Strong convergence theorem for approximation of solutions of equations of Hammerstein type

Chidume, C. E.; Shehu, Yekini

doi:10.1016/j.na.2012.05.014

Cited by 9 publications

(10 citation statements)

References 23 publications

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“…Theorem 4.1 (Chidume and Shehu (CS12) [24]) Let H be a real Hilbert space, and let F, K : H → H be bounded, continuous, and monotone mappings. Let {u n } and {v n } be sequences defined iteratively from arbitrary u 1 , v 1 ∈ H by…”

Section: Corollary 32 Let H Be a Real Hilbert Space And Let F : H → Cb(h) K : H → H Be Maximal Monotone And Bounded Maps For Arbitrarymentioning

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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Section: Corollary 32 Let H Be a Real Hilbert Space And Let F : H → Cb(h) K : H → H Be Maximal Monotone And Bounded Maps For Arbitrarymentioning

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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“…(2) strongly monotone (see, e.g., Alber and Ryazantseva [1], p. 25), if hx − y, Tx − Tyi ≥ η∥x − y∥ 2 (3) η-strongly pseudomonotone if hx − y, Tyi ≥ 0 ⟹ hx − y, Txi ≥ η∥x − y∥ 2 (4) ðp, ηÞ-strongly monotone if hx − y, Tx − Tyi ≥ η∥x − y∥ p (see, e.g., Chidume and Djitté [2], Chidume and Shehu [3], and Aibinu and Mewomo [4,5]).…”

Section: Introductionmentioning

confidence: 99%

“…Abstract and Applied Analysis Example 19. An example of a function which is coercive is a real valued function f : ℝ 2 ⟶ ℝ which is defined by f ðu, vÞ = u 4 − 7uv + v 3 :…”

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confidence: 99%

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Algorithm for Solutions of Nonlinear Equations of Strongly Monotone Type and Applications to Convex Minimization and Variational Inequality Problems

Aibinu

Thakur

Moyo

2020

Abstract and Applied Analysis

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Real-life problems are governed by equations which are nonlinear in nature. Nonlinear equations occur in modeling problems, such as minimizing costs in industries and minimizing risks in businesses. A technique which does not involve the assumption of existence of a real constant whose calculation is unclear is used to obtain a strong convergence result for nonlinear equations of p,η-strongly monotone type, where η>0,p>1. An example is presented for the nonlinear equations of p,η-strongly monotone type. As a consequence of the main result, the solutions of convex minimization and variational inequality problems are obtained. This solution has applications in other fields such as engineering, physics, biology, chemistry, economics, and game theory.

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“…Recall that we will establish global solvability of integral equations of Hammerstein type by stating sufficient conditions for Hammerstein operator to be a diffeomorphism. For references on Hammerstein equations see, for example, among others, [13][14][15][16][17][18][19] and references therein. Interest in Hammerstein equation, being the special case of Fredholm equation, stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems, whose linear parts possess the inverse defined via the Green's function, can, as a rule, be transformed into equation involving Hammerstein integral operator.…”

Section: Introductionmentioning

confidence: 99%

Global Solvability of Hammerstein Equations with Applications to BVP Involving Fractional Laplacian

Bors

2013

Abstract and Applied Analysis

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Some sufficient conditions for the nonlinear integral operator of the Hammerstein type to be a diffeomorphism defined on a certain Sobolev space are formulated. The main result assures the invertibility of the Hammerstein operator and in consequence the global solvability of the nonlinear Hammerstein equations. The applications of the result to nonlinear Dirichlet BVP involving the fractional Laplacian and to some specific Hammerstein equation are presented.

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Strong convergence theorem for approximation of solutions of equations of Hammerstein type

Cited by 9 publications

References 23 publications

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

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

Algorithm for Solutions of Nonlinear Equations of Strongly Monotone Type and Applications to Convex Minimization and Variational Inequality Problems

Global Solvability of Hammerstein Equations with Applications to BVP Involving Fractional Laplacian

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