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: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/6/400 |
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| Summary: | 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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| ISSN: | 2075-1680 |