Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsTo study common fixed point theorems in complete Hausdroff uniform spaces
Volume 3(3) · Pages 270–273 · DOI: 10.22271/maths.2018.v3.i3c.606 ↗
Research areasFixed Point Theorems Analysis
Abstract
Author-writtenThere are several fixed point theorems in Mathematics having numerous applications. In this paper, we will discuss some fixed point theorems that are applicable in Complete Hausdroff uniform space. Some common fixed points of point-valued mappings on a sequentially complete Hausdroff uniform space satisfying contractive type conditions have been obtained.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Fixed point theorems are a significant area of mathematics with a wide range of applications. View evidence ↗
- There is limited existing literature regarding common fixed points of point-valued mappings specifically within Uniform Hausdorff Spaces. View evidence ↗
The question
- The study aims to discuss and establish fixed point theorems applicable to complete Hausdorff uniform spaces. View evidence ↗
- The research seeks to generalize existing fixed point theorems from metric spaces to sequentially complete Hausdorff uniform spaces. View evidence ↗
What they did
- The study utilizes a uniform space (X, υ) defined by an associated family of pseudometrics {𝑑𝜆} that forms a sub-base for the uniformity. View evidence ↗
- The analysis employs an augmented associated family of pseudometrics, which ensures that for any two pseudometrics, a third exists that is greater than or equal to the maximum of the first two. View evidence ↗
- The study defines weakly commutative mappings S and T as those where the distance between STx and TSx is less than or equal to the distance between Tx and Sx for all x and λ. View evidence ↗
- Mappings are defined as compatible if the limit of the distance between STxn and TSxn is zero whenever the limits of Sx n and Tx n are equal. View evidence ↗
What they found
- Corollary 1 proves that a single mapping f satisfying a specific contractive condition has a unique fixed point in X. View evidence ↗
- Corollary 3 shows that mappings f, g, and h have a unique common fixed point given that h is continuous, h and f are compatible, and the range of f is in h. View evidence ↗
- Corollary 4 establishes that f, j, and h have a unique common fixed point if h and j are continuous, f is compatible with both, and the range of f is in the intersection of the ranges of h and j. View evidence ↗
What it means
- The authors conclude that they have successfully obtained several common fixed point theorems within sequentially complete Hausdorff spaces. View evidence ↗
- The authors state that this area of research will serve as a powerful tool for finding existence solutions. View evidence ↗
Visual summary
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Study overview
AI-generated conceptual illustration (A mathematical representation of a Hausdorff uniform space using a set of overlapping geometric spheres and a mapping矢). It explains the idea and does not show study data.
View source evidence ↗Introduction
This paper analyzes some common fixed point theorems in sequentially complete Hausdroff uniform spaces in such a way that they generalize some theorems which have already been proved in metric spaces. Let (X, υ) is a uniform space. A family {𝑑 𝜆 : λ 𝜖 Г, Г is an index set} of pseudometrics is called an associated family for the uniformity u if the family β = {V(I, r): i 𝜖 Г, r > 0}, where V(I, r) = {(x, y): x, y 𝜖 X, d(x, y) < r} is a sub-base for the uniformity u. A family {𝑑 𝜆 }, λ 𝜖 Г, Г is an index set of pseudometrics on X is called an augmented associated family for u if {𝑑 𝜆 : λ 𝜖 Г} is an associated family for u and has the additional property: Given a, b 𝜖 Г, there is a v 𝜖 Г such that 𝑑 𝑣 (x, y) ≥ max {𝑑 𝑎 (x, y), 𝑑 𝑏 (x, y)}. An associated family and an augmented family will be denoted by P and 𝑃 * respectively. For details one can see Kelley [2] , Thorn [5] , etc.
