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...

In this article, we investigate sufficient conditions for existence of maximal and minimal solutions to a coupled system of highly nonlinear differential equations of fractional order with mixed type boundary conditions. To achieve this goal, we apply monotone iterative technique together with the method of upper and lower solutions. Also an error estimation is given to check the accuracy of th...

We study maximal monotone inclusions from the perspective of (convex) gap functions. We propose a very natural gap function and will demonstrate how this function arises from the Fitzpatrick function — a convex function used effectively to represent maximal monotone operators. • This approach allows us to use the powerful strong Fitzpatrick inequality to analyse solutions of the inclusion. – We...

0 ∈ Tx + λCx, where T : D(T )⊂ X → 2X is a strongly quasibounded maximal monotone operator and C : D(C)⊂ X → X∗ satisfies the condition (S+)D(C) with L⊂ D(C). The method of approach is to use a topological degree theory for (S+)L-perturbations of strongly quasibounded maximal monotone operators, recently developed by Kartsatos and Quarcoo. Moreover, applying degree theory, a variant of the Fred...

Maximal monotone operators on a Banach space into its dual can be represented by convex functions bounded below by the duality product. It is natural to ask under which conditions a convex function represents a maximal monotone operator. A satisfactory answer, in the context of reflexive Banach spaces, has been obtained some years ago. Recently, a partial result on non-reflexive Banach spaces w...

We propose a new discrete version of nonlinear oscillator with damping dynamical system governed by a general maximal monotone operator. We show the weak convergence of solutions and their weighted averages to a zero of a maximal monotone operator A. We also prove some strong convergence theorems with additional assumptions on A. This iterative scheme gives also an extension of the proximal poi...

Let X be an infinite-dimensional real reflexive Banach space with dual space X∗ and G⊂ X open and bounded. Assume that X and X∗ are locally uniformly convex. Let T : X ⊃ D(T) → 2X be maximal monotone and C : X ⊃ D(C) → X∗ quasibounded and of type (S̃+). Assume that L ⊂ D(C), where L is a dense subspace of X , and 0 ∈ T(0). A new topological degree theory is introduced for the sum T +C. Browder’s...

The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A+B provided that A,B are maximally monotone and A is a linear relation, as soon as Rockafellar’s constraint qualification holds: domA ∩ int domB 6...

We introduce an iterative sequence for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for three inversestrongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply...