‘₯)))=1forπ‘₯>0.(3.16) It follows from (3.15) and (3.16) that limπ‘›β†’βˆžπΉπ‘‡π‘›π‘,π‘‡π‘šπ‘(π‘₯)=1forπ‘βˆˆπ‘ˆ,π‘₯>0.(3.17) This implies that {𝑇𝑛𝑝} is a Cauchy sequence for π‘βˆˆπ‘ˆ. This is the proof of (c). Since 𝑋 is 𝑇 orbitally complete, and {𝑇𝑛𝑝} is a Cauchy sequence for π‘βˆˆπ‘ˆ, {𝑇𝑛𝑝} has a limit 𝑒 in 𝑋. To prove (d), let us consider the following inequality; 𝐹𝑇𝑒,𝑇𝑛+1𝑝𝐹(πœ‘(π‘₯))β‰₯min𝑒,𝑇𝑛𝑝(π‘₯),𝐹𝑒,𝑇𝑒(π‘₯),𝐹𝑇𝑛𝑝,𝑇𝑛+1𝑝(π‘₯),𝐹𝑒,𝑇𝑛+1𝑝(π‘₯),𝐹𝑇𝑛𝑝,𝑇𝑒(π‘₯)βˆ€π‘₯>0.(3.18) Since limπ‘›β†’βˆžπ‘‡π‘›π‘=𝑒, from (3.18), we get 𝐹𝑇𝑒,𝑒(πœ‘(π‘₯))β‰₯𝐹𝑇𝑒,𝑒(π‘₯)βˆ€π‘₯>0.(3.19) In terms of (3.19), (i), (ii), (iii), and Definition 2.2, we deduce that 𝑇𝑒=𝑒, that is, 𝑒 is a fixed point of 𝑇. To prove uniqueness of a fixed point of 𝑇, let 𝑀 be another fixed point of 𝑇. Then 𝑇𝑀=𝑀. Putting 𝑝=𝑒 and π‘ž=𝑀 in (*2), we get 𝐹𝑇𝑒,𝑇𝑀(πœ‘(π‘₯))=𝐹𝑒,𝑀𝐹(πœ‘(π‘₯))β‰₯min𝑒,𝑀(π‘₯),𝐹𝑒,𝑇𝑒(π‘₯),𝐹𝑀,𝑇𝑀(π‘₯),𝐹𝑒,𝑇𝑀(π‘₯),𝐹𝑀,𝑇𝑒(π‘₯)=𝐹𝑒,𝑀(π‘₯)βˆ€π‘₯>0,(3.20) which gives 𝑒=𝑀. Thus, 𝑒 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 π‘˜βˆˆ(0,1) such that 𝑑(𝑇𝑝,π‘‡π‘ž)β‰€π‘˜β‹…max{𝑑(𝑝,π‘ž),𝑑(𝑝,𝑇𝑝),𝑑(π‘ž,π‘‡π‘ž),𝑑(𝑝,π‘‡π‘ž),𝑑(π‘ž,𝑇𝑝)},βˆ€π‘,π‘žβˆˆπ‘‹.(3.21) Suppose that 𝑋 is 𝑇 orbitally complete. Then, 𝑇 has a unique fixed point in 𝑋.

