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...

In this paper, we consider the monotone inclusion problem consisting of the sum of a continuous monotone map and a point-to-set maximal monotone operator with a separable two-block structure and introduce a framework of block-decomposition prox-type algorithms for solving it which allows for each one of the single-block proximal subproblems to be solved in an approximate sense. Moreover, by sho...

‎In this paper‎, ‎we generalize the proximal point algorithm to complete CAT(0) spaces and show‎ ‎that the sequence generated by the proximal point algorithm‎ $w$-converges to a zero of the maximal‎ ‎monotone operator‎. ‎Also‎, ‎we prove that if $f‎: ‎Xrightarrow‎ ‎]-infty‎, +‎infty]$ is a proper‎, ‎convex and lower semicontinuous‎ ‎function on the complete CAT(0) space $X$‎, ‎then the proximal...

We are concerned with the study of a class of forward-backward penalty schemes for solving variational inequalities 0 ∈ Ax+NC(x) where H is a real Hilbert space, A : H ⇉ H is a maximal monotone operator, and NC is the outward normal cone to a closed convex set C ⊂ H. Let Ψ : H → R be a convex differentiable function whose gradient is Lipschitz continuous, and which acts as a penalization functi...

We propose two inexact parallel splitting algorithms for computing the resolvent of a weighted sum of maximal monotone operators in a Hilbert space and show their strong convergence. We start by establishing new results on the asymptotic behavior of the Douglas-Rachford splitting algorithm for the sum of two operators. These results serve as a basis for the first algorithm. The second algorithm...

In this paper, we study the existence of generalized solutions for the infinite dimensional nonlinear stochastic differential inclusions $dx(t) in F(t,x(t))dt +G(t,x(t))dW_t$ in which the multifunction $F$ is semimonotone and hemicontinuous and the operator-valued multifunction $G$ satisfies a Lipschitz condition. We define the It^{o} stochastic integral of operator set-valued stochastic pr...

we establish a sharp maximal function estimate for some vector-valued multilinear singular integral operators. as an application, we obtain the $(l^p, l^q)$-norm inequality for vector-valued multilinear operators.

The classical Loewner’s theorem states that operator monotone functions on real intervals are described by holomorphic the upper half-plane. We characterize local order isomorphisms domains biholomorphic automorphisms of generalized half-plane, which is collection all operators with positive invertible imaginary part. describe such maps in an explicit manner, and examine properties maximal isom...