Strong convergence theorem for integral equations of Hammerstein type in Hilbert spaces

Shehu, Yekini

doi:10.1016/j.amc.2013.12.157

Cited by 8 publications

(5 citation statements)

References 14 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: 62%

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: 62%

Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

Chidume

Adamu

Okereke

2020

Fixed Point Theory Appl

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“…Theorem 4.3 (Shehu (S14) [41]) Let H be a real Hilbert space, and let F : H → H be a bounded, coercive, and maximal monotone mapping. Let F : H → H be a bounded and maximal monotone mapping.…”

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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“…[22][23][24][25] concerning existence and uniqueness results for the Hammerstein Equation ( 20) involving monotone mappings. Recently, Chidume et al [10] established existence result for (20) involving accretive maps and concerning approximation of solutions of the Hammerstein Equation (20), see, e.g., [22,[26][27][28][29][30][31][32] and the references therein. Now, we use Theorem 4 to approximate solutions of Equation (20).…”

Section: Approximating Solutions Of Hammerstein Equations Definitionmentioning

confidence: 99%

An Inertial Algorithm for Solving Hammerstein Equations

2021

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An inertial algorithm for solving Hammerstein equations is presented. This algorithm is obtained as a consequence of a new inertial algorithm proposed and studied for solving nonlinear equations involving operators that are m-accretive. Some strong convergence theorems are proved in real Banach spaces that are uniformly smooth. Furthermore, comparisons of the numerical performance of our algorithms with the numerical performance of some recent important algorithms are presented.

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Strong convergence theorem for integral equations of Hammerstein type in Hilbert spaces

Cited by 8 publications

References 14 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

An Inertial Algorithm for Solving Hammerstein Equations

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