Some Fixed Point Results for Generalized Weak Contraction Mappings in Modular Spaces

Abstract

We prove the existence theorem of fixed points for a generalized weak contractive mapping in modular spaces.

1. Introduction

In 1997, Alber and Guerre-Delabriere [1] introduced the concept of weak contraction in Hilbert spaces. Later, Rhoades [2] proved that the result which Alber et al. is also valid in complete metric spaces, the result of Rhoades in the following: A mapping 𝑇∶𝑋→𝑋 where (𝑋,𝑑) is a metric space, is said to be weakly contractive if 𝑑𝑇(𝑥),𝑇𝑦≤𝑑𝑥,𝑦−𝜙𝑑𝑥,𝑦, (1) where 𝜙∶[0,∞)→[0,∞) is continuous and nondecreasing function such that 𝜙(𝑡)=0 if and only if 𝑡=0. In 2008, Dutta and Choudhury [3] introduced a new generalization of contraction in metric spaces and proved the following theorem.

Theorem 1. Let (𝑋,𝑑) be a complete metric space, and let 𝑇∶𝑋→𝑋 be a self-mapping satisfying the following inequality: 𝜓𝑑𝑇𝑥,𝑇𝑦≤𝜓𝑑𝑥,𝑦−𝜙𝑑𝑥,𝑦, (2) where 𝜓,𝜙∶[0,∞)→[0,∞) are both continuous and monotone nondecreasing function with 𝜓(𝑡)=𝜙(𝑡)=0 if and only if 𝑡=0. Then 𝑇 has a unique fixed point.

We note that, if one takes 𝜓(𝑡)=𝑡, then (2) reduces to (1).

Recall that the theory of modular on linear spaces and the corresponding theory of modular linear spaces were founded by Nakano [4, 5] and redefined by Musielak and Orlicz [6]. Furthermore, the most complete development of these theories is due to Mazur, Luxemburg, and Turpin [7–9]. In the present time, the theory of modular and modular spaces is extensively applied, in particular, in the study of various Orlicz spaces which in their turn have broad applications [10–14]. In many cases, particularly in applications to integral operators, approximation and fixed point theory, modular-type conditions are much more natural as modular-type assumptions can be more easily verified than their metric or norm counterparts. Even though a metric is not defined, many problems in metric fixed point theory can be reformulated in modular spaces. For instance, fixed point theorems are proved in [15, 16] for nonexpansive mappings. The existences for contraction mapping in modular spaces has been studied in [3, 17–24] and the references therein.

From the above mentioned, we will study the existence of fixed point theorems for mappings satisfying generalized weak contraction mappings in modular spaces.

2. Preliminaries

First, we start with a brief recollection of basic concepts and facts in modular spaces.

Definition 2. Let 𝑋 be a vector space over ℝ (or ℂ). A functional 𝜌∶𝑋→[0,∞] is called a modular if for arbitrary 𝑥 and 𝑦, elements of 𝑋, it satisfies the following conditions: (1)𝜌(𝑥)=0 if and only if 𝑥=0; (2)𝜌(𝛼𝑥)=𝜌(𝑥) for all scalar 𝛼 with |𝛼|=1; (3)𝜌(𝛼𝑥+𝛽𝑦)≤𝜌(𝑥)+𝜌(𝑦), whenever 𝛼,𝛽≥0 and 𝛼+𝛽=1.  If one replaces (3) by (4)𝜌(𝛼𝑥+𝛽𝑦)≤𝛼^𝑠𝜌(𝑥)+𝛽^𝑠𝜌(𝑦), for 𝛼,𝛽≥0, 𝛼^𝑠+𝛽^𝑠=1 with an 𝑠∈(0,1], then the modular 𝜌 is called 𝑠-convex modular, and if 𝑠=1, 𝜌 is called convex modular.

If 𝜌 is modular in 𝑋, then the set 𝑋_𝜌=𝑥∈𝑋∶𝜌(𝜆𝑥)⟶0as𝜆⟶0, (3) is called a modular space. 𝑋_𝜌 is a vector subspace of 𝑋.

Definition 3. A modular 𝜌 is said to satisfy the Δ₂-condition if 𝜌(2𝑥_𝑛)→0, whenever 𝜌(𝑥_𝑛)→0 as 𝑛→∞.

