Nonexpansive Sequences Research Articles

Accelerated iterative splitting methods on Hadamard manifolds

This paper aims to solve the monotone inclusion problem, minimization problem of multiple summands and the generalized Heron problem. We present an innovative approach, the modified normal S-iteration method, designed to approximate common fixed points of nearly nonexpansive sequences and families of operators via the property ( A ) . Some deductions of our results improve some existing results in the literature. To show the applicability of our result, we give application to the inclusion problem via forward–backward splitting method version of our algorithm and minimization problem via Douglas–Rachford splitting method version of our algorithm. To demonstrate the practical utility of the algorithm, we apply it to the generalized Heron problem.

Applications of a variable anchoring iterative method to equation and inclusion problems on Hadamard manifolds

In this paper, we introduce a new iterative technique with a variable anchoring operator for reckoning the solution of a variational inequality problem over the set of the common fixed points of a nearly nonexpansive sequence of operators in the framework of Hadamard manifolds. We also establish a convergence result on the proposed algorithm for approximating a solution of the problem, under suitable assumptions. We apply our results for finding the solutions of a system of nonlinear equations, and of inclusion problems to support their utility. Our work improves results in the recent literature. Numerical simulations are given for a better understanding of the effectiveness of our outcomes.

Asymptotic Behaviour of -Nonexpansive Sequences and Mappings in Banach Spaces

In this paper, we introduce the notion of -nonexpansive sequences and maps in a Banach space E, and prove an ergodic theorem, a fixed point theorem, and some weak and strong convergence theorems for such sequences and maps in E, with rather mild assumptions on the Banach space E, weaker than those assumed by previous authors for nonexpansive maps. These results give a partial answer to the still open problem raised in 1981 by Djafari Rouhani for nonexpansive sequences in a Banach space.

Ergodic and fixed point theorems for Bregman nonexpansive sequences and mappings in Banach spaces

Ergodic and fixed point theorems for Bregman nonexpansive sequences and mappings in Banach spaces

New Construction of Strongly Relatively Nonexpansive Sequences by Firmly Nonexpansive-Like Mappings

In recent works, many authors generated strongly relatively nonexpansive sequences of mappings by the sequences of firmly nonexpansive-like mappings. In this paper, we introduce a new method for construction of strongly relatively nonexpansive sequences from firmly nonexpansive-like mappings.

Approximation of common fixed points of strongly nonexpansive sequences in a Banach space

The aim of this paper is to establish a strong convergence theorem for a strongly nonexpansive sequence in a Banach space. We also deal with some applications of the convergence theorem.

Nonhomogeneous nonlinear oscillator with damping: asymptotic analysis in continuous and discrete time

Abstract We consider the following second order evolution equation modelling a nonlinear oscillator with damping ü ( t ) + γ u ˙ ( t ) + A u ( t ) = f ( t ) , ( SEE ) $$\ddot{u} (t) + \gamma \dot u(t) + Au\left( t \right) = f\left( t \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\rm{SEE}}} \right)$$ where A is a maximal monotone and α-inverse strongly monotone operator in a real Hilbert space H. With suitable assumptions on γ and f(t) we show that A −1(0) ≠ ∅, if and only if (SEE) has a bounded solution and in this case we provide approximation results for elements of A −1(0) by proving weak and strong convergence theorems for solutions to (SEE) showing that the limit belongs to A −1(0). As a discrete version of (SEE), we consider the following second order difference equation u n + 1 - u n - α n ( u n - u n - 1 ) + λ n A u n + 1 ∋ f ( t ) , $${u_{n + 1}} - {u_n} - {\alpha _n}\left( {{u_n} - {u_{n - 1}}} \right) + {\lambda _n}A{u_{n + 1}\ni} f\left( t \right),$$ where A is assumed to be only maximal monotone (possibly multivalued). By using the results in [Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417], we prove ergodic, weak and strong convergence theorems for the sequence un , and show that the limit is the asymptotic center of un and belongs to A −1(0). This again shows that A −1(0) ≠ ∅ if and only if un is bounded. Also these results solve an open problem raised in [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11], namely the study of the convergence results for the inexact inertial proximal algorithm. Our paper is motivated by the previous results in [Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl., 1990, 147, 465–476; Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226–235; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math., 2010, 40, 1289–1311; Djafari Rouhani B., Khatibzadeh H., A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation, J. Math. Anal. Appl., 2010, 363, 648–654; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations, Nonlinear Anal., 2009, 70, 4369–4376; Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417] and significantly improves upon the results of [Attouch H., Maingé P. E., Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects, ESAIM Control Optim. Calc. Var., 2011, 17(3), 836–857], and [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11].

