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In this manuscript, a new concept of multivalued contraction is defined from a combination of Jaggi-type contractions, interpolative-type contraction and Pata-type inequality in the framework of metric space, and we analyze the existence of fixed points for such contractions equipped with some suitable hypotheses. One of the motivations forming the background of this paper is the fact that fixed point of a single-valued mapping satisfying the interpolative contractive condition is not necessarily unique, and thereby making the notions more appropriate for fixed point theorems of multi-valued mappings. A few salient consequences, including the single-valued cases are highlighted and discussed to indicate the significance of our proposed ideas. Also we give a comparative example. [ABSTRACT FROM AUTHOR]

In this paper, we propose criteria for unboundedness of the images of set-valued mappings having closed graphs in Euclidean spaces. We focus on mappings whose domains are non-closed or whose values are connected. These criteria allow us to see structural properties of solutions in vector optimization, where solution sets can be considered as the images of solution mappings associated to specific scalarization methods. In particular, we prove that if the domain of a certain solution mapping is non-closed, then the weak Pareto solution set is unbounded. Furthermore, for a quasi-convex problem, we demonstrate two criteria to ensure that if the weak Pareto solution set is disconnected then each connected component is unbounded. [ABSTRACT FROM AUTHOR]

In this work, the Banach contraction principle is generalized to a new formula. Then, it is used to give the guarantee conditions for the existence of common fixed point for set valued mappings defined on complete partial b-metric spaces. In addition, some results in connection with the existence of a fixed point for a setvalued mapping defined on a partial b-metric space are given. Some examples are established to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we study three fixed point problems. The first one is a fixed point problem with a set-valued mapping, while the second and third problems are convex feasibility problems with infinitely many constraints solved by the subgradient projection algorithm. We show that an approximate solution is reached after a finite number of iterations under the presence of computational errors. [ABSTRACT FROM AUTHOR]

This paper deals with a type of fixed point obtained from a family of generalized graph contractions that we adapt in the framework of a -metric space. We state and prove many new results in the underlying spaces. We provide a few applications of our results through some examples in order to support and illustrate the main results. To the best of our knowledge, the results derived herein improve and generalize many results in the cited references. [ABSTRACT FROM AUTHOR]

In the present paper, we obtain three results on the existence of a fixed point for nonexpansive mappings. Two of them are generalizations of the result for F-contraction, while third one is a generalization of a recent result for set-valued contractions. [ABSTRACT FROM AUTHOR]

In this paper we prove several turnpike results for trajectories of discrete disperse dynamical systems introduced in 1980 by A. M. Rubinov, which have a prototype in mathematical economics. [ABSTRACT FROM AUTHOR]

Probability functions appear in constraints of many optimization problems in practice and have become quite popular. Understanding their first-order properties has proven useful, not only theoretically but also in implementable algorithms, giving rise to competitive algorithms in several situations. Probability functions are built up from a random vector belonging to some parameter-dependent subset of the range of that given random vector. In this paper, we investigate first order information of probability functions specified through a convex-valued set-valued application. We provide conditions under which the resulting probability function is indeed locally Lipschitzian. We also provide subgradient formulæ. The resulting formulæ are made concrete in a classic optimization setting and put to work in an illustrative example coming from an energy application. [ABSTRACT FROM AUTHOR]

This study is concerned with the stability of Nash equilibria for a class of nn-person noncooperative differential games. More precisely, due to a preorder induced by a convex cone on a real linear normed space, we define a new concept called ordering stability of equilibria against the perturbation of the right-hand side functions of state equations for the differential game. Moreover, using the set-valued analysis theory, we present the sufficient conditions of the ordering stability for such differential games.

In this paper, we prove the existence of fixed point results for set-valued mappings in Menger probabilistic metric spaces equipped with an amorphous binary relation and a Hadžić -type t-norm. For the usability of such findings we present a Kelisky-Rivlin type result for a class of Bernstein type special operators introduced by Deo et. al. [Appl. Math. Comput. 201, (2008), 604-612 ] on the space C([ 0, n/n+1]). In this way, these investigations extend, modify and generalize some prominent recent fixed point results of the existing literature.