Proof. Define πΉβˆΆπ‘‹Γ—π‘‹β†’πΏ by 𝐹(𝑝,π‘ž)=𝐹𝑝,π‘ž for all 𝑝,π‘žβˆˆπ‘‹ and 𝐹𝑝,π‘ž(π‘₯)=𝐻(π‘₯βˆ’π‘‘(𝑝,π‘ž)) for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯βˆˆπ‘…, where 𝐻 and 𝐿 are the same as in Definition 2.1. Let π‘‘βˆΆ[0,1]Γ—[0,1]β†’[0,1] be defined by 𝑑(π‘Ž,𝑏)=min{π‘Ž,𝑏} for all π‘Ž,π‘βˆˆ[0,1]. Let πœ‘βˆΆ[0,∞)β†’[0,∞) be defined by πœ‘(π‘₯)=π‘˜π‘₯,0<π‘˜<1.(3.22) Then from Lemma 2.8, (𝑋,𝐹,𝑑) 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=[βˆ’1,1] be the closed interval with the usual metric and π‘‡βˆΆπ‘‹β†’π‘‹ and πœ‘βˆΆπ‘…+→𝑅+ be mappings defined as follows: ⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π‘π‘‡π‘=0,βˆ’1≀𝑝<0,41+𝑝,0≀𝑝<57or8βˆ’1<𝑝≀1,416𝑝,57≀𝑝≀8,⎧βŽͺ⎨βŽͺβŽ©π‘‘(3.23)πœ‘(𝑑)=π‘‘βˆ’287,0≀𝑑≀1,8𝑑,1<𝑑.(3.24) Define 𝐹𝑝,π‘žβˆΆπ‘…β†’π‘…+ by 𝐹𝑝,π‘žξ€·||||ξ€Έ(π‘₯)=𝐻π‘₯βˆ’π‘βˆ’π‘ž(3.25) for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯βˆˆπ‘…, where 𝐹𝑝,π‘ž and 𝐻 are the same as in Definition 2.1 and Definition 2.2. Let π‘‘βˆΆ[0,1]Γ—[0,1]β†’[0,1] be defined by 𝑑(π‘Ž,𝑏)=min{π‘Ž,𝑏} for all π‘Ž,π‘βˆˆ[0,1]. Then from Lemma 2.8, (𝑋,𝐹,𝑑) is a 𝑇 orbitally complete Menger space, and πœ‘ is continuous function on 𝑅+ which satisfy (i), (ii), and (iii). Clearly 𝑝=0βˆˆπ‘‹ satisfies (*3). To show that condition (*2) is satisfied, we need to consider several possible cases.

Case 1. Let 𝑝,π‘žβˆˆ[βˆ’1,0). Then ||||𝑑(𝑇𝑝,π‘‡π‘ž)=π‘‡π‘βˆ’π‘‡π‘ž=0β‰€πœ‘(𝑑(𝑝,π‘ž)).(3.26)

Case 2. Let π‘βˆˆ[βˆ’1,0) and π‘žβˆˆ[0,4/5)βˆͺ(7/8,1]. Then ||||=||||=π‘žπ‘‘(𝑇𝑝,π‘‡π‘ž)=π‘‡π‘βˆ’π‘‡π‘žπ‘‡π‘žπ‘ž1+π‘žβ‰€π‘žβˆ’28ξ€·||||ξ€Έ=πœ‘(π‘ž)=πœ‘π‘žβˆ’π‘‡π‘=πœ‘(𝑑(π‘ž,𝑇𝑝)).(3.27)