Definition 4. Let 𝑋_𝜌 be a modular space. (1)The sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋_𝜌 is said to be 𝜌-convergent to 𝑥∈𝑋_𝜌if 𝜌(𝑥_𝑛−𝑥)→0, as 𝑛→∞. (2)The sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋_𝜌 is said to be 𝜌-Cauchy if 𝜌(𝑥_𝑛−𝑥_𝑚)→0, as 𝑛,𝑚→∞. (3)A subset 𝐶 of 𝑋_𝜌 is said to be 𝜌-closed if the 𝜌-limit of a 𝜌-convergent sequence of 𝐶 always belong to 𝐶. (4)A subset 𝐶 of 𝑋_𝜌 is said to be 𝜌-complete if any 𝜌-Cauchy sequence in 𝐶 is 𝜌-convergent sequence and its 𝜌-limit is in 𝐶. (5)A subset 𝐶 of 𝑋_𝜌 is said to be 𝜌-bounded if any 𝛿_𝜌(𝐶)=sup{𝜌(𝑥−𝑦);𝑥,𝑦∈𝐶}<∞.

Definition 5. Let 𝑋 be a nonempty set and 𝑇∶𝑋→𝑋. A point 𝑥∈𝑋 is a fixed point of 𝑇 if and only if 𝑇𝑥=𝑥.

Definition 6. Let 𝐶 be a subset of a real numbers ℝ. A mapping 𝑇∶𝐶→ℝ is called monotone increasing (or monotone nondecreasing) 𝑥≤𝑦 if and only if 𝑇(𝑥)≤𝑇(𝑦), for all 𝑥 and 𝑦 are elements in 𝐶. A mapping 𝑇∶𝐶→ℝ is called monotone decreasing (or monotone nonincreasing), 𝑥≥𝑦 if and only if 𝑇(𝑥)≥𝑇(𝑦) for all 𝑥 and 𝑦 are elements in 𝐶.

Definition 7. A sequence {𝑎_𝑛} of a real number is said to be monotone increasing (or monotone nondecreasing), if it satisfies 𝑎₁≤𝑎₂≤𝑎₃≤…. It is also said to be monotone decreasing (or monotone nonincreasing) if it satisfies 𝑎₁≥𝑎₂≥𝑎₃≥….

3. A Generalized Weak Contraction in Modular Spaces

In this section, we prove fixed point theorems for mappings satisfying generalized weak contractions in modular spaces.

Proposition 8. Let 𝜌 be a modular space on 𝑋. If 𝑎,𝑏∈ℝ⁺ with 𝑏≥𝑎, then 𝜌(𝑎𝑥)≤𝜌(𝑏𝑥).

Proof. In case 𝑎=𝑏, we are done. Suppose 𝑏>𝑎, and then one has 𝑎/𝑏<1 and 𝜌()𝑎𝑎𝑥=𝜌𝑏𝑎𝑏𝑥=𝜌𝑏𝑎𝑏𝑥+1−𝑏0)()(0)().(≤𝜌𝑏𝑥+𝜌=𝜌𝑏𝑥 (4)

Proposition 9. Let 𝑋_𝜌 be a modular space in which 𝜌 satisfies the Δ₂-condition and let {𝑥_𝑛}_𝑛∈ℕ be a sequence in 𝑋_𝜌. If 𝜌(𝑐(𝑥_𝑛−𝑥_𝑛−1))→0 as 𝑛→∞, then 𝜌(𝛼𝑙(𝑥_𝑛−𝑥_𝑛−1))→0 as 𝑛→∞, where 𝑐,𝑙,𝛼∈ℝ⁺ with 𝑐>𝑙 and 𝑙/𝑐+1/𝛼=1.

