Set-valued Mappings Research Articles

On the core of a low dimensional set-valued mapping

Let \mathfrak{M}=(\mathcal{M},\rho) be a metric space and let X be a Banach space. Let F be a set-valued mapping from \mathcal{M} into the family \mathcal{K}_m(X) of all compact convex subsets of X of dimension at most m . The main result in our recent joint paper with Charles Fefferman (which is referred to as a “finiteness principle for Lipschitz selections”) provides efficient conditions for the existence of a Lipschitz selection of F , i.e., a Lipschitz mapping f\colon \mathcal{M}\to X such that f(x)\in F(x) for every x\in\mathcal{M} . We give new alternative proofs of this result in two special cases. When m=2 , we prove it for X=\mathbb{R}^{2} , and when m=1 we prove it for all choices of X . Both of these proofs make use of a simple reiteration formula for the “core” of a set-valued mapping F , i.e., for a mapping G\colon \mathcal{M}\to\mathcal{K}_m(X) which is Lipschitz with respect to the Hausdorff distance, and such that G(x)\subset F(x) for all x\in\mathcal{M} .

ON THE STUDY OF CHAINS SOFT SETS, SOFT ORDERED SETS AND SOFT SEMILATTICES

In this paper, we study the concept of chains soft sets and set-valued function of chains soft sets. The definitions of chains soft sets or linear order or total order soft set are given. The notions of binary relation of comparability of the elements of set-valued functions are also discussed. Linearization’s of a partial order soft set are also defined. The definition and some algebraic structure of soft semilattice are also given.

Solvability for a Class of Integro-Differential Inclusions Subject to Impulses on the Half-Line

In this paper, we study a semilinear integro-differential inclusion in Banach spaces, under the action of infinitely many impulses. We provide the existence of mild solutions on a half-line by means of the so-called extension-with-memory technique, which consists of breaking down the problem in an iterate sequence of non-impulsive Cauchy problems, each of them originated by a solution of the previous one. The key that allows us to employ this method is the definition of suitable auxiliary set-valued functions that imitate the original set-valued nonlinearity at any step of the problem’s iteration. As an example of application, we deduce the controllability of a population dynamics process with distributed delay and impulses. That is, we ensure the existence of a pair trajectory-control, meaning a possible evolution of a population and of a feedback control for a system that undergoes sudden changes caused by external forces and depends on its past with fading memory.

From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability

We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.

Some periodic and fixed point theorems on quasi-b-gauge spaces

The notions of a quasi-b-gauge space (U,textsl{Q}_{s ; Omega }) and a left (right) mathcal{J}_{s ; Omega }-family of generalized quasi-pseudo-b-distances generated by (U,textsl{Q}_{s ; Omega }) are introduced. Moreover, by using this left (right) mathcal{J}_{s ; Omega }-family, we define the left (right) mathcal{J}_{s ; Omega }-sequential completeness, and we initiate the Nadler type contractions for set-valued mappings T:Urightarrow Cl^{mathcal{J}_{s ; Omega }}(U) and the Banach type contractions for single-valued mappings T: U rightarrow U, which are not necessarily continuous. Furthermore, we develop novel periodic and fixed point results for these mappings in the new setting, which generalize and improve the existing fixed point results in the literature. Examples validating our obtained results are also given.

Openness, Hölder Metric Regularity, and Hölder Continuity Properties of Semialgebraic Set-Valued Maps

Openness, Hölder Metric Regularity, and Hölder Continuity Properties of Semialgebraic Set-Valued Maps

Atriodic tree-like continua as inverse limits on [0,1] with interval-valued functions

For an inverse sequence on [0,1] with interval-valued functions, it is shown that chainability of the graphs of the bonding functions, and chainability of the partial graphs associated with the inverse sequence are necessary conditions for an atriodic tree-like inverse limit. An analogous result is proved for inverse sequences with functions fi where each fi−1 is interval-valued. It follows that non-chainable, atriodic tree-like continua cannot be realized as inverse limits on [0,1] with functions from either of these subclasses of set-valued functions.

An analogue of Nadler's result in Hardy-Rogers type iterated multifunction system

The iterated multifunction system (IMS) provides a primary way of deriving a class of set-valued functions constructed on a complete metric space. An exemplification of the generalisation of single-valued function to set-valued function is the Hardy-Rogers type iterated function system (HR-IFS). In this regard, set-valued functions show their efficiency in various domains like robotics, preventive maintenance, control systems, and energy. We extend the notion of HR-IFS to a class of Hardy-Roger type iterated multifunction system (HR-IMS). Moreover, concentrating on tremendous applicability of fixed point theory in real-life scenario, we have obtained a fixed point of the newly constructed IMS with the aid of the Hutchinson-Barnsley theory. In the main result, the attractor of the HR-IMS is constructed in an unconventional way. Consequently, a common idea of the Banach contraction principle, known as Nadler's type result, is gleaned for our HR-IMS.

