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In this paper, we propose a new projection method (extragradient method) for solving the generalized variational inequality problem. Our method is well-defined and is proven to be globally convergent to a solution of the problem under the assumptions that the multivalued mapping is continuous and satisfies a condition that is strictly weaker than the pseudomonotonicity. Finally, to support our results and to illustrate the numerical behavior of the proposed algorithm, some fundamental experiments are provided and also compared with recent works. [ABSTRACT FROM AUTHOR]

In this paper, we propose a random and cyclic projection algorithm for solving variational inequality problems with special structure where the underlying mapping is strongly pseudomonotone and L-Lipschitz continuous and the constraint set is the intersection of a large number of simple closed convex sets. Compared with some existing incremental constraint projection algorithms, the proposed algorithm has two notable advantages: Its global convergence can be guaranteed under the assumption that F is strongly pseudomonotone, not strongly monotone or monotone plus; It just computes one projection onto a halfspace rather than two or more times projections onto the full or single constraint set at each iteration. Computational experiments are also reported to illustrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this work, we present a refined convergence analysis of the Popov's projection algorithm for solving pseudo-monotone variational inequalities in Hilbert spaces. Our analysis results in a larger range of stepsize, which is achieved by using a new Lyapunov function. Furthermore, when the operator is strongly pseudo-monotone and Lipschitz continuous, we establish the linear convergence of the sequence generated by the Popov's algorithm. As a by-product of our analysis, we extend the range of stepsize in the projected reflected gradient algorithm for solving unconstrained pseudo-monotone variational inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of solving strongly monotone variational inequality problems over the solution set of the split common fixed point problem with multiple output sets for demicontractive mappings in real Hilbert spaces. The strong convergence of the iterative sequence generated by the algorithm method is established without the prior knowledge of the norms of the given bounded linear operators. In addition, using our method, we do not require any information of the Lipschitz and strongly monotone constants of the mappings. Several corollaries of our main result are also presented. Finally, we include several numerical examples that serve to illustrate the performance of our proposed algorithm. [ABSTRACT FROM AUTHOR]

The inertial forward–backward splitting algorithm can be considered as a modified form of the forward–backward algorithm for variational inequality problems with monotone and Lipschitz continuous cost mappings. By using parallel and inertial techniques and the forward–backward splitting algorithm, in this paper, we propose a new parallel inertial forward–backward splitting algorithm for solving variational inequality problems, where the constraints are the intersection of common solution sets of a finite family of variational inclusion problems. Then, strong convergence of proposed iteration sequences is showed under standard assumptions imposed on cost mappings in a real Hilbert space. Finally, some numerical experiments demonstrate the reliability and benefits of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper introduces and analyzes a self-adaptive inertial subgradient-like extragradient method designed to solve the bilevel split pseudomonotone variational inequality problem within the context of a common fixed-point problem, constrained by finite Bregman relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. The method incorporates a strongly monotone mapping for the upper-level problem and a pseudomonotone operator for the lower-level. We establish the strong convergence of the proposed method under mild conditions on the algorithm parameters without requiring prior knowledge of the operator norm or the coefficient of the underlying operator. Finally, we present numerical experiments to demonstrate the practicality and applicability of the proposed method. Our findings extend and improve existing results in the literature. [ABSTRACT FROM AUTHOR]

With application point of view, Gopal et al. [D. Gopal, W. Sintunavarat, A.S. Ranadive, S. Shukla, The investigation of κ-fuzzy metric spaces with the first contraction principle in such spaces, Soft Comput., 27:11081-11089, 2023] generalized the conceptions of a fuzzy metric space and introduced the definition of κ-fuzzy metric space. Here a fuzzy set defined in κ-fuzzy metric space is a membership function FY: X × X × (0,+∞)k → [0, 1], that is, the fuzzy distance between two points of the set depends on more than one parameter, and then also introduced first contraction principle in this space. In this sequel, we extend the work on κ-fuzzy metric spaces by generalizing Banach contraction principle by introducing various type of inequalities. Here we introduce Tirado-type κ-fuzzy contraction condition and prove fixed point theorem for Tirado-type contractive mapping. We also discuss the κ-fuzzy ѱ-contractive mapping, where ѱ Є ψ, and ψis a class of mappings defined from ѱ: [0; 1] → [0; 1] that has certain properties, and also obtained fixed point for such class of mappings. Later, we define Ćirić-type contraction inequalities to prove fixed point results by restricting ourselves on l-natural property of the fuzzy space to ensure the existence of fixed point. Between all results, a set of supportive examples are also produced to validate the results. In application section, we discuss the solutions of Volterra-type integral equations and second-order nonlinear ordinary differential equation. [ABSTRACT FROM AUTHOR]

