Some fixed point results under generalized distance

In this paper, the historical account of distances in metric space have introduced with generalize metric space which have applications to obtaining the solutions of various new important problems in nonlinear analysis.

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Introduction

In 1996, w-distance introduced a notion of on a metric space by Kada et al. 1996 [4] and using this notion to improve the Caristi’s fixed point theorem, and Ekeland variational principle. Using this notion of w-distance, Suzuki and Takahashi [9] have introduced notions of single- valued, Takahashi minimization theorem and multivalued weakly contractive i.e. w-contractive mappings and proved fixed point results for two or more mappings. There are two useful lemma as follows Lemma 1.1 [4]: Let (𝑋, 𝑑) be a metric space and let 𝑝 be a 𝑤-distance on 𝑋. Let {𝑥𝑛} and {𝑦𝑛} be sequences in 𝑋 and let { 𝛼𝑛} and {𝛽𝑛} be sequences in [0, ∞)converging to 0.

Then, the following statements hold for every 𝑥, 𝑦, 𝑧∈𝑋: (a) If 𝑝(𝑥𝑛, 𝑦) ≤ 𝛼𝑛 and 𝑝(𝑥𝑛, 𝑧) ≤ 𝛽𝑛for any 𝑛∈ 𝑁, then 𝑦 = 𝑧, in particular, if 𝑝 (𝑥, 𝑦) = 0 and 𝑝(𝑥, 𝑧) = 0, then 𝑦 = 𝑧; (b) If 𝑝(𝑥𝑛, 𝑦𝑛) ≤𝛼𝑛 and 𝑝(𝑥𝑛, 𝑧) ≤ 𝛽𝑛 for any 𝑛∈𝑁, Then 𝑦𝑛converges to 𝑧; (c) If 𝑝 (𝑥𝑛, 𝑥𝑚) ≤ 𝛼𝑛for any 𝑛, 𝑚∈𝑁 With 𝑚> 𝑛, then {𝑥𝑛} is a Cauchy sequence; (d) If 𝑝 (𝑦, 𝑥𝑛) ≤ 𝛼𝑛 for any 𝑛∈𝑁, then 𝑥𝑛 is a Cauchy sequence. Lemma 1.2: Let (𝑋, 𝑑) be a metric space and let 𝑝 be a 𝑤-distance on 𝑋. Let 𝐾 be a closed subset of 𝑋. Suppose that there exists 𝑢∈𝑋 Such that 𝑝(𝑢, 𝑢) = 0.

Then, 𝑝(𝑢, 𝐾) = 0 if and only if 𝑢∈𝐾, Where 𝑝(𝑢, 𝐾) = 𝑖𝑛𝑓𝑦∈𝐾 𝑝(𝑢, 𝑦). In 1996, Kada, Suzuki and Takahashi improved Caristi’s fixed point theorem as follows; Theorem 1.1: [4] Let (𝑋, 𝑑) be a complete metric space and 𝑝 be a w-distance on 𝑋. Then, each Caristi mapping 𝑇 on 𝑋 with respect to 𝑝 has a fixed point 𝑥𝑛∈𝑋 and 𝑝(𝑥0, 𝑥0) = 0. In 1996, Suzuki and Takahashi improved the Banach Contraction Principle as follows; Theorem 1.2: [5] Let (𝑋, 𝑑) be a complete metric space and 𝑝 be a w-distance on X.

Then, each contraction mapping 𝑇 on 𝑋 with respect to 𝑝 has a unique fixed point 𝑥0 ∈𝑋 and 𝑝(𝑥𝑜, 𝑥0) = 0. Maths 2022; 7(6): 35-38 Pavan Kumar Jha Research Centre for Applied Science Technology (RECAST), T. U., Kirtipur, Nepal Khageshwar Mandal Department of Mathematics and Statistics, Padma Kanya Multiple Campus, T. U., Nepal Pavan Kumar Jha Research Centre for Applied Science Technology (RECAST), T.