Case 3. Let π‘βˆˆ[βˆ’1,0) and π‘žβˆˆ[4/5,7/8]. Then ||||=||||=1𝑑(𝑇𝑝,π‘‡π‘ž)=π‘‡π‘βˆ’π‘‡π‘žπ‘‡π‘žπ‘ž16π‘žβ‰€π‘žβˆ’28ξ€·||||ξ€Έ=πœ‘(π‘ž)=πœ‘π‘žβˆ’π‘‡π‘=πœ‘(𝑑(π‘ž,𝑇𝑝)).(3.28)

Case 4. Let 𝑝,π‘žβˆˆ[0,4/5)βˆͺ(7/8,1]. Then, by simple calculation, 𝑑||||=||||𝑝(𝑇𝑝,π‘‡π‘ž)=π‘‡π‘βˆ’π‘‡π‘žβˆ’π‘ž1+𝑝||||≀||||βˆ’||||1+π‘žπ‘βˆ’π‘žπ‘βˆ’π‘ž28ξ€·||||ξ€Έ=πœ‘π‘βˆ’π‘ž=πœ‘(𝑑(𝑝,π‘ž)).(3.29)

Case 5. Let π‘βˆˆ[0,4/5)βˆͺ(7/8,1] and π‘žβˆˆ[4/5,7/8]. Then 𝑑||||=||||𝑝(𝑇𝑝,π‘‡π‘ž)=π‘‡π‘βˆ’π‘‡π‘žβˆ’ξ‚€βˆ’11+π‘π‘žξ‚||||=𝑝16+11+π‘π‘žβ‰€1162+1Γ—7168=71πœ‘ξ€·||||ξ€Έξ‚€|||ξ‚€βˆ’1128=πœ‘(𝑑(𝑝,π‘ž)),(3.30)(𝑑(π‘ž,π‘‡π‘ž))=πœ‘π‘žβˆ’π‘‡π‘ž=πœ‘π‘žβˆ’π‘žξ‚|||16=πœ‘17π‘žξ‚ξ‚€416β‰₯πœ‘5Γ—17=1617Γ—20143>16071.128(3.31) Thus, 𝑑(𝑇𝑝,π‘‡π‘ž)≀71<12817Γ—20143160<πœ‘(𝑑(π‘ž,π‘‡π‘ž)).(3.32)

Case 6. Let 𝑝,π‘žβˆˆ[4/5,7/8]. Then, ||||=|||ξ‚€βˆ’1𝑑(𝑇𝑝,π‘‡π‘ž)=π‘‡π‘βˆ’π‘‡π‘žπ‘ξ‚βˆ’ξ‚€βˆ’116π‘žξ‚|||=116||||ξ€·||||ξ€Έ16π‘βˆ’π‘žβ‰€πœ‘π‘βˆ’π‘ž=πœ‘(𝑑(𝑝,π‘ž)).(3.33) Hence, we obtain [],𝑑(𝑇𝑝,π‘‡π‘ž)β‰€πœ‘(𝑀(𝑝,π‘ž))βˆ€π‘,π‘žβˆˆβˆ’1,1(3.34) where 𝑀(𝑝,π‘ž)=max{𝑑(𝑝,π‘ž),𝑑(𝑝,𝑇𝑝),𝑑(π‘ž,π‘‡π‘ž),𝑑(𝑝,π‘‡π‘ž),𝑑(π‘ž,𝑇𝑝)}.(3.35) From (3.25) and (3.34), we have 𝐹𝑇𝑝,π‘‡π‘žξ€½πΉ(πœ‘(π‘₯))β‰₯min𝑝,π‘ž(π‘₯),𝐹𝑝,𝑇𝑝(π‘₯),πΉπ‘ž,π‘‡π‘ž(π‘₯),𝐹𝑝,π‘‡π‘ž(π‘₯),πΉπ‘ž,𝑇𝑝(π‘₯)(3.36) for all 𝑝,π‘žβˆˆπ‘‹ and π‘₯>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 π‘˜βˆˆ(0,1) be any fixed number. Then, for 𝑝=0βˆˆπ‘‹ and π‘žβˆˆπ‘‹ with 0<π‘ž<min{4/5,(1/π‘˜)βˆ’1}, we have 1π‘˜β‹…max{𝑑(𝑝,π‘ž),𝑑(𝑝,𝑇𝑝),𝑑(π‘ž,π‘‡π‘ž),𝑑(𝑝,π‘‡π‘ž),𝑑(π‘ž,𝑇𝑝)}=π‘˜β‹…π‘‘(𝑝,π‘ž)<π‘ž1+π‘žπ‘‘(𝑝,π‘ž)=1+π‘ž=𝑑(0,π‘‡π‘ž)=𝑑(𝑇𝑝,π‘‡π‘ž).