The present study investigates a class of variational inequality problems under the framework of the parabolic Kirchhoff operator from the financial contract problem. This particular issue stems from the financial contract problem. By utilizing the energy inequality of the obtained solutions, the energy inequality of the solution gradients, and the Caffarelli–Kohn–Nirenberge inequality, an estimation of the infinite norm of the solution gradients is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel approach to approximate the solution of variational inequality problems without relying on the monotonicity assumption. We propose a two-step inertial modified projection and contraction method for solving quasi-monotone and without-monotone variational inequalities in real Hilbert spaces. We establish a weak convergence result for the proposed method under suitable conditions. Additionally, numerical examples and a network equilibrium flow problem are provided to illustrate the effectiveness of our method and compare it with recent related methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study an averaged Halpern-type algorithm for approximating the solution of a common fixed-point problem for a couple of nonexpansive and demicontractive mappings with a variational inequality constraint in the setting of a Hilbert space. The strong convergence of the sequence generated by the algorithm is established under feasible assumptions on the parameters involved. In particular, we also obtain the common solution of the fixed point problem for nonexpansive or demicontractive mappings and of a variational inequality problem. Our results extend and generalize various important related results in the literature that were established for two pairs of mappings: (nonexpansive, nonspreading) and (nonexpansive, strongly quasi-nonexpansive). Numerical tests to illustrate the superiority of our algorithm over the ones existing in the literature are also reported. [ABSTRACT FROM AUTHOR]

In this paper, our main interest is to propose a viscosity iterative method for approximating solutions of variational inequality problems, resolvents of monotone operators and fixed points of ρ-demimetric mappings with multiple output sets in Hadamard spaces. We prove a strong convergence result for approximating the solutions of the aforementioned problems under some mild conditions. Also, we present an application of our main result to a convex minimization problem. Our results improve and generalize many related results in the literature. [ABSTRACT FROM AUTHOR]

We propose and study two new methods based on the golden ratio technique for approximating solutions to variational inequality problems in Hilbert space. The first method combines the golden ratio technique with the subgradient extragradient method. In the second method, we incorporate the alternating golden ratio technique into the subgradient extragradient method. Both methods use self-adaptive step sizes which are allowed to increase during the execution of the algorithms, thus limiting the dependence of our methods on the starting point of the scaling parameter. We prove that under appropriate conditions, the resulting methods converge either weakly or R-linearly to a solution of the variational inequality problem associated with a pseudomonotone operator. In order to show the numerical advantage of our methods, we first present the results of several pertinent numerical experiments and then compare the performance of our proposed methods with that of some existing methods which can be found in the literature. [ABSTRACT FROM AUTHOR]

This paper deals with the problem of finding a common solution for a fixed point problem for strictly pseudocontractive mappings and for a certain variational inequality problem. We propose a projection-type implicit averaged algorithm and establish the strong convergence of the sequences generated by this method to the common solution for the fixed point problem and the variational inequality problem. In order to illustrate the feasibility of the hypotheses and the superiority of our theoretical results over the existing literature, an example is also presented. [ABSTRACT FROM AUTHOR]

We present and study the iteration-complexity of a relative-error inexact proximal-Newton extragradient algorithm for solving smooth monotone variational inequality problems in real Hilbert spaces. We removed a search procedure from Monteiro and Svaiter (2012) by introducing a novel approach based on homotopy, which requires the resolution (at each iteration) of a single strongly monotone linear variational inequality. For a given tolerance ρ > 0, our main algorithm exhibits pointwise O(1/ρ) and ergodic O(1/ρ2/3) iteration-complexities. From a practical perspective, preliminary numerical experiments indicate that our main algorithm outperforms some previous proximal-Newton schemes. [ABSTRACT FROM AUTHOR]

