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This paper is devoted to strict efficiency in set optimization studied with the set approach. Strict efficient solutions are defined with respect to the l-type less order relation and the possibly less order relation. Scalar characterization and necessary and/or sufficient conditions for such solutions are obtained. In particular, we establish some conditions expressed in terms of a high-order directional derivative of set-valued maps and the (convex or limiting) subdifferentials, normal cones and coderivatives. Various illustrating examples are presented. [ABSTRACT FROM AUTHOR]

In this article, we establish that any uniformly local Mizoguchi-Takahashi contraction is actually a set-valued contraction due to Feng and Liu on a metrically convex complete metric space. Through an example, we demonstrate that this result need not hold on any arbitrary metric space. Furthermore, when the metric space is compact, we derive that any Mizoguchi-Takahashi local contraction and Nadler local contraction are equivalent. Moreover, a result related to invariant best approximation is established.

Let Y be a real vector metric space and K ⊂ Y be a closed convex cone such that K ∩ (− K) = {0}. We prove that a convex compact-valued map F : ℝ → 2Y ∖ {∅} is K-continuous and K-additive if and only if there are non-empty convex compact sets A, B ⊂ Y such that 0 ∈ A − B ⊂ K and F is equal "modulo K" to the continuous set-valued map G (t) = t A , t ≥ 0 , t B , t < 0. Next, we use this result to characterize convex compact-valued maps F : ℝN → 2Y ∖ {∅}. [ABSTRACT FROM AUTHOR]

We discuss about the lower semicontinuity of set-valued maps, which is a crucial concept in parametric optimization and game theory. The focus is on the intersection of set-valued maps and the preservation of lower semicontinuity under this operation. We present new results based on some minimal properties that ensure the lower semicontinuity of the intersection with other lower semicontinuous maps. Additionally, the lower semicontinuity of the intersection of an infinite family of set-valued maps is considered, which has not been studied in the literature. We provide two examples to illustrate the applications of this concept in bilevel games, including a single-leader multi-follower game with robust followers and a secured multi-leader-follower game. [ABSTRACT FROM AUTHOR]

In this article, we establish that any uniformly local Mizoguchi-Takahashi contraction is actually a set-valued contraction due to Feng and Liu on a metrically convex complete metric space. Through an example, we demonstrate that this result need not hold on any arbitrary metric space. Furthermore, when the metric space is compact, we derive that any Mizoguchi-Takahashi local contraction and Nadler local contraction are equivalent. Moreover, a result related to invariant best approximation is established. [ABSTRACT FROM AUTHOR]

In this study, we describe higher-order (p; r)-ρ-cone arcwise connectedness as an entirely new type of generalized higher-order arcwise connectedness for set-valued optimization problems. Under the higher-order contingent epiderivative and higher-order (p; r)-ρ-cone arcwise connectedness suppositions, we demonstrate the higher-order sufficient Karush-Kuhn-Tucker (KKT) optimality requirements. We also investigate higherorder Mond-Weir (??) form of duality and establish the corresponding higher-order weak, strong, and converse theorems of duality between the primary (??) and the corresponding dual problems by employing the higher-order (p; r)-ρ-cone arcwise connectedness supposition. In order to demonstrate that higher-order (p; r)-ρ-cone arcwise connectedness is more widespread than higher-order cone arcwise connectedness, an example is also constructed. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics (including shadowing property, expansiveness, topological stability and entropy) of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view. It is shown that (1) if f is a hyperbolic endomor-phism then for each ε> 0 there exists a C1-neighborhood U of f such that the induced set-valued map F f , U has the ε-shadowing property, and moreover, if f is an expanding endomorphism then there exists a C1-neighborhood U of f such that the induced set-valued map F f , U has the Lipschitz shadowing property; (2) when a set-valued map F is generated by finite expanding endomorphisms, it has the shadowing property, and moreover, if the collection of the generators has no coincidence point then F is expansive and hence is topologically stable; (3) if f is an expanding endomorphism then for each ε> 0 there exists a C1-neighborhood U of f such that h (F f , U , ε) = h (f) (4) when F is generated by finite expanding endomorphisms with no coincidence point, the entropy formula of F is given. Furthermore, the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered. [ABSTRACT FROM AUTHOR]