Analyzing Feedback Interconnections of Maximal Monotone Systems Using Dissipativity Approach

We consider interconnections of two dynamical systems in feedback configuration. The dynamics of the individual systems are modeled by a differential inclusion, and the corresponding set-valued mapping is (anti-) maximal monotone with respect to the state of the system for each fixed value of the external signal that defines the interconnection. We provide conditions on these mappings under which the dynamics of the resulting interconnected system are (anti-) maximal monotone. An interpretation of our main result is provided: firstly, by considering dynamical systems defined by the gradient of a saddle function, and secondly, by considering an interconnection of incrementally passive systems. In the same spirit, when we associate more structure to the individual systems by considering linear complementarity systems, we allow for more flexibility in describing the interconnections and derive more specific sufficient conditions in terms of system matrices that result in the overall system being described by (anti-) maximal monotone operator.

Interpolative contractions and intuitionistic fuzzy set-valued maps with applications

<abstract><p>Over time, the interpolative approach in fixed point theory (FPT) has been investigated only in the setting of crisp mathematics, thereby dropping-off a significant amount of useful results. As an attempt to fill up the aforementioned gaps, this paper initiates certain hybrid concepts under the names of interpolative Hardy-Rogers-type (IHRT) and interpolative Reich-Rus-Ciric type (IRRCT) intuitionistic fuzzy contractions in the frame of metric space (MS). Adequate criteria for the existence of intuitionistic fuzzy fixed point (FP) for such contractions are examined. On the basis that FP of a single-valued mapping obeying interpolative type contractive inequality is not always unique, and thereby making the ideas more suitable for FP theorems of multi-valued mappings, a few special cases regarding point-to-point and non-fuzzy set-valued mappings which include the conclusions of some well-known results in the corresponding literature are highlighted and discussed. In addition, comparative examples which dwell on the generality of our obtained results are constructed.</p></abstract>

Fixed points of non-linear set-valued $ \left(\alpha _{\ast }, \phi _{M}\right) $-contraction mappings and related applications

<abstract><p>The aim of this manuscript is to prove some fixed point results for non-linear set-valued maps with new approach of $ \left(\alpha _{\ast }, \phi _{M}\right) $-contraction in complete $ M $-metric space. Also, we prove some fixed point results in ordered $ M $-metric space. As an presented work which are the extension and improves the current study of set-valued mappings. Finally, we also give an non-trivial extensive examples and application to homotopy theory and the existence solution of functional equations to show that our concepts are meaningful and to support our results.</p></abstract>

The value does not exist! A motivation for extremal analysis

<p style='text-indent:20px;'>Standard mathematical economics studies the production, exchange, and consumption of goods “<i>provided with units of measurement</i>,” as in physics, in order to be enumerated, quantified, added, etc. Therefore, “baskets of goods,” which should describe subsets of goods, are mathematically represented as commodity vectors of a vector space, linear combination of units of goods, evaluated by prices, which are linear numerical functions. Therefore, in this sense, mathematical economics is a branch of physics.</p><p style='text-indent:20px;'>However, economics, and many other domains of life sciences, investigate also what will be called <i>entities</i>, defining <i>elements deprived of units of measure</i>, which thus cannot be enumerated.</p><p style='text-indent:20px;'>(1) Denoting by <inline-formula><tex-math id="M1">\begin{document}$X$\end{document}</tex-math></inline-formula> the set of entities <inline-formula><tex-math id="M2">\begin{document}$x \in X$\end{document}</tex-math></inline-formula> <i>deprived of units of measurement</i>, a “basket of goods” is actually a <i>subset</i> <inline-formula><tex-math id="M3">\begin{document}$K \subset X$\end{document}</tex-math></inline-formula> of the set entities, i.e., an element of the “<i>hyperset</i>” <inline-formula><tex-math id="M4">\begin{document}${\cal{P}}(X)$\end{document}</tex-math></inline-formula>, the family of subsets of <inline-formula><tex-math id="M5">\begin{document}$X$\end{document}</tex-math></inline-formula>, and no longer a commodity vector of the vector space of commodities;</p><p style='text-indent:20px;'>(2) Entities can be “gathered” instead of being “added”;</p><p style='text-indent:20px;'>(3) Entities can still be evaluated by a <i>family</i> of functions <inline-formula><tex-math id="M6">\begin{document}$A: x \in X \mapsto A(x) \in \mathbb{R}$\end{document}</tex-math></inline-formula> regarded as a “valuators,” in lieu and place of linear “prices” evaluating the units of economic goods.</p><p style='text-indent:20px;'>(4) Subsets of entities can be evaluated by an “<i>interval of values</i>” <i>between two extremal ones</i>, the minimum and the maximum, instead of the sum of values of units of goods weighted by their quantities.</p><p style='text-indent:20px;'>Life sciences dealing with intertwined relations among many combinations of entities, hypersets offer metaphors of “Lamarckian complexity” that keeps us away from binary relations, graphs of functions, and set-valued maps, to focus our attention on “<i>multinary relations</i>” between families of hypersets. Even deprived of units of measurement, these “proletarian” entities still enjoy enough properties for this pauperization to be mathematically consistent.</p><p style='text-indent:20px;'>This is the object of this <i>extremal manifesto</i>: <i>in economics and other domains of life sciences, vector spaces should yield their imperial status of “state space” to hypersets and linear prices to hypervaluators</i>.</p><p style='text-indent:20px;'>We no longer have to add goods which can only be gathered, prices do not have to be linear, although it costs some effort to deprive oneself of the powerful and luxurious charms of convex and linear functional analysis motivated by physics.</p><p style='text-indent:20px;'>These sacrifices concern only economics and other fields of life science, since physicists deal with experimental observations of objects endowed with unit of measurement by adequate processes of measurement. They can happily live in vector spaces without any guilt. This is not the case of life scientists, who have mainly history to support and validate their observations, with, sometimes, the privilege to statistically measure the frequency of some of them.</p>

