Approximation of a common minimum-norm fixed point of a finite family of \(\sigma\)-asymptotically quasi-nonexpansive mappings with applications

Volume 9, Issue 5, pp 3240--3254 http://dx.doi.org/10.22436/jnsa.009.05.112

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Authors

Hemant Kumar Pathak - School of Studies in Mathematics, Pt. Ravishankar Shukla University, Raipur (C.G.) 492010, India. Vinod Kumar Sahu - Department of Mathematics, Govt. V.Y.T. P.G. Autonomous College, Durg(C.G.) 491001, India. Yeol Je Cho - Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Korea. - Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia.

Abstract

In this paper, we use the iterative method proposed by Zegeye and Shahzad [H. Zegeye, N. Shahzed, Fixed Point Theory Appl., 2013 (2013), 12 pages] which converges strongly to the common minimum-norm fixed point of a finite family of \(\sigma\)-asymptotically quasi-nonexpansive mappings. As consequence, convergence results to a common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings is proved. Our result generalize and improve a recent result of Zegeye and Shahzad [H. Zegeye, N. Shahzed, Fixed Point Theory Appl., 2013 (2013), 12 pages]. In the sequel, we apply our main result to find solution of minimizer of a continuously Frechet-differentiable convex functional which has the minimum norm in Hilbert spaces.

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Keywords

• Asymptotically quasi-nonexpansive mappings
• asymptotically nonexpansive mappings
• nonexpansive mappings
• minimum-norm fixed point
• strong convergence.

MSC

•  47H09
•  54H25
•  47J25
•  65J15.

References

• [1] F. E. Browder , Convergence of approximats to fixed points of nonexpansive non-linear mappings in Banach spaces, Arch. Ration. Mech. Anal., 24 (1967), 82-90.


• [2] F. E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z., 100 (1967), 201-225.


• [3] C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453.


• [4] Y. Cai, Y. Tang, L. Liu, Iterative algorithms for minimum-norm fixed point of non-expansive mapping in Hilbert space, Fixed point Theory Appl., 2012 (2012), 10 pages.


• [5] Y. Censor, T. Bortfeld, B. Martin, A. Trofimov, A unified approach for inversion problem in intensity-modulated radiation therapy, Phys. Med. Biol., 51 (2006), 2353-2365.


• [6] Y. Censor, T. Elfving, A multiprojection algorithm using Bregman projection in a product space, Numer. Algorithms, 8 (1994), 221-239.


• [7] S.-S. Chang, Y. J. Cho, H. Zhou, Demi-closed principle and weak convergence problems for asymptotically non- expansive mappings, J. Korean Math. Soc., 38 (2001), 1245-1260.


• [8] Y. J. Cho, X. Qin, Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces, J. Comp. Appl. Math., 228 (2009), 458-465.


• [9] Y. L. Cui, X. Liu, Notes on browder's and halpern's methods for nonexpansive mappings, Fixed Point Theory, 10 (2009), 89-98.


• [10] J. B. Diaz, F. B. Metcalf, On the structure of the set of subsequential limit points of successive approximation, Bull. Amer. Math. Soc., 73 (1967), 516{519.


• [11] M. Edelstein, On nonexpansive mappings, Proc. Amer. Math. Soc., 15 (1964), 689-695.


• [12] M. K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl., 207 (1997), 96-103.


• [13] K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174.


• [14] B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961.


• [15] P.-L. Lions, Approximation de points fixes de contractions, C. R. Acad. Sci. Paris Sr. A-B, 284 (1977), 1357-1359.


• [16] P.-E. Mainge, Strong convergence of projected subgradient methods and nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912.


• [17] E. M. Mazcunan-Navarro, Three-dimensional convexity and the fixed point property for nonexpansive mappings, Nonlinear Anal., 71 (2009), 587-592.


• [18] J. G. O'Hara, P. Pillay, H.-K. Xu, Iterative approaches to convex feasibility problem in Banach spaces, Nonlinear Anal., 64 (2006), 2022-2042.


• [19] W. V. Petryshyn, T. E. Williamson, Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl., 43 (1973), 459-497.


• [20] L. Qihou, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 259 (2001), 1-7.


• [21] M. Qing, Nonexpansive nonselfmaps and generalized quasi-nonexpansive multivalued maps, (Chinese), Qufu Shifan Dexue Xuebao Ziran Kexue Ban, 13 (1987), 148-153.


• [22] N. Shahzad, H. Zegeye, Strong convergence of an implicit iteration process for a nite family of generalized asymptotically quasi-nonexpansive maps, Appl. Math. Comput., 189 (2007), 1058-1065.


• [23] W. Takahashi , Nonlinear Functional Analysis: Fixed Points and Applications, Yokohama Publ., Yokahama (2000)


• [24] F. Tricomi , Un teorema sull convergenza delle successioni formate delle successive iterate di una funzione di una variable reale, Giorn. Mat. Battaglini, 54 (1916), 1-9.


• [25] H.-K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109-113.


• [26] H.-K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279-291.


• [27] X. Yang, Y.-C. Liou, Y. Yao, Finding minimum-norm fixed point of nonexpansive mappings and applications , Math. Probl. Eng., 2011 (2011), 13 pages.


• [28] Y. Yao, H.-K. Xu, Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications, Optimization, 60 (2011), 645-658.


• [29] H. Zegeye, A hybrid iteration scheme for equilibrium problems, variational inequality problems and common fixed point problems in Banach spaces, Nonlinear Anal., 72 (2010), 2136-2146.


• [30] H. Zegeye, N. Shahzed, Approximation of the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings, Fixed Point Theory Appl., 2013 (2013), 12 pages.