The generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping. In this article, by assuming that the operators are maximal monotone of Gossez type (D), we provide sufficient conditions of both interiorityand...

where F is a monotone operator. Recently, Lu et al. [] were concerned with a special class of variational inequalities in which the mapping F is the complement of a nonexpansive mapping and the constraint set is the set of fixed points of another nonexpansive mapping. Namely, they considered the following type of monotone variational inequality (VI) problem: Find x∗ ∈ Fix(T), such that 〈(I –V ...

and Applied Analysis 3 2. Main Results In this paper, we are interested in the following classes of mappings. Definition 8. Let T : H → H be a mapping.Then T is called (a) β-strongly monotone if there exists β > 0 such that ⟨T (x) − T (y) , x − y⟩ ≥ β 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ H, (12) (b) ξLipschitz if there exists ξ > 0 such that 󵄩󵄩󵄩󵄩T (x) − T (y) 󵄩󵄩󵄩󵄩 ≤ ξ 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 , ∀x, y ∈ H. (13) Tha...

1. Overview This paper talks about that monotone circuit is P-Complete. Decision problem that include P-Complete is mapping that classify input with a similar property. Therefore equivalence relation of input value is important for computation. But monotone circuit cannot compute the equivalence relation of the value because monotone circuit can compute only monotone function. Therefore, I make...

We introduce and study a new system of nonlinear variational-like inclusions involving sG, η maximal monotone operators, strongly monotone operators, η-strongly monotone operators, relaxed monotone operators, cocoercive operators, λ, ξ -relaxed cocoercive operators, ζ, φ, g-relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique ...

A new monotonicity, M-monotonicity, is introduced, and the resolvant operator of an M-monotone operator is proved to be single-valued and Lipschitz continuous. With the help of the resolvant operator, the positively semidefinite general variational inequality (VI) problem VI (S+,F +G) is transformed into a fixed point problem of a nonexpansive mapping. And a proximal point algorithm is construc...

We prove strong convergence theorems for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a hemirelatively nonexpansive mapping in a Banach space by using monotone hybrid iteration method. By using these results, we obtain new convergence results for resolvents of maximal monotone operators and hemirelatively nonexpansive mappings in a Ban...

We introduce and study the Split Common Null Point Problem (SCNPP) for set-valued maximal monotone mappings in Hilbert space. This problem generalizes our Split Variational Inequality Problem (SVIP) [Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms, accepted for publication, DOI 10.1007/s11075-011-9490-5]. The SCNPP with only two s...

‎A new approximation method for the set of common fixed points of‎ ‎nonexpansive mappings and the set of solutions of systems of‎ ‎variational inequalities is introduced and studied‎. ‎Moreover‎, ‎we‎ ‎apply our main result to obtain strong convergence theorem to a‎ ‎common fixed point of a nonexpannsive mapping and solutions of ‎a ‎system of variational inequalities of an inverse strongly mono...