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We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k/ log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log2 k/ log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1930s. Using this connection, we obtain a lower bound of $$\Omega \left( {\sqrt {\log k/\log \log k} } \right)$$ for flow sparsifiers and a lower bound of $$\Omega \left( {\sqrt {\log k} /\log \log k} \right)$$ for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist $$\tilde O\left( {\sqrt {\log k} } \right)$$ cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than $$\tilde \Omega \left( {\sqrt {\log k} } \right)$$ would imply a negative answer to this question.
10Stability of iterative procedures with errors for approximating common fixed points of a couple of q-contractive-like mappings in Banach spaces
Stability of iterative procedures with errors for approximating common fixed points of a couple of q-contractive-like mappings in Banach spaces
Read more74Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces
Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces
1Generalized Projection Algorithms for Maximal Monotone Operators and Relatively Nonexpansive Mappings in Banach Spaces
In this paper, we prove strong convergence theorems of modified Halpern’s iteration for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping in a Banach space by using two hybrid methods. Using these results, we obtain new convergence results for resolvents of maximal monotone operators and relatively nonexpansive mappings in Banach spaces.
Read more12Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications
Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications
11Iterative algorithm for generalized quasi-variational-like inclusions with fuzzy mappings in Banach spaces
Iterative algorithm for generalized quasi-variational-like inclusions with fuzzy mappings in Banach spaces
45Fixed point theorems and ergodic theorems for nonlinear mappings in Banach spaces
In this paper, we first introduce a class of nonlinear mappings called generalized nonspreading which contains the class of nonspreading mappings in a Banach space and then prove a fixed point theorem, a nonlinear mean convergence theorem of Baillon’s type and a weak convergence theorem of Mann’s type for such nonlinear mappings in a Banach space. Using these theorems, we obtain some fixed point theorems, nonlinear mean convergence theorems and weak convergence theorems in a Banach space.
Read more45Modified block iterative algorithm for Quasi- ϕ-asymptotically nonexpansive mappings and equilibrium problem in banach spaces
Modified block iterative algorithm for Quasi- ϕ-asymptotically nonexpansive mappings and equilibrium problem in banach spaces
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9On the Stability of Generalized Additive Functional Inequalities in Banach Spaces
We study the following generalized additive functional inequality , associated with linear mappings in Banach spaces. Moreover, we prove the Hyers-Ulam-Rassias stability of the above generalized additive functional inequality, associated with linear mappings in Banach spaces.
Read more7A hybrid method for solving variational inequalities over the common fixed point sets of infinite families of nonexpansive mappings in Banach spaces
In this paper, we introduce a hybrid method, a combination of the steepest-descent method and the Krasnosel'skii-Mann one, for solving a variational inequality over the set of common fixed points of an infinite family of nonexpansive mappings in Banach spaces under two different conditions on the Banach space, either a uniformly smooth Banach space or a reflexive and strictly convex one with a uniformly Gâteaux differentiable norm, without imposing the sequential weak continuity of the normalized duality mapping. The method is an improvement and extension of some other published results. We also give a numerical example to illustrate the convergence analysis of the proposed method.
Read moreOn the existence of common fixed points for a pair of lipschitzian mappings in banach spaces
On the existence of common fixed points for a pair of lipschitzian mappings in banach spaces
33Fixed point and weak convergence theorems for point-dependent λ-hybrid mappings in Banach spaces
The purpose of this article is to study the fixed point and weak convergence problem for the new defined class of point-dependent λ-hybrid mappings relative to a Bregman distance D f in a Banach space. We at first extend the Aoyama-Iemoto-Kohsaka-Takahashi fixed point theorem for λ-hybrid mappings in Hilbert spaces in 2010 to this much wider class of nonlinear mappings in Banach spaces. Secondly, we derive an Opial-like inequality for the Bregman distance and apply it to establish a weak convergence theorem for this new class of nonlinear mappings. Some concrete examples in a Hilbert space showing that our extension is proper are also given.2010 MSC: 47H09; 47H10.
Read more5A note on “Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces” [Nonlinear Anal. 67 (2007) 1211–1225
A note on “Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces” [Nonlinear Anal. 67 (2007) 1211–1225
Read more50Approximating fixed points of enriched nonexpansive mappings in Banach spaces by using a retraction-displacement condition
In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij-Mann typeassociated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generaliza-tions of the corresponding ones in [Berinde, V.,Approximating fixed points of enriched nonexpansive mappings byKrasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304.], from the setting of Hilbertspaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G.,Approximating fixed pointsof nonexpansive mappings, Proc. Amer. Math. Soc.,44(1974), No. 2, 375–380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.], byconsidering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related resultsin literature can be obtained as particular instances of our results.
Read more1A strong convergence theorem for approximation of a zero of the sum of two maximal monotone mappings in Banach spaces
The purpose of this paper is to study the method of approximation for a zero of the sum of two maximal monotone mappings in Banach spaces and prove strong convergence of the proposed method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.
Read more19Convergence of three-step iterations for asymptotically nonexpansive mappings
Convergence of three-step iterations for asymptotically nonexpansive mappings
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