Abstract
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 (S, T) 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 (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.
Similar content being viewed by others
References
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 595–597 (1967)
Falset, J.-G., Liorens-Fuster, E., Marino, G., Rugiano, A.: On strong convegence of Halpern’s method for quasi-nonexpansive mappings in Hilbert spaces. Math. Modell. Anal. 21(1), 63–82 (2016)
Bregman, L.M.: The relation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)
Butnariu, D., Iusem, A.N.: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic Publishers, Dordrecht (2000)
Censor, Y., Lent, A.: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321–358 (1981)
Matsushita, S., Takahashi, W.: A strong convegence theorem for relatively nonexpansive mappings in a Banch space. J. Approx. Theroy 134, 257–266 (2005)
Iiduka, H., Takahashi, W.: Strong convergence theorem by a hybrid method for nonlinear mappings of nonexpansive and monotone type and applications. Adv. Nonlinear. Var. Inequal. 9, 1–10 (2006)
Matsushita, S., Takahashi, W.: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2004, 37–47 (2004)
Takahashi, Wataru, Zembayashi, Kei: Strong and weak convergence theorems for equlibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45–57 (2009)
Takahashi, W.: Nonlinear Functional Analysis, Fixed Point Theory and its Applications. Yokahama Publishers, Yokahama (2000)
Reich, S.: A weak convergence theorem for the altering method with Bregman distances. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York, pp. 313–318 (1996)
Halpern, B.: Fixed points of nonexpanding mappings. Bull. Am. Math. Soc. 73, 957–961 (1967)
Reich, S.: Weak convergence theorems for nonexpansive mappings in Banch spaces. J. Math. Anal. Appl. 67, 274–276 (1979)
Moudafi, A.: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Problems 23, 1635–1640 (2007)
Kohsaka, F., Takahashi, W.: Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Arch. Math. (Basel) 91, 166–177 (2008)
Kohsaka, F., Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 19, 824–835 (2008)
Kurokawa, Y., Takahashi, W.: Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces. Nonlinear Anal. 73, 1562–1568 (2010)
Iemoto, S., Takahashi, W.: Approximating commom fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Anal. 71, e2080–e2089 (2009)
Osilike, M.O., Isiogugu, F.O.: Weak and strong convergence theorems of nonspreading-type mappings. Nonlinear Anal. 74, 1814–1822 (2011)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker Inc, New York (1984)
Xu, H.-K., Kim, T.K.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory. Appl. 119(1), 185–201 (2003)
Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set Valued Anal. 16, 899–912 (2008)
Cho, Y.J., Zhou, H.Y., Guo, G.: Weak and converence theorems for three-step iterations with with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 47, 707–717 (2004)
Xu, H.-K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Naraghirad, E. Approximation of Common Fixed Points of Nonlinear Mappings Satisfying Jointly Demi-closedness Principle in Banach Spaces. Mediterr. J. Math. 14, 162 (2017). https://doi.org/10.1007/s00009-017-0962-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00009-017-0962-2