Ergodic and fixed point theorems for sequences and nonlinear mappings in a Hilbert space

Abstract In this paper, we introduce the notion of 2-generalized hybrid sequences, extending the notion of nonexpansive and hybrid sequences introduced and studied in our previous work [Djafari Rouhani B., Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. thesis, YaleUniversity, 1981; and other published in J. Math. Anal. Appl., 1990, 2002, and 2014; Nonlinear Anal., 1997, 2002, and 2004], and prove ergodic and convergence theorems for such sequences in a Hilbert space H. Subsequently, we apply our results to prove new fixed point theorems for 2-generalized hybrid mappings, first introduced in [Maruyama T., Takahashi W., Yao M., Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal., 2011, 12, 185-197] and further studied in [Lin L.-J., Takahashi W., Attractive point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces, Taiwanese J. Math., 2012, 16, 1763-1779], defined on arbitrary nonempty subsets of H.

Convergence of Inexact Mann Iterations Generated by Nearly Nonexpansive Sequences and Applications

ABSTRACTThe Mann iterates behave well for nonexpansive mappings for any initial guess in the domain. Our aim in this article is to extend this method to a broad class of inexact fixed point algorithms generated by nearly nonexpansive sequences in Banach spaces and to locate the weak limit of the iterates by its initial guesses. Due to the inexactness, our algorithms become efficiently applicable for a wider class of problems. As applications, we give convergence theorems for finding solutions of variational inclusion problems and constrained multiple-sets split feasibility problems. Our results are significant refinements and improvements of the corresponding results in the literature.

On the Halpern iteration in CAT(0) spaces

In this paper, strong convergence of Halpern iteration is shown for a quasi-strongly nonexpansive sequence of multivalued mappings in complete CAT(0) spaces.

Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings

We show that a strongly relatively nonexpansive sequence of mappings can be constructed from a given sequence of firmly nonexpansive-like mappings in a Banach space. Using this result, we study the problem of approximating common fixed points of such a sequence of mappings. MSC:47H09, 47H05, 65J15.

Convergence theorems for a system of equilibrium problems and fixed point problems of a strongly nonexpansive sequence

The purpose of this paper is to prove a strong convergence theorem of an iterative scheme associated to a strongly nonexpansive sequence for finding a common element of the set of equilibrium problems and the set of fixed point problems of a pair of sequences of nonexpansive mappings where one of them is a strongly nonexpansive sequence. Moreover, in the last section, by using our main result, we obtain a strong convergence theorem of an iterative scheme associated to a strongly nonexpansive sequence for finding a common element of the set of a finite family of equilibrium problems and the set of fixed point problems of a pair of sequences of nonexpansive mappings where one of them is a strongly nonexpansive sequence in a Hilbert space, and we also give some examples to support our main result.

Ergodic theorems for hybrid sequences in a Hilbert space with applications

In this paper, we introduce the notion of generalized hybrid sequences, extending the notion of nonexpansive sequences introduced and studied in our previous work Djafari Rouhani (1981, 1990, 1990, 1997, 2002, 2004, 2002) [2–8], and prove ergodic and convergence theorems for such sequences in a Hilbert space H. Subsequently, we apply our results to prove new fixed point theorems for generalized hybrid mappings, first introduced in Kocourek et al. (2010) [14], Takahashi and Takeuchi (2011) [20], defined on arbitrary nonempty subsets of H.

Existence and asymptotic behaviour of solutions to first- and second-order difference equations with periodic forcing

By using previous results of Djafari Rouhani for non-expansive sequences in Refs (Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale University, Part I (1981), pp. 1–76; Djafari Rouhani, J. Math. Anal. Appl. 147 (1990), pp. 465–476; Djafari Rouhani, J. Math. Anal. Appl. 151 (1990), pp. 226–235), we study the existence and asymptotic behaviour of solutions to first-order as well as second-order difference equations of monotone type with periodic forcing. In the first-order case, our result extends to general maximal monotone operators, the discrete analogue of a result of Baillon and Haraux (Rat. Mech. Anal. 67 (1977), 101–109) proved for subdifferential operators. In the second-order case, our results extend among other things, previous results of Apreutesei (J. Math. Anal. Appl. 288 (2003), 833–851) to the non-homogeneous case, and show the asymptotic convergence of every bounded solution to a periodic solution.

Strong convergence theorems for strongly nonexpansive sequences

The aim of this paper is to prove a strong convergence theorem for a pair of sequences of nonexpansive mappings in a Hilbert space, where one of them is a strongly nonexpansive sequence, and provide some applications of the theorem.

On the asymptotic properties of non-autonomous systems

By giving a new method, we study asymptotic behavior of weakly almost nonexpansive sequences and curves introduced by Djafari Rouhani (J Differ Equ 229:412–425, 2006) in a reflexive Banach space X. Subsequently, we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system \({du/dt +Au\ni f}\), where A is an accretive (possibly multivalued) operator in X × X, and for some \({f_{\infty}\in X}\) and 1 ≤ p 1)}\). Our results extend and improve many previously known results. Moreover, we answer an open question raised by B. Djafari Rouhani.

