On some midpoint-type algorithms.

Year 2018, Volume: 2 Issue: 1, 42 - 61, 25.03.2018

Giuseppe Marino Roberta Zaccone

https://doi.org/10.31197/atnaa.407069

Cited By: 1

Abstract

We introduce iterative methods approximating fixed points for nonlinear operators defined on infinite-dimensional spaces. The starting points are the Implicit and Explicit Midpoint Rules, which generate polygonal functions approximating a solution for an ordinary differential equation infinite-dimensional spaces.

The purpose is to determine suitable conditions on the mapping and the underlying space, in order to get strong convergence of the generated sequence to a common solution of a fixed point problem and a variational inequality. 

Keywords

Nonexpansive mappings, quasi-nonexpansive mappings, iterative algorithms, fixed points, midpoint rules, viscosity approximation, p-uniformly convex Banach spaces.

References

• R.P. Agarwal, D. O’Regan, D.R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings withApplications, Series Topological Fixed Point Theory and Its Applications, Springer, New York,2009;
• Aoyama K, Iemoto S, Kohsaka F, W. Takahashi, Fixed point and ergodic theorems for $\lambda$-hybrid mappingsin Hilbert spaces, J. Nonlinear Convex Anal. 2010;11:335–343;
• M.A. Alghamdi, M.A. Alghamdi, N. Shahzad, H.K. Xu, The implicit midpoint rule for nonexpansive mappings, Fixed PointTheory Appl. 2014, 96 (2014);
• E. Asplund, Positivity of duality mappings, Bull. Amer. Math. Soc, 73 (1967), 200-203;
• S. Banach, Sur les op\'erations dans les ensembles abstraits et leur application aux \'equations integrals , Fundamenta Mathematicae, (1922);
• V. Berinde, Iterative Approximation of Fixed Points, second ed., in: Lecture Notes in Mathematics, vol. 1912, Springer, Berlin, 2007;
• A. Beurling and A. E. Livingston, A theorem on duality mappings in Banach spaces, Ark. Mat. 4 (1961), 405-411;
• F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A., 54 (1965),1041–1044;
• F.E. Browder, Fixed point theorems for nonlinear semicontractive mappings in Banach paces, Arch. Rational Mech. Anal. 21 (1966), 259-269;
• Browder, F.E., Convergence of approximants to fixed points of nonexpansivenonlinear maps in Banach spaces, Arch. Rat. Mech. Anal., 24,82-90 (1967);
• F. E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Amer.Math. Soc. 73 (1967), 875-882;
• F.E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660-665;
• F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Sot. 72 (1966), 571-575.;
• F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197-228;
• R. E. Bruck, Asymptotic behavior of nonexpansive mappings, Contemporary Math. 18 (1983), 1-47;
• C. Byrne, A unified treatment of some iterative algorithms in signal processing andimage reconstruction, Inverse Problems, 20 (2004), 103-120;
• Carnahan, B.; Luther, A. H.; and Wilkes, J., Applied NumericalMethods, John Wiley and Sons, New York, 1969;
• C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009;
• C. E. Chidume and C. O. Chidume, Iterative approximation offixed points of nonexpansive mappings, Journal of MathematicalAnalysis and Applications, vol. 318, no. 1, pp. 288-295, 2006;
• Chidume, C.E., Mutangadura, S.A., An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc., 129,No. 8, 2359-2363 (2001);
• F. Cianciaruso, G. Marino, A. Rugiano, B. Scardamaglia, On strong convergence of viscosity type method usingaveraged type mappings, J. Nonlinear Convex Anal., {\bf 16} (2015), 1619-1640;
• I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990;
• Deimling, K., Zeros of accretive operators, Manuscripta Math., 13, 365-374 (1974);
• F. Deutsch, I. Yamada, Minimizing certain convex functions over the intersection of the fixed point sets ofnonexpansive mappings, Numer. Funct. Anal. Optim. 19 (1998) 33–56;
• J. B. Diaz and F. T. Metcalf, On the set of subsequential limit points of successiveapproximations, Trans. Amer. Math. Soc. 135 (1969), 459-485;
• W.G. Dotson, On the Mann iterative process, Trans. Amer. Math. Soc. 149 (1970)65-73;
• W. G. Dotson, Jr., Fixed points of quasi-nonexpansive mappings, J. Austral. Math.Soc. 13 (1972), 167-170;
