Abstract
This paper deals with a general formalism which consists in approximating a point in a nonempty set \(S\), in a real Hilbert space \(H\), by a sequence \((x_n) \subset H\) such that \(x_{{n + 1}} : = {\user1{\mathcal{T}}}_{n} {\left( {x_{n} + \theta _{n} {\left( {x_{n} - x_{{n - 1}} } \right)}} \right)}\), where \({\left( {\theta _{n} } \right)} \subset \left[ {0,} \right.\left. 1 \right)\), \(x_0\) \(x_1\) are in \(H\) and \({\left( {{\user1{\mathcal{T}}}_{n} } \right)}_{{n \geqslant 0}}\) is a sequence included in a certain class of self-mappings on \(H\), such that every fixed point set of \({\user1{\mathcal{T}}}_{n}\) contains \(S\). This iteration method is inspired by an implicit discretization of the second order ‘heavy ball with friction’ dynamical system. Under suitable conditions on the parameters and the operators \({\left( {{\user1{\mathcal{T}}}_{n} } \right)}\), we prove that this scheme generates a sequence which converges weakly to an element of \(S\). In particular, by appropriate choices of \({\left( {{\user1{\mathcal{T}}}_{n} } \right)}\), this algorithm works for approximating common fixed points of infinite countable families of a wide class of operators which includes \(\alpha\)-averaged quasi-nonexpansive mappings for \(\alpha \in {\left( {0,\;1} \right)}\).
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References
Alvarez, F., Attouch, H.: An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)
Bauschke, H.H.: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 202(1), 150–159 (1996)
Bauschke, H.H., Combettes, P.L.: A weak-to-strong convergence principle for Fejer monotone methods in Hilbert space. Math. Oper. Res. 26, 248–264 (2001)
Brezis, H.: Operateurs maximaux monotones. North Holland Math. Stud. 5, (1973)
Browder, F.E.: Convergence of approximants to fixed points of non-expansive maps in Banach spaces. Arch. Rational Mech. Anal. 24, 82–90 (1967)
Combettes, P.L.: Construction d’un point fixe commun d’une famille de contractions fermes (Construction of a common fixed point for firmly nonexpansive mappings). C. R. Acad. Sci. Paris Sér. I Math. 320(11), 1385–1390 (1995)
Combettes, P.L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradients projections. IEEE Trans. Image Process. 6(4), 493–506 (1997)
Combettes, P.L., Pesquet, J.C.: Image restoration subject to a total variation constraint. IEEE Trans. Image Process. 13, 1213–1222 (2004)
Ciric, LJ.B., Ume, J.S., Khan, M.S.: On the convergence of the Ishikawa iterates to a common fixed point of two mappings. Arch. Math. (BRNO) 39, 123–127 (2003)
Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. (N.S.) 73, 957–961 (1967)
Jules, F., Maingé, P.E.: Numerical approaches to a stationary solution of a second order dissipative dynamical system. Optimization 51(2), 235–255 (2002)
Lions, P.L.: Approximation de points fixes de contractions. C.R. Acad. Sci. Paris, Sér. I Math. A 284, 1357–1359 (1977)
Moudafi, A., Elisabeth, E.: An approximate inertial proximal method using enlargement of a maximal monotone operator. Intern. J. Pure Appl. Math. 5(3), 283–299 (2003)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967)
Polyak, B.T.: Some methods of speeding up the convergence of iterative methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 1–17 (1964)
Vasin, V.V., Agreev, A.L.: Ill-posed problems with a priori information VSP. (1995)
Wittman, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992)
Yamada, I., Ogura, N.: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25(7–8), 619–655 (2004)
Yamada, I., Ogura, N.: Adaptive projective subgradient method for asymptotic minimization of sequence of nonnegative convex functions. Numer. Funct. Anal. Optim. 25(7–8), 593–617 (2004)
Yamada, I., Slavakis, K., Yamada, K.: An efficient robust adaptive filtering algorithm based on parallel subgradient projection techniques. IEEE Trans. Signal Process. 50(5), 1091–1101 (2002)
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Maingé, PE. Inertial Iterative Process for Fixed Points of Certain Quasi-nonexpansive Mappings. Set-Valued Anal 15, 67–79 (2007). https://doi.org/10.1007/s11228-006-0027-3
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DOI: https://doi.org/10.1007/s11228-006-0027-3