Approximation of Common Fixed Points of Nonlinear Mappings Satisfying Jointly Demi-closedness Principle in Banach Spaces

  • Eskandar Naraghirad


In this paper we first introduce the notion of jointly demi-closedness principle, extending the notion of demi-closedness principle introduced and studied in Opial (Bull Am Math Soc, 73: 595–597, 1967). Given a Banach space \((E,\Vert .\Vert )\) and a nonempty subset C of E, a pair (ST) of mappings \(S,T:C\rightarrow C\) is said to satisfy the jointly demi-closedness principle if \(\{x_n\}_{n\in {\mathbb {N}}}\subset C\) converges weakly to a point \(z\in C\) and \(\lim _{n\rightarrow \infty }\Vert Sx_n-Tx_n\Vert =0\), then \(S(z)=z\) and \(T(z)=z\). We then introduce new modified Halpern’s type iterations to approximate common fixed points of a pair (ST) of quasi-nonexpansive mappings defined on a closed convex subset C of a Banach space E satisfying the jointly demi-closedness principle in a real Banach space E. Finally, we prove strong convergence theorems of the sequences generated by our algorithms and show that the new convergence for methods known from the literature follows from our general results. We modify Halpern’s iterations for finding common fixed points two quasi-nonexpansive mappings and provide an affirmative answer to an open problem posed by Kurokawa and Takahashi (Nonlinear Anal 73:1562–1568, 2010) in their final remark for nonspreading mappings. Our results improve and generalize many known results in the current literature.


Bregman function jointly demi-closedness principle fixed point strong convergence 

Mathematics Subject Classification

47H10 37C25 


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsYasouj UniversityYasoujIran

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