The Rademacher theorem asserts that Lipschitz continuous functions between Euclidean spaces are differentiable almost everywhere. In this work we extend this result to set-valued maps using an adequate notion of set-valued differentiability relating to convex processes. Our approach uses the Rademacher theorem but also recovers it as a special case. [ABSTRACT FROM AUTHOR]

In this paper, an optimization problem (DP) is studied where the objective maps and the constraints are the difference of set-valued maps (abbreviated as SVMs). The higher-order σ-cone arcwise connectedness is described as an entirely new type of generalized higher-order arcwise connectedness for set-valued optimization problems. Under the higher-order contingent epiderivative and higher-order σ-cone arcwise connectedness suppositions, the higher-order sufficient Karush–Kuhn–Tucker (KKT) optimality requirements are demonstrated for the problem (DP). The higher-order Wolfe (WD) form of duality is investigated and the corresponding higher-order weak, strong, and converse theorems of duality are established between the primary (DP) and the corresponding dual problem by employing the higher-order σ-cone arcwise connectedness supposition. In order to demonstrate that higher-order σ-cone arcwise connectedness is more generalized than higher-order cone arcwise connectedness, an example is also constructed. As a special case, the results coincide with the existing ones available in the literature.

In this paper, firstly, we define a kind of new nonlinear functional with variable ordering structures and investigate its properties. Secondly, we establish Ekeland's variational principle with variable ordering structures. As the applications of Ekeland's variational principle, we obtain Caristi's fixed point theorem for set-valued maps and Takahashi's minimization theorem for set-valued maps with variable ordering structures. [ABSTRACT FROM AUTHOR]

Let X be an abelian metric group with an invariant metric, Y be a real normed space and K be a convex cone in Y. We prove that a K-subadditive (K-superadditive) compact- and convex-valued map F : X → C C (Y) , for which the functionals f y ∗ (x) = inf y ∗ (F (x)) are lower (upper, resp.) semicontinuous for any real continuous and non-negative on K functional y ∗ , has to be locally K-bounded on X. Our results refer to the papers Banakh and Jabłońska (Israel J Math 230:361–386, 2019), Jabłońska and Nikodem (Math Inequal Appl 22:1081–1089, 2019) and Nikodem (Aequationes Math 62:175–183, 2001). [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of approximate convexity to setvalued maps and obtain some relations between approximate convexity and approximate monotonicity of their normal subdifferential. [ABSTRACT FROM AUTHOR]

In this paper, we determine the sufficient Karush-Kuhn-Tucker (KKT) conditions of optimality of a set-valued fractional programming problem (in short, SVFP) $\rm (FP)$ under the suppositions of contingent epidifferentiation and $ \sigma $-arcwisely connectivity. We additionally explore the results of duality of parametric $\rm (PD)$, Mond-Weir $\rm (MWD)$, Wolfe $\rm (WD)$, and mixed $\rm (MD)$ kinds for the problem $\rm (FP)$.

This paper gives verifiable conditions for the existence of solutions of some set quasi-optimization problems involving Minkowski difference, where the objective maps are defined in complete metric spaces. Our proof method is not based on any scalarizing approach, and our existence results are written in terms of the given data of the problems. Several examples are provided. As applications of the main results of this paper, new results on the existence of robust solutions for uncertain multi-objective quasi-optimization problems with set-valued maps are formulated. [ABSTRACT FROM AUTHOR]

The fixed-time synchronization problems are solved for fractional-order memristive neural networks. According to the voltage and current characteristics of memristors, the model of fractional-order memristive neural networks with time-varying delays is established. Different from the traditional calculation method of memristive synaptic weights based on the maximum absolute value, by introducing some transformations, using differential inclusion theory and set-valued maps, the fractional-order memristive neural networks are transformed into a type of fractional-order systems with uncertain parameters in the framework of Filippov solution. Based on the fixed-time stability theory and the theory of measurable selection, the sufficient conditions of fixed-time synchronization are given by constructing Lyapunov function and using inequality techniques, and the calculation formula of the upper bound for the synchronization time is given. By designing an appropriate state feedback controller, the master-slave systems can reach fixed-time synchronization, and the upper bound for the synchronization time is independent of the initial state for the systems. The simulation example shows that the designed controller makes the systems achieve synchronization quickly. [ABSTRACT FROM AUTHOR]