In this work, we study an iterative process induced by a contractive type set-valued mapping in a complete metric space and show its convergence, taking into account computational errors. [ABSTRACT FROM AUTHOR]

The paper presents various derivatives of set-valued mappings, their main properties and how they are related to each other. Next, we consider Cauchy problems with linear homogeneous set-valued differential equations with different types of derivatives (Hukuhara derivative, PSderivative and BG-derivative). It is known that such initial value problems with PS-derivative and BG-derivative have infinitely many solutions. Two of these solutions are called basic. These are solutions such that the diameter function of the solution section is a monotonically increasing (the first basic solution) or monotonically decreasing (the second basic solution) function. However, the second basic solution does not always exist. We provide conditions for the existence of basic solutions of such initial value problems. It is shown that their existence depends on the type of derivative, the matrix of coefficients on the right-hand and the type of the initial set. Model examples are considered. [ABSTRACT FROM AUTHOR]

In this paper, we present the generalized iterated function system for the construction of common fractals of generalized contractive mappings in the setup of dislocated metric spaces. The well-posedness of attractors' problems of rational contraction maps in the framework of dislocated metric spaces is also established. Moreover, the generalized collage theorem is also obtained in dislocated metric spaces. [ABSTRACT FROM AUTHOR]

The countably paracompact normal spaces were characterised by Dowker and Katětov in terms of an insertion property. Dowker also described them by normality of their product with the closed unit interval. Michael used the Dowker-Katětov insertion property to motivate his selection characterisation of these spaces. Morita extended in a natural way Dowker's product characterisation to all τ-paracompact normal spaces. In this paper, we look at these results from the point of view of open relations. Insertions and selections are equivalent for such relations. Furthermore, we obtain a natural characterisation of τ-paracompact normal spaces in terms of selections for convex-valued open relations. Based on this, we give simple alternative proofs of the above mentioned results. Other applications are obtained as well. [ABSTRACT FROM AUTHOR]

In this paper, we are devoted to investigating the existence of the Generalized Vector Quasi-Variational-Like Inequality Problem by using a new approach. Here we use admissibility of the function space topology and convergence of nets to establish the required existence results. We also introduce the concept of adic maps. Moreover, we provide an example to show that our solutions are independent of the solutions available in the literature. [ABSTRACT FROM AUTHOR]

We consider a guaranteed deterministic approach to the discrete-time superreplication problem in which it is required to cover a contingent liability on a written option in all feasible scenarios. These scenarios are described by a priori given compact sets depending on the price history: at each time instant, the price increments must lie in the corresponding compact sets. We assume no transaction costs. The problem statement is game-theoretic and leads to the Bellman-Isaacs equations in pure and mixed "market" strategies. In the case of no trading constraints, we study the relationship between the Bellman functions in the "deterministic" and "probabilistic" statements of the superhedging problem. As established under very general conditions, the "probabilistic" Bellman function does not exceed the "deterministic" counterpart. Sufficient conditions for their coincidence are found. [ABSTRACT FROM AUTHOR]

This paper focuses on solving a class of equilibrium problems, namely, the fixed point problem of set-valued mappings with second-order cone double constraints. Under certain conditions, the variational inequality form of the fixed point problem of set-valued mappings with second-order cone double constraints is obtained by using the generalized saddle point theory three times. The alternating direction method is used to solve the fixed point problem of set-valued mappings with second-order cone double constraints, and the global convergence of the algorithm is proved. Finally, numerical results of solving five examples with an inexact alternating direction method are given, and the feasibility and effectiveness of the algorithm are demonstrated by comparing with other algorithms.