(3.37) Thus, 𝑑(𝑇𝑝,π‘‡π‘ž)>π‘˜β‹…max{𝑑(𝑝,π‘ž),𝑑(𝑝,𝑇𝑝),𝑑(π‘ž,π‘‡π‘ž),𝑑(𝑝,π‘‡π‘ž),𝑑(π‘ž,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,π‘žβˆˆπ‘‹ and π‘₯>0 such that 𝐹𝑇𝑝,π‘‡π‘ž(πœ‘(π‘₯))<𝐹𝑝,π‘ž(π‘₯). Let 𝑝=4/5, π‘ž=(4/5)βˆ’(1/100), and ξ‚™π‘₯=4βˆ’ξ‚€116βˆ’8Γ—+2079179.(3.39) Then, 1/100<π‘₯<1, β€‰πœ‘(π‘₯)βˆ’|π‘‡π‘βˆ’π‘‡π‘ž|=0, and π‘₯βˆ’|π‘βˆ’π‘ž|>0. Hence, 𝐻(πœ‘(π‘₯)βˆ’|π‘‡π‘βˆ’π‘‡π‘ž|)=0 and 𝐻(π‘₯βˆ’|π‘βˆ’π‘ž|)=1. Thus, 𝐹𝑇𝑝,π‘‡π‘ž(πœ‘(π‘₯))<𝐹𝑝,π‘ž(π‘₯). 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,𝑑) be a Menger space with continuous 𝑑 norm and 𝑑(π‘₯,π‘₯)β‰₯π‘₯ for all π‘₯∈[0,1], and let π»βˆΆπ‘…β†’π‘…+ be the same as in Definition 2.1. Suppose that π‘”βˆΆπ‘‹Γ—π‘‹β†’π‘…+ and π‘“βˆΆπ‘‹β†’(βˆ’βˆž,∞] are mappings satisfying the following conditions: (1)𝑔(𝑒,𝑀)≀𝑔(𝑒,𝑣)+𝑔(𝑣,𝑀) for all 𝑒,𝑣,π‘€βˆˆπ‘‹, (2)𝑓 is a proper function which is bounded from below, (3)for any sequence {𝑒𝑛}βˆžπ‘›=1 in 𝑋 satisfying limπ‘›β†’βˆžξ€½ξ€Ίπ‘”ξ€·π‘’sup𝑛,π‘’π‘šξ€ΈβˆΆπ‘š>𝑛=0,(4.1)there exists 𝑒0βˆˆπ‘‹ such that limπ‘›β†’βˆžπ‘’π‘›=𝑒0,𝑔𝑒𝑛,𝑒0≀liminfπ‘šβ†’βˆžπ‘”ξ€·π‘’π‘›,π‘’π‘šξ€Έ,𝑓𝑒0≀liminfπ‘›β†’βˆžπ‘“ξ€·π‘’π‘›ξ€Έ,(4.2)(4)for any π‘’βˆˆπ‘‹ with infπ‘£βˆˆπ‘‹π‘“(𝑣)<𝑓(𝑒), there exists π‘€βˆˆπ‘‹βˆ’{𝑒} such that 𝐹𝑔(𝑒,𝑀)≀𝑓(𝑒)βˆ’π‘“(𝑀),(4.3)𝑒,𝑀1(π‘˜π‘₯)β‰₯𝐻π‘₯βˆ’π‘˜[]𝑓(𝑒)βˆ’π‘“(𝑀),βˆ€π‘₯>0andsomeπ‘˜βˆˆ(0,1),(4.4)where 𝐿 is the set of all distribution functions,𝐹𝑝,π‘žξ€ΎβˆΆπ‘,π‘žβˆˆπ‘‹βŠ†πΏ,πΉβˆΆπ‘‹Γ—π‘‹β†’πΏ(4.5)is defined by 𝐹(𝑝,π‘ž)=𝐹𝑝,π‘ž for all 𝑝,π‘žβˆˆπ‘‹. Then, there exists 𝑀0βˆˆπ‘‹ such that infπ‘£βˆˆπ‘‹π‘“(𝑣)=𝑓(𝑀0).