The projection onto the epigraph or a level set of a closed proper convex function can be achieved by finding a root of a scalar equation that involves the proximal operator as a function of the proximal parameter. This paper develops the variational analysis of this scalar equation. The approach is based on a study of the variational-analytic properties of general convex optimization problems that are (partial) infimal projections of the sum of the function in question and the perspective map of a convex kernel. When the kernel is the Euclidean norm squared, the solution map corresponds to the proximal map, and thus, the variational properties derived for the general case apply to the proximal case. Properties of the value function and the corresponding solution map—including local Lipschitz continuity, directional differentiability, and semismoothness—are derived. An SC1 optimization framework for computing epigraphical and level-set projections is, thus, established. Numerical experiments on one-norm projection illustrate the effectiveness of the approach as compared with specialized algorithms. Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC). M.P. Friedlander was supported by NSERC discovery grant [Grant RGPIN-2017-04461]. T. Hoheisel was supported by NSERC discovery grant [Grant RGPIN-2017-04035]. A. Goodwin's work was partially supported by an NSERC summer research stipend. [ABSTRACT FROM AUTHOR]

Nonstationary fractional-order systems represent a new class of dynamic systems characterized by time-varying parameters as well as memory effect and hereditary properties. Differential game described by system of linear nonstationary differential equations of fractional order is treated in the paper. The game involves two players, one of which tries to bring the system's trajectory to a terminal set, whereas the other strives to prevent it. Using the technique of set-valued maps and their selections, sufficient conditions for reaching the terminal set in a finite time are derived. Theoretical results are supported by a model example. [ABSTRACT FROM AUTHOR]

In this paper the right upper semicontinuity at p = 1 and continuity at p = ∞ of the set-valued map p → B Ω , 풳 , p ⁢ (r) , p ∈ [ 1 , ∞ ] , are studied where B Ω , 풳 , p ⁢ (r) is the closed ball of the space L p ⁢ (Ω , Σ , μ ; 풳) centered at the origin with radius r, (Ω , Σ , μ) is a finite and positive measure space, 풳 is a separable Banach space. It is proved that the considered set-valued map is right upper semicontinuous at p = 1 and continuous at p = ∞ . An application of the obtained results to the set of integrable outputs of the input-output system described by the Urysohn-type integral operator is discussed. [ABSTRACT FROM AUTHOR]

In this paper we provide a characterization theorem for set-valued solutions of the functional equation for the case of compact, convex valued maps. We prove that every solution of the functional equation is the sum of an additive single-valued function and a nonempty compact and convex set. [ABSTRACT FROM AUTHOR]

An existence result is established for a generalized bifunction-set optimization problem, under some verifiable assumptions. Its proof is based on a Browder fixed point theorem and the KKMFan-Lemma. As applications of the main result of this paper, we obtain new existence results for some generalized versions of Kuroiwa set optimization problems and strong vector quasiequilibrium problems. Examples are provided. One of them is a non-cooperative generalized game with utility multifunctions. [ABSTRACT FROM AUTHOR]

This paper is devoted to new derivative criterion of metric regularity for set-valued map from a Fréchet-Montel space to a Fréchet space. Such type of criteria have been well studied in Banach spaces. Our work extends recent studies motivated by Nash-Moser-Ekeland approach. As a consequence of our criterion, an implicit mapping result is also given in the paper. [ABSTRACT FROM AUTHOR]

This article discusses the fixed-time stability (FTS) of a kind of delayed discontinuous system (DS) in Filippov sense. Based on the set-valued map, the FTS analysis of the general solution is first transformed into the zero solution of the differential inclusion. Second, the new criteria of the Lyapunov–Krasovskii functional (LKF) are given and LKF is proved to possess the indefinite derivatives by using the simple integral inequalities. In addition, the FTS of the considered delayed DS is achieved and the new settling time is estimated. Third, to demonstrate the applicability of the new FTS theorems, the FTS control of a class of discontinuous inertial neural networks (DINNs) with time-varying delays is solved. Finally, two numerical examples are given to examine the theoretical results and simulations are also provided to make some illustrations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain some characterizations (or representations) of set-valued solutions defined on an Abelian monoid with values in a Hausdorff topological vector space of a new class of functional equations. [ABSTRACT FROM AUTHOR]