• M. Edelstein, A remark on a theorem of M. A. Krasnoselski, Amer. Math. Monthly 73 (1966),509-510;
• R. L. Franks and R. P. Marzec, A theorem on mean value iterations, Proc. Amer.Math. Soc. 30 (1971), 324-326;
• J.Garcia-Falset, W. Kaczor, T. Kuczumow, S. Reich, Weak convergence theorems for asymptotically nonexpansivemappings and semigroups, Nonlinear Anal., {\bf 43} (2001) 377-401;
• J. Garcia-Falset, Existence of fixed points for the sum of two operators, Math. Nachr. 283 (12) (2010) 1726-1757;
• Garcia-Falset, K. Latrach, E. Moreno-Gálvez, M. A Taoudi, Shaefer–Krasnoselskii fixed points theorems using a usual measure of weaknoncompactness, J. Diff. Equ. 352 (2012) 3436-3452;
• J. Garcia-Falset, O. Muniz-Perez, Fixed point theory for 1-set contractive and pseudocontractive mappings, Applied Mathematics and Computation, 219 (2013), 6843-6855;
• J. Garcia-Falset, G. Marino, R. Zaccone, An explicit midpoint algorithm in Banach spaces, to appear in J. Nonlinear and Convex Anal. (2017);
• A. Genel and J. Lindenstrauss, An example concerning fixed points, Israel J. Math., 22 (1975), 81-86;
• M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl. 207 (1997) 96-103;
• D. Gohde, Zum Prinzip der kontraktiven Abbildung, (German) Math. Nachr., 30 (1965), 251–258;
• K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., vol. 28, Cambridge Univ. Press, 1990;
• B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc.73 (1967), 957-961;
• S. He, C. Yang , Solving the variational inequality problem defined on intersection of finite level sets, Abstr. Appl. Anal. Vol. 2013 (2013), Article ID 942315, 8 pages;
• Z. He, W-S Du, Nonlinear algorithms approach to split common solution problems, Fixed Point Theory Appl. 2012,Article ID 130 (2012);
• Hicks, T.L., Kubicek, J.R., On the Mann iteration process in Hilbertspace, J. Math. Anal. Appl., 59, 498-504 (1977);
• N. Hussain, G. Marino, L. Muglia, L. Alamri, On some Mann's type iterative algorithms, Fixed Point Theory Appl 2015 (2015), Article ID 17;
• S. Iemoto and W. Takahashi, Approximating common fixed points of nonexpansive mappingsand nonspreading mappings in a Hilbert space, Nonlinear Anal., 71 (2009), 2082-2089;
• Ishikawa, S., Fixed points and iterations of a nonexpansive mapping in a Banachspace, Proc. Amer. Math. Soc. 59, pp. 65-71 (1976);
• S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974) 147–150;
• J.S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509-520;
• Y. Ke. C. Ma, The generalized viscosity implicit rules of nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl. 2015, 190 (2015), 21 pages;
• T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal., {\bf 61} (2005) 51-60;
• W. A. Kirk, A fixed point theorem for mappings which do not increase distance,Amer. Math. Monthly 72 (1965), 1004-1006;
• F. Kohsaka, W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Archiv der Mathematik, vol. 91, no. 2, pp. 166-177, 2008;
• M. A. Krasnoselskii, Two observations about the method of successive approximations,Usp. Math. Nauk. 10(1955), 123-127;
• K. Latrach, M. Aziz Taoudi, A. Zeghal, Some fixed point theorems of Schauder and the Krasnosel’skii type and application to nonlinear transportequations, J. Diff. Equ. 221 (2006) 256-271;
• Lions, P.-L., Approximation de points fixes de contractions, C.R. Acad.Sci. Ser. A-B Paris, 284, 1357-1359 (1977);
• P.E. Maing\'e Strong convergence of projected subgradient methods for nonsmooth and nonstrictlyconvex minimization, Set-Valued Anal. 16 (2008), 899-912;
• P-E. Mainge, The viscosity approximation process for quasi-nonexpansive mappings in Hilbert ´spaces, Computers and Mathematics with Applications, vol. 59, no. 1, pp. 74–79, 2010;
• W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-510;
• G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (1) (2006) 43-52;
• G. Marino, B. Scardamaglia, R. Zaccone, A general viscosity explicit midpoint rule for quasi-nonexpansive mappings, J. Nonlinear and Convex Anal., vol. 18, n. 1 (2017), 137-148;
• G. Marino, R. Zaccone, On strong convergence of some midpoint type methods for nonexpansive mappings, J. Nonlinear Var. Anal., vol. 1 (2017), n. 2, 159-174;
• C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400-2411;
