Some Problems and Results in the Study of Nonlinear Analysis

On the Existence and Iterative Approximation Problems of Solutions for Set-Valued Variational Inclusions in Banach Spaces

“…By using Michael's selection theorem [16] and Nadler's theorem [18] some existence theorems and some iterative algorithms for solving this kind of set-valued variational inclusions in Banach spaces are established and suggested. The results presented in this paper generalize, improve, and unify the corresponding results of Noor [19][20][21][22][23][24], Ding [8], Huang [10], Kazmi [12], Jung and Morales [11], Liu [14], Hassouni and Mouafi [9], Zeng [28], and Chang [2][3][4][5].…”

“…Throughout this paper, we assume that E is a real Banach space, E * is the topological dual space of E, CB E is the family of all nonempty closed and bounded subset of E, · · is the dual pair between E and E * , and D · · is the Hausdorff metric on CB E defined by (2) The mapping A is said to be φ-strongly accretive if, for any x y ∈ D A , there exists j x − y ∈ J x − y such that for any u ∈ Ax, v ∈ Ay,…”

“…(ii) Theorems 4.1-4.4 generalize, improve, and unify the corresponding recent results of Noor [19][20][21][22][23][24], Liu [14], Ding [8], Huang [10], Kazmi [12], Chang [2][3][4][5], Jung and Morales [11], Hassouni and Moudafi [9], and Zeng [28].…”

Iterative solution of nonlinear equations with $ \phi $-strongly accretive operators

“…Theorem 3.1 generalizes, improves and unifies Theorem 5.2 of Chang [2], Theorem 5.2 of Chang et al [3], Theorem 1 of Chidume [4], Theorem 2 of Chidume [5], Theorems 1 and 3 of Chidume and Osilike [7], Theorem 1 of Deng [9], Theorems 1 and 3 of Deng [10], Theorem 1 of Deng [11], Theorem 2 of Deng and Ding [12], Theorem 1 of Liu [15], Theorem 2 of Osilike [17], Theorem 1 of Osilike [18], Theorem 1 of Osilike [19], Theorem 1 of Osilike [20], Theorem 4.1 of Tan and Xu [21] and Theorems 1 and 2 of Zeng [24] in the following ways:…”

“…In [4], Chidume proved that, if X = L p (or l p ), p м 2, and an operator T : X → X is Lipschitzian and strongly accretive, then the Mann iteration method converges strongly to the solution of (1.2). The results of Chidume [4] have been extended to both the Ishikawa iteration process and the Ishikawa iteration process with errors introduced by Liu [15] and all q-uniformly smooth Banach spaces, 1 < q < ∞ (see, for example, [2], [3], [5]- [7], [9]- [12], [15], [17], [19], [21], [22], [24]). On the other hand, if (1.2) has a solution and T : X → X is a Lipschitzian and φ-strongly accretive operator, Osilike [18], [20] have applied the Ishikawa iteration scheme to approximate solutions of (1.2).…”