Algorithm for approximating solutions of Hammerstein integral equations with maximal monotone operators

Uba, Markjoe Olunna; Uzochukwu, M. I.; Onyido, Maria Amarakristi

doi:10.1007/s13226-017-0232-9

Cited by 5 publications

(7 citation statements)

References 18 publications

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“…In this paper, an iterative algorithm that extends the results of Chidume and Shehu [24], and complements the results of Uba et al [44] is constructed. Strong convergence of the sequence generated by the algorithm is proved in a uniformly convex and uniformly smooth real Banach space.…”

Section: Resultsmentioning

confidence: 88%

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Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

Chidume

Adamu

Okereke

2020

Fixed Point Theory Appl

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Let B be a uniformly convex and uniformly smooth real Banach space with dual space B *. Let F : B → B * , K : B * → B be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation u + KFu = 0. This theorem is a significant improvement of some important recent results which were proved in real Hilbert spaces under the assumption that F and K are maximal monotone continuous and bounded. The continuity and boundedness restrictions on K and F have been dispensed with, using our new method, even in the more general setting considered in our theorems. Finally, numerical experiments are presented to illustrate the convergence of the sequence of our algorithm.

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Section: Resultsmentioning

confidence: 88%

“…In 2017, Uba et al [44] proved the following theorem: Theorem 1.4 Let E be a uniformly convex and uniformly smooth real Banach space and F : E → E * , K : E * → E be maximal monotone and bounded maps. For u 1 ∈ E, v 1 ∈ E * define the sequences {u n } and {v n } in E and E * , respectively, by…”

Section: Theorem 12 Let H Be a Real Hilbert Space Let K Fmentioning

confidence: 99%

Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

Chidume

Adamu

Okereke

2020

Fixed Point Theory Appl

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“…Until now, no method, which finds the closed form solutions to these nonlinear equations, is known. Thus, iterative algorithms which estimate these solutions are of great interest (see e.g., [21,11,12,14,34,35,36] and also Chapter 13 of [9]).…”

Section: Application To Hammerstein Integral Equationsmentioning

confidence: 99%

New Method for Computing zeros of monotone maps in Lebesgue spaces with applications to integral equations, fixed points, optimization, and variational inequality problems

Bello¹,

Uba²,

Omojola³

et al. 2021

Preprint

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“…Remark 3 Algorithm (4.7) (Inertial Algorithm 2) will be compared with Algorithm (4.8) of Uba et al [44] and Algorithm (4.9) of Chidume et al [11] below. We state the theorems for completeness.…”

Section: Theorem 45 Let E Be a Uniformly Convex And Uniformly Smooth ...mentioning

confidence: 99%

“…Theorem 4.7 (Uba et al [44]) Let E be a uniformly convex and uniformly smooth real Banach space and and F : E → E * , K : E * → E be maximal monotone and bounded maps.…”

Section: Theorem 45 Let E Be a Uniformly Convex And Uniformly Smooth ...mentioning

confidence: 99%

Strong convergence of an inertial algorithm for maximal monotone inclusions with applications

Chidume

Adamu

Nnakwe

2020

Fixed Point Theory Appl

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An inertial iterative algorithm is proposed for approximating a solution of a maximal monotone inclusion in a uniformly convex and uniformly smooth real Banach space. The sequence generated by the algorithm is proved to converge strongly to a solution of the inclusion. Moreover, the theorem proved is applied to approximate a solution of a convex optimization problem and a solution of a Hammerstein equation. Furthermore, numerical experiments are given to compare, in terms of CPU time and number of iterations, the performance of the sequence generated by our algorithm with the performance of the sequences generated by three recent inertial type algorithms for approximating zeros of maximal monotone operators. In addition, the performance of the sequence generated by our algorithm is compared with the performance of a sequence generated by another recent algorithm for approximating a solution of a Hammerstein equation. Finally, a numerical example is given to illustrate the implementability of our algorithm for approximating a solution of a convex optimization problem.

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Algorithm for approximating solutions of Hammerstein integral equations with maximal monotone operators

Cited by 5 publications

References 18 publications

Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

New Method for Computing zeros of monotone maps in Lebesgue spaces with applications to integral equations, fixed points, optimization, and variational inequality problems

Strong convergence of an inertial algorithm for maximal monotone inclusions with applications

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