Search Results (26)

In this paper, we propose a penalty approach for solving generalized absolute value equations (GAVEs) of the type $Ax-B\left|x\right|=b$, $(A,B\in {\mathbb{R}}^{n\times n},b\in {\mathbb{R}}^n)$. Firstly, we reformulate the GAVEs as variational inequality problems passing through an equivalent horizontal linear complementarity problem. To approximate the resulting variational inequality, a sequence of nonlinear equations containing a penalty term is then defined. Under a mild assumption, we show that the solution of the considered sequence converges to that of GAVE if the penalty parameter tends to infinity. An algorithm is developed where its corresponding theoretical arguments are well established. Finally, some numerical experiments are presented to show that our approach is quite appreciable. Full article

This paper addresses the challenge of solving quasi-variational inequalities (QVIs) by developing and analyzing a forward–backward–forward algorithm from a continuous and iterative perspective. QVIs extend classical variational inequalities by allowing the constraint set to depend on the decision variable, a formulation that is particularly useful in modeling various problems. A critical computational challenge in these settings is the expensive nature of projection operations, especially when closed-form solutions are unavailable. To mitigate this, we consider the moving set case and propose a forward–backward–forward algorithm that requires only one projection per iteration. Under the assumption that the operator is strongly monotone, we establish that the continuous trajectories generated by the corresponding dynamical system converge exponentially to the unique solution of the QVI. We extend Tseng’s well-known forward–backward–forward algorithm for variational inequalities by adapting it to the more complex framework of QVIs. We prove that it converges when applied to strongly monotone QVIs and derive its convergence rate. We perform numerical implementations of our proposed algorithm and give numerical comparisons with other related gradient projection algorithms for quasi-variational inequalities in the literature. Full article

Let K be a closed convex subset of a real Banach space X, and let F be a map from X to its dual $X^{*}$. We study the so-called variational inequality problem: given $y\in X^{*,},$ does there exist $x_0\in K$ such that (in standard notation) $⟨F\left(x_0\right)-y,x-x_0⟩\ge 0$ for all $x\in K?$ After a short exposition of work in this area, we establish conditions on F sufficient to ensure a positive answer to the question of whether F is a gradient operator. A novel feature of the proof is the key role played by the well-known Ekeland variational principle. Full article

The mechanical properties of rubber-like materials, such as their high flexibility and durability, make them widely applicable across different industrial fields, from aerospace to healthcare and, most notably, the automotive sector. In operative conditions, these materials experience large deformations and repeated loadings, which may result in inelastic and dissipative phenomena. The aim of this study is to investigate the mechanical properties of two thermoplastic elastomeric materials manufactured with the Fused Filament Fabrication (FFF) technique: unfilled thermoplastic polyurethane (TPU) and TPU reinforced with carbon nanotubes (CNTs). Several experimental tests were performed to assess the response of both materials under monotonic and cyclic loadings. The addition of CNTs led to improved stiffness and strength without compromising elasticity. Under repeated loadings, both materials were characterized by the Mullins and viscous effects. However, the presence of CNTs was found to slightly amplify these inelastic phenomena. The integration of additive manufacturing technologies, combined with the use of innovative fillers, can offer design and performance optimization to all those components that strongly rely on elastomers. Full article

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed-point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, with special cases being able to be found in many important engineering practical applications, such as image recognizing, signal processing, and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results. Full article

In this paper, we investigate the solvability of a boundary value problem for a heat and mass transfer model with the spatially averaged Rayleigh function. The considered model describes the 3D steady-state non-isothermal flow of a generalized Newtonian fluid (with shear-dependent viscosity) in a bounded domain with Lipschitz boundary. The main novelty of our work is that we do not neglect the viscous dissipation effect in contrast to the classical Boussinesq approximation, and hence, deal with a system of strongly nonlinear partial differential equations. Using the properties of the averaging operation and d-monotone operators as well as the Leray–Schauder alternative for completely continuous mappings, we prove the existence of weak solutions without any smallness assumptions for model data. Moreover, it is shown that the set of all weak solutions is compact, and each solution from this set satisfies some energy equalities. Full article

