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In this paper, we investigate the interval coupled Sylvester matrix equations which include the well-known (generalized) Sylvester and Lyapunov matrix equations in both real and interval forms. Assuming that certain matrices are simultaneously diagonalizable, we present a fast and efficient approach for enclosing the solution set of the interval coupled Sylvester matrix equations. Our approach, which is a modification of the Krawczyk operator, enables us to reduce the computational complexity considerably. Some numerical tests are given to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

The primary focus of this paper is threefold: first, to investigate the existence of mild solutions; second, to analyze the topological and geometrical structure of the solution sets; and third, to determine the continuous dependence of the solution for second-order semilinear integro-differential inclusion. In this study, we employ Bohnenblust–Karlin’s fixed point theorem in conjunction with the theory of resolvent operators, as presented by Grimmer. An illustrative example is employed to showcase the achieved outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for solving the system of fuzzy relation equations (SFRE) with extended max–min composition for inverse inference problems. The properties of interval and constrained solutions with granular and relational structure of the solution set are investigated. The extended max–min SFRE can be represented in the form of the max–min subsystems aggregated using the min operator or dual min–max subsystems aggregated using the max operator. When decomposing the SFRE, the set of solutions can be decomposed into the lower and upper subsets bounded by the same aggregating solutions. Each lower (upper) subset is defined by the unique greatest (least) or aggregating solution and the set of minimal (maximal) solutions. Following Bartl et al. (Fuzzy Sets Syst 187:103–112, 2012), to avoid excessive granularity and ensure interpretability of the interval solutions when restoring causes through observed effects, the constraints in the form of linguistic modifiers are imposed on the measures of causes significances. The interval solutions are modeled by the complete crisp solutions, that is, the maximum solutions for the vectors of binary weights of the linguistic modifiers. The search for approximate solutions of the SFRE amounts to solving the optimization problem using the genetic algorithm. Due to the properties of the solution set, the genetic search for the lower and upper subsets is parallelized for each aggregating solution. The developed method makes it possible to simplify the search for the solution set based on the constraints on accuracy (interpretability) of the applied problem. [ABSTRACT FROM AUTHOR]

This article combines (k,Θ)-Hilfer fractional calculus with glucose molecular graph, defines fractional differential and inclusion systems on each edge of a glucose molecular graph by the assumption that 0 or 1 marks the vertices, and studies the single-valued and multi-valued (k,Θ)-Hilfer type fractional boundary value problems on the glucose molecular graph. On the one hand, the existence and uniqueness of solutions in the single-valued case are proved by using several fixed point theorems. On the other hand, in the multi-valued case, we consider that the right side of the inclusion has convex valued and non-convex value. By applying Leray-Schauder nonlinear alternative method of multi-valued maps as well as Covitz-Nadler fixed point theorem of multi-valued contractions, two existence results are obtained respectively. On this basis, we also get the topological structure of the solution set, which is a pioneering work for (k,Θ)-Hilfer fractional differential inclusion on the glucose graph. Finally, several examples are provided to verify the reliability of our proposed results.

By using the fixed point theory for condensing multivalued maps, we prove the continuous dependence on parameters and initial data of a solution set of the Cauchy type problem for fractional order differential inclusions. We demonstrate the application of the averaging principle to the investigation of the continuous dependence of the solution set on a parameter in the case when the right-hand side of the inclusion is rapidly oscillating. [ABSTRACT FROM AUTHOR]

In convex optimization problems, characterizations of the solution set in terms of the classical subdifferentials have been investigated by Mangasarian. In quasiconvex optimization problems, characterizations of the solution set for quasiconvex programming in terms of the Greenberg–Pierskalla subdifferentials were given by Suzuki and Kuroiwa. In this paper, our attention focuses on the class of tangentially convex functions. Indeed, we study characterizations of the solution set for tangentially convex optimization problems in terms of subdifferentials. For this purpose, we use tangential subdifferentials and the Greenberg-Pierskalla subdifferentials and present necessary and sufficient optimality conditions for tangentially convex optimization problems. As a consequence, we investigate characterizations of the solution set in terms of tangential subdifferentials and the Greenberg–Pierskalla subdifferentials for tangentially convex optimization problems. Moreover, we compare our results with previous ones. [ABSTRACT FROM AUTHOR]