Proof. Suppose that infπ‘£βˆˆπ‘‹π‘“(𝑣)<𝑓(𝑒)βˆ€π‘’βˆˆπ‘‹.(4.6) For each π‘’βˆˆπ‘‹, let 𝑆(𝑒)={π‘€βˆˆπ‘‹βˆ£π‘”(𝑒,𝑀)≀𝑓(𝑒)βˆ’π‘“(𝑀)}.(4.7) Then, by (4), (4.6), and (4.7) 𝑆(𝑒) is nonempty for each π‘’βˆˆπ‘‹. From (1) and (4.7), we obtain 𝑆(𝑀)βŠ†π‘†(𝑒),foreachπ‘€βˆˆπ‘†(𝑒).(4.8) For each π‘’βˆˆπ‘‹, let 𝑐(𝑒)=inf{𝑓(𝑀)βˆ£π‘€βˆˆπ‘†(𝑒)}.(4.9) Choose π‘’βˆˆπ‘‹ with 𝑓(𝑒)<∞. Then from (4.8) and (4.9), there exists a sequence {𝑒𝑛}βˆžπ‘›=1 in 𝑋 such that for all π‘›βˆˆπ‘π‘’1=𝑒,𝑒𝑛+1ξ€·π‘’βˆˆπ‘†π‘›ξ€Έξ€·π‘’,π‘†π‘›ξ€Έξ€·π‘’βŠ†π‘†(𝑒),𝑓𝑛+1𝑒<𝑐𝑛+1𝑛.(4.10) In virtue of (4.7), (4.9), and (4.10), we have 𝑔𝑒𝑛,𝑒𝑛+1ξ€Έξ€·π‘’β‰€π‘“π‘›ξ€Έξ€·π‘’βˆ’π‘“π‘›+1𝑓𝑒,(4.11)𝑛+1ξ€Έβˆ’1𝑛𝑒<𝑐𝑛𝑒≀𝑓𝑛+1ξ€Έ(4.12) for all π‘›βˆˆπ‘. In view of (4.11), {𝑓(𝑒𝑛)}βˆžπ‘›=1 is a nonincreasing sequence of real numbers, and so it converges to some π›½βˆˆπ‘…. Therefore, due to (4.12), 𝛽=limπ‘›β†’βˆžπ‘ξ€·π‘’π‘›ξ€Έ=limπ‘›β†’βˆžπ‘“ξ€·π‘’π‘›ξ€Έ.(4.13) Combining (1) and (4.11), we get 𝑔𝑒𝑛,π‘’π‘šξ€Έξ€·π‘’β‰€π‘“π‘›ξ€Έξ€·π‘’βˆ’π‘“π‘šξ€Έβˆ€π‘›,π‘šβˆˆπ‘with𝑛<π‘š.(4.14) On account of (4.13) and (4.14), we have limπ‘›β†’βˆžξ€½ξ€Ίπ‘”ξ€·π‘’sup𝑛,π‘’π‘šξ€ΈβˆΆπ‘š>𝑛=0.(4.15) Thus, by virtue of (3), (4.13), (4.14), and (4.15), there exists 𝑒0βˆˆπ‘‹ such that limπ‘›β†’βˆžπ‘’π‘›=𝑒0,𝑓𝑒(4.16)0≀limπ‘›β†’βˆžπ‘“ξ€·π‘’π‘›ξ€Έπ‘”ξ€·π‘’=𝛽,(4.17)𝑛,𝑒0≀liminfπ‘šβ†’βˆžπ‘”ξ€·π‘’π‘›,π‘’π‘šξ€Έ.(4.18) Using (4.14), (4.17), and (4.18), we obtain 𝑓𝑒0≀𝛽=limsupπ‘šβ†’βˆžπ‘“ξ€·π‘’π‘šξ€Έβ‰€limsupπ‘šβ†’βˆžξ€½π‘“ξ€·π‘’π‘›ξ€Έξ€·π‘’βˆ’π‘”π‘›,π‘’π‘šξ€·π‘’ξ€Έξ€Ύ=𝑓𝑛+limsupπ‘šβ†’βˆžξ€½ξ€·π‘’βˆ’π‘”π‘›,π‘’π‘šξ€·π‘’ξ€Έξ€Ύ=π‘“π‘›ξ€Έβˆ’liminfπ‘šβ†’βˆžπ‘”ξ€·π‘’π‘›,π‘’π‘šξ€Έξ€·π‘’β‰€π‘“π‘›ξ€Έξ€·π‘’βˆ’π‘”π‘›,𝑒0ξ€Έ.(4.19) Combining (4.7), (4.9), and (4.19), it follows that 𝑒0βˆˆπ‘†(𝑒𝑛) and, hence, 𝑐𝑒𝑛𝑒≀𝑓0ξ€Έ,βˆ€π‘›βˆˆπ‘.