Proof. Since 𝜌(𝑐(𝑥_𝑛−𝑥_𝑛−1))→0 as 𝑛→∞, by the Δ₂-condition, we get 𝜌2^𝑚𝑐𝑥_𝑛−𝑥_𝑛−1⟶0as𝑛⟶∞, (5) for 𝑚∈ℕ. Using (5), Proposition 8, and the sandwich theorem, we conclude that 𝜌2𝑁𝑐𝑥_𝑛−𝑥_𝑛−1⟶0as𝑛⟶∞, (6) for 𝑁∈ℕ. From the fact that 𝑙/𝑐+1/𝛼=1, we get 𝛼𝑙=(𝛼−1)𝑐≥𝑐, then there exist 𝑁_𝛼∈ℕ such that 2𝑁_𝛼−1𝑐≤(𝛼−1)𝑐≤2𝑁_𝛼𝑐. (7) By Proposition 8, we get 𝜌2𝑁_𝛼−1𝑐𝑥_𝑛−𝑥_𝑛−1≤𝜌(𝛼−1)𝑐𝑥_𝑛−𝑥_𝑛−1≤𝜌2𝑁_𝛼𝑐𝑥_𝑛−𝑥_𝑛−1. (8) From (6) and (8), we obtain lim_𝑛→∞𝜌𝛼𝑙𝑥_𝑛−𝑥_𝑛−1=lim_𝑛→∞𝜌(𝛼−1)𝑐𝑥_𝑛−𝑥_𝑛−1=0. (9)

Theorem 10. Let 𝑋_𝜌 be a 𝜌-complete modular space, where 𝜌 satisfies the Δ₂-condition. Let 𝑐,𝑙∈ℝ⁺, 𝑐>𝑙, and 𝑇∶𝑋_𝜌→𝑋_𝜌 be a mapping satisfying the inequality 𝜓𝜌𝑐𝜌𝑙𝜌𝑙,𝑇𝑥−𝑇𝑦≤𝜓𝑥−𝑦−𝜙𝑥−𝑦 (10) for all 𝑥,𝑦∈𝑋_𝜌, where 𝜓,𝜙∶[0,∞)→[0,∞) are both continuous and monotone nondecreasing functions with 𝜓(𝑡)=𝜙(𝑡)=0 if and only if 𝑡=0. Then, 𝑇 has a unique fixed point.