Korovkin-type approximation of set-valued and vector-valued functions

<p style='text-indent:20px;'>We establish some general Korovkin-type results in cones of set-valued functions and in spaces of vector-valued functions. These results constitute a meaningful extension of the preceding ones.</p>

Liapounoff type inequality for pseudo-integral of interval-valued function

<abstract><p>In this paper, two new Liapounoff type inequalities in terms of pseudo-analysis dealing with set-valued functions are given. The first one is given for a pseudo-integral of set-valued function where pseudo-operations are given by a generator $ g:[0, \infty]\to [0, \infty] $ and the second one is given for the semiring $ ([0, \infty], \sup, \odot) $ with generated pseudo-multiplication. The interval Liapounoff inequality is applied for estimation of interval-valued central $ g $-moment of order $ n $ for interval-valued functions in a $ g $-semiring.</p></abstract>

An algorithm based on a set-valued nonexpansive mapping

An algorithm based on a set-valued nonexpansive mapping

Common fixed points of fuzzy set-valued contractive mappings on metric spaces with a directed graph

<abstract><p>We introduce a new class of generalized graphic fuzzy $ F $- contractive mappings on metric spaces and establish the existence of common fuzzy coincidence and fixed point results for such contractions. It is significant to note that we do not use any form of continuity of mappings to prove these results. Some examples are provided to verify our proven results. Various developments in the existing literature are generalized and extended by our results. It is aimed that the initiated concepts in this work will encourage new research aspects in fixed point theory and related hybrid models in the literature of fuzzy mathematics.</p></abstract>

Stability of Generalized Minimax Regret Equilibria with Scalar Set Payoff

In this paper, we first introduce the notion and model of generalized minimax regret equilibria with scalar set payoffs. After that, we study its general stability theorem under the conditions that the existence theorem of generalized minimax regret equilibrium point with scalar set payoffs holds. In other words, when the scalar set payoffs functions and feasible constraint mappings are slightly disturbed, by using Fort theorem and continuity results of set-valued mapping optimal value functions, we obtain a general stability theorem for generalized minimax regret equilibria with scalar set payoffs. At the same time, an example is given to illustrate our result.

Existence of solutions for non-autonomous second-order stochastic inclusions with Clarke’s subdifferential and non instantaneous impulses

This manuscript explores a new class of non-autonomous second-order stochastic inclusions of Clarke?s subdifferential form with non-instantaneous impulses (NIIs), unbounded delay, and the Rosenblatt process in Hilbert spaces. The existence of a solution is deduced by employing a fixed point strategy for a set-valued map together with the evolution operator and stochastic analysis approach. An example is analyzed for theoretical developments.

On Hermite-Hadamard inequalities for (k,h)-convex set-valued maps

On Hermite-Hadamard inequalities for (k,h)-convex set-valued maps

$H(.,.,.,.)$-$varphi$-$eta$-cocoercive Operator with an Application to Variational Inclusions

In this work, we study generalized $ H(.,.,.,.)$-$varphi$-$eta-$ cocoercive operator to find the solution of variational like inclusion involving an infinite family of set-valued mappings in semi-inner product spaces via resolvent equation approach. Furthermore, we established an equivalence between the set-valued variational-like inclusion problem and fixed point problem by employing generalized resolvent operator technique involving generalized $H(.,.,.,.)$-$varphi$-$eta$-cocoercive operator. Using the equivalent formulation of set-valued variational-like inclusion problem and resolvent equation problem, an iterative algorithm is developed that approximate the uniqueness of solution of the resolvent equation problem.