We show that two different notions of solutions to the obstacle problem for the porous medium equation, a potential theoretic notion and a notion based on a variational inequality, coincide for regular enough compactly supported obstacles. [ABSTRACT FROM AUTHOR]

In this paper, we study a variant of the extragradient method for solving quasi-variational inequality problems in Hilbert spaces. In fact, we assume that the given operator is pseudo-monotone and Lipschitz continuous, and the given multivalued mapping is quasi-nonexpansive with nonempty, closed and convex values. Then by modifying the so-called extragradient method (an explicit discretization of a dynamical system) and using the Halpern's regularization method, we show that the generated sequence is strongly convergent to a solution of the quasi-variational inequality problem, without any knowledge of the Lipschitz constant of the operator. Finally, we give some examples of applications and numerical experiments of our main result. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel dual inertial Tseng's extragradient method for solving variational inequality problems in real Hilbert spaces, particularly those involving pseudomonotone and Lipschitz continuous operators. Our secondary method incorporates variable step‐size, updated at each iteration based on some previous iterates. A notable advantage of these algorithms is their ability to operate without prior knowledge of Lipschitz‐type constants and without the need for any line‐search procedure. We establish the convergence theorem of the proposed algorithms under mild assumptions. To illustrate the numerical behavior of the algorithms and to make comparisons with other methods, we conduct several numerical experiments. The results of these evaluations are showcased and thoroughly examined to exemplify the practical significance and effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified Tseng's extragradient method with a new step-length rule to solve pseudo-monotone variational inequalities in real Hilbert spaces. Under suitable conditions, the sequence generated by this algorithm strongly converges to the common elements of the solution set of pseudo-monotone variational inequality problems and the fixed point set of k-demicontractive mappings. Finally, we give some numerical experiments to illustrate the effectiveness of the proposed algorithm. The main results of this paper generalize and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

This paper proposes a Fourier-Legendre spectral method to find the minimizers of a variational problem, called \sigma _{2,p}-energy, in polar coordinates. Let {\mathbb {X}}\subset \mathbb {R}^n be a bounded Lipschitz domain and consider the energy functional (1.1) whose integrand is defined by {\mathbf {W}}(\nabla u(x))≔(\sigma _2(u))^{\frac {p}{2}}+\Phi (\det \nabla u) over an appropriate space of admissible maps, \mathcal {A}_p({\mathbb {X}}). Using Fourier and Legendre interpolation errors, we obtain an error estimate for the energy functional and prove a convergence theorem for the proposed method. Furthermore, we apply the gradient descent method to solve a nonlinear algebraic system which is obtained by discretizing the Euler-Lagrange equations. The numerical experiments are performed to demonstrate the accuracy and effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper, we propose an adaptive algorithm with multi-step inertia for solving a convex bilevel optimization problem. Under suitable parameter conditions, we prove that our algorithm converges strongly to some solution of the problem, which is the unique solution to some variational inequality problem. The effectiveness of the proposed algorithm is verified by numerical experiments and compared with other algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study a class of non-Lipschitz quasi-monotone variational inequalities. We introduce a new inertial algorithm with self-adaptive step sizes for finding the minimum-norm solutions of the variational inequalities. Unlike the existing results on non-Lipschitz variational inequalities, our algorithm does not require any linesearch technique. We prove that the sequence generated by our proposed algorithm converges strongly to the minimum-norm solutions of the variational inequalities in Hilbert spaces. In addition, we also consider a class of non-Lipschitz variational inequalities without monotonicity. Finally, we present several numerical examples to illustrate our algorithm by comparing it with some existing methods. [ABSTRACT FROM AUTHOR]