Definitions
Let S and T be self-mapping of a sequentially complete Hausdroff uniform space (X, υ) defined by {𝑑 𝜆 : λ 𝜖 Г} = 𝑃 * . S and Tare said to be weakly commutative on X if 𝑑 𝜆 ((STx, TSx) ≤ 𝑑 𝜆 (Tx, Sx) for all x 𝜖 X and λ 𝜖 Г. Further S and T are said to be compatible if lim 𝑑 𝜆 (ST𝑥 𝑛 , TS𝑥 𝑛 ) = 0 for all λ 𝜖 Г whenever {𝑥 𝑛 } is a sequence in x and that lim 𝑑 𝜆 (S𝑥 𝑛 , t) = 𝑑 𝜆 (T𝑥 𝑛 , t) for all λ 𝜖 Г and for some t 𝜖 λ. The above two definitions are analogous to the definitions as introduced by Sessa [4] and Jungck [1] in metric spaces while proving some common fixed point theorems in metric spaces. It is to be noted that weakly commutative mappings are compatible but the converse is not true by Sastry [3] . Let w: [0, ∞) →[0, ∞) be such that w is continuous and 0 < w(r) < r for r > 0. Now, we prove the following theorems: Theorem 1: Let (X, υ) be a sequentially complete Hausdroff uniform space defined by {𝑑 𝜆 : λ 𝜖 Г} = 𝑃 * . Let f, g, h, and jss are four mappings on X satisfying. 𝑑 𝜆 (fx, gy) ≤ max {𝑑 𝜆 (jx, hy), 𝑑 𝜆 (fx, gx), 𝑑 𝜆 (gy, hy) for all x, y 𝜖 X, λ 𝜖 Г.
Let h and j be continuous, h and g be compatible, f and j be compatible. If f(x) ⊂ h(x) and g(x) ⊂ j(x), then f, g, h, and j have a unique common fixed point in X.
Proof: Suppose f(x) ⊂ h(x) and g(x) ⊂ j(x), Let 𝑥 0 𝜖 X. Then there exists a sequence {𝑥 𝑛 } such that 𝑦 2𝑛 = f𝑥 2𝑛 = h𝑥 2𝑛+1 and 𝑦 2𝑛+1 = g𝑥 2𝑛+1 = j𝑥 2𝑛+2 for all n = 0, 1, 2,…. Now for all x, y 𝜖 X, λ 𝜖 Г. We have
Which is a contradiction and so 𝑑 𝜆 (𝑦 2𝑛 , 𝑦 2𝑛+1 ) ≤ 𝑑 𝜆 (𝑦 2𝑛-1 , 𝑦 2𝑛 ). We next show that lim 𝑛→∞ 𝑑 𝜆 (𝑦 𝑛-1 , 𝑦 𝑛 ) = 0 for each λ 𝜖 Г.
Since 𝑑 𝜆 (𝑦 𝑛-1 , 𝑦 𝑛 ) is a decreasing sequence of non-negative terms, then lim We now show that {𝑦 𝑛 } is a Cauchy sequence. Then for every positive number 𝜀 and for every positive integer K, there exists two positive integers 2m(K) and 2n(K) such that 2m(K) > 2n(K) > K and 𝑑 𝜆 (𝑦 2𝑚(𝐾) , 𝑦 2𝑛(𝐾) ) > 𝜀 and 𝑑 𝜆 (𝑦 2𝑚(𝐾)-2 , 𝑦 2𝑛(𝐾) ) ≤ 𝜀. Further let 2m(K) denote the smallest even integer for which 2m(K) > 2n(K) > k, 𝑑 𝜆 (𝑦 2𝑚(𝐾) , 𝑦 2𝑛(𝐾) ) > 𝜀 and 𝑑 𝜆 (𝑦 2𝑚(𝐾)-2 , 𝑦 2𝑛(𝐾) ) ≤ 𝜀 Now, 𝜀 < 𝑑 𝜆 (𝑦 2𝑚(𝐾) , 𝑦 2𝑛(𝐾) ) ≤ 𝑑 𝜆 (𝑦 2𝑚(𝐾)-2 , 𝑦 2𝑛(𝐾) ) + 𝑑 𝜆 (𝑦 2𝑚(𝐾)-2 , 𝑦 2𝑛(𝐾)-1 ) + 𝑑 𝜆 (𝑦 2𝑚(𝐾)-1 , 𝑦 2𝑛(𝐾) ).
Letting K→∞, we get 𝑑 𝜆 (𝑦 2𝑚(𝐾) , 𝑦 2𝑛(𝐾) ) = 𝜀 for each λ 𝜖 Г. Letting K→∞, we get, 𝜀 ≤ 𝜀 -ω(𝜀) < 𝜀.
