We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued operators are processed individually vi...

We shall prove the equivalence bewteen the convergences of Mann and implicit Mann iterations dealing with various classes of non-Lipschitzian operators.

We show that the convergence of Mann iteration is equivalent to the convergence of Ishikawa iteration for various classes of non-Lipschitzian operators.

In this paper, we introduce and study a new class of generalized nonlinear mixed quasi-variational inequalities involving maximal η-monotone mapping. Using the resolvent operator technique for maximal η-monotone mapping, we prove the existence of solution for this kind of generalized nonlinear multi-valued mixed quasi-variational inequalities without compactness and the convergence of iterative...

and Applied Analysis 3 Lemma 7 (see [15]). LetC be a nonempty closed convex subset of a real Hilbert space H. Let T : C 󳨃→ C be a k-strict pseudo-contractive mapping. Let γ and δ be two nonnegative real numbers such that (γ + δ)k ≤ γ; then 󵄩󵄩󵄩γ (x − y) + δ (Tx − Ty) 󵄩󵄩󵄩 ≤ (γ + δ) 󵄩󵄩󵄩x − y 󵄩󵄩󵄩 , ∀x, y ∈ C. (21) Lemma 8 (see [16]). Let H be a Hilbert space and C a nonempty convex subset of H. Let...

Let E be a real q-uniformly smooth Banach space whose duality map is weakly sequentially continuous. Let T : E → E be a nonexpansive mapping with F (T ) 6= ∅. Let A : E → E be an η-strongly accretive map which is also κ-Lipschitzian. Let f : E → E be a contraction map with coefficient 0 < α < 1. Let a sequence {yn} be defined iteratively by y0 ∈ E, yn+1 = αnγf(yn) + (I − αnμA)Tyn, n ≥ 0, where ...

Assume that F is a nonlinear operator which is Lipschitzian and stronglymonotone on a nonempty closed convex subset C of a real Hilbert space H. Assume also thatΩ is the intersection of the fixed point sets of a finite number of Lipschitzian pseudocontractive self-mappings on C. By combining hybrid steepest-descent method, Mann’s iteration method and projection method, we devise a hybrid iterat...

Abstract. Many applications give rise to separable parameterized equations, which have the form A(y, μ)z + b(y, μ) = 0, where z ∈ R , y ∈ R, μ ∈ R, and the (N + n)×N matrix A(y, μ) and (N + n) vector b(y, μ) are C-Lipschitzian in (y, μ) ∈ Ω ⊂ R × R. We present a technique which reduces the original equation to the form f(y, μ) = 0, where f : Ω → R is C-Lipschitzian in (y, μ). This reduces the d...

Certain generalized Banach's contraction mapping principles on metric spaces are unified and/or extended to Hausdorff uniform spaces. Also given are some relationships between the set of all cluster points of the Picard iterates and the set of all fixed points for the mapping. These are obtained by assuming that the latter set is nonempty and by considering certain "quasi"-contractive condition...