Proof. Let 𝑥₀∈𝑋_𝜌, and we construct the sequence {𝑥_𝑛}_𝑛∈ℕ by 𝑥_𝑛=𝑇𝑥_𝑛−1, 𝑛=1,2,3,…. First, we prove that the sequence {𝜌(𝑐(𝑇𝑥_𝑛−𝑇𝑥_𝑛−1))} converges to 0. Indeed 𝜓𝜌𝑐𝑥_𝑛−𝑥_𝑛+1≤𝜓𝜌𝑙𝑥_𝑛−1−𝑥_𝑛−𝜙𝜌𝑙𝑥_𝑛−1−𝑥_𝑛≤𝜓𝜌𝑙𝑥_𝑛−1−𝑥_𝑛. (11) By monotone nondecreasing of 𝜓 and Proposition 8, we have 𝜌𝑐𝑥_𝑛−𝑥_𝑛+1≤𝜌𝑙𝑥_𝑛−1−𝑥_𝑛≤𝜌𝑐𝑥_𝑛−1−𝑥_𝑛. (12)
This means that the sequence {𝜌(𝑐(𝑥_𝑛−𝑥_𝑛−1))} is monotone decreasing and bounded below. Hence there exists 𝑟≥0 such that lim_𝑛→∞𝜌𝑐𝑥_𝑛−𝑥_𝑛−1=𝑟. (13) If 𝑟>0, taking 𝑛→∞ in the inequality (11), we get 𝜓(𝑟)(𝑟)(𝑟)(𝑟),≤𝜓−𝜙<𝜓 (14) which is a contradiction, thus 𝑟=0. So we have 𝜌𝑐𝑥_𝑛−𝑥_𝑛−1→0as𝑛→∞. (15) Next, we prove that the sequence {𝑐𝑥_𝑛}_𝑛∈ℕ is a 𝜌-Cauchy. Suppose that {𝑐𝑥_𝑛}_𝑛∈ℕ is not 𝜌-Cauchy, then there exist 𝜀>0 and subsequence {𝑥_𝑚𝑘}, {𝑥_𝑛𝑘} with 𝑚_𝑘>𝑛_𝑘≥𝑘 such that 𝜌𝑐𝑥_𝑚𝑘−𝑥_𝑛𝑘≥𝜀,𝜌𝑐𝑥_𝑚𝑘−1−𝑥_𝑛𝑘<𝜀. (16) Now, let 𝛼∈R⁺ such that 𝑙/𝑐+1/𝛼=1, then we get 𝜓𝜌𝑐𝑥_𝑚𝑘−𝑥_𝑛𝑘≤𝜓𝜌𝑙𝑥_𝑚𝑘−1−𝑥_𝑛𝑘−1−𝜙𝜌𝑙𝑥_𝑚𝑘−1−𝑥_𝑛𝑘−1≤𝜓𝜌𝑙𝑥_𝑚𝑘−1−𝑥_𝑛𝑘−1 (17) which implies that 𝜌𝑐𝑥_𝑚𝑘−𝑥_𝑛𝑘≤𝜌𝑙𝑥_𝑚𝑘−1−𝑥_𝑛𝑘−1. (18) We have 𝜌𝑙𝑥_𝑚𝑘−1−𝑥_𝑛𝑘−1=𝜌𝑙𝑥_𝑚𝑘−1−𝑥_𝑛𝑘+𝑥_𝑛𝑘−𝑥_𝑛𝑘−1𝑙=𝜌𝑐𝑐𝑥_𝑚𝑘−1−𝑥_𝑛𝑘+1𝛼𝛼𝑙𝑥_𝑛𝑘−𝑥_𝑛𝑘−1≤𝜌𝑐𝑥_𝑚𝑘−1−𝑥_𝑛𝑘+𝜌𝛼𝑙𝑥_𝑛𝑘−𝑥_𝑛𝑘−1<𝜀+𝜌𝛼𝑙𝑥_𝑛𝑘−𝑥_𝑛𝑘−1. (19) By (16), (18), and (19), we get 𝜀≤𝜌𝑐𝑥_𝑚𝑘−𝑥_𝑛𝑘≤𝜌𝑙𝑥_𝑚𝑘−1−𝑥_𝑛𝑘−1<𝜀+𝜌𝛼𝑙𝑥_𝑛𝑘−𝑥_𝑛𝑘−1. (20) Using (15) and Proposition 9, we have lim_𝑘→∞𝜌𝛼𝑙𝑥_𝑛𝑘−𝑥_𝑛𝑘−1=0. (21) From (20) and (21), we obtain lim_𝑘→∞𝜌𝑐𝑥_𝑚𝑘−𝑥_𝑛𝑘=lim_𝑘→∞𝜌𝑙𝑥_𝑚𝑘−1−𝑥_𝑛𝑘−1=𝜀. (22)
Letting 𝑘→∞ in (17), by property of 𝜓 and (22), we get 𝜓(𝜀)≤𝜓(𝜀)−𝜙(𝜀)<𝜓(𝜀) (23) which is a contradiction. Therefore, {𝑐𝑥_𝑛}_𝑛∈ℕ is 𝜌-Cauchy. Since 𝑋_𝜌 is 𝜌-complete there exists a point 𝑢∈𝑋_𝜌 such that 𝜌(𝑐(𝑥_𝑛−𝑢))→0 as 𝑛→∞. Consequently, 𝜌(𝑙(𝑥_𝑛−𝑢))→0 as 𝑛→∞. Next, we prove that 𝑢 is a unique fixed point of 𝑇. Putting 𝑥=𝑥_𝑛−1 and 𝑦=𝑢 in (10), we obtain 𝜓𝜌𝑐𝑥_𝑛−𝑇𝑢≤𝜓𝜌𝑙𝑥_𝑛−1𝜌𝑙𝑥−𝑢−𝜙_𝑛−1−𝑢. (24) Taking 𝑛→∞ in the inequality (24), we have 𝜓𝜌(𝑐(𝑢−𝑇𝑢))≤𝜓(0)−𝜙(0)=0, (25) which implies that 𝜌(𝑐(𝑇𝑢−𝑢))=0 and 𝑇𝑢=𝑢. Suppose that there exists 𝑣∈𝑋_𝜌 such that 𝑇𝑣=𝑣 and 𝑣≠𝑢, and then we have 𝜓𝜌(𝑐(𝜌(𝑐(𝜌(𝑙(𝜌(𝑙(𝜌(𝑙(𝜌(𝑐(,𝑢−𝑣))=𝜓𝑇𝑢−𝑇𝑣))≤𝜓𝑢−𝑣))−𝜙𝑢−𝑣))<𝜓𝑢−𝑣))≤𝜓𝑢−𝑣)) (26) which is a contradiction. Hence 𝑢=𝑣 and the proof is complete.