In this paper, we consider two common scalarization functions and their applications via asymptotic analysis. We mainly analyze the recession and asymptotic properties of translation invariant function and oriented distance function, and discuss their monotonicity and Lipschitz continuity in terms of recession functions. Finally, we apply these scalarization functions to the characterization of the nonemptiness and boundedness of the solution set for a general constrained nonconvex optimization problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of mixed variational inequalities on nonempty closed convex subsets of real Euclidean spaces. One of the mappings involved is lower semicontinuous and the other is weakly homogeneous. After discussing the boundedness for the solution set (if it is nonempty) of the problem, we focus on the nonemptiness and compactness of the solution set. Two new results on the nonemptiness and compactness of the solution set of the problem are established, and some examples are used to compare the results with those in the literature. It can be seen that new results improve some known related results. [ABSTRACT FROM AUTHOR]

This paper focuses on low computational efficiency in simulated annealing (SA) inversion of Transient Electromagnetic (TEM) data. Asynchronous multiple Markov chains (MMC) parallel strategy is a very promising SA acceleration method, which can be accelerated almost linearly. However, this method also reduces the accuracy of the solution. To overcome this problem, we added the solution set strategy to the asynchronous MMC parallel simulated annealing (PSA) algorithm for the first time. In this new algorithm, each thread independently searches for direction and exchanges data with the solution set in the shared memory. We used both the synthetic and field data to test the new algorithm. The synthetic data tests showed that the MMC PSA results are better than those of the original MMC PSA. We analyzed the efficiency of the new algorithm. Compared with the sequential VFSA, the maximum speedup of the new algorithm is approximately 10 times. The field data test also showed that the improved MMC PSA algorithm has good practicability. These tests demonstrate that the improved algorithm is effective, showing that its convergence speed is greatly improved without reducing the accuracy. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the topological structure of solution sets for a Volterra-type nonautonomous evolution inclusion with impulsive effects in a Fréchet space. The evolution family generated by the operator in the principal part is of no equicontinuity. Our attention is paid to establish the R δ -structure of the solution set and geometric features of the corresponding solution map. Moreover, the long-time existence of the nonlocal Cauchy problem is also considered. Finally, we present an example to illustrate the applicability of our abstract results. [ABSTRACT FROM AUTHOR]

We study systems of relations of the form A x σ b , where σ is a vector of binary relations with the components " = ", " ≥ ", and " ≤ ", while the parameters (elements of the matrix A and right-hand side vector b) are uncertain and can take values from prescribed intervals. What is considered to be the set of its solutions depends on which logical quantifier is associated with each interval-valued parameter and what is the order of the quantifier prefixes for specific parameters. For solution sets that correspond to the quantifier prefix of a general form, we present equivalent quantifier-free analytical descriptions in the classical interval arithmetic, in Kaucher complete interval arithmetic and in the usual real arithmetic. [ABSTRACT FROM AUTHOR]

This paper aims to an initial value problem for an impulsive fractional differential inclusion with the Riemann-Liouville fractional derivative. We apply Covitz and Nadler theorem concerning the study of the fixed point for multivalued maps to obtain the existence results for the given problems. We also obtain some topological properties about the solution set. [ABSTRACT FROM AUTHOR]

We study the topological regularity of solutions to the Cauchy problem of a (spatial) third-order partial differential equation with a multivalued perturbation and an impulsive effect. In the framework of the functional space, the principal part of the differential operator corresponds to an Airy operator generating a noncompact C0-group of unitary operators. Our attention is concerned with the Rδ-structure of the solution set for the Cauchy problem. Geometric aspects of the corresponding solution map are also considered. In our main results, no any compactness condition on the impulsive functions is needed. Moreover, we give illustrating examples for the nonlinearity and impulsive functions. [ABSTRACT FROM AUTHOR]

We study systems of relations of the form [Formula: see text], where [Formula: see text] is a vector of binary relations with the components “[Formula: see text]”, “[Formula: see text]”, and “[Formula: see text]”, while the parameters (elements of the matrix A and right-hand side vector b) are uncertain and can take values from prescribed intervals. What is considered to be the set of its solutions depends on which logical quantifier is associated with each interval-valued parameter and what is the order of the quantifier prefixes for specific parameters. For solution sets that correspond to the quantifier prefix of a general form, we present equivalent quantifier-free analytical descriptions in the classical interval arithmetic, in Kaucher complete interval arithmetic and in the usual real arithmetic.