In this article, the problem of solving a strongly monotone variational inequality problem over the solution set of a monotone inclusion problem in the setting of real Hilbert spaces is considered. To solve this problem, two methods, which are improvements and modifications of the Tseng splitting method, and projection and contraction methods, are presented. These methods are equipped with inertial terms to improve their speed of convergence. The strong convergence results of the suggested methods are proved under some standard assumptions on the control parameters. Also, strong convergence results are achieved without prior knowledge of the operator norm. Finally, the main results of this research are applied to solve bilevel variational inequality problems, convex minimization problems, and image recovery problems. Some numerical experiments to show the efficiency of our methods are conducted. Full article

A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, $(\theta ,h-m)-p$-convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, $(k-p)$-Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and $(p,q)$-calculus are also included. Full article

The purpose of the research is the study of a nonconstant gradient constrained problem for nonlinear monotone operators. In particular, we study a stationary variational inequality, defined by a strongly monotone operator, in a convex set of gradient-type constraints. We investigate the relationship between the nonconstant gradient constrained problem and a suitable double obstacle problem, where the obstacles are the viscosity solutions to a Hamilton–Jacobi equation, and we show the equivalence between the two variational problems. To obtain the equivalence, we prove that a suitable constraint qualification condition, Assumption S, is fulfilled at the solution of the double obstacle problem. It allows us to apply a strong duality theory, holding under Assumption S. Then, we also provide the proof of existence of Lagrange multipliers. The elements in question can be not only functions in $L^2$, but also measures. Full article

The purpose of the current study is to investigate a special case of art gallery problem, namely a sculpture garden problem. In this problem, for a given polygon P, the ultimate goal is to place the minimum number of guards (landmarks) to define the interior polygon P by applying a monotone Boolean formula composed of the guards. Using this problem, it can replace the operation-based method with time-consuming, pixel-based algorithms. So, the processing time of some problems in the fields of machine vision, image processing and gamification can be strongly reduced. The problem has also many applications in mobile device localization in the Internet of Things (IoT). An open problem in this regard is the proof of Eppstein’s conjecture, which has remained an open problem since 2007. According to his conjecture, in the worst case, $n-2$ vertex guards are required to describe any n-gon. In this paper, a lower bound is introduced for the special case of this problem (natural vertex guard), which shows that if a polygon can be defined with natural vertex guards, then $n-2$ is a lower bound. Full article

In this paper, under some new appropriate conditions imposed on the parameters and mappings involved in the proximal mapping associated with a general H-monotone operator, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. The main contribution of this work is the establishment of a new equivalence relationship between the graph convergence of a sequence of general strongly H-monotone mappings and their associated proximal mappings, respectively, to a given general strongly H-monotone mapping and its associated proximal mapping by using the notions of graph convergence and proximal mapping concerning a general strongly H-monotone mapping. By employing the concept of proximal mapping relating to general strongly H-monotone mapping, some iterative algorithms are proposed, and as an application of the obtained equivalence relationship mentioned above, a convergence theorem for approximating a common element of the set of solutions of a system of generalized variational inclusions involving general strongly H-monotone mappings and the set of fixed points of an $(\left\{a_n\right\},\left\{b_n\right\},\varphi )$-total uniformly L-Lipschitzian mapping is proved. It is significant to emphasize that our results are new and improve and generalize many known corresponding results. Full article

