<p>In this paper, new classes of nonlinear maps, namely strictly-asymptotically pseudocontractive-type and asymptotically pseudocontractive-type multivalued maps, are introduced in the setting of a real Hilbert space. The paper also provides weak and strong convergence theorems of modified Mann and Ishikawa sequence of iterates for these classes of multivalued maps. Moreover, the paper establishes the demiclosedness property for these multivalued maps ⅁, under the condition that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3398_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <msup> <mi mathvariant="normal">⅁</mi> <mi>n</mi> </msup> </msub> </math></EquationSource> <EquationSource Format="TEX">$P_{\Game ^{n}}$</EquationSource> </InlineEquation> is not asymptotically nonexpansive, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3398_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="297" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <msup> <mi mathvariant="normal">⅁</mi> <mi>n</mi> </msup> </msub> <mo>=</mo> <mo stretchy="false">{</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo>∈</mo> <msup> <mi mathvariant="normal">⅁</mi> <mi>n</mi> </msup> <mo>:</mo> <mo stretchy="false">∥</mo> <mi mathvariant="normal">℘</mi> <mo>−</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">∥</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">℘</mi> <mo>,</mo> <msup> <mi mathvariant="normal">⅁</mi> <mi>n</mi> </msup> <mi mathvariant="normal">℘</mi> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> </math></EquationSource> <EquationSource Format="TEX">$P_{\Game ^{n}}=\{u_{n}\in \Game ^{n}:\Vert \wp -u_{n}\Vert =d(\wp , \Game ^{n}\wp )\}$</EquationSource> </InlineEquation>. The obtained results improve, extend and complement several existing results of single-valued and multivalued maps in the contemporary literature.</p>

Advances in multivalued mapping theory: fixed points and demiclosedness for asymptotically pseudocontractive-type mappings in real Hilbert spaces

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2025年11月13日

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Imo Kalu Agwu,
Hüseyin Işık,
Donatus Ikechi Igbokwe

摘要

In this paper, new classes of nonlinear maps, namely strictly-asymptotically pseudocontractive-type and asymptotically pseudocontractive-type multivalued maps, are introduced in the setting of a real Hilbert space. The paper also provides weak and strong convergence theorems of modified Mann and Ishikawa sequence of iterates for these classes of multivalued maps. Moreover, the paper establishes the demiclosedness property for these multivalued maps ⅁, under the condition that $P_{⅁^{n}}$ $P_{\Game ^{n}}$ is not asymptotically nonexpansive, where $P_{⅁^{n}} = {u_{n} \in ⅁^{n} : ∥ ℘ - u_{n} ∥ = d (℘, ⅁^{n} ℘)}$ $P_{\Game ^{n}}=\{u_{n}\in \Game ^{n}:\Vert \wp -u_{n}\Vert =d(\wp , \Game ^{n}\wp )\}$ . The obtained results improve, extend and complement several existing results of single-valued and multivalued maps in the contemporary literature.