Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces

Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces

Let nonempty subsets <i>E</i> and <i>F</i> of a Banach space <i>X</i> be given, along with a mapping <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant=&...

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Main Authors: Moosa Gabeleh, Maggie Aphane
Format: Article
Language:English
Published: MDPI AG 2025-05-01
Series:Axioms
Subjects:
optimal pair of fixed points
strictly convex Banach space
noncyclic relatively nonexpansive
proximal projection map
Online Access:https://www.mdpi.com/2075-1680/14/6/400
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author Moosa Gabeleh
Maggie Aphane
author_facet Moosa Gabeleh
Maggie Aphane
author_sort Moosa Gabeleh
collection DOAJ
description Let nonempty subsets <i>E</i> and <i>F</i> of a Banach space <i>X</i> be given, along with a mapping <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>:</mo><mi>E</mi><mo>∪</mo><mi>F</mi><mo>→</mo><mi>E</mi><mo>∪</mo><mi>F</mi></mrow></semantics></math></inline-formula> defined as noncyclic when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><mi>E</mi><mo>)</mo><mo>⊆</mo><mi>E</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>F</mi></mrow></semantics></math></inline-formula>. In this case, an optimal pair of fixed points is defined as a point <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>∈</mo><mi>E</mi><mo>×</mo><mi>F</mi></mrow></semantics></math></inline-formula> where <i>p</i> and <i>q</i> are fixed points of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula> that estimate the distance between <i>E</i> and <i>F</i>. This article explores an extended version of Göhde’s fixed point problem to identify optimal fixed point pairs for noncyclic relatively nonexpansive maps in strictly convex Banach spaces, while introducing new classes of noncyclic Kannan contractions, noncyclic relatively Kannan nonexpansive contractions using the proximal projection mapping defined on union of proximal pairs, and proving additional existence results with supporting examples.
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spelling doaj-art-0b46c16228794d0891eb0b1a6a2b0a692025-08-20T03:27:11ZengMDPI AGAxioms2075-16802025-05-0114640010.3390/axioms14060400Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach SpacesMoosa Gabeleh0Maggie Aphane1Department of Mathematics, Faculty of Basic Sciences, Ayatollah Boroujerdi University, Boroujerd 6919969737, IranDepartment of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Pretoria 0204, South AfricaLet nonempty subsets <i>E</i> and <i>F</i> of a Banach space <i>X</i> be given, along with a mapping <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>:</mo><mi>E</mi><mo>∪</mo><mi>F</mi><mo>→</mo><mi>E</mi><mo>∪</mo><mi>F</mi></mrow></semantics></math></inline-formula> defined as noncyclic when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><mi>E</mi><mo>)</mo><mo>⊆</mo><mi>E</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">S</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>F</mi></mrow></semantics></math></inline-formula>. In this case, an optimal pair of fixed points is defined as a point <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>∈</mo><mi>E</mi><mo>×</mo><mi>F</mi></mrow></semantics></math></inline-formula> where <i>p</i> and <i>q</i> are fixed points of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula> that estimate the distance between <i>E</i> and <i>F</i>. This article explores an extended version of Göhde’s fixed point problem to identify optimal fixed point pairs for noncyclic relatively nonexpansive maps in strictly convex Banach spaces, while introducing new classes of noncyclic Kannan contractions, noncyclic relatively Kannan nonexpansive contractions using the proximal projection mapping defined on union of proximal pairs, and proving additional existence results with supporting examples.https://www.mdpi.com/2075-1680/14/6/400optimal pair of fixed pointsstrictly convex Banach spacenoncyclic relatively nonexpansiveproximal projection map
spellingShingle Moosa Gabeleh
Maggie Aphane
Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces
Axioms
optimal pair of fixed points
strictly convex Banach space
noncyclic relatively nonexpansive
proximal projection map
title Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces
title_full Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces
title_fullStr Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces
title_full_unstemmed Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces
title_short Extensions of Göhde and Kannan Fixed Point Theorems in Strictly Convex Banach Spaces
title_sort extensions of gohde and kannan fixed point theorems in strictly convex banach spaces
topic optimal pair of fixed points
strictly convex Banach space
noncyclic relatively nonexpansive
proximal projection map
url https://www.mdpi.com/2075-1680/14/6/400
work_keys_str_mv AT moosagabeleh extensionsofgohdeandkannanfixedpointtheoremsinstrictlyconvexbanachspaces
AT maggieaphane extensionsofgohdeandkannanfixedpointtheoremsinstrictlyconvexbanachspaces

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