Strongly relatively nonexpansive sequences in Banach spaces and applications

We introduce the concept of a strongly relatively nonexpansive sequence in a Banach space and investigate its properties. Then we apply our results to the problem of approximating a common fixed point of a countable family of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space.

Asymptotic behavior of bounded solutions to a nonhomogeneous second-order evolution equation of monotone type

By using previous results of Djafari Rouhani [B. Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale university, 1981, part I, pp. 1–76; B. Djafari Rouhani, Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl. 147 (1990) 465–476; B. Djafari Rouhani, Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl. 151 (1990) 226–235] for dissipative systems, we study the asymptotic behavior of solutions to the following system of second-order nonhomogeneous evolution equations: { u ″ ( t ) ∈ A u ( t ) + f ( t ) a.e. t ∈ ( 0 , + ∞ ) u ( 0 ) = u 0 , sup t ≥ 0 | u ( t ) | < + ∞ where A is a maximal monotone operator in a real Hilbert space H , and f : R + → H is a suitable function. We investigate weak and strong convergence theorems for solutions to this system. Our results extend previous results of Morosanu [G. Morosanu, Nonlinear Evolution Equations and Applications, Editura Academiei Romane (and D. Reidel publishing Company, Bucharest, 1988; G. Morosanu, Asymptotic behaviour of solutions of differential equations associated to monotone operators, Nonlinear Anal. 3 (1979) 873–883] and Mitidieri [E. Mitidieri, Some remarks on the asymptotic behaviour of the solutions of second order evolution equations, J. Math. Anal. Appl. 107 (1985) 211–221] who studied the case f ≡ 0 by assuming that A − 1 ( 0 ) ≠ ϕ , as well as previous results of the authors [B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math. (in press)] who studied the homogeneous case without this additional assumption.

Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations

By using previous results of Djafari Rouhani [B. Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph. D. Thesis, Yale University, 1981, part I, pp. 1–76; B. Djafari Rouhani, Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl. 147 (1990) 465–476; B. Djafari Rouhani, Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl. 151 (1990) 226–235] for dissipative systems, we study the weak and strong convergence of solutions to the following class of second order nonhomogeneous evolution equations { u ″ ( t ) − c u ′ ( t ) ∈ A u ( t ) + f ( t ) a.e. t ∈ ( 0 , + ∞ ) u ( 0 ) = u 0 , sup t ≥ 0 | u ( t ) | < + ∞ where A is a monotone operator in a real Hilbert space H , c ≥ 0 , and f : R + → H is a given function. Our results extend previous results by Morosanu [G. Morosanu, Nonlinear Evolution Equations and Applications, Editura Academiei Romane (and D. Reidel Publishing Company), Bucharest, 1988; G. Morosanu, Asymptotic behaviour of solutions of differential equations associated to monotone operators, Nonlinear Anal. 3 (1979) 873–883] and Mitidieri [E. Mitidieri, Some remarks on the asymptotic behaviour of the solutions of second order evolution equations, J. Math. Anal. Appl. 107 (1985) 211–221] who studied the case c = 0 and f ≡ 0 by assuming that A is maximal monotone and A − 1 ( 0 ) ≠ ϕ , as well as previous results by Véron [L. Véron, Problèmes d’évolution du second ordre associés à des opérateurs monotones, C. R. Acad. Sci. Paris Sér. A 278 (1974) 1099–1101; L. Véron, Equations d’évolution du second ordre associées à des opérateurs maximaux monotones, Proc. Roy. Soc. Edinburgh Sect. A 75 (1975–76) 131–147] and the authors [B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of solutions to some homogeneous second order evolution equations of monotone type, J. C Inequal. Appl. (2007) Art. ID 72931, 8 pp; B. Djafari Rouhani, H. Khatibzadeh, Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math. (in press)] who studied the case f ≡ 0 . Some applications are also presented.

Iterative approximation to common fixed points of a countable family of nonexpansive mappings

We proved several strong convergence results by using the conception of a uniformly asymptotically regular sequence {T n } of nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping J ϕ(l p (1 < p < ∞) has a weakly continuous duality map with gauge ϕ(t) = t p−1. The results presented develop and complement the corresponding ones by Song, Y. and Chen, R., 2007 [Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces. Nonlinear Analysis, 66, 591–603], Song, Y., Chen, R. and Zhou, H., 2007 [Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Analysis, 66, 1016–1024] and O'Hara, J.G., Pillay, P. and Xu, H.K., 2006 [Iterative approaches to convex feasibility problem in Banach Space. Nonlinear Analysis, 64, 2022–2042], O'Hara, J.G., Pillay, P. and Xu, H.K., 2003 [Iterative approaches to fineding nearest common fixed point of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis, 54, 1417–1426] and Jung, J.S., 2005 [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 302, 509–520] and many other existing literatures.