‎In this paper‎, ‎some properties of pre-monotone operators are proved‎. ‎It is shown that in a reflexive Banach space‎, ‎a full domain multivalued $sigma$-monotone operator with sequentially norm$times$weak$^*$ closed graph is norm$times$weak$^*$ upper semicontinuous‎. ‎The notion of $sigma$-convexity is introduced and the‎ ‎relations between the $sigma$-monotonicity and $sigma$-convexity is i...

For a maximal monotone operator T on a Hilbert space H and a closed subspace A of H, we consider the problem of finding (x, y ∈ T (x)) satisfying x ∈ A and y ∈ A⊥. An equivalent formulation of this problem makes use of the partial inverse operator of Spingarn. The resulting generalized equation can be solved by using the proximal point algorithm. We consider instead the use of hybrid proximal m...

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let α > 0, and let A be an α-inverse strongly-monotone mapping of C into H. Let T be a generalized hybrid mapping of C into H. Let B andW be maximal monotone operators on H such that the domains of B andW are included in C. Let 0 < k < 1, and let g be a k-contraction of H into itself. Let V be a γ -strongly monoto...

We propose and analyze the convergence of a novel stochastic algorithm for monotone inclusions that are sum of a maximal monotone operator and a single-valued cocoercive operator. The algorithm we propose is a natural stochastic extension of the classical forward-backward method. We provide a non-asymptotic error analysis in expectation for the strongly monotone case, as L. Rosasco DIBRIS, Univ...

In this paper, we consider a proximal point algorithm for finding a common zero of a finite family of maximal monotone operators in real Hilbert spaces. Also, we give a necessary and sufficient condition for the common zero set of finite operators to be nonempty, and by showing that in this case, this iterative sequence converges strongly to the metric projection of some point onto the set of c...

and Applied Analysis 3 4 strictly η-monotone if 〈 u − v, ηx, y > 0, ∀x, y ∈ H, x / y, u ∈ M x , v ∈ M ( y ) ; 2.5 5 strongly monotone if there exists a constant r > 0 satisfying 〈 u − v, x − y ≥ r∥x − y∥2, ∀x, y ∈ H, u ∈ M x , v ∈ My; 2.6 6 strongly η-monotone if there exists a constant r > 0 satisfying 〈 u − v, ηx, y ≥ r∥x − y∥2, ∀x, y ∈ H, u ∈ M x , v ∈ My; 2.7 7 maximal monotone resp., maxim...

This paper is concerned with the solvability of some abstract differential equation of type u̇(t) + Au(t) + Bu(t) ∋ f(t), t ∈ (0, T ], u(0) = 0, where A is a linear selfadjoint operator and B is a nonlinear (possibly multi-valued) maximal monotone operator in a real Hilbert space H with the normalization 0 ∈ B(0). We use the concept of variational sum introduced by H. Attouch, J.-B. Baillon, and...

and Applied Analysis 3 nonexpansive mapping such that F S ∩ A−10/ ∅. Let {xn} be a sequence generated by x0 x ∈ C and un J−1 ( βnJxn ( 1 − βn ) JSJrnxn ) , Cn { z ∈ C : φ z, un ≤ φ z, xn } , Qn {z ∈ C : 〈xn − z, Jx0 − Jxn〉 ≥ 0}, xn 1 ΠCn∩Qnx0 1.6 for all n ∈ N ∪ {0}, where J is the duality mapping on E, {βn} ⊂ 0, 1 , and {rn} ⊂ a,∞ for some a > 0. If lim infn→∞ 1 − βn > 0, then {xn} converges s...

This paper is devoted to the study of a perturbed differential inclusion governed by nonconvex sweeping process in Hilbert space. The sum an integral forcing term which integrand depends on two time-variables and maximal monotone operator. By using semiregularization method combined with new Pachpatte inequality Gronwall-like we prove solvability initial value problem.

If L is a selfdual Lagrangian L on a reflexive phase space X ×X∗, then the vector field x → ∂̄L(x) := {p ∈ X∗; (p, x) ∈ ∂L(x, p)} is maximal monotone. Conversely, any maximal monotone operator T on X is derived from such a potential on phase space, that is there exists a selfdual Lagrangian L on X ×X∗ (i.e, L∗(p, x) = L(x, p)) such that T = ∂̄L. This solution to problems raised by Fitzpatrick can...