In this article we define a notion of metric entropy for an invariant measure associated to a set-valued map F on a compact metric space X. Besides, we describe its main properties and prove the Half Variational Principle , which relates the metric entropy with the notion of topological entropy given in [ 13 ] for this class of maps. [ABSTRACT FROM AUTHOR]

In the paper, we introduce two new concepts on differentiability for set-valued maps, named by the higher-order generalized tangent epiderivative and the higher-order weak tangent epiderivative. Then, we study some basic properties and calculus rules for these notions. Finally, we mention their applications to two topics in set-valued optimization: optimality conditions and sensitivity analysis. More precisely, optimality conditions for some particular optimization problems are established and sensitivity analysis for the solution map of the parametric inclusion is given using higher-order generalized (weak) tangent epiderivatives. Several examples are proposed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to prove the existence result for a class of differential inclusions governed by the difference of two subdifferentials of nonconvex functions in Hilbert spaces. More precisely, by using the Moreau-Yosida regularization, the existence of local solutions for the following differential inclusion u ˙ (t) + ∂ Φ 1 (u (t)) − ∂ Φ 2 (u (t)) ∋ f (t) for almost every t ∈ [ 0 , T 0 ] , u (0) = u 0 is proved, where both functions Φ 1 and Φ 2 are assumed to be primal lower nice uniformly with respect to subgradients. [ABSTRACT FROM AUTHOR]

We study a feedback optimal control problem for a three-dimensional model of a stationary flow of a non-Newtonian fluid (with variable viscosity) in a pipeline network with complex geometry. The control parameter is the dynamic pressure on connection surfaces of pipes to nodes. The flow model is a mixed boundary-value problem for a system of strongly nonlinear partial differential equations in a netlike domain with Kirchhoff-type transmission conditions at interior nodes of the network. The solvability of the optimization problem in the weak formulation is proved; namely, we establish sufficient conditions for the existence of a weak solution which minimizes a lower semicontinuous cost functional. [ABSTRACT FROM AUTHOR]

We study the stability of set-valued optimization problems by using epi-convergence coupled with asymptotic analysis. To do this, we recall the notion of epi-convergence and introduce the notion of total epi-convergence for set-valued maps. We characterize them by means of epi-limits and horizon epi-limits. We use these epi-limits to study the behavior of vector/set type solutions and level/colevel sets under variations of the whole data. We also introduce several stronger epi-convergence notions and use them to study the stability of solution sets and minimal solution sets. We extend and generalize various results from the literature since we deal with unbounded feasible sets and objective maps. Finally, to illustrate our main results, we apply them to various classes of set-valued maps. [ABSTRACT FROM AUTHOR]

In this paper, we establish second-order KKT conditions of a set-valued optimization problem and study second-order Mond-Weir, Wolfe, and mixed types duals with the help of second-order contingent epiderivative and second-order generalized cone convexity assumptions.

We prove two existence results for functional differential inclusions governed by sweeping process. We consider the class of subsmooth moving sets. The perturbations depend on all the variables and their values are nonconvex. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the existence result of viable solutions for the differential inclusion ...(t) ∈ F(t, T(t)x) where F is a set-valued map with closed graph. We consider the case when the constraint is moving. [ABSTRACT FROM AUTHOR]

In this paper, we consider a set-valued minimax fractional programming problem (MFP), where the objective as well as constraint maps are set-valued. We introduce the notion of ρ-cone arcwise connectedness of set-valued maps as a generalization of cone arcwise connected set-valued maps. We establish the sufficient Karush-Kuhn-Tucker (KKT) conditions for the existence of minimizers of the problem (MFP) under ρ-cone arcwise connectedness assumption. Further, we study the Mond-Weir (MWD), Wolfe (WD), and mixed (MD) types of duality models and prove the corresponding weak, strong, and converse duality theorems between the primal (MFP) and the corresponding dual problems under ρ-cone arcwise connectedness assumption. [ABSTRACT FROM AUTHOR]