A topological structure of the solution set to a class of fractional differential inclusions with (or impulses at variable times) is investigated. It is shown that the solution set is an R δ -set under some assumptions by the well-known theorem Bothe, D.: Multivalued perturbations of m-accretive differential inclusions. Israel J. Math. 108, 109–138 (1998) and the generalized Gronwall inequality under suitable Banach space. One example is listed for illustrating the main results. [ABSTRACT FROM AUTHOR]

This article combines ( k , Θ ) -Hilfer fractional calculus with glucose molecular graph, defines fractional differential and inclusion systems on each edge of a glucose molecular graph by the assumption that 0 or 1 marks the vertices, and studies the single-valued and multi-valued ( k , Θ ) -Hilfer type fractional boundary value problems on the glucose molecular graph. On the one hand, the existence and uniqueness of solutions in the single-valued case are proved by using several fixed point theorems. On the other hand, in the multi-valued case, we consider that the right side of the inclusion has convex valued and non-convex value. By applying Leray-Schauder nonlinear alternative method of multi-valued maps as well as Covitz-Nadler fixed point theorem of multi-valued contractions, two existence results are obtained respectively. On this basis, we also get the topological structure of the solution set, which is a pioneering work for ( k , Θ )-Hilfer fractional differential inclusion on the glucose graph. Finally, several examples are provided to verify the reliability of our proposed results., Competing Interests: The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Guotao Wang reports article publishing charges was provided by Elsevier Inc., Cell Press. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author(s).)

In this paper, we establish the existence of mild solutions for nonlocal fractional semilinear differential inclusions with noninstantaneous impulses of order α ∈ (1,2) and generated by a cosine family of bounded linear operators. Moreover, we show the compactness of the solution set. We consider both the case when the values of the multivalued function are convex and nonconvex. Examples are given to illustrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the topological structure of solution sets of monotone vector variational inequalities (VVIs). We show that if the weak Pareto solution set of a monotone VVI is disconnected, then each connected component of the set is unbounded. Similarly, this property holds for the proper Pareto solution set. Two open questions on the topological structure of the solution sets of (symmetric) monotone VVIs are raised at the end of the paper. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the characterization of the solution set for nonconvex semi-infinite programming problems related to tangential subdifferentials. We give a necessary optimality condition for the solution set of the nonconvex semi-infinite programming problem. We also prove that the Lagrangian function associated with a fixed Lagrange multiplier is constant on the solution set for semi-infinite programming problems. Finally, by using Dini pseudoconvexity, we obtain two characterizations of the solution set of the problem considered in this paper. Some examples are given to illustrate our results. [ABSTRACT FROM AUTHOR]

The present paper discusses an innovative approach providing the solution sets of engineering structures with uncertain parameters. The approach is based on the properties of convex sets and can be applied to structures described by the system of algebraic equations. The present paper focuses on trusses and frames applications, but in general it can be applied to various structures made of thin and thick bars and some plate and shell problems. The uncertain parameters are assumed to be independent. In addition, calculations are valid for any level of uncertainty and the obtained solution sets are exact within the assumed theory and are insensitive for perturbed data. Furthermore, solutions obtained by the present approach can be considered as benchmark solutions and can be used as a reference for other algorithms. The presented formulae allow the analysis of the influence of uncertain parameters on the behaviour of the structure. The presented considerations are illustrated by calculation of two truss examples. [ABSTRACT FROM AUTHOR]

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.