The paper aims to propose a two-step iterated method, which is derived from a regularized dynamical system of extragradient-type in terms of time discretizing, for solving an equilibrium problem. We prove that the iterative sequence generated by the method converges strongly to a solution of the equilibrium problem. Some numerical experiments are given to illustrate and compare the behavior of the new method with several other methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a forward-backward splitting dynamical system designed to address the inclusion problem of the form 0 ∈ G (x) + F(x), where G is a multi-valued operator and F is a single-valued operator in Hilbert spaces. The involved operators are required to satisfy a generalized monotonicity condition, which is less restrictive than standard monotone assumptions. Also, the maximality property does not impose on our involved operators. With mild conditions on parameters, we demonstrate the finite-time stability of the proposed dynamical system. We also present some applications to other optimization problems, such as Constrained Optimization Problems (COPs), Mixed Variational Inequalities (MVIs), and Variational Inequalities (VIs). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of solving strongly monotone variational inequalities over the solution sets of split feasibility problems with multiple output sets in real Hilbert spaces. We present two new iterative algorithms when the involved subsets are given as the level sets of convex functions. In our algorithms, the projection to the half-space is replaced by the one to the intersection of two half-spaces. The algorithms are accelerated using the inertial technique and eliminate the need for calculating or estimating the norms of linear operators by employing self-adaptive step size criteria. We give convergence of the sequence generated by our algorithms under some suitable assumptions. Some applications to monotone variational inequalities over the solution set of the split feasibility problem are also reported. Finally, we present some numerical examples to illustrate the efficiency and implementation of our algorithms in comparison with existing algorithms in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose two self-adaptive extragradient-like algorithms for solving pseudomonotone variational inequalities. We consider two cases: the mapping is Lipschitz continuous (with unknown modulus) and is not Lipschitz continuous. The step size in our algorithms can either increase or decrease at each step. This feature makes our algorithms more efficient. We also give some numerical examples to compare the performance of our algorithm with the existing one. [ABSTRACT FROM AUTHOR]

This paper presents two new algorithms for solving multivalued variational inequality problems in a real Hilbert space. By combining the nonexpansiveness of proximal operators associated with the proper lower semicontinuous convex function of the problems and inertial techniques, we demonstrate the weak convergence of the iteration sequences generated by our first algorithm under monotone and Lipschitz continuous assumptions of the cost mappings. Next, we use Mann iteration technique to obtain the second algorithm and show its strong convergence. Finally, we give some numerical results for two proposed algorithms and comparison with some other known algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we intend to consider infinitely many high energy solutions for a kind of superlinear Klein–Gordon–Maxwell systems. Under some suitable assumptions on the potential function and nonlinearity, by using variational methods and the method of Nehari manifold, we obtain the existence result of infinitely many high energy solutions for this kind of systems, which improves many relative results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose an extragradient type projection algorithm for solving quasimonotone variational inequalities in $ \mathbb {R}^n $ R n . Under the assumption that f is a quasimonotone and continuous mapping and the solution set of the dual variational inequality problem is nonempty, the globally convergence of the algorithm is established. Numerical experiments are also reported. [ABSTRACT FROM AUTHOR]

With the help of the Meir–Keeler contraction method and the Mann‐type method, two adaptive inertial iterative schemes are introduced for finding solutions of the split variational inclusion problem in Hilbert spaces. The strong convergence of the suggested algorithms is guaranteed by a new stepsize criterion that does not require calculation of the bounded linear operator norm. Some numerical experiments and applications in signal recovery problems are given to demonstrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

A survey of the existing results in the literature shows that several of the results on variational inequality problem were established under some stringent conditions and employed some form of linesearch technique even in the framework of Hilbert spaces. However, due to the loop nature of the linesearch technique, the implementation of such algorithms might be economically nonviable. In this article, we study the pseudomonotone variational inequality problem. We propose a new inertial‐like Tseng's extragradient method with non‐monotonic self‐adaptive step size that does not involve any linesearch technique for finding the solution of the problem in reflexive Banach spaces. Under some mild conditions, we prove a strong convergence result for the proposed method without the sequentially weakly continuity condition often assumed by authors to guarantee convergence when studying the pseudomonotone variational inequality problems. Moreover, we apply our result to study a constrained convex minimization problem. Finally, we present several theoretical and real applications to optimal control numerical experiments to illustrate and show the performance of our method in comparison with related methods in the literature. [ABSTRACT FROM AUTHOR]

