Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
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POSITIVE SOLUTIONS FOR A NONLINEAR MATRIX EQUATION USING FIXED POINT RESULTS IN EXTENDED BRANCIARI b-DISTANCE SPACES

  • Reena, Jain (Mathematics Division, SASL, VIT Bhopal University) ;
  • Hemant Kumar, Nashine (Mathematics Division, SASL, VIT Bhopal University, Department of Mathematics and Applied Mathematics, University of Johannesburg, Kingsway Campus) ;
  • J.K., Kim (Department of Mathematics Education, Kyungnam University)
  • Received : 2021.03.17
  • Accepted : 2022.03.19
  • Published : 2022.12.06
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Abstract

We consider the nonlinear matrix equation (NMEs) of the form 𝓤 = 𝓠 + Σki=1 𝓐*iℏ(𝓤)𝓐i, where 𝓠 is n × n Hermitian positive definite matrices (HPDS), 𝓐1, 𝓐2, . . . , 𝓐m are n × n matrices, and ~ is a nonlinear self-mappings of the set of all Hermitian matrices which are continuous in the trace norm. We discuss a sufficient condition ensuring the existence of a unique positive definite solution of a given NME and demonstrate this sufficient condition for a NME 𝓤 = 𝓠 + 𝓐*1(𝓤2/900)𝓐1 + 𝓐*2(𝓤2/900)𝓐2 + 𝓐*3(𝓤2/900)𝓐3. In order to do this, we define 𝓕𝓖w-contractive conditions and derive fixed points results based on aforesaid contractive condition for a mapping in extended Branciari b-metric distance followed by two suitable examples. In addition, we introduce weak well-posed property, weak limit shadowing property and generalized Ulam-Hyers stability in the underlying space and related results.

Keywords

Acknowledgement

The second author is thankful to SERB, INDIA for providing fund under the project-CRG/2018/000615, and the third author was supported by the Basic Science Research Program through the National Research Foundation(NRF) Grant funded by Ministry of Education of the republic of Korea (2018R1D1A1B07045427).

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