(4.20) Taking the limit in inequality (4.20) when 𝑛 tends to infinity, we have limπ‘›β†’βˆžπ‘ξ€·π‘’π‘›ξ€Έξ€·π‘’β‰€π‘“0ξ€Έ.(4.21) In terms of (4.13), (4.17), and (4.21), we deduce that 𝑒𝛽=𝑓0ξ€Έ.(4.22) On the other hand, from (4), (4.6), (4.7), and (4.16), we have the following property: thereexists𝑀1ξ€½π‘’βˆˆπ‘‹βˆ’0ξ€Ύsatisfying𝑀1ξ€·π‘’βˆˆπ‘†0ξ€Έ.(4.23) In terms of (4.7), (4.8), (4.9), (4.20), and (4.23), we deduce that 𝑀1ξ€·π‘’βˆˆπ‘†π‘›ξ€Έπ‘ξ€·π‘’,βˆ€π‘›βˆˆπ‘,(4.24)𝑛𝑀≀𝑓1ξ€Έ.(4.25) In view of (4.7), (4.13), (4.22), (4.23), and (4.25), we have 𝑀𝛽=𝑓1ξ€Έ.(4.26) Due to (4), (4.22), (4.23), and (4.26), we have the following: 𝐹𝑒0,𝑀1ξ‚€1(π‘˜π‘₯)β‰₯𝐻π‘₯βˆ’π‘˜ξ€Ίπ‘“ξ€·π‘’0ξ€Έξ€·π‘€βˆ’π‘“1=𝐻(π‘₯)β‰₯𝐹𝑒0,𝑀1(π‘₯)βˆ€π‘₯>0.(4.27) By virtue of (4.27), we obtain 𝐹𝑒0,𝑀1(π‘₯)β‰₯𝐹𝑒0,𝑀1ξ€·π‘˜βˆ’1π‘₯ξ€Έ,βˆ€π‘₯>0.(4.28) By repeating the application of inequality (4.28), we get 𝐹𝑒0,𝑀1(π‘₯)β‰₯𝐹𝑒0,𝑀1(π‘˜βˆ’π‘šπ‘₯),βˆ€π‘₯>0,π‘šβˆˆπ‘.(4.29) In terms of (4.29), we deduce that 𝐹𝑒0,𝑀1(π‘˜βˆ’π‘šπ‘₯) converges to 1 as π‘šβ†’βˆž and, hence, 𝐹𝑒0,𝑀1(π‘₯)=1,βˆ€π‘₯>0.(4.30) From (4.30) and Definition 2.2, we have 𝑒0=𝑀1. This is a contradiction from (4.23). Therefore, there exists 𝑀0βˆˆπ‘‹ such that infπ‘£βˆˆπ‘‹π‘“ξ€·π‘€(𝑣)=𝑓0ξ€Έ.(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 π‘’βˆˆπ‘‹ with infπ‘£βˆˆπ‘‹π‘“(𝑣)<𝑓(𝑒), there exists π‘€βˆˆπ‘‹βˆ’{𝑒} such that ξ‚€1𝑔(𝑒,𝑀)≀𝑓(𝑒)βˆ’π‘“(𝑀),(4.32)𝐻(π‘˜π‘₯βˆ’π‘‘(𝑒,𝑀))β‰₯𝐻π‘₯βˆ’π‘˜[]𝑓(𝑒)βˆ’π‘“(𝑀)(4.33) for all π‘₯>0 and some π‘˜βˆˆ(0,1). Thus, there exists 𝑀0βˆˆπ‘‹ such that infπ‘£βˆˆπ‘‹π‘“ξ€·π‘€(𝑣)=𝑓0ξ€Έ.(4.34)

Proof. The proof follows from Lemma 4.1 by considering the induced Menger space (𝑋,𝐹,𝑑), where 𝑑(π‘Ž,𝑏)=min{π‘Ž,𝑏} and 𝐹𝑝,π‘ž(π‘₯)=𝐻(π‘₯βˆ’π‘‘(𝑝,π‘ž)),βˆ€π‘,π‘žβˆˆπ‘‹,π‘₯βˆˆπ‘….(4.35)