Here, the solvability of the following inclusion elliptic problem − Δ H n , p u ∈ F (ξ , u) in Ω , u = 0 on ∂ Ω , is proved, via variational technique, where Ω is a Korányi ball in the Heisenberg Lie group ℍn and 퓕: Ω × ℝ → 풫(ℝ) is a real set-valued mapping. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of fixed point results for set-valued mappings in Menger probabilistic metric spaces equipped with an amorphous binary relation and a Hadzic-type t-norm. For the usability of such findings we present a Kelisky-Rivlin type result for a class of Bernstein type special operators introduced by Deo et. al. [Appl. Math. Comput. 201, (2008), 604-612] on the space C([0, n+i]). In this way, these investigations extend, modify and generalize some prominent recent fixed point results of the existing literature. [ABSTRACT FROM AUTHOR]

The article deals with the correspondence between soft ideal topological spaces and ideal topological ones. Investigation of soft ideal topological spaces is based on methods of general topology, and the application of results for soft omega open and strongly soft omega open sets is given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of hybrid inertial and contraction proximal point algorithm for the variational inclusion problem of the sum of two mappings in Hilbert spaces. We prove that the proposed algorithm converges strongly to a solution of the variational inclusion problem whenever its solution set is nonempty and the single-valued mapping f is Lipschitz continuous, monotone, and the set-valued mapping A is maximal monotone in infinite-dimensional real Hilbert spaces. Our work generalize and extend some related existing results in the literature. Finally, we illustrate the numerical performance of our Algorithm 1 and we give an application to the split feasibility problem. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a generalization of a recent result of Tam (2018) which is for iterates of a set-valued paracontracting operators in a finite-dimensional space. Our result in this paper is obtained for operators in a metric space such that every its bounded and closed set is compact. We establish a convergence of inexact iterates of the set-valued operator by taking into account computational errors. [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence and uniqueness of the solution for multi–term fractional boundary value problem under Riemann–Liouville fractional operators by using Banach's fixed point theorem. Afterward, we investigate some existence results for a semilinear fractional differential inclusions multi–term problem by using some notions and properties on set‐valued maps together with classical fixed point theorems due to Leray–Schauder. The cases when the set‐valued function has convex as well as nonconvex values are considered. Finally, some examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we study a fixed-point problem with a set-valued mapping by using an algorithm based on unions of nonexpansive mappings. We show that an approximate solution is reached after a finite number of iterations in the presence of computational errors. This result is an extension of the results known in the literature. [ABSTRACT FROM AUTHOR]

The authors consider a nonstationary game problem of control of moving objects in the case of violations in their dynamics caused by a breakdown or failure of the control devices. A game situation is analyzed, where the moment of failure of control devices is a priori unknown, and the time required to eliminate it is given. The sufficient conditions for bringing the trajectory of the conflict-controlled process to the terminal set in a certain finite time are established. The results are illustrated using a model example with simple motion. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of non-empty fixed point sets of newly introduced generalized set-valued F-contractions of b-metric spaces. Some illustrative examples show that the new results in this paper generalize properly, unify and extend some related results in the existing literature. Moreover, we extract some important consequences of the results in b-metric spaces. Particularly, by setting b-metric constant equal to one, we obtain some specific cases showing notable enhancement of existing results yet in metric spaces.

The focus of this paper is to employ a new topological condition, based on admissibility of the function space topology, to provide existence results for a generalized vector quasi-variational inequality problem and its stronger form as well. The inequality problems involve a set-valued function, and the existence results are proved without using any monotonicity or convexity assumptions on the functions. Further, the solution sets of the inequality problems are shown to be closed and compact. [ABSTRACT FROM AUTHOR]

In our recent paper we obtained an extension of a theorem of Tam (2018), which was established for iterates of a set-valued paracontracting operators in a finite-dimensional Euclidean space. This extension was obtained for exact iterates of maps in a metric space. In the present paper we prove two analogs of this result for inexact iterates under the presence of computational errors. In the first result, the errors are summable, while for the second one they converge to zero. In a particular case of a Banach space, if all the operators are symmetric, then the set of all limit points of iterates is symmetric too. [ABSTRACT FROM AUTHOR]

LetMD.M; be a metric space and let X be a Banach space. Let F be a set-valued mapping fromM into the family Km. 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 WM ! X such that f .x/2 F.x/for every x 2 M. We give new alternative proofs of this result in two special cases. When m D 2, we prove it for X D R2, and when m D 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 GWM ! Km.X/which is Lipschitz with respect to the Hausdorff distance, and such that G.x/- F.x/for all x 2 M. [ABSTRACT FROM AUTHOR]

