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Resolvents of Set-Valued Monotone Vector Fields in Hadamard Manifolds

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Abstract

Firmly nonexpansive mappings are introduced in Hadamard manifolds, a particular class of Riemannian manifolds with nonpositive sectional curvature. The resolvent of a set-valued vector field is defined in this setting and by means of this concept, a strong relationship between monotone vector fields and firmly nonexpansive mappings is established. This fact is then used to prove that the resolvent of a maximal monotone vector field has full domain. The Yosida approximation of a set-valued vector field is also introduced, analyzing its properties from which the asymptotic behavior of the resolvent is studied. Regarding the singularities of a set-valued monotone vector field, existence results are proved under certain boundary condition. As a consequence, the existence of fixed points for continuous pseudo-contractive mappings is obtained.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Chong Li

  2. Departamento de Análisis Matemático, Universidad de Sevilla, 41080, Sevilla, Spain

    Genaro López & Victoria Martín-Márquez

  3. Department of Mathematics, Zhejiang University of Technology, Hangzhou, 310032, People’s Republic of China

    Jin-Hua Wang

Authors
  1. Chong Li
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  2. Genaro López
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  3. Victoria Martín-Márquez
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  4. Jin-Hua Wang
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Correspondence to Victoria Martín-Márquez.

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Li, C., López, G., Martín-Márquez, V. et al. Resolvents of Set-Valued Monotone Vector Fields in Hadamard Manifolds. Set-Valued Anal 19, 361–383 (2011). https://doi.org/10.1007/s11228-010-0169-1

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