Corollary 11. Let 𝑋_𝜌 be a 𝜌-complete modular space, where 𝜌 satisfies the Δ₂-condition. Let 𝑐,𝑙∈ℝ⁺, 𝑐>𝑙, and 𝑇∶𝑋_𝜌→𝑋_𝜌 be a mapping satisfying the inequality 𝜌𝑐𝑙𝜌𝑙,𝑇𝑥−𝑇𝑦≤𝜌𝑥−𝑦−𝜙𝑥−𝑦 (27) for all 𝑥,𝑦∈𝑋_𝜌, where 𝜙∶[0,∞)→[0,∞) is continuous and monotone nondecreasing function with 𝜙(𝑡)=0 if and only if 𝑡=0. Then, 𝑇 has a unique fixed point.

Proof. Take 𝜓(𝑡)=𝑡, and then we obtain the Corollary 11.

Theorem 12. Let 𝑋_𝜌 be a 𝜌-complete modular space, where 𝜌 satisfies the Δ₂-condittion and let 𝑇∶𝑋_𝜌→𝑋_𝜌 be a mapping satisfying the inequality 𝜓𝜌𝑚𝑚𝑇𝑥−𝑇𝑦≤𝜓𝑥,𝑦−𝜙𝑥,𝑦 (28) for all 𝑥,𝑦∈𝑋_𝜌,where 𝑚𝜌(),𝜌1𝑥,𝑦=max𝑥−𝑦,𝜌𝑥−𝑇𝑥,𝜌𝑦−𝑇𝑦21𝑥−𝑇𝑦+𝜌2𝑦−𝑇𝑥(2)⁻¹ (29) and 𝜓,𝜙∶[0,∞)→[0,∞) are both continuous and monotone nondecreasing functions with 𝜓(𝑡)=𝜙(𝑡)=0 if and only if 𝑡=0. Then, 𝑇 has a unique fixed point.