Corollary 4.3 (see [12]). Let (𝑋,𝑑) be a complete metric space, and let π‘“βˆΆπ‘‹β†’(βˆ’βˆž,∞] be a proper lower semicontinuous function, bounded from below. Assume that for any π‘’βˆˆπ‘‹ with infπ‘£βˆˆπ‘‹π‘“(𝑣)<𝑓(𝑒), there exists π‘€βˆˆπ‘‹ with 𝑀≠𝑒 and 𝑓(𝑀)+𝑑(𝑒,𝑀)≀𝑓(𝑒). Then there exists π‘€π‘œβˆˆπ‘‹ such that infπ‘£βˆˆπ‘‹π‘“(𝑣)=𝑓(𝑀0).

Proof. Let (𝑋,𝑑) be a complete metric space, and let π‘”βˆΆπ‘‹Γ—π‘‹β†’π‘…+,π»βˆΆπ‘…β†’π‘…+, 𝐹𝑝,π‘žβˆΆπ‘…β†’[0.1](𝑝,π‘žβˆˆπ‘‹), and π‘‘βˆΆ[0,1]Γ—[0,1]β†’[0,1] be mappings such that 𝐹𝑔(𝑝,π‘ž)=𝑑(𝑝,π‘ž)βˆ€π‘,π‘žβˆˆπ‘‹,𝐻(π‘₯)=0ifπ‘₯≀0,𝐻(π‘₯)=1ifπ‘₯>0,𝑝,π‘ž[].(π‘₯)=𝐻(π‘₯βˆ’π‘‘(𝑝,π‘ž))βˆ€π‘,π‘žβˆˆπ‘‹,π‘₯>0,𝑑(π‘Ž,𝑏)=min{π‘Ž,𝑏}βˆ€π‘Ž,π‘βˆˆ0,1(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,β€‰β€‰π‘˜=1/2, and let π‘”βˆΆπ‘‹Γ—π‘‹β†’π‘…+,β€‰β€‰πΉβˆΆπ‘‹Γ—π‘‹β†’πΏ, and π‘“βˆΆπ‘‹β†’(βˆ’βˆž,∞] be mappings defined as follows: 𝑔(𝑒,𝑀)=π‘€βˆ€π‘’,π‘€βˆˆπ‘‹,(4.37)[]𝐹(𝑒,𝑀)(π‘₯)≑𝐹𝑒,𝑀(π‘₯)=𝐻(π‘₯βˆ’|π‘’βˆ’π‘€|)βˆ€π‘’,π‘€βˆˆπ‘‹,π‘₯>0,(4.38)⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π‘“(𝑒)=0,0≀𝑒≀1βˆ’3𝑒+7,1<𝑒<22π‘’βˆ’2,2≀𝑒≀3.(4.39) Then, for any π‘’βˆˆπ‘‹ with infπ‘£βˆˆπ‘‹π‘“(𝑣)<𝑓(𝑒), there exists 𝑀=1βˆˆπ‘‹βˆ’{𝑒} such that ξ‚€1𝑔(𝑒,𝑀)≀𝑓(𝑒)βˆ’π‘“(𝑀),(4.40)𝐻(π‘˜π‘₯βˆ’|π‘’βˆ’π‘€|)β‰₯𝐻π‘₯βˆ’π‘˜[]𝑓(𝑒)βˆ’π‘“(𝑀)βˆ€π‘₯>0.(4.41) Let {𝑒𝑛} be a sequence of 𝑋 such that limπ‘›β†’βˆžπ‘’π‘›=0. Then, clearly conditions (1), (2), (3), and (4) in Lemma 4.1 are satisfied. Thus there exists 0βˆˆπ‘‹ such that infπ‘£βˆˆπ‘‹π‘“(𝑣)=𝑓(0).(4.42) Therefore, all conditions of Theorem 4.2 are satisfied. Since 𝑓 is not lower semicontinuous at 𝑒=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(𝑒,𝑇𝑒)≀𝑓(𝑒)βˆ’π‘“(𝑇𝑒)βˆ€π‘’βˆˆπ‘‹,𝑒,𝑇𝑒1(π‘˜π‘₯)β‰₯𝐻π‘₯βˆ’π‘˜[]𝑓(𝑒)βˆ’π‘“(𝑇𝑒)βˆ€π‘’βˆˆπ‘‹,π‘₯>0andsomeπ‘˜βˆˆ(0,1).(4.43) Then, 𝑇 has a fixed point in 𝑋.