In this paper, the continuity of the set-valued map , , is proved where is the closed ball of radius in the space centered at the origin, is a finite and positive measure space, and is a separable Banach space. An application to input-output systems described by Urysohn type integral operators is discussed. [ABSTRACT FROM AUTHOR]

Epi-convergence plays an important role when studying the behaviour of a set optimization problem under variations of its data. In this paper, we obtain properties for this notion and compare it with other convergence notions for set-valued maps that have been used recently to study the stability of set optimization problems. [ABSTRACT FROM AUTHOR]

In the paper, we propose the higher order Hadamard directional derivative, a new kind of generalized derivatives for set-valued maps. Then we establish optimality conditions for Pareto efficient solutions of equilibrium problems with mixed constraints using this derivative. For applications, we obtain optimality conditions for some mathematical models in optimization. Several examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

Sufficient conditions are given for the existence of solutions of bifunction-set optimization problems, whose underlying sets are complete metric spaces. The main result of this paper is formulated in Theorem 3.1, with the help of some new data. The advantage of introducing the new data is that, by choosing suitable new data, we can obtain existence results with assumptions, that are imposed directly on the objective maps, or that are formulated via scalarization. The main result is applied successfully to Kuroiwa set optimization problems and Ky Fan vector inequality problems. We also discuss equivalences between the existence of solutions of the bifunction-set optimization problem, an Ekeland-type variational principle and a Caristi-type fixed point theorem. Examples are provided. [ABSTRACT FROM AUTHOR]

We prove that every K–subadditive set–valued map weakly K–upper bounded on a "large" set (e.g. not null–finite, not Haar–null or not a Haar–meager set), as well as any K–superadditive set–valued map K–lower bounded on a "large" set, is locally K–lower bounded and locally weakly K–upper bounded at every point of its domain. [ABSTRACT FROM AUTHOR]

In the current paper, some new set-valued Meir–Keeler, Geraghty and Edelstein type fixed point theorems are presented in a b-metric space. Some new contractive conditions are introduced and the results are proved using different techniques than their single-valued analogues. Applicability of these newly established theorems is demonstrated with some examples. [ABSTRACT FROM AUTHOR]

In this paper, we prove a theorem concerning the existence and uniqueness of a solution for a degenerate sweeping process with a Lipschitz single valued perturbation in Hilbert space. An application to differential complementarity problems is given. [ABSTRACT FROM AUTHOR]

In the paper, we introduce higher-order tangent epiderivatives for set-valued maps. Then, we study some basic properties of these concepts. Finally, we establish some results on duality in set-valued optimization. Several examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Lower and upper semicontinuity results for the approximate solution mapping of the parametric bifunction-set optimization problem are established under new assumptions that are quite different from the ones used previously in the case of exact solution mapping. Applications to the stability study of the approximate solution mapping of a parametric Kuroiwa set optimization problem and a parametric vector Ky Fan inequality problem are given. To our knowledge, our stability results are original. Several examples are provided. [ABSTRACT FROM AUTHOR]

A generalized Nash equilibrium problem corresponds to a noncooperative interaction between a finite set of players in which the cost function and the feasible set of each player depend on the decisions of the others. The classical existence result for generalized equilibria due to Arrow and Debreu requires continuity of the cost functions. In this work, we provide an existence of solutions transferring this hypothesis to a "continuity-like" condition over the sublevel sets of the aforementioned functions. Comparison with Reny's approach for discontinuous games is also considered. [ABSTRACT FROM AUTHOR]

In this paper, some properties of the lower (upper) semi-continuity for set-valued maps taking values in an abstract pre-ordered set are showed, which are then applied to study the existence of solutions for abstract set optimization problems. Moreover, some relationships between minimal solutions for abstract set-valued optimization problems with the vector and set criteria are given under mild conditions. As applications, existence results of solutions for set optimization problems are applied to obtain the existence of saddle points for the set-valued map taking values in the pre-ordered set with the set criterion and of solutions for vector optimization problems whose image space is a real vector space not necessarily endowed with a topology. [ABSTRACT FROM AUTHOR]