U., Kirtipur, Nepal https://www.mathsjournal.com Theorem 1.3: Let (𝑋, 𝑑) be a metric space. Then, 𝑋 is complete if and only if every contraction mapping 𝑇 on 𝑋 with respect to 𝑝 has a fixed point in 𝑋. In 1998, Ume established the result as follows; Theorem 1.4 [11]: Let (𝑋, 𝑑) be a complete metric space and let 𝑝 be a w-distance on X. Let 𝑇∶ 𝑋 →𝑋 be a mapping such that for a fixed constant ℎ < 1 and for all 𝑥, 𝑦∈𝑋, 𝑝(𝑇(𝑥), 𝑇(𝑦)) ≤𝑚𝑎𝑥 {𝑝(𝑥, 𝑦), 𝑝(𝑥, 𝑇𝑥), 𝑝(𝑦, 𝑇𝑦), 𝑝(𝑥, 𝑇𝑦), 𝑝(𝑦, 𝑇𝑥)}, And 𝑖𝑛𝑓{𝑝(𝑥, 𝑢) + 𝑝(𝑥, 𝑇𝑥) ∶ 𝑥∈𝑋} > 0, for every 𝑢∈𝑋.

With 𝑢≠ 𝑓(𝑢). Then, 𝑇 has a unique fixed point 𝑥0 ∈𝑋 and 𝑝(𝑥0, 𝑥0) = 0. In 1998, Shioji, Suzuki and Takahashi proved the following lemma; Lemma 1.3 [6]: Let (𝑋, 𝑑) be a metric space with metric, let 𝑝 be a w-distance on 𝑋, let 𝑇 be a mapping of 𝑋 into itself and let 𝑢 be a point in 𝑋 Such that lim 𝑚,𝑛→∞𝑝(𝑇𝑚𝑢, 𝑇𝑛𝑢) = 0. Then for every 𝑥 ∈ 𝑋, lim 𝑘→∞𝑝(𝑇𝑘𝑢, 𝑥) = 0.

And lim 𝑚,𝑛→∞𝑝(𝑥, 𝑇𝑘𝑢) = 0 exist. Moreover, let 𝛽 and 𝛾 be functions from 𝑋 to [0, ∞) defined by 𝛽(𝑥) = lim 𝑘→∞𝑝(𝑇𝑘𝑢, 𝑥) = 0. And 𝛾(𝑥) = lim 𝑘→∞𝑝(𝑥, 𝑇𝑘𝑢) = 0. Then the following hold: (i) 𝛽 is lower semicontinuous on 𝑋; (ii) for every 𝜀 > 0, there exists 𝛿 > 0 such that 𝛽(𝑥) ≤ 𝛿 and 𝛽(𝑦) ≤ 𝛿 imply 𝑑(𝑥, 𝑦) ≤ 𝜀.

In particular, the set {𝑥 ∈ 𝑋∶ 𝛽(𝑥) = 0} consists of at most one point; (iii) The functions 𝑞0 and 𝑞1from 𝑋 × 𝑋 𝑡𝑜 [0, ∞) defined by 𝑞0(𝑥, 𝑦) = 𝛽(𝑥) + 𝛽(𝑦) And 𝑞1(𝑥, 𝑦) = 𝛾(𝑥) + 𝛽(𝑦) are w-distances on 𝑋. In 1998, Shioji, Suzuki and Takahashi proved the following theorem by using lemma 1.2; Theorem 1.5 [6]: Let (𝑋, 𝑑) be a metric space. Then 𝑊𝐶1(𝑋) = 𝑊𝐶0(𝑋) = 𝑊𝐾1(𝑋) = 𝑊𝐾0(𝑋) ⊂𝑊𝐶2(𝑋) = 𝑊𝐾2(𝑋). In 2010, Iemoto, Takahashi and Yingtaweesittikul proved the following result by the help of above lemma; Theorem 1.6 [3]: Let (𝑋, 𝑑) be a metric space and let 𝑝 be a w-distance on 𝑋 such that 𝑝(𝑥, 𝑥) = 0 for all 𝑥 ∈ 𝑋.