We establish an Rδ structure theorem for the fixed point set of the Krasnosel'skii type operator on Frechet space. Applying this result, a topological structure for the set of all mild solutions of fractional neutral evolution equations with infinite delay on half-line is investigated. We show that the solution set is an Rδ-set. [ABSTRACT FROM AUTHOR]

This article presents a survey of several properties of the set of solutions for a differential inclusion involving a time-delayed component and with right-hand side parametrized by either an upper semicontinuous or lower semicontinuous multifunction. Our results include: existence of solutions, compactness and contractibility of the solution and reachable sets in the upper semicontinuous case, precompactness and connectedness of the solution and reachable sets in the lower semicontinuous case, regularity of the solution and reachable mappings with respect to parameters, and existence of solutions for a dynamic optimization problem. [ABSTRACT FROM AUTHOR]

Solution sets of systems of homogeneous linear equations over fields are characterized as being subspaces, i.e., sets that are closed under linear combinations. Our goal is to characterize solution sets of systems of equations over arbitrary finite algebras by a similar closure condition. We show that solution sets are always closed under the centralizer of the clone of term operations of the given algebra; moreover, the centralizer is the only clone that could characterize solution sets. If every centralizer-closed set is the set of all solutions of a system of equations over a finite algebra, then we say that the algebra has Property (SDC) . Our main result is the description of finite lattices and semilattices with Property (SDC) : we prove that a finite lattice has Property (SDC) if and only if it is a Boolean lattice, and a finite semilattice has Property (SDC) if and only if it is distributive. [ABSTRACT FROM AUTHOR]

This paper deals with the max-algebraic linear equation system A ⊗ x = b. As in the conventional linear algebra such a linear system may have none, exactly one or infinitely many solutions. When the number of solutions is exactly one, Cramer's rule is given as an analogue of the classical linear algebra. When the number of solutions is infinite, the existence of a minimal solution is shown and the formula of minimal solution is given. Furthermore, it is proved that every solution can be expressed as a linear combination of a respective minimal solution and some special vectors. Finally, an algorithm to describe all the solutions of a given max-algebraic linear equation system is proposed when its number of solutions is infinite. AMS classification: 15A80,15A06 [ABSTRACT FROM AUTHOR]

Entropy principles based on thermodynamic consistency requirements are widely used for constitutive modeling in continuum mechanics, providing physical constraints on a priori unknown constitutive functions. The well-known Müller–Liu procedure is based on Liu's lemma for linear systems. While the Müller–Liu algorithm works well for basic models with simple constitutive dependencies, it cannot take into account nonlinear relationships that exist between higher derivatives of the fields in the cases of more complex constitutive dependencies. The current contribution presents a general solution set-based procedure, which, for a model system of differential equations, respects the geometry of the solution manifold, and yields a set of constraint equations on the unknown constitutive functions, which are necessary and sufficient conditions for the entropy production to stay nonnegative for any solution of the model. Similarly to the Müller–Liu procedure, the solution set approach is algorithmic, its output being a set of constraint equations and a residual entropy inequality. The solution set method is applicable to virtually any physical model, allows for arbitrary initially postulated forms of the constitutive dependencies, and does not use artificial constructs like Lagrange multipliers. A Maple implementation makes the solution set method computationally straightforward and useful for the constitutive modeling of complex systems. Several computational examples are considered, in particular models of gas, anisotropic fluid, and granular flow dynamics. The resulting constitutive function forms are analyzed, and comparisons are provided. It is shown how the solution set entropy principle can yield classification problems, leading to several complementary sets of admissible constitutive functions; such classification problems have not previously appeared in the constitutive modeling literature. [ABSTRACT FROM AUTHOR]

In this paper, we establish explicit upper bounds for the number of connected components in the proper Pareto solution sets and the weak Pareto solution sets of polynomial vector variational inequalities. Consequently, upper bounds for the numbers of connected components in the stationary point sets, the proper stationary point sets, and the weak Pareto solution sets of polynomial vector optimization problems are obtained. [ABSTRACT FROM AUTHOR]

This paper establishes several upper and lower estimates for the maximal number of the connected components of the solution sets of monotone affine vector variational inequalities. Our results give a partial solution to Question 2 in Yen and Yao (Optimization 60:53-68, 2011) and point out that the number depends not only on the number of the criteria but also on the number of variables of the vector variational inequality under investigation. [ABSTRACT FROM AUTHOR]

In this paper we investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. One of the aims of the paper is to give sufficient conditions to ensure that a regulated solution of an impulsive problem is globally defined.