This paper extends Berge's maximum theorem for possibly noncompact action sets and unbounded cost functions to minimax problems and studies applications of these extensions to two-player zero-sum games with possibly noncompact action sets and unbounded payoffs. For games with perfect information, also known under the name of turn-based games, this paper establishes continuity properties of value functions and solution multifunctions. For games with simultaneous moves, it provides results on the existence of lopsided values (the values in the asymmetric form) and solutions. This paper also establishes continuity properties of the lopsided values and solution multifunctions. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of non-empty fixed point sets of newly introduced generalized set-valued F-contractions of b-metric spaces. Some illustrative examples show that the new results in this paper generalize properly, unify and extend some related results in the existing literature. Moreover, we extract some important consequences of the results in b-metric spaces. Particularly, by setting b-metric constant equal to one, we obtain some specific cases showing notable enhancement of existing results yet in metric spaces. [ABSTRACT FROM AUTHOR]

We consider a guaranteed deterministic approach to the discrete-time superreplication problem in which it is required to cover a contingent liability on a written option in all feasible scenarios. These scenarios are described by a priori given compact sets depending on the price history: at each time instant, the price increments must lie in the corresponding compact sets. We assume no transaction costs. The problem statement is game-theoretic and leads to the Bellman-Isaacs equations in pure and mixed "market" strategies. In the case of no trading constraints, we study the relationship between the Bellman functions in the "deterministic" and "probabilistic" statements of the superhedging problem. As established under very general conditions, the "probabilistic" Bellman function does not exceed the "deterministic" counterpart. Sufficient conditions for their coincidence are found. [ABSTRACT FROM AUTHOR]

In this paper, we obtain two extensions of a recent result of Tam, which was proved for iterates of set-valued paracontracting operators in a finite-dimensional space. Our results are obtained for operators in an arbitrary metric space. In the first result, we study exact iterates of the set-valued mapping, while in the second one, we deal with its inexact iterates, taking computational errors into account. In a particular case of a Banach space, if all the operators are symmetric, then the set of all limit points of iterates is symmetric too. [ABSTRACT FROM AUTHOR]

We present the uplift of a binary operation on a subposet of a complete lattice as a generic way of extending this operation to the entire complete lattice. It involves the use of a set-valued mapping between that complete lattice and the subposet. We show that the uplift of a binary operation is increasing when this set-valued mapping is increasing, and that it is a proper extension when the given binary operation is also increasing and the set-valued mapping satisfies an additional natural condition. If some technical cofinality condition is met, then uplifting also preserves associativity. Moreover, if we consider a canonical set-valued mapping, then the uplift of a t-subnorm on a subposet is a t-subnorm as well. It is then easy to modify the uplift of a t-norm on a bounded subposet to turn it from a t-subnorm into a t-norm. Several existing t-norm construction methods turn out to be instantiations of this (modified) uplifting process. Finally, we formulate the dual results on the downlift of a binary operation. [ABSTRACT FROM AUTHOR]

It is always a goal to fully exploit difference features and their importance in infrared polarization and intensity images to drive selection of fusion algorithms rather than using fixed algorithm, to improve pertinence and effectiveness of image fusion. However, it is difficult to obtain a better and more reasonable result for fused image due to the varied difference features and uncertain relationship between difference features and fusion algorithms. The study aims to investigate the two-tuple linguistic cloud model for possibility distribution to solve the fusion algorithm selection problem. Firstly, fusion validity distributions of difference feature amplitudes are constructed through giving consideration to both fuzziness and randomness. Secondly, in order to build set-valued mapping between difference feature amplitudes and fusion algorithms, a novel fusion validity degree transformation method into two-tuple linguistic cloud variable is proposed based on possibility theory. Thirdly, difference feature weights are calculated based on nonparametric estimation for the frequency of each corresponding feature in images. Next, a cloud weighted arithmetic averaging operator is constructed to rank the algorithms. Finally, a case of the fusion algorithm selection is provided to illustrate the implementation process and applicability of the method proposed in this paper. In addition, the proposed method can be utilized the comprehensive and multi-constrained optimal problem with clear and effective management process.