Which is a contradiction. Thus {𝑦 𝑛 } 𝑖𝑠 𝑎 Cauchy sequence. Since X is sequentially complete, there is a point ξ 𝜖 X such that ξ = lim 𝑛→∞ (𝑦 𝑛 ). Consequently, {f𝑥 2𝑛 } = { h𝑥 2𝑛+1 } and {g𝑥 2𝑛+1 } = { j𝑥 2𝑛+2 } converges to ξ. The mapping j is continuous. Then we have for all λ 𝜀 Г,
Since the mapping f and j are compatible, then we have, 𝑑 𝜆 (jξ, ξ) ≤ max {𝑑 𝜆 (jξ, ξ), 𝑑 𝜆 (jξ, jξ), 𝑑 𝜆 (ξ, ξ), 𝑑 𝜆 (jξ, ξ)} -w[max 𝑑 𝜆 (jξ, ξ) ≤ max {𝑑 𝜆 (jξ, ξ), 𝑑 𝜆 (jξ, jξ), 𝑑 𝜆 (ξ, ξ), 𝑑 𝜆 (jξ, ξ)}] = 𝑑 𝜆 (jξ, ξ)w(𝑑 𝜆 (jξ, ξ)). Now, we consider ξ ≠ jξ. Since(X, υ) is a Hausdroff space and ξ ≠ jξ, there is an index μ 𝜀 Г such that 𝑑 𝜇 (ξ, jξ) ≠ 0.
Therefore we have, 𝑑 𝜇 (jξ, ξ) ≤ 𝑑 𝜇 (jξ, ξ)w(𝑑 𝜇 (jξ, ξ)) < 𝑑 𝜇 (jξ, ξ) which is a contradiction. Hence jξ = ξ. Taking limit as n→∞, we have, 𝑑 𝜆 (fξ, ξ) ≤ max[𝑑 𝜆 (ξ, ξ), 𝑑 𝜆 (fξ, ξ), 𝑑 𝜆 (ξ, ξ), i.e. 𝑑 𝜆 (fξ, ξ) ≤ 𝑑 𝜆 (fξ, ξ)w(𝑑 𝜆 (fξ, ξ)) for each λ 𝜖 Г, which is a contradiction. Hence fξ = jξ = ξ. Since h is continuous, we can show that gξ = hξ = ξ and ξ is a common fixed point of f, g, h, and j. To prove uniqueness of ξ, if possible let 𝜉 ′ be another fixed point of f, g, h, and j and let ξ ≠ 𝜉 ′ . Then there is an index ν 𝜖 Г such that 𝑑 ν (ξ, 𝜉
ξ), 𝑑 ν (ξ, 𝜉 ′ )}] = 𝑑 ν (ξ, 𝜉 ′ )w[𝑑 ν (ξ, 𝜉 ′ )) < 𝑑 ν (ξ, 𝜉 ′ ) which is a contradiction. Hence ξ = 𝜉 ′ and ξ is the unique common fixed point off, g, h and j.
Corollary 1
Let (X, υ) be a sequentially complete Hausdroff Uniform Space defined by {𝑑 𝜆 : λ 𝜖 Г} = 𝑃 * . Let f be a mapping on X satisfying the condition 𝑑 𝜆 (fx, gy) ≤ max{ 𝑑 𝜆 (x, y), 𝑑 𝜆 (hx, x), 𝑑 𝜆 (hy, y), 𝑑 𝜆 (x,hy)+𝑑 𝜆 (y,fx) 2 } -w[max 𝑑 𝜆 (fx, gy) ≤ max{ 𝑑 𝜆 (x, y), 𝑑 𝜆 (hx, x), 𝑑 𝜆 (hy, y), 𝑑 𝜆 (x,hy)+𝑑 𝜆 (y,fx) 2 }] for all x, y 𝜖 X, λ 𝜖 Г. Then f has a unique fixed point in X.
Proof: Put f = g and j = h = I, identity mapping in theorem 1, Corollary 1 follows.
Corollary 2
Let (X, υ) be a sequentially complete Hausdroff Uniform Space defined by {𝑑 𝜆 : λ 𝜖 Г} = 𝑃 * . Let f, h be mappings on X satisfying the condition
References
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- Thorn WJ. Topological structure, Rinehart and Winston, New York 1966.