Search

The purpose of this paper is to introduce and analyze an extragradient method with regularization for finding a common element of the solution set @C of the split feasibility problem and the set Fix(S) of fixed points of a nonexpansive mapping S in the setting of infinite-dimensional Hilbert spaces. Combining the regularization method and the extragradient method due to Nadezhkina and Takahashi [N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128 (2006) 191-201], we propose an iterative algorithm for finding an element of Fix(S)@[email protected] We prove that the sequences generated by the proposed algorithm converge weakly to an element of Fix(S)@[email protected] under mild conditions.

Let H be a real Hilbert space. For each i=1,2,...,m, let F"i,K"i:H->H be bounded and monotone mappings. Assume that the generalized Hammerstein equation u+@?"i"="1^mK"iF"iu=0 has a solution in H. We construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein equation.

In this paper, we prove some coupled fixed point theorems involving a (@j,@f)-weakly contractive condition for mapping having the mixed monotone property in ordered partial metric spaces. These results are analogous to theorems of Van Luong and Xuan Thuan (2011) [10] on the class of ordered partial metric spaces. Also, an application is given to support our results.

In this paper, we introduce and study a general iterative method with strongly positive operators for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. The explicit and implicit iterative algorithms are proposed by virtue of the general iterative method with strongly positive operators. Under two sets of quite mild conditions, we prove the strong convergence of these explicit and implicit iterative algorithms to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively.

In this paper, we introduce a new class of accretive operators-(H(@?,@?),@h)-accretive operators, which generalize many existing monotone or accretive operators. The resolvent operator associated with an (H(@?,@?),@h)-accretive operator is defined and its Lipschitz continuity is presented. By using the new resolvent operator technique, we also introduce and study a new class of set-valued variational inclusions involving (H(@?,@?),@h)-accretive operators and construct a new algorithm for solving this class of set-valued variational inclusions. These results are new, and improve and generalize many known corresponding results.

In this paper, we employ partial order method, cone theory and monotone iterative technique to prove several new theorems on the existence and uniqueness of fixed points for decreasing operators without compactness. Some applications are also given.

Generalized nonlinear set-valued mixed quasi-variational inequalities with maximal monotone mappings have been studied in the literature. Since the maximal monotone operators are known as special cases of the H-monotone operators, we generalize the study for these inequality problems with H-monotone mappings in this paper. Precisely, we establish the existence of solutions for the generalized nonlinear set-valued mixed quasi-variational inequality problem with H-monotone mappings. An iterative method for finding their approximate solutions are proposed. Furthermore, a new convergence criteria for this iterative method is imposed.

The aim of this paper is to establish the equivalence of convergence for H-monotone mappings of variational inclusions between the one-step iteration algorithm of the paper [Y.P. Fang, N.J. Huang, H-monotone opertor and resolvent operator technique for variational inclusions, Appl. Math. Comput. 145 (2003) 795-803] and the two-step iteration algorithm of the paper [L.C. Zeng, S.M. Guu, J.C. Yao, Characterization of H-monotone operators with applications to variational inclusions, Comput. Math. Appl. 50 (2005) 329-337]. It is clear from this paper that the one-step iteration algorithm with easy implement and small computational workload has more advantages than the two-step iteration algorithm.

In the paper, we obtain the existence of monotone positive solutions and establish acorresponding iterative scheme for the following multipoint boundary value problem with p-Laplacian operator, (φp(u′))′(t)+q(t)f(t,u(t))=0,0

This paper contains two contributions of the theory of T-monotone operators introduced by Chen. First, we prove a new fixed-point theorem for a discontinuous T-monotone mapping. After, we use this theory to obtain the solution of a classical continuous problem, for which the usual iterative methods fail. © 2004 Elsevier Ltd. All rights reserved.

Some new results are obtained for the bounded and monotone properties of solutions of second-order quasi-linear forced difference equations δ(P n −1(δY n −1) α )=q n Y n β +r n , n≥n 0 , where n 0 ϵ N = {0,1,2,3,…}, { p n } n = n 0 −1 ∞ is a positive sequence, { q n } n = n 0 ∞ is a nonnegative real sequence with q n ≢ 0, { r n } n = n 0 ∞ is a real sequence, and α and β are quotients of odd positive integers. Some errors in [1] are pointed out and addressed. Examples are given to illustrate the advantages of the new results.

Some sufficient conditions for the existence of fixed points of increasing operators in fuzzy real line R L are given. We also establish some coupled quasi-fixed-point theorems of mixed monotone operators in fuzzy real line R L.

Let H be a real Hilbert space, K a nonempty closed convex subset of H and T : K → K a Lipschitz pseudocontraction with a nonempty fixed-point set. Weak and strong convergence theorems for the iterative approximation of fixed points of T are proved. Furthermore, if T : H → H is a Lipschitz monotone operator and f ∈ R(T), where R(T) denotes the range of T, weak and strong convergence theorems for the iterative approximation of solutions of the operator equation Tx = f are proved.

In this paper, two existence theorems of solutions for a kind of quasi-variational inequalities with monotone type multivalued mapping are obtained.

Comparison theorems and existence criteria are summarized and new ones are derived for the monotone solutions of a class of nonlinear difference equations which arises from Hardy's inequality.