Proof. Suppose 𝑒≠𝑇𝑒 for all π‘’βˆˆπ‘‹. Then by Lemma 4.1, there exists 𝑀0βˆˆπ‘‹ such that 𝑓𝑀0ξ€Έ=infπ‘£βˆˆπ‘‹π‘“(𝑣).(4.44) Since 𝑔(𝑀0,𝑇𝑀0)+𝑓(𝑇𝑀0)≀𝑓(𝑀0), we have 𝑓𝑇𝑀0𝑀=𝑓0ξ€Έ=infπ‘£βˆˆπ‘‹π‘“πΉ(𝑣),(4.45)𝑀0,𝑇𝑀0ξ‚€1(π‘˜π‘₯)β‰₯𝐻π‘₯βˆ’π‘˜ξ€Ίπ‘“ξ€·π‘€0ξ€Έξ€·βˆ’π‘“π‘‡π‘€0=𝐻(π‘₯)β‰₯𝐹𝑀0,𝑇𝑀0(π‘₯),βˆ€π‘₯>0andsomeπ‘˜βˆˆ(0,1).(4.46) By the same method as in proof of Lemma 4.1, it follows that 𝑀0=𝑇𝑀0. But this contradicts our assumption that 𝑒≠𝑇𝑒 for all π‘’βˆˆπ‘‹. The proof of Theorem 4.5 is complete.

Theorem 4.6. Let (𝑋,𝑑) be a metric space, π‘‡βˆΆπ‘‹β†’π‘‹, and let 𝐻,𝑔, and 𝑓 be satisfied conditions (1), (2), and (3) in Lemma 4.1. Suppose that ξ‚€1𝑔(𝑒,𝑇𝑒)≀𝑓(𝑒)βˆ’π‘“(𝑇𝑒)βˆ€π‘’βˆˆπ‘‹,(4.47)𝐻(π‘˜π‘₯βˆ’π‘‘(𝑒,𝑇𝑒))β‰₯𝐻π‘₯βˆ’π‘˜[]𝑓(𝑒)βˆ’π‘“(𝑇𝑒)βˆ€π‘₯>0andsomeπ‘˜βˆˆ(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 π‘“βˆΆπ‘‹β†’(βˆ’βˆž,∞] is a proper lower semicontinuous function bounded from below. Let 𝑇 be a mapping from 𝑋 into itself such that 𝑑(𝑒,𝑇𝑒)≀𝑓(𝑒)βˆ’π‘“(𝑇𝑒)βˆ€π‘’βˆˆπ‘‹.(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 π‘˜ be the same as in Example 4.4. Suppose that π‘“βˆΆπ‘‹β†’(βˆ’βˆž,∞] and π‘‡βˆΆπ‘‹β†’π‘‹ are mappings defined as follows: ⎧βŽͺ⎨βŽͺ⎩3𝑓(π‘₯)=21π‘₯,if0≀π‘₯<2,2π‘₯,if2≀π‘₯≀3,𝑇π‘₯=2π‘₯βˆ€π‘₯βˆˆπ‘‹.(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 π‘₯=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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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.


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