Let 𝑇 be a p-contractively non spreading mapping of 𝑋 into itself. Then 𝑇 is in 𝑊𝐶0(𝑋). In 2011, Hasegawa, Komiya and Takahashi proved the following theorem, Theorem 1.7 [2]: Let (𝑋, 𝑑) be a complete metric space and let 𝑇 be a mapping of 𝑋 into itself. Suppose that there exist a real number 𝑟 with 0 ≤ 𝑟 < 1 And an element 𝑥 ∈ 𝑋 such that {𝑇𝑛𝑥} is bounded and 𝜇𝑛𝑑(𝑇𝑛𝑥, 𝑇𝑦) ≤ 𝑟𝜇𝑛𝑑(𝑇𝑛𝑥, 𝑦), ∀𝑦 ∈ 𝑋 For some mean µ on 𝑙∞.

Then, the following results hold (i) 𝑇 has a unique fixed point 𝑢 in 𝑋; (ii) For every 𝑧 ∈ 𝑋, the sequence {𝑇𝑛𝑧} converges to 𝑢. In 2013, Takahashi, Wong and Yao proved following fixed point a theorem by using lemma 1.2 with w-distances in complete metric spaces. Which is direct consequence of above theorem; Theorem 1.8 [10]: Let (𝑋, 𝑑) be a complete metric space, let 𝑝 ∈𝑤0(𝑋) and let {𝑥𝑛} be a sequence in X such that {𝑝(𝑥𝑛, 𝑥)} is bounded for some 𝑥 ∈ 𝑋. Let 𝑇 be a mapping of 𝑋 into itself.

Suppose that there exist a real number 𝑟 ∈ [0,1) and a mean µ on ℓ∞. Such that 𝜇𝑛𝑝(𝑥𝑛, 𝑇𝑦) ≤ 𝑟𝜇𝑛𝑝(𝑥𝑛, 𝑦), ∀𝑦 ∈ 𝑋. Then, the following hold: (i) 𝑇 has a unique fixed point 𝑢 in 𝑋; (ii) For every 𝑧 ∈ 𝑋, the sequence {𝑇𝑛𝑧} converges to 𝑢. In 2013, the following theorems established by Takahashi, Wong and Yao; https://www.mathsjournal.com Theorem 1.9 [10]: Let (𝑋, 𝑑) be a complete metric space and let 𝑝 be a w-distance on 𝑋.

Let 𝑇∶ 𝑋 → 𝑋 be a p-contractive mapping, i.e., there exists a real number 𝑟 with 0 ≤ 𝑟 < 1 such that 𝑝 (𝑇𝑥, 𝑇𝑦) ≤ 𝑟𝑝(𝑥, 𝑦) For all 𝑥, 𝑦 ∈ 𝑋. Then, the following results hold (i) 𝑇 has a unique fixed point 𝑢 in 𝑋; (ii) For every 𝑧 ∈ 𝑋, the sequence {𝑇𝑛𝑧} converges to 𝑢. Theorem 1.10: Let (𝑋, 𝑑) be a complete metric space and let 𝑝 be aw-distance on 𝑋. Let 𝑇 ∈ 𝑊𝐾1(𝑋), i.e., there exists 𝛼 ∈[0,1/2) such that 𝑝(𝑇𝑥, 𝑇𝑦) ≤ 𝛼{𝑝(𝑇𝑥, 𝑥) + 𝑝(𝑇𝑦, 𝑦)} for all 𝑥, 𝑦 ∈ 𝑋.