This paper investigates the relationship between approximation error and complexity. A variety of complexity measures are used, including: the number of alternating strictly monotone segments; computation time; and for piecewise linear approximations, the number of linear segments. The results apply to piecewise monotone functions and to finite maps from reals to reals, i.e., real data. We provide a theoretical framework expressing the exact relationship between approximation error and the number of alternating strictly monotone segments. We provide a linear-time algorithm taking an error bound as input and returning a minimal segmentation of the approximated function's domain such that there exists an approximation, alternatingly strictly monotone on the segments, with error less than the given bound. For real data, we provide a suboptimal tradeoff between approximation error and number of linear segments in piecewise linear approximation. The results are obtained by extending the theory of best piecewise monotone approximation to piecewise monotone functions, and by application of a new concept, scale-dependent monotonicity.

We consider the second-order advanced functional difference equation Δ(a(n)Δχ(n)) + p(n)χ(g(n)) = 0 , where a ( n ) > 0, ∑ ∞ s=n 0 ( 1 a(s) ) = ∞ , p ( n ) ≥ 0, p ( n ) ≡ 0, g ( n ) ≥ n + 1, { g ( n )} is a monotone increasing integer sequence. We obtain some new oscillation criteria through an appropriate Riccati equation.

In this paper, we obtain a successively iterative scheme of positive solution for the nonlinear third-order two-point boundary value problem u‴+q(u″)f(t,u)=0,u(0)=A,u(1)=B,u″(0)=C, where f is a Carathéodory function, f and q satisfy some additional monotone conditions. The iterative scheme starts off with zero function and is therefore useful for computation purpose. The main tool is monotone iterative technique on Banach space. Moreover, the iterative scheme is independent of the existence of lower and upper solutions.

We consider monotone arithmetic circuits with restricted depths to compute monotone multivariate polynomials such as the elementary symmetric functions, convolution of several vectors and raising a matrix to a power. We develop general lower- and upper-bound techniques that seem to generate almost-matching bounds for all the functions considered. These results imply exponential lower bounds for circuits of bounded depths which compute any of these functions. We also obtain several examples for which negation can reduce the size exponentially.

This article presents algorithms of linear time complexity (O(n)) for computation of optimal solutions to the two problems of convex and monotone approximation where data points are approximated, respectively, by convex and monotone (nondecreasing) functions on a grid of (n + 1) points. For the convex approximation case, the algorithms are based on a linear programming approach which exploits the structure of matrices involved and uses a special pivoting procedure to obtain a “maximal” optimal solution. Analogously, in the monotone approximation case, the algorithms compute “maximal” and “minimal” optimal solutions which also “enclose” any other optimal solution between them. Computational results are presented.

In this paper, we introduce and study a new system of variational inclusions involving (H, η)-monotone operators in Hilbert space. Using the resolvent operator associated with (H, η)-monotone operators, we prove the existence and uniqueness of solutions for this new system of variational inclusions. We also construct a new algorithm for approximating the solution of this system and discuss the convergence of the sequence of iterates generated by the algorithm.

In this paper, we introduce a new concept of (A, η)-accretive mappings, which generalizes the existing monotone or accretive operators. We study some properties of (A, η)-accretive mappings and define resolvent operators associated with (A, η)-accretive mappings. By using the new resolvent operator technique, we also construct a new perturbed iterative algorithm with mixed errors for a class of nonlinear relaxed Cocoercive variational inclusions involving (A, η)-accretive mappings and study applications of (A, η)-accretive mappings to the approximation-solvability of this class of nonlinear relaxed Cocoercive variational inclusions in q-uniformly smooth Banach spaces. Our results improve and generalize the corresponding results of recent works.

A Boolean response to a random binary input of length n can be modeled as a {;0, 1}- valued function v defined on a discrete probability space Ω of all subsets of a finite set of size n. An ω ∈ Ω represents the locations of 1's in the input. For a particular jth location, 1 ≤ j ≤ n, we assume that 1 appears with probability ρj independently of other locations. Then, for ρ = (ρ1,…, ρn), we will investigate Pρ(v = 1) as a function of ρ. Using the sharp version of the Khinchin inequality, we give an upper estimate for the ℓ2 norm of the gradient of Pρ(v = 1) evaluated at ρ = (12,…, 12) (cf. (5) below). For monotone functions, the estimate applies also to vector of influences of Boolean functions. We also provide a handy expansion of P(·)(v = 1) based on a Fourier expansion of v (cf. (4) below).Numerical analysis of the bounds leads to the conjecture about the sharp bound that depends on cardinality of the underlying set; the sharp version of the Khinchin inequality is also conjectured.

A general design for the Eckstein–Bertsekas proximal point algorithm, using the notion of the A-maximal monotonicity, is developed. Convergence analysis for the generalized Eckstein–Bertsekas proximal point algorithm in the context of solving a class of nonlinear inclusion problems is explored. Some auxiliary results of interest involving A-maximal monotone mappings are also included. The obtained results generalize investigations on general maximal monotonicity and beyond.

In Bor (2011) [5], we proved a main theorem dealing with absolute Nörlund summability factors of infinite series. In the present paper, this result has been further generalized under weaker conditions by using δ-quasi-monotone sequences