Recently, the authors studied the connection between each maximal monotone operator T and a family H(T ) of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities. The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a uniq...

We propose and analyze the convergence of a novel stochastic algorithm for solving monotone inclusions that are sum maximal operator monotone, Lipschitzian operator. The requires only unbiased estimations obtain rate $${\mathcal {O}}(log(n)/n)$$ in expectation strongly case, as well almost sure general case. Furthermore, context application to convex–concave saddle point problems, we derive pri...

In this paper, we provide a regularization method for finding a solution of Noor’s variational inequality problem induced by a hemicontinuous monotone operator. Moreover, such a solution is related to the set of zero of inverse strongly monotone mappings. Consequently, since we do not assume the strong monotonicity of the considered operator, our results are general and extend some well-known r...

We consider the learning algorithms under general source condition with the polynomial decay of the eigenvalues of the integral operator in vector-valued function setting. We discuss the upper convergence rates of Tikhonov regularizer under general source condition corresponding to increasing monotone index function. The convergence issues are studied for general regularization schemes by using...

In a recent paper by Monteiro and Svaiter, a hybrid proximal extragradient (HPE) framework was used to study the iteration-complexity of a first-order (or, in the context of optimization, second-order) method for solving monotone nonlinear equations. The purpose of this paper is to extend this analysis to study a prox-type first-order method for monotone smooth variational inequalities and incl...

In this paper a reduction and equivalence theorems for the boundedness of the composition of a quasilinear operator T with the Hardy and Copson operators in weighted Lebesgue spaces are proved. New equivalence theorems are obtained for the operator T to be bounded in weighted Lebesgue spaces restricted to the cones of monotone functions, which allow to change the cone of non-decreasing function...

In this paper, a new monotonicity, M(·, ·)-monotonicity,is introduced in Banach spaces, and the resolvent operator of an M(·, ·)monotone operator is proved to be single valued and Lipschitz continuous. By using the resolvent operator technique associated with M(·, ·)monotone operators, we construct a proximal point algorithm for solving a class of variational inclusions. And we prove the conver...

In this paper we provide a splitting algorithm for solving coupled monotone inclusions in real Hilbert space involving the sum of normal cone to vector subspace, maximally monotone, monotone-Lipschitzian, and cocoercive operator. The proposed method takes advantage intrinsic properties each operator generalizes partial inverses forward-backward-half forward splitting, among other methods. At it...