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In this paper, we study turnpike properties for discrete disperse dynamical systems generated by set-valued mappings which were introduced by A. M. Rubinov in 1980. [ABSTRACT FROM AUTHOR]

In this paper, we present a rough set model based on fuzzy ideals of distributive lattices. In fact, we consider a distributive lattice as a universal set and we apply the concept of a fuzzy ideal for definitions of the lower and upper approximations in a distributive lattice. A novel congruence relation induced by a fuzzy ideal of a distributive lattice is introduced. Moreover, we study the special properties of rough sets which can be constructed by means of the congruence relations determined by fuzzy ideals in distributive lattices. Finally, the properties of the generalized rough sets with respect to fuzzy ideals in distributive lattices are also investigated.

A method for solving the game problem of the trajectory of a quasi-linear non-stationary system approaching a cylindrical terminal set that varies with time is proposed. A case is considered where the Pontryagin condition (the condition of the first player’s advantage) is not satisfied. The time dilation function is introduced, which postpones the time of the game termination, and with its help a modified Pontryagin’s condition, which allows making a measurable choice of control. The basic method is the method of resolving functions. Using the technique of set-valued mappings and their selectors, strategies are generated, which guarantee the problem solution. The process of the trajectory approaching the terminal set consists of two sections: active and passive, where the control of the first player is selected, using the control of the second player with a certain time delay, which depends on the function of time dilation. The scheme of the method is outlined and sufficient conditions for the game termination in a finite time are obtained. [ABSTRACT FROM AUTHOR]

In this paper, in the context of quasi-metric spaces, we obtain two set-valued versions of the Ekeland variational-type principle by means of lower and upper set less relations, for the case where the perturbations need not satisfy the triangle inequality. An application in terms of migration problems and quality of life is given. [ABSTRACT FROM AUTHOR]

This paper is concerned with the system (NEESSVMEP) obtained by mixing a nonlinear evolutionary equation and a strong set-valued mixed equilibrium problem (SSVMEP). In our setting, the set of constraints is not necessarily compact and the problem is driven by a not necessarily monotone set-valued mapping. We show that the solution set for (SSVMEP) is nonempty, closed, convex and bounded. We then establish the upper semicontinuity and the measurability properties of the involved functions in the nonlinear evolutionary equation. Utilizing these results, we prove that the solution set for (NEESSVMEP) is nonempty and compact. [ABSTRACT FROM AUTHOR]

We present several fixed point and convergence results for nonexpansive set-valued mappings which map a closed subset of a complete metric space into the space. [ABSTRACT FROM AUTHOR]

In this work, we establish several results concerning the existence of solutions for set-valued and single-valued equilibrium problems in real Hausdorff topological vector spaces. Firstly we introduce some generalizations of convexity and continuity conditions to set-valued mappings and then apply them to special dense subsets of the domain to obtain the existence of local dense solutions of equilibrium problems. Then the existence of the global solutions follows from a condition that is weaker than semistrict quasiconvexity. Specifically, we give an existence theorem for noncooperative n-person games, under assumptions imposed on a locally segment-dense subset of the strategy set of each player. [ABSTRACT FROM AUTHOR]

Abstract We study an approximate coincidence point and a common fixed point problem for a hybrid pair of mappings with constraints in Menger PM-spaces, and obtain some new results. We derive interesting consequences of the main results by using the properties of a Menger–Hausdorff metric, and analogous results based on graphs instead of partial orders can be similarly formulated. Moreover, we construct two examples to reveal that the main results are valid, and show that the main results can be used to explore the existence of solutions to a system of nonlinear integral equations.

Let $X$ and $Y$ be Banach spaces. Let $\Omega$ be an open subset of $X$. Suppose that $f:X\to{Y}$ is Fr\'{e}chet differentiable in $\Omega$ and $\mathcal F:X\rightrightarrows2^Y$ is a set-valued mapping with closed graph. In the present paper, a modified superquadratic method (MSQM) is introduced for solving the generalized equations $0\in{f(x)+\mathcal F(x)}$, and studied its convergence analysis under the assumption that the second Fr\'{e}chet derivative of $f$ is H\"{o}lder continuous. Indeed, we show that the sequence, generated by MSQM, converges super-quadratically in both semi-locally and locally to the solution of the above generalized equation whenever the second Fr\'{e}chet derivative of $f$ satisfies a H\"{o}lder-type condition.