Proof. First, we prove that the sequence {𝜌(𝑐(𝑇^𝑛𝑥−𝑇^𝑛−1𝑥))} converges to 0. Since, 𝜓𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥≤𝜓𝑚𝑇^𝑛−1𝑥,𝑇^𝑛−2𝑥−𝜙𝑚𝑇^𝑛−1𝑥,𝑇^𝑛−2𝑥.≤𝜓𝑚𝑇^𝑛−1𝑥,𝑇^𝑛−2𝑥. (30)
By monotone nondecreasing of 𝜓, we have 𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥≤𝑚𝑇^𝑛−1𝑥,𝑇^𝑛−2𝑥. (31) From the definition of 𝑚(𝑥,𝑦), we get 𝑚𝑇^𝑛−1𝑥,𝑇^𝑛−2𝑥=max𝜌𝑇^𝑛−1𝑥−𝑇^𝑛−2𝑥,𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥,𝜌(1/2)𝑇^𝑛𝑥−𝑇^𝑛−2𝑥2=max𝜌𝑇^𝑛−1𝑥−𝑇^𝑛−2𝑥,𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥,𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥+𝜌𝑇^𝑛−1𝑥−𝑇^𝑛−2𝑥2=max𝜌𝑇^𝑛−1𝑥−𝑇^𝑛−2𝑥,𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥. (32) If 𝜌(𝑇^𝑛𝑥−𝑇^𝑛−1𝑥)>𝜌(𝑇^𝑛−1𝑥−𝑇^𝑛−2𝑥)≥0, then 𝑚(𝑇^𝑛−1𝑥,𝑇^𝑛−2𝑥)=𝜌(𝑇^𝑛𝑥−𝑇^𝑛−1𝑥). Furthermore it is implied that 𝜓𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥≤𝜓𝑚𝑇^𝑛−1𝑥,𝑇^𝑛−2𝑥−𝜙𝑚𝑇^𝑛−1𝑥,𝑇^𝑛−2𝑥≤𝜓𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥−𝜙𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥<𝜓𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥 (33) which is a contradiction, and, hence, 𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥≤𝑚𝑇^𝑛−1𝑥−𝑇^𝑛−2𝑥=𝜌𝑇^𝑛−1𝑥−𝑇^𝑛−2𝑥. (34) So, we have that the sequence {𝜌(𝑇^𝑛𝑥−𝑇^𝑛−1𝑥)} is monotone decreasing and bounded below. Hence there exists 𝑟≥0 such that lim_𝑛→∞𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥=𝑟. (35) If 𝑟>0, taking 𝑛→∞ in the inequality (30), we get 𝜓(𝑟)(𝑟)(𝑟)(𝑟)≤𝜓−𝜙<𝜓 (36) which is a contradiction, and thus 𝑟=0. So, we have lim_𝑛→∞𝜌𝑇^𝑛𝑥−𝑇^𝑛−1𝑥=0. (37) Next, we prove that the sequence {𝑇^𝑛(𝑥)}_𝑛∈ℕ is 𝜌-Cauchy. Suppose {𝑇^𝑛(𝑥)}_𝑛∈ℕ is not 𝜌-Cauchy, and there exist 𝜀>0 and sequence of integers {𝑚_𝑘}, {𝑛_𝑘} with {𝑚_𝑘}>{𝑛_𝑘}≥𝑘 such that 𝜌𝑇^𝑚𝑘𝑥−𝑇^𝑛𝑘𝑥≥𝜀,𝜌2𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥<𝜀. (38) Since, 𝜓𝜌𝑇^𝑚𝑘𝑥−𝑇^𝑛𝑘𝑥≤𝜓𝑚𝑇^𝑚𝑘−1𝑥,𝑇^𝑛𝑘−1𝑥−𝜙𝑚𝑇^𝑚𝑘−1𝑥,𝑇^𝑛𝑘−1𝑥≤𝜓𝑚𝑇^𝑚𝑘−1𝑥,𝑇^𝑛𝑘−1𝑥 (39) which implies that 𝜌𝑇^𝑚𝑘𝑥−𝑇^𝑛𝑘𝑥≤𝑚𝑇^𝑚𝑘−1𝑥,𝑇^𝑛𝑘−1𝑥. (40) On the other hand, 𝑚𝑇^𝑚𝑘−1𝑥,𝑇^𝑛𝑘−1𝑥=max𝜌𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘−1𝑥,𝜌𝑇^𝑚𝑘𝑥−𝑇^𝑚𝑘−1𝑥,𝜌𝑇^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥,𝜌12𝑇^𝑚𝑘𝑥−𝑇^𝑛𝑘−1𝑥1+𝜌2𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥(2)⁻¹,𝜌𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘−1𝑥=𝜌𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥+𝑇^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥≤𝜌2𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥+𝜌2𝑇^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥<𝜀+𝜌2𝑇^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥. (41) For the last term in 𝑚(𝑇^𝑚𝑘−1𝑥,𝑇^𝑛𝑘−1𝑥), by Proposition 8, we have 