Iterative algorithms for solutions of Hammerstein integral inclusions

Minjibir, Ma’aruf Shehu; Mohammed, I.

doi:10.1016/j.amc.2017.09.041

Cited by 6 publications

(4 citation statements)

References 25 publications

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“…With this, Chidume and Zegeye [25] were able to prove strong convergence of an iterative algorithm defined in the Cartesian product space E to a solution of the Hammerstein equation (1.1). Extensions of these early results of Chidume and Zegeye [25] were obtained by several authors (see, e.g., Chidume and Zegeye [26,27], Chidume and Djitte [15][16][17], Chidume and Ofoedu [21], Chidume and Shehu [22,23], Chidume et al [12,20] Djitte and Sene [29], Ofoedu and Onyi [39], Ofoedu and Malonza [38], Shehu [43], Minjibir and Mohammed [34], and the references contained therein).…”

Section: Theorem 11 Let H Be a Separable Hilbert Space And C Be A Clmentioning

confidence: 61%

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: Theorem 11 Let H Be a Separable Hilbert Space And C Be A Clmentioning

confidence: 61%

Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

Chidume

Adamu

Okereke

2020

Fixed Point Theory Appl

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“…We construct these examples on R p , p ≥ 1, 2 (R), and L 2 R ( ), where is a closed bounded interval of R. Numerical experiments were carried out on MATLAB R2013a and R2020 versions. Then the subdifferential of f at x (see, e.g., [36]) is given by…”

Section: Numerical Examplesmentioning

confidence: 99%

“…(See, e.g., Chidume and Djitte [17][18][19], Chidume and Ofeodu [20], Chidume and Shehu [23], Chidume and Osilike [22], and Chidume and Bello [15].) Recently, Minjibir and Muhammad [36] proved a strong convergence result for Hammerstein inclusion under the setting of Hilbert spaces. They proved the following theorem with the sequences {α n }, {θ n } ⊂ (0, 1) satisfying some appropriate conditions.…”

Section: Introductionmentioning

confidence: 99%

An inertial-type algorithm for approximation of solutions of Hammerstein integral inclusions in Hilbert spaces

Bello

Omojola

Yahaya

2021

Fixed Point Theory Algorithms Sci Eng

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Let H be a real Hilbert space. Let $F:H\rightarrow 2^{H}$ F : H → 2 H and $K:H\rightarrow 2^{H}$ K : H → 2 H be two maximal monotone and bounded operators. Suppose the Hammerstein inclusion $0\in u+KFu$ 0 ∈ u + K F u has a solution. We construct an inertial-type algorithm and show its strong convergence to a solution of the inclusion. As far as we know, this is the first inertial-type algorithm for Hammerstein inclusions in Hilbert spaces. We also give numerical examples to compare the new algorithm with some existing ones in the literature.

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“…Hence, the conclusion follows from Theorem 3.1. where, without loss of generality, we have taken w ≡ 0. Interest in Hammerstein integral equations stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Green's function can, as a rule, be put in the form ( [42], and the references contained in them. We now apply Theorem 3.1 to approximate a solution of equation (5.3).…”

Section: Theorem 44 Let X Be a Uniformly Convex And Uniformly Smoothmentioning

confidence: 99%

A strong convergence theorem for generalized-Φ-strongly monotone maps, with applications

Chidume

Nnakwe

Adamu

2019

Fixed Point Theory Appl

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Let X be a uniformly convex and uniformly smooth real Banach space with dual space X *. In this paper, a Mann-type iterative algorithm that approximates the zero of a generalized-Φ-strongly monotone map is constructed. A strong convergence theorem for a sequence generated by the algorithm is proved. Furthermore, the theorem is applied to approximate the solution of a convex optimization problem, a Hammerstein integral equation, and a variational inequality problem. This theorem generalizes, improves, and complements some recent results. Finally, examples of generalized-Φ-strongly monotone maps are constructed and numerical experiments which illustrate the convergence of the sequence generated by our algorithm are presented.

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Iterative algorithms for solutions of Hammerstein integral inclusions

Cited by 6 publications

References 25 publications

Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

An inertial-type algorithm for approximation of solutions of Hammerstein integral inclusions in Hilbert spaces

A strong convergence theorem for generalized-Φ-strongly monotone maps, with applications

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