Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type

Abstract

Suppose that H is a real Hilbert space and F,K:H→H are bounded monotone maps with D(K)=D(F)=H. Let u* denote a solution of the Hammerstein equation u+KFu=0. An explicit iteration process is shown to converge strongly to u*. No invertibility or continuity assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our result is a significant improvement on the Galerkin method of Brézis and Browder.

Strong Convergence Theorems for Solutions of Equations of Hammerstein Type

“…where is a real Hilbert space, and : → are bounded monotone mappings satisfying the range condition, ∈ , and { } ∈N and { } ∈N are sequences in (0, 1). Chidume and Zegeye [8] show that this sequence converges strongly to the solution of (1) under suitable conditions. In 2011, Chidume and Ofoedu [9] introduced a coupled explicit iterative process as follows:…”

“…Chidume and Ofoedu [9] gave a strong convergence theorem for approximation of the solution of (1) under suitable conditions. In 2012, Chidume and Djitté [10] consider the following iterative process:…”

“…where is a real Hilbert space, and are bounded and maximal monotone mappings, { } ∈N and { } ∈N are sequences in (0, 1) and Chidume and Djitté [10] show that this iterative process converges to an approximate solution of nonlinear equations of Hammerstein type under suitable conditions. Motivated by the previous works, in this paper, we consider an auxiliary operator, defined in a real Hilbert space in terms of and , that is monotone and Lipschitz mapping, or monotone and bounded mappings.…”

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

“…(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]).…”

Algorithms for Hammerstein inclusions in certain Banach spaces

“…Following this, Chidume and Djitte studied this explicit coupled iterative algorithms and proved several strong convergence theorems (see, Chidume and Djitte [23,24] ). For recent results using these recursion formulas or their modifications, the reader may consult any of the following references Chidume and Djitte [25][26][27], Djitte and Sene [37,38], Chidume and Ofeodu [28], Chidume and Shehu [30] and also Chapter 13 of [21]. For Hammerstein equations involving monotone mappings from E to E * , very little has been achieved.…”