Arcwise Connectedness Research Articles

Higher-order [formula omitted]-cone arcwisely connectedness in optimization problems associated with difference of set-valued maps

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.

Set-valued parametric optimization problems with higher-order $$\rho $$-cone arcwise connectedness

Set-valued parametric optimization problems with higher-order $$\rho $$-cone arcwise connectedness

On a fractional differential inclusion involving a generalized Caputo type derivative with certain fractional integral boundary conditions

We study a class of fractional differential inclusions defined by Caputo-Katugampola fractional derivativeinvolving a nonconvex set-valued map in the presence of certain fractional integral boundary conditions.Using a technique developed by Filippov we establish an existence result for the problem considered underthe hypothesis that the set-valued map is Lipschitz in the state variable. Also, based on a result concerningthe arcwise connectedness of the fixed point set of a class of set-valued contractions, we prove the arcwiseconnectedness of the solution set of the problem considered. The paper is the first in literature which containssuch kind of results in the framework of the problem studied.

Using ρ-cone arcwise connectedness on parametric set-valued optimization problems

Within the inquiry about work, we explore a parametric set-valued optimization problem, where the objective as well as constraint maps are set-valued. A generalization of cone arcwise associated set-valued maps is presented named ρ-cone arcwise connectedness of set-valued maps. We set up adequate Karush–Kuhn–Tucker optimality conditions for the problem beneath contingent epiderivative and ρ-cone arcwise connectedness presumptions. Assist, Mond–Weir, Wolfe, and blended sorts duality models are examined. We demonstrate the related theorems between the primal and the comparing dual problems beneath the presumption.

On constrained set-valued optimization problems with $$\rho $$-cone arcwise connectedness

On constrained set-valued optimization problems with $$\rho $$-cone arcwise connectedness

Set-valued minimax fractional programming problems under ρ-cone arcwise connectedness

Abstract 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.

Density and connectedness of optimal points with respect to improvement sets

Based on the improvement set E, this paper aims at investigating density and connectedness of the sets of E-optimal points, weak E-optimal points, E-quasi-optimal points, E-Benson proper optimal points, E-super optimal points and E-strictly optimal points. By virtue of the separation theorem for convex sets, we obtain the scalarizaiton results for the sets of E-optimal points, weak E-optimal points, E-Benson proper optimal points, E-super optimal points and E-strictly optimal points. Then we make a new attempt to establish some density theorems for the sets of these optimal points. Finally, we investigate connectedness and arcwise connectedness of the sets of various notions of E-optimality by using the scalarization method and the density theorems.

On a second-order differential inclusion with certain integral and multi-strip boundary conditions

We study a second-order differential inclusion with integral and multi-strip boundary conditions defined by a set-valued map with nonconvex values. We obtain an existence result and we prove the arcwise connectedness of the solution set of the considered problem.

On Constrained Set-Valued Semi-Infinite Programming Problems with ρ-Cone Arcwise Connectedness

In this paper, we establish sufficient Karush–Kuhn–Tucker (for short, KKT) conditions of a set-valued semi-infinite programming problem (SP) via the notion of contingent epiderivative of set-valued maps. We also derive duality results of Mond–Weir (MWD), Wolfe (WD), and mixed (MD) types of the problem (SP) under ρ-cone arcwise connectedness assumptions.

Arcwise Connectedness of the Solution Sets for Generalized Vector Equilibrium Problems

In this research, by means of the scalarization method, arcwise connectedness results were established for the sets of globally efficient solutions, weakly efficient solutions, Henig efficient solutions and superefficient solutions for the generalized vector equilibrium problem under suitable assumptions of natural quasi cone-convexity and natural quasi cone-concavity.

Connectedness of the approximate solution sets for set optimization problems

ABSTRACT In this paper, we deal with connectedness of the approximate solution sets for set optimization problems by using the scalarization method. We first obtain two scalarization results for the sets of l-minimal approximate solutions and weak l-minimal approximate solutions for set optimization problems without using the convexity of the objective mapping and the separation theorem of convex sets. Then, we establish connectedness of the sets of l-minimal approximate solutions and weak l-minimal approximate solutions for set optimization problems by utilizing the scalarization results. Finally, we derive arcwise connectedness of the set of weak l-minimal approximate solutions for set optimization problems.