• D. P. Milman, On some criteria for the regularity of spaces of the type (B)} (Russian), Dokl. Akad. Nauk SSSR 20 (1938);
• A. Moudafi, \emph{Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000),46-55;
• K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,J. Math. Anal. Appl., {\bf 279} (2003), 372-379;
• J.G. O'Hara, P. Pillay, H.K. Xu, Iterative approaches to finding nearest common fixed points of nonexpansivemappings in Hilbert spaces, Nonlinear Anal., {\bf 54} (2003), 1417-1426;
• E. Hairer, S. P. N\o rsett and G. Wanner, Solving Ordinary Differential EquationsI: Nonstiff Problems, Springer Series in Computational Mathematics, 2nd edn.(Springer-Verlag, 1993);
• Z. Opial, Weak convergence of the sequence of successive approximations fornonexpansive mappings, Bull. Amer. Math. Sot. 73 (1967), 591-597;
• J.G. O'Hara, P. Pillay, H.K. Xu, Iterative approaches to finding nearest common fixed points of nonexpansivemappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417-1426;
• W. V. Petryshyn, Construction of fixed points of demicompact mappings inHilbert space, Math. Anal. Appl. 14 (1966), 276-284;
• B. J. Pettis, A proof that every uniformly convex space is reflexive, Duke Math. J. 5 (1939), no. 2, 249--253;
• S. Reich, Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl. 44 (1973) 57-70;
• S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), 274-276;
• S. Reich, A limit theorem for projections, Linear and Multilinear Algebra, vol. 13, no. 3, pp. 281–290,1983;
• S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal.Appl. 75 (1980) 287-292;
• H. Schaefer, Uber die Methode sukzessive Approximationen, Jahre Deutsch Math.Verein 59(1957), 131-40;
• N. Shioji and W. Takahashi, Strong convergence of approximatedsequences for nonexpansive mappings in Banach spaces, Proc. Amer.Math. Soc., 125(12) (1997), 3641-3645;
• Y. Song and X. Chai, Halpern iteration for firmly type nonexpansivemappings, Nonlinear Analysis: Theory, Methods And Applications, vol. 71, no. 10, pp. 4500-4506, 2009;
• Y. Song, R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Appl. 321 (2006) 316-326;
• T. Suzuki, A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings,Proceedings of the American Mathematical Society, vol.135, no. 1, pp. 99-106, 2007;
• M.A. Taoudi, N. Salhi, B. Ghribi, Integrable solutions of a mixed type operator equation, Appl. Math. Comput. 216 (2010) 1150-1157;
• M. Tian, A general iterative algorithm for nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 73 (2010) 689-694;
• F. Tricomi, Un teorema sulla convergenza delle successioni formate delle successive iterate diuna funzione di una variabile reale, Giorn. Mat. Battaglini 54 (1916), 1-9;
• W. V. Petryshyn and T. E. Williamson, Strong and weak convergence of the sequence ofsuccessive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43Ž . 1973 , 459-497;
• Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992);
• Wongchan, K, Saejung, S, On the strong convergence of viscosity approximation process for quasinonexpansivemappings in Hilbert spaces, Abstr. Appl. Anal. 2011;
• H. K. Xu, \emph{Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991),pp. 1127-1138;
• H.K. Xu, Iterative algorithms for nonlinear operators}, J. London Math. Soc. 66 (2002), 240-256;
• H.-K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc. 65 (2002), 109-113;
• H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291;
• H. K. Xu, T. H. Kim, X. Yin, Weak continuity of the normalized duality map, Journal of Nonlinear and Convex Analysis, Vol. 15, no 3, 2014, 595-604;
• H. K. Xu, M. A. Alghamdi, N. Shahzad, The viscosity technique for the implicit midpoint rule of nonexpansive mappingsin Hilbert spaces, Fixed Point Theory Appl. 41 (2015), 12 pages;
• C. Yang, S. He, Convergence of explicit iterative algorithms for solving a class of variational inequalities, Wseas Trans. Math., (2014), 830-839;
• H. Zhou, Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractionsin Banach spaces, Nonlinear Analysis, 68 (2008), 2977-2983;
• H.Y. Zhou, P.Y. Wang, A simpler explicit iterative algorithm for a class of variational inequalities in Hilbert spaces, J. Optim. Theory Appl. 161 (2014), 716-727.