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,‖.‖)\documentclass[12pt]{minimal}
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\begin{document}$$(E,\Vert .\Vert )$$\end{document} and a nonempty subset C of E, a pair (S, T) of mappings S,T:C→C\documentclass[12pt]{minimal}
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\begin{document}$$S,T:C\rightarrow C$$\end{document} is said to satisfy the jointly demi-closedness principle if {xn}n∈N⊂C\documentclass[12pt]{minimal}
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\begin{document}$$\{x_n\}_{n\in {\mathbb {N}}}\subset C$$\end{document} converges weakly to a point z∈C\documentclass[12pt]{minimal}
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\begin{document}$$z\in C$$\end{document} and limn→∞‖Sxn-Txn‖=0\documentclass[12pt]{minimal}
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\begin{document}$$\lim _{n\rightarrow \infty }\Vert Sx_n-Tx_n\Vert =0$$\end{document}, then S(z)=z\documentclass[12pt]{minimal}
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\begin{document}$$S(z)=z$$\end{document} and T(z)=z\documentclass[12pt]{minimal}
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\begin{document}$$T(z)=z$$\end{document}. We then introduce new modified Halpern’s type iterations to approximate common fixed points of a pair (S, T) 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.