SET-VALUED OPTIMIZATION PROBLEMS WITH (p, r)-ρ -CONE ARCWISE CONNECTEDNESS

In this paper, we introduce a new type of generalized arcwise connectedness, namely (p,r)-ρ-cone arcwise connectedness, for set-valued optimization problems. We establish the sufficient Karush-Kuhn-Tucker (KKT) optimality conditions under contingent epiderivative and (p,r)-ρ-cone arcwise connectedness assumptions. Further, we study Mond-Weir (MWD), Wolfe (WD), and mixed (MD) types duality models and prove the corresponding weak, strong, and converse duality theorems between the primal (P) and the corresponding dual problems under (p,r)-ρ-cone arcwise connectedness assumption. We also construct an example to ensure that (p,r)-ρ-cone arcwise connectedness is more general than cone arcwise connectedness.

Further results on the arcwise connectedness of solution sets of discontinuous quantum stochastic differential inclusions

In the framework of the Hudson–Parthasarathy quantum stochastic calculus, we employ a recent generalization of the Michael selection results in the present noncommutative settings to prove that the function space of the matrix elements of solutions to discontinuous quantum stochastic differential inclusion (DQSDI) is arcwise connected.

The Hyperspace $$F_n^K(X)$$

Given a metric continuum X and a positive integer n, $$F_{n}(X)$$ denotes the hyperspace of all nonempty subsets of X with at most n points endowed with the Hausdorff metric. For any $$K\in F_{n}(X)$$ , we define $$F_{n}(K,X)$$ as the collection of elements of $$F_{n}(X)$$ containing K and we consider $$F_{n}^K(X)$$ as the quotient space obtained from $$F_{n}(X)$$ by shrinking $$F_{n}(K,X)$$ to one point set, endowed with the quotient topology. In this paper, we report the first results of the investigation related with this hyperspace. We focus our attention on proving results regarding to aposyndesis, local connectedness, arcwise connectedness, unicoherence, and cut points of $$F_{n}^K(X)$$ .

Fixed point sets of isotopies on surfaces

We consider a self-homeomorphism h of some surface S . A subset F of the fixed point set of h is said to be unlinked if there is an isotopy from the identity to h that fixes every point of F . With Le Calvez’ transverse foliations theory in mind, we prove the existence of unlinked sets that are maximal with respect to inclusion. As a byproduct, we prove the arcwise connectedness of the space of homeomorphisms of the 2-sphere that preserve orientation and pointwise fix some given closed connected set F .

Arcwise connectedness of the solution sets for set optimization problems

In this paper, we establish the arcwise connectedness of the sets of l-minimal solutions and weak l-minimal solutions for set optimization problems under the assumption of the strictly natural quasi cone-convexity. Finally, our main result is applied to the arcwise connectedness of the solution sets for vector optimization problems.

On the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras

Let V be a Euclidean Jordan algebra of rank n. A set E in V is said to be a spectral set if there exists a permutation invariant set Q in Rn such that E=λ−1(Q), where λ:V→Rn is the eigenvalue map that takes x∈V to λ(x) (the vector of eigenvalues of x written in the decreasing order). If the above Q is also a convex cone, we say that E is a spectral cone. This paper deals with connectedness and arcwise connectedness properties of spectral sets. By relying on the result that in a simple Euclidean Jordan algebra, every eigenvalue orbit [x]:={y:λ(y)=λ(x)} is arcwise connected, we show that if a permutation invariant set Q is connected (arcwise connected), then λ−1(Q) is connected (respectively, arcwise connected). A related result is that in a simple Euclidean Jordan algebra, every pointed spectral cone is irreducible.

The hyperspace of regular subcontinua

Given a metric continuum X, we define the hyperspace of regular subcontinua of X as the collection of all regular closed subcontinua of X. This is a new hyperspace for X. In this paper we study the connectedness, compactness, arcwise connectedness and contractibility of this hyperspace. Also we raise some open questions.

Whitney blocks

Let C(X) be the hyperspace of subcontinua of a continuum X. A Whitney block in C(X) is a set of the form μ−1([s,t]), where μ:C(X)→[0,1] is a Whitney map and 0≤s<t<1. In this paper we study topological properties P satisfying the implication: if X has property P, then each Whitney block in C(X) has property P. We also consider the converse implication.We study properties related to local connectedness, arcwise connectedness, property of Kelley, contractibility, aposyndesis, unicoherence, etc.

Arcwise Connectedness of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

In the framework of the Hudson - Parthasarathy quantum stochastic calculus, we employ some recent selection results to prove that the function space of the matrix elements of solutions to quantum stochastic differential inclusion (QSDI) is arcwise connected both locally and globally.