The demand for high-strength steel welds, as observed in civil and transport engineering, is related to a mass reduction in vehicles. Container-type trucks are examples of this kind of transport means because their boxes are able to be produced using Hardox grade steels. Therefore, this study reflects on the properties of welds in the MAG welding of Hardox 450, obtained through an innovative micro-jet cooling process with helium. This joining technology aims to reduce the formation of defects and to obtain a joint with very good assumed mechanical properties. Structural components of grade steel require welds with acceptable mechanical parameters with respect to operational loading conditions. That is, this study focuses on selecting welding parameters for the Hardox 450 steel and determining the weld quality with respect to microstructural observations and mechanical tests, such as the Charpy, tensile and fatigue tests. Weld fracturing under increasing monotonic force was examined and was strongly related to both stress components, i.e., axial and shear. The joint response under fatigue was expressed through differences in the fracture zones, i.e., at a stress value lower than the proportional limit, and weld degradation occurred in the shear and axial stress components. The data indicate that the hourglass specimen, with the weld in the centre zone of the measurement section, can be directly used to determine a weld response under cyclic loading. The impact test results showed attractive behaviour in the tested joint, as represented by 47 J at −20 °C. The recommended MAG welding parameters for Hardox 450 steel are low-oxygen when using an Ar + 18% CO₂ shielding mixture. The collected results can be directly used as a guide to weld thin-walled structures (6 mm) made of Hardox grade steel, while the data from mechanical tests can support the modelling, designing and manufacturing of components made from this kind of steel grade. Full article

This study aims to consider new kinds of generalized convex fuzzy mappings (convex-FM), which are called strongly α-preinvex fuzzy mappings. We investigated the characterization of preinvex-FMs using α-preinvex-FMs, which can be viewed as a novel and innovative application. Some different types of strongly α-preinvex-FMs are introduced, and their properties are investigated. Under appropriate conditions, we establish the relationship between strongly α-invex-FMs and strongly αj-monotone fuzzy operators. Then, the minimum of strongly α-preinvex-FMs are characterized by strongly fuzzy α-variational-like inequalities. Results obtained in this paper can be viewed as a refinement and improvement of previously known results. Full article

The main aim of this paper is to investigate the solvability of the steady-state flow model for low-concentrated aqueous polymer solutions with a damping term in a bounded domain under the no-slip boundary condition. Mathematically, the model under consideration is a boundary value problem for the system of strongly nonlinear partial differential equations of third order with the zero Dirichlet boundary condition. We propose the concept of a full weak solution (velocity–pressure pair) in the distributions sense. Using the method of introduction of auxiliary viscosity, the acute angle theorem for generalized monotone nonlinear operators, and the Krasnoselskii theorem on the continuity of the superposition operator in Lebesgue spaces, we obtain sufficient conditions for the existence of a full weak solution satisfying some energy inequality. Moreover, it is shown that the obtained solutions of the original problem converge to a solution of the steady-state damped Navier–Stokes system as the relaxation viscosity tends to zero. Full article

Low-voltage cable systems in nuclear power plants are key components that have a crucial role in the safe operation of nuclear facilities. Thus, the aging management of cable systems is of utmost importance as they cannot easily or economically be replaced or upgraded. Therefore, there is a continuous need to develop reliable non-destructive condition monitoring techniques, mostly based on the measurement of the dielectric properties of cable insulation. This paper introduces the changing of dielectric and mechanical properties of XLPE insulated and CSPE jacketed unshielded low-voltage nuclear power plant power cable in case of simultaneous mechanical and radiation aging. The cable samples were bent and exposed to 400 kGy gamma irradiation with a 0.5 kGy/hr dose rate. Dielectric response (real and imaginary permittivity) in the 0.1 Hz−1 kHz frequency range, extended voltage response (EVR), and the Shore D hardness test techniques were measured to track aging. The electrical and mechanical parameters have increased monotonically with aging, except the imaginary permittivity, which increased only at frequencies higher than 10 Hz. Furthermore, different quantities were deducted based on the frequency and permittivity data. The electrical parameters and deducted quantities correlation with aging and mechanical parameters were investigated. Since the deducted quantities and the electrical parameters are strongly correlated with absorbed dose and mechanical properties, the electrical measurements can be applied as a non-destructive aging indicator for XLPE/CSPE unshielded low-voltage nuclear power cables. Full article