In this article, a generalized variational inequality problem in the setting of Hadamard spaces is introduced and analyzed. For approximating a solution of the problem when the underlined mapping is monotone, an adaptive algorithm that requires the computation of only one proximal operator at each iteration is proposed. The work gives a step towards investigating monotone variational inequality problems in geodesic settings. [ABSTRACT FROM AUTHOR]

This paper was focused on the solvability of a class of doubly critical sub-Laplacian problems on the Carnot group : Here, , , , , and . By means of variational techniques, we extended the arguments developed in [1]. In addition, we also established the existence result for the subelliptic system which involved sub-Laplacian and critical homogeneous terms. [ABSTRACT FROM AUTHOR]

In this paper, we study an optimal control problem for a stationary Navier–Stokes hemivariational inequality with control constraints and its numerical approximation. The hemivariational inequality is the weak formulation of a stationary incompressible fluid flow problem, modeled by the Navier-Stokes equations subject to a nonleak boundary condition and a subdifferential condition of friction type. We investigate the stability of the solution of the hemivariational inequality with respect to perturbations in the density of external force and superpotential, and demonstrate the existence of a solution for the optimal control problem with the external force density as the control. Moreover, we consider the numerical solution of the optimal control problem and show its convergence. As an example and for definiteness, the numerical solution is defined through the finite element method. [ABSTRACT FROM AUTHOR]

In this paper, we introduced a new double inertial subgradient extragradient method for solving a variational inequality problem in Hilbert space. In our method, the mapping needed not to satisfy any assumption of monotonicity and two different self-adaptive step sizes were used for avoiding the need of Lipschitz constant of the mapping. The strong convergence of the proposed method was proved under some new conditions. Finally, some numerical examples were presented to illustrate the convergence of our method and compare with some related methods in the literature. [ABSTRACT FROM AUTHOR]

This paper aims to find generalizations of inequalities that hold for unified integral operators by applying strongly exponentially (α, ℏ − m) − p‐convex functions. These inequalities generate results for several fractional integral operators and simultaneously hold for convex, strongly convex, and exponentially convex functions. The particular cases of presented inequalities are directly connected with many recently published results. [ABSTRACT FROM AUTHOR]

We devise an iterative algorithm incorporating inertial techniques to approximate the shared solution of a generalized equilibrium problem, a fixed point problem for a finite family of relatively nonexpansive multivalued mappings, and a variational inequality problem. Our discussion encompasses the strong convergence of the proposed algorithm and highlights specific outcomes derived from our theorem. Additionally, we provide a computational analysis to underscore the significance of our findings and draw comparisons. The results presented in this paper serve to extend and unify numerous previously established outcomes in this particular research domain. [ABSTRACT FROM AUTHOR]

In this paper, we present an inertial projection and contraction algorithm (IPCA for short) for variational inequality problems (VIP for short) without Lipschitz continuity of the underlying mapping in Euclidean space. To obtain a larger step, the next iterate point of IPCA is generated by projecting the current iterate point onto a selected half-space, which is selected from finite half-spaces (one half-space can strictly separate the current iterate point from the solution set of VIP and other half-spaces are all contain the solution set of VIP) and has the largest distance to the current iterate point. Moreover, a new initial step-size strategy is used to accelerate IPCA. The global convergence of the sequence generated by IPCA is established. Numerical experiments show that IPCA can accelerate PCA both from the number of iterations point of view and the number of projections point of view. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three new inertial-like Bregman projection methods with a nonmonotone adaptive step-size for solving quasi-monotone variational inequalities in real Hilbert spaces. Under some suitable conditions, the weak convergence of these methods is proved without the prior knowledge of the Lipschitz constant of the operator and the strong convergence of some proposed methods under a strong quasi-monotonicity assumption of the mapping is also provided. Finally, several numerical experiments and applications in image restoration problems are provided to illustrate the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

We study the problem of finding a common solution to the variational inequality problem with a pseudomonotone and Lipschitz continuous operator and the fixed point problem with a demicontractive mapping in real Hilbert spaces. Inspired by the inertial method and the subgradient extragradient method, two improved viscosity-type efficient iterative methods with a new adaptive non-monotonic step size criterion are proposed. We prove that the strong convergence theorems of these new methods hold under some standard and mild conditions. Numerical examples in finite-and infinite-dimensional spaces are provided to illustrate the effectiveness and potential applicability of the suggested iterative methods compared to some known ones. [ABSTRACT FROM AUTHOR]