𝜌12𝑇^𝑚𝑘𝑥−𝑇^𝑛𝑘−1𝑥1+𝜌2𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥(2)⁻¹=𝜌12𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥1+𝜌2𝑇^𝑚𝑘𝑥−𝑇^𝑚𝑘−1𝑥+𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥+12𝑇^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥(2)⁻¹≤𝜌12𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥+𝜌𝑇^𝑚𝑘𝑥−𝑇^𝑚𝑘−1𝑥+𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥+𝜌𝑇^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥(2)⁻¹≤𝜌12𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥+𝜌2𝑇^𝑚𝑘𝑥−𝑇^𝑚𝑘−1𝑥+𝜌2𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥+𝜌𝑇^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥(2)⁻¹𝜌𝑇<𝜀+^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥+𝜌2𝑇^𝑚𝑘𝑥−𝑇^𝑚𝑘−1𝑥2. (42) It follow from (41) and (42) that 𝑚𝑇^𝑚𝑘−1𝑥,𝑇^𝑛𝑘−1𝑥=max𝜌𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘−1𝑥,𝜌𝑇^𝑚𝑘𝑥−𝑇^𝑚𝑘−1𝑥,𝜌𝑇^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥,𝜌12𝑇^𝑚𝑘𝑥−𝑇^𝑛𝑘−1𝑥1+𝜌2𝑇^𝑚𝑘−1𝑥−𝑇^𝑛𝑘𝑥(2)⁻¹<max𝜀+𝜌2𝑇^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥,𝜌𝑇^𝑚𝑘𝑥−𝑇^𝑚𝑘−1𝑥,𝜌𝑇^𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥+𝜌𝑇,𝜀_𝑛𝑘𝑥−𝑇^𝑛𝑘−1𝑥+𝜌2𝑇^𝑚𝑘𝑥−𝑇^𝑚𝑘−1𝑥2. (43) By (37), (38), (40), (43), and the Δ₂-condition of 𝜌, we have lim_𝑘→∞𝜌𝑇^𝑚𝑘𝑥−𝑇^𝑛𝑘𝑥=lim_𝑘→∞𝑚𝑇^𝑚𝑘−1𝑥,𝑇^𝑛𝑘−1𝑥=𝜀. (44) Taking 𝑘→∞ in (39), by (44) and the continuity of 𝜓, we get 𝜓(𝜀)≤𝜓(𝜀)−𝜙(𝜀)<𝜓(𝜀) (45) which is a contradiction. Hence, {𝑇^𝑛(𝑥)}_𝑛∈ℕ is 𝜌-Cauchy. Since 𝑋_𝜌 is 𝜌-complete, there exists a point 𝑢∈𝑋_𝜌 such that 𝜌(𝑇^𝑛𝑥−𝑢)→0 as 𝑛→∞. Next, we prove that 𝑢 is a unique fixed point of 𝑇. Suppose that 𝑇𝑢≠𝑢, then 𝜌(𝑢−𝑇𝑢)>0.
Since, 𝜓𝜌𝑇^𝑛𝑥−𝑇𝑢≤𝜓𝑚𝑇^𝑛−1𝑚𝑇𝑥,𝑢−𝜙^𝑛−1,𝑚𝑇𝑥,𝑢_𝑛−1𝜌𝑇𝑥,𝑢=max^𝑛−1𝑥−𝑢,𝜌𝑇^𝑛−1𝑥−𝑇^𝑛𝑥,𝜌(),𝜌1𝑢−𝑇𝑢2𝑇^𝑛−11𝑥−𝑇𝑢+𝜌2𝑢−𝑇^𝑛𝑥(2)⁻¹(),𝜌⟶max0,0,𝜌𝑢−𝑇𝑢((1/2)(𝑢−𝑇𝑢))2=𝜌(𝑢−𝑇𝑢)as𝑛⟶∞. (46)(47) Taking 𝑛→∞ in (46), by using (47), we get 𝜓𝜌()𝜌()𝜌()𝜌()𝑢−𝑇𝑢≤𝜓𝑢−𝑇𝑢−𝜙𝑢−𝑇𝑢<𝜓𝑢−𝑇𝑢 (48) which is a contradiction. Hence, 𝜌(𝑢−𝑇𝑢)=0 and 𝑇𝑢=𝑢. If there exists point 𝑣∈𝑋_𝜌 such that 𝑇𝑣=𝑣 and 𝑢≠𝑣, then using an argument similar to the above we get 𝜓𝜌()𝜌()(𝑚((𝑚(𝜌()𝜌()𝜌()𝑢−𝑣=𝜓𝑇𝑢−𝑇𝑣≤𝜓𝑢,𝑣))−𝜙𝑢,𝑣))≤𝜓𝑢−𝑣−𝜙𝑢−𝑣<𝜓𝑢−𝑣 (49) which is a contradiction. Hence, 𝑢=𝑣 and the proof is complete.

Corollary 13. Let 𝑋_𝜌 be a 𝜌-complete modular space, where 𝜌 satisfies the Δ₂-condition, and let 𝑇∶𝑋_𝜌→𝑋_𝜌 be a mapping satisfying the inequality 𝜌𝑇𝑥−𝑇𝑦≤𝑚𝑥,𝑦−𝜙𝑚𝑥,𝑦 (50) for all 𝑥,𝑦∈𝑋_𝜌, where 𝑚(𝑥,𝑦)=max{𝜌(𝑥−𝑦), 𝜌(𝑥−𝑇𝑥), 𝜌(𝑦−𝑇𝑦), (𝜌((1/2)(𝑥−𝑇𝑦))+𝜌((1/2)(𝑦−𝑇𝑥)))/2} and 𝜙∶[0,∞)→[0,∞) is continuous and monotone nondecreasing function with 𝜙(𝑡)=0 if and only if 𝑡=0. Then, 𝑇 has a unique fixed point.

Proof. Taking 𝜓(𝑡)=𝑡, we obtain the Corollary 13.

Acknowledgments

This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission. The authors would like to thank the referee for his comments and suggestion. C. Mongkolkeha was supported from the Thailand Research Fund through the the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0029/2553).