A new generalization of Ćirić's multi-valued operators

We consider general variational inequalities with a multivalued maximal monotone operator in a Hilbert space. For solving these problems, Cohen developed several years ago the auxiliary problem method. Perturbed versions of this method have been already studied in the literature for the single-valued case. They allow to consider for example, barrier functions and interior approximations of the feasible domain. In this paper, we present a relaxation of these perturbation methods by using the concept of e-enlargement of a maximal monotone operator. We prove that, under classical assumptions, the sequence generated by this scheme is bounded and weakly convergent to a solution of the problem. Strong convergence is also obtained under additional conditions.

In this paper, we will consider the following problem: if a fixed point inclusion x ? B(x) (where B is a ?bad? multi-valued operator) is replaced by another fixed point inclusion x ? G(x) (where G is now a ?good/better? multi-valued operator), then which properties (existence, data dependence, stability) can be obtained for our initial problem in terms of some conditions on the new operator G? We will do this by the help of the admissible perturbation technique, which will generate a ?good? multi-operator G related to the ?bad? multi-valued operator B by the fact that the fixed point set of B and respectively of G coincide.

We present a generalization of the Uniform Boundedness Principle valid for random multivalued linear operators, i.e., multivalued linear operators taking values in the space L0(Ω,Y) of random variables defined on a probability space \((\Omega,{\mathcal{A}},P)\) with values in the Banach space Y. Namely, for a family of such operators that are continuous with positive probability, if the family is pointwise bounded with probability at least δ>0, then the operators are uniformly bounded with a probability that in each case can be estimated in terms of δ and the index of continuity of the operator. To achieve this result, we develop the fundamental theory of multivalued linear operators on general topological vector spaces. In particular, we exhibit versions of the Closed Graph Theorem, the Open Mapping Theorem, and the Uniform Boundedness Principle for multivalued operators between F-spaces.

Existence of Urysohn and Atangana–Baleanu fractional integral inclusion systems solutions via common fixed point of multi-valued operators

In this work, in the setting of a vector-valued metric space X, the fixed point inclusion x∈G(x),x∈X governed by a multi-valued operator G:X⊸X satisfying a weak contraction type condition on its graph is studied. Some stability results are obtained and the coincidence point problem involving two multi-valued operators is also studied. The best proximity point problem in vector-valued metric spaces is finally discussed.

Algebraic hyperstructures are algebraic systems whose objects equipped with the hyperoperations or multivalued operations. In a paper entitled ”Pre-semihyperadditive Categories” we introduced pre-semihyperadditive categories as the categories in which for objects A and B, the class of all morphisms from A to B denoted by Mor(A, B), admits an algebraic hyperstructure, such as semihypergroup or hypergroup. Moreover, we constructed some categories formed by the (general Krasner) hypermodules. In this paper we present some properties related to the morphisms of a category of (general Krasner) hypermodules denoted by RG.mod.Algebraic hyperstructures are algebraic systems whose objects equipped with the hyperoperations or multivalued operations. In a paper entitled ”Pre-semihyperadditive Categories” we introduced pre-semihyperadditive categories as the categories in which for objects A and B, the class of all morphisms from A to B denoted by Mor(A, B), admits an algebraic hyperstructure, such as semihypergroup or hypergroup. Moreover, we constructed some categories formed by the (general Krasner) hypermodules. In this paper we present some properties related to the morphisms of a category of (general Krasner) hypermodules denoted by RG.mod.

The present paper represents a continuation of [3]. There, we studied a new class of variational inequalities involving a pseudomonotone univalued operator and a multivalued operator, for which we obtained an existence result, among others. In the current paper we prove that this result remains valid under significantly weaker assumption on the multivalued operator. Then, we consider a new mathematical model which describes the equilibrium of an elastic body attached to a nonlinear spring on a part of its boundary. We use our abstract result to prove the weak solvability of this elastic model.

In this paper, we introduce the concept of generalized contraction for multivalued operators defined on ordered complete metric spaces. We analyze the existence of fixed points for generalized multivalued operators. Moreover, as an application of our main theorem, we give an existence theorem for the solution of a hyperbolic differential inclusion problem.

In this paper two versions of the relaxed proximal point schemes for solving variational inequalities with maximal monotone and multi-valued operators are investigated. The first one describes an algorithm with an adaptive choice of the relaxation parameter and is combined with the use of ϵ–enlargements of multi-valued operators. The second one makes use of Bregman functions in order to construct relaxed proximal point algorithms with an interior point effect. For both algorithms convergence is proved under a numerically tractable error summability criterion, based on the ideas in [R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res. 1(2) (1976), pp. 97–116.] [R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14(5) (1976), pp. 877–898.] Finally, some numerical aspects are discussed and test examples show the performance of the first algorithm when applying to non-smooth optimization problems.

In this paper we propose the method of solving of the variational inequalities with multivalued operators by some extremal problems. These extremal problems consist the inclusion with para-meter and the penalty function. As examples we consider the free boundary problems on W p 1 (Ω), p ≥ 3.

We establish new results on the existence of positive solutions for a kind multi-point boundary value problem with multivalued operator. Our results are based on a recent Leggett-Williams theorem for coincidences of multivalued operators due to O’Regan and Zima. The most interesting point is the acquisition of positive solutions for the resonance case. And an example is constructed to show that our result here is valid.

Let X, Y be Banach spaces and M be a linear subspace in X × Y = {{x, y}|x ∈ X, y ∈ Y}. We may view M as a multi-valued linear operator from X to Y by taking M(x) = {y|{x, y} ∈ M}. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M. The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.

The upper and lower semi-Fredholm perturbations of multivalued linear operators were discussed in a previous paper by Álvarez et al. (J. Math Methods Appl Sci. 2014;37(5):620–644). The purpose of the present study is to investigate the and -Atkinson perturbation of linear relations. After that, we use the reached findings to establish a survey of results concerning various essential spectra of linear relations. We finish this paper by establishing some results of the essential spectra of a transport equations with abstract boundary conditions in -spaces.

Conventional binary logic based operations restrict the speed of operations as well as information handling capacity. A way to overcome these limitations is the implementation of multivalued logic operations in the optical domain. Multivalued logic operations not only enhance the data handling capacities but also increase the speed of processing. integrating enormous potential bandwidth of optical fiber as information carrying medium and faster optoelectronic/optical switches with no hardware complexity. A new method is proposed for the implementation of all-optical quaternary inversion, MAX, MIN, and equality operations using frequency-encoded data. Cross phase modulation-based frequency conversion, polarization switch (PSW) characteristics of a semiconductor optical amplifier (SOA), frequency routing by a wave division multiplexer (MUX), and a demultiplexer (DMUX) have been exploited to implement the desired quaternary logic operations. Simulation results support the feasibility of the proposed scheme.

The first of these seven steps is to eliminate multivalued attributes, composite attributes, and composite operations. The second is to eliminate the partial dependency and transitive dependency among the attributes, as well as shared operations. The third is to eliminate homogeneous operations among the classes to meet the requirements of inheritance and polymorphism. The fourth and fifth steps involve establishing classes for encapsulation, and the sixth and seventh steps eliminate multivalued dependency and operations with multivalued dependency. These steps create normalised concrete classes and control classes in the object-oriented class diagram, and they also maintain favourable consistency, completeness, and accuracy of the data. As such, they can provide effective guidelines and practical reference for system analysis and development.