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.

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...

We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivi...

For constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an associated domain of starting points from which a family of Newton-type methods is well-defined and necessarily converges to this specific solution (des...

Suppose that E is a real normed linear space, C is a nonempty convex subset of E, T : C → C is a Lipschitzian mapping, and x∗ ∈ C is a fixed point of T . For given x0 ∈ C, suppose that the sequence {xn} ⊂ C is the Mann iterative sequence defined by xn 1 1−αn xn αnTxn, n ≥ 0, where {αn} is a sequence in 0, 1 , ∑∞ n 0 α 2 n < ∞, ∑∞ n 0 αn ∞. We prove that the sequence {xn} strongly converges to x...