Then, the following hold (i) 𝑇 has a unique fixed point 𝑢 in 𝑋; (ii) For every 𝑧 ∈ 𝑋, the sequence {𝑇𝑛𝑧} converges to u. Theorem 1.11: Let (𝑋, 𝑑) be a complete metric space and let 𝑝 be aw-distance on 𝑋 such that 𝑝(𝑥, 𝑥) = 0 for all 𝑥 ∈ 𝑋. Let 𝑇∶ 𝑋 → 𝑋 be p-contractively nonspreading, i.e., there exists a real number 𝛾 with 0 ≤ 𝛾 < 1 /2 such that 𝑝(𝑇𝑥, 𝑇𝑦) ≤ 𝛾{𝑝(𝑇𝑥, 𝑦) + 𝑝(𝑥, 𝑇𝑦)} for all 𝑥, 𝑦 ∈ 𝑋. Then, the following results hold: (i) 𝑇 has a unique fixed point 𝑢 in 𝑋; (ii) For every 𝑧 ∈ 𝑋, the sequence {𝑇𝑛𝑧} converges to 𝑢.

Lemma 1.4: Let (𝑋, 𝑑) be a complete metric space and let 𝑝 be a symmetric w-distance on 𝑋. Let 𝑇∶ 𝑋 → 𝑋 be a p-contractively generalized hybrid mapping. Then 𝑇 has a fixed point in 𝑋 if and only if {𝑝(𝑇𝑛𝑥, 𝑥)}is bounded for some 𝑥 ∈ 𝑋. In this case, the following results hold: (i) 𝑇 has a unique fixed point 𝑢 in 𝑋; (ii) For every 𝑧 ∈ 𝑋, the sequence {𝑇𝑛𝑧} converges to 𝑢.

In general, a τ -distance 𝑝 does not necessarily satisfy 𝑝(𝑥, 𝑥) = 0. The metric 𝑑 is a τ-distance on 𝑋. Each w-distance on a metric space 𝑋 is also a τ- distance on 𝑋.Using the concept of τ -distance, Susuki improved the Banach Contraction Principle and Caristi’s fixed point theorem as follows; Theorem 1.12 [7]: Let (𝑋, 𝑑) be a complete metric space and 𝑝 be aτ –distance on 𝑋. Then, each contraction mapping 𝑇 on 𝑋 with respect to 𝑝 has a unique fixed point 𝑥0 ∈𝑋 and 𝑝(𝑥0, 𝑥0) = 0.

Theorem 1.13 [7]: Let (𝑋, 𝑑) be a complete metric space and 𝑝 be aτ -distance on 𝑋. Then, each Caristi’s mapping 𝑇 on 𝑋 with respect to 𝑝 has a fixed point 𝑥0 ∈𝑋 and 𝑝(𝑥0, 𝑥0) = 0. On the basis of this result, Suzuki obtained generalized Caristi’s fixed point theorem as follows; Theorem 1.14 [8]: Let (𝑋, 𝑑) be a complete metric space, 𝑝 be aτ -distance on 𝑋 and Let g : 𝑋 →(0, ∞) be a function such that for some𝑟> 0, 𝑠𝑢𝑝{𝑔(𝑥) ∶ 𝑥∈𝑋, 𝜓 (𝑥) ≤ 𝜓 (𝑧) 𝑧∈𝑋 𝑖𝑛𝑓 + 𝑟} < ∞, where 𝜓∶ 𝑋 → (0, ∞) is a lower semicontinuous function. Let 𝑇∶ 𝑋 →𝑋 be a map such that for each 𝑥∈𝑋, 𝑝(𝑥, 𝑇(𝑥)) ≤𝑔(𝑥)(𝜓 (𝑥) − 𝜓 (𝑇(𝑥))).

Then, there exists 𝑥0 ∈𝑋 such that 𝑇(𝑥0) = 𝑥0 𝑎𝑛𝑑 𝑝(𝑥0, 𝑥0) = 0.

Conclusion

Now a days, distances in metric space have been introduced and generalize metric space and its applications to obtaining the solutions of several new important problems in nonlinear analysis.

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