In this paper, we have introduced some viscosity-type inertial iterative methods for solving fixed point and variational inclusion problems in Hilbert spaces. Our methods calculated the viscosity approximation, fixed point iteration, and inertial extrapolation jointly in the starting of every iteration. Assuming some suitable assumptions, we demonstrated the strong convergence theorems without computing the resolvent of the associated monotone operators. We used some numerical examples to illustrate the efficiency of our iterative approaches and compared them with the related work. [ABSTRACT FROM AUTHOR]

In this paper, we establish some upper bound estimates for the maximum modulus of the higher order derivative of a polynomial with restricted zeros on a circle of |z| = R, R ≥ 1, in terms of the maximum modulus of the same polynomial on the unit circle |z| = 1. These results generalize and sharpen some known results in this direction. [ABSTRACT FROM AUTHOR]

Let Lp be the subspace of the space Lp(R) consisting of the restriction to the real axis of all entire functions of exponential type ≤ π. In this paper, for any function f ∈ Lp (1 ≤ p ≤ ∞), we obtain estimates for the norm of f in terms of the sequence (f(n/2))n∈Z, namely... where... Here J:Lp → lp(Z) is the linear operator given by the formula (Jf)(n) := (-1)nf(n), n ∈ Z, and Tτ is the shift by τ ∈ R of the function f, (Tτ f)(z):= f(z + τ), z ∈ C. [ABSTRACT FROM AUTHOR]

Several new classes of general quasi variational inequalities involving two arbitrary operators are introduced and considered in this paper. Some important cases are discussed, which can be obtained by choosing suitable and appropriate choice of the operators. It is shown that the implicit obstacle boundary value can be studied via these quasi variational inequalities. Projection technique is applied to establish the equivalent between the general quasi variational inequalities and fixed point problems. This alternative formulation is used to discuss the uniqueness of the solution as well as to propose a wide class of proximal point algorithms. Convergence criteria of the proposed methods is considered. Asymptotic stability of the solution is studied using the first order dynamical system associated with variational inequalities. Second order dynamical systems associated with general quasi variational inequalities are applied to suggest some inertial type methods. Some special cases are discussed as applications of the main results. Several open problems are indicated for future research work. [ABSTRACT FROM AUTHOR]

In this article, we propose an inertial and a viscosity iterative method for solving variational inequality problem on Hadamard manifolds. The iterative algorithm is inspired by Tseng's extragradient method with a self-adaptive procedure which generates dynamic stepsizes converging to a positive constant. The proposed method does not require the knowledge of the Lipschitz constant as well as the sequential weak continuity of the corresponding operator. Under a pseudomonotone assumption on the underlying vector field, we establish a convergence result for solving a pseudomonotone variational inequality and fixed point problems of nonexpansive mapping under some mild assumptions. Finally, we present some fundamental experiment to illustrate the numerical behavior of our proposed method. The result discussed in this article extends and complements many related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we prove an existence result for a quasilinear elliptic variational inequality of the form u ∈ K ⊂ V : 0 ∈ - Δpu+aF(u) + ...IK(u) ∈ V* in the whole ℝ N under bilateral constraints K given by K = {v ∈ V : ϕ (x) ≤ v(x) ≤ Ψ(x) a.e. in ℝ N}, where Δp is the p-Laplacian, the underlying solution space V is the homogeneous Sobolev space (also called Beppo-Levi space) V = D1,p(ℝN) with 1 < p < N, and IK : V → ℝ U {+∞} is the indicator functional corresponding to K with its subdifferential ...IK. The lower order Nemytskij operator F is generated by a Carathéodory function f : ℝN x ℝ → ℝ, and the measurable and bounded coefficient α is supposed to decay like |x|- (N+α) at infinity. The growth conditions that we impose on f are such that the operator -- Δp + aF : V → V*, in general, is not coercive with respect to K which prevents us from applying standard existence results. Another difficulty, which arises due to the lack of compact embedding of V into Lq(ℝN) spaces, needs to be overcome in an appropriