Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical banach spaces

Chidume, C. E.; Bello, A. U.; Usman, Basiru

doi:10.1186/s40064-015-1044-1

Cited by 14 publications

(19 citation statements)

References 38 publications

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“…Finally, we apply our results to solve a convex minimization problem and to approximate a solution of a Hammerstein integral equation. The results obtained in this paper mainly generalized, extended and improved those in Chidume, Bello and Usman [5].…”

Section: Introductionsupporting

confidence: 77%

“…In general Banach spaces, if A is accretive, zeros of A correspond to equilibrium points of some dynamical system whereas if A is monotone, they correspond to minimizers of some convex functionals. Iterative methods have been utilized to approximate zeros of monotone and accretive maps and their types assuming existence, by numerous authors in various Banach spaces (see, e.g., [5], [6], [7], [9], [12], [17] and [18] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

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A Krasnoselskii-type algorithm for approximating zeros of monotone maps in Banach spaces with applications

2018

J. Nonlinear Var. Anal.

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In this paper, a Krasnoselskii-type algorithm that approximates the unique zero of a Hölder continuous and strongly monotone map is constructed. Strong convergence of the sequence generated by this algorithm is established in strictly convex, reflexive and smooth real Banach spaces. Results obtained in this paper are applied to approximate a minimizer of some convex functionals and a solution of a Hammerstein integral equation.

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Section: Introductionsupporting

confidence: 77%

Section: Introductionmentioning

confidence: 99%

A Krasnoselskii-type algorithm for approximating zeros of monotone maps in Banach spaces with applications

2018

J. Nonlinear Var. Anal.

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“…Observe that at equilibrium, u is independent of time so that the equation reduces to Approximation of solutions of Eq. (1.1) has been studied extensively by various authors (see, e.g., Aoyama et al [4], Blum and Oettli [6], Censor, Gibali, Reich and Sabach [12], Censor, Gibali and Reich [9][10][11], Chidume [14], Chidume et al [15,16,18,23,25,26], Gibali, Reich and Zalas [27], Iiduka and Takahashi [29], Iiduka et al [31], Kassay, Reich and Sabach [33], Kinderlehrer and Stampacchia [34], Lions and Stampacchia [36], Liu [37], Liu and Nashed [38], Ofoedu and Malonza [43], Osilike et al [44], Reich and Sabach [48], Reich [46], Rockafellar [49], Su and Xu [51], Zegeye et al [58], Zegeye and Shahzad [57], and the references therein).…”

Section: Introductionmentioning

confidence: 99%

New algorithms for approximating zeros of inverse strongly monotone maps and J-fixed points

Chidume

Ezea

2020

Fixed Point Theory Appl

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Let E be a real Banach space with dual space E *. A new class of relatively weak J-nonexpansive maps, T : E → E * , is introduced and studied. An algorithm to approximate a common element of J-fixed points for a countable family of relatively weak J-nonexpansive maps and zeros of a countable family of inverse strongly monotone maps in a 2-uniformly convex and uniformly smooth real Banach space is constructed. Furthermore, assuming existence, the sequence of the algorithm is proved to converge strongly. Finally, a numerical example is given to illustrate the convergence of the sequence generated by the algorithm.

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“…Recently, motivated by approximating zeros of monotone mappings, Chidume et al [16] proposed a Krasnoselskii-type scheme and proved a strong convergence theorem in L p , 2 ≤ p < ∞. In fact, they obtained the following result.…”

Section: Introductionmentioning

confidence: 99%

Computation of Zeros of Monotone Type Mappings: On Chidume’s Open Problem

Djitte¹,

Mendy²,

Sow³

2020

J. Aust. Math. Soc.

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For $p\geq 2$, let $E$ be a 2-uniformly smooth and $p$-uniformly convex real Banach space and let $A:E\rightarrow E^{\ast }$ be a Lipschitz and strongly monotone mapping such that $A^{-1}(0)\neq \emptyset$. For given $x_{1}\in E$, let $\{x_{n}\}$ be generated by the algorithm $x_{n+1}=J^{-1}(Jx_{n}-\unicode[STIX]{x1D706}Ax_{n})$, $n\geq 1$, where $J$ is the normalized duality mapping from $E$ into $E^{\ast }$ and $\unicode[STIX]{x1D706}$ is a positive real number in $(0,1)$ satisfying suitable conditions. Then it is proved that $\{x_{n}\}$ converges strongly to the unique point $x^{\ast }\in A^{-1}(0)$. Furthermore, our theorems provide an affirmative answer to the Chidume et al. open problem [‘Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical Banach spaces’, SpringerPlus4 (2015), 297]. Finally, applications to convex minimization problems are given.

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Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical banach spaces

Abstract: Let , , and be a strongly monotone and Lipschitz mapping. A Krasnoselskii-type sequence is constructed and proved to converge strongly to the unique solution of . Furthermore, our technique of proo f is of independent interest.

Cited by 14 publications

References 38 publications

A Krasnoselskii-type algorithm for approximating zeros of monotone maps in Banach spaces with applications

A Krasnoselskii-type algorithm for approximating zeros of monotone maps in Banach spaces with applications

New algorithms for approximating zeros of inverse strongly monotone maps and J-fixed points

Computation of Zeros of Monotone Type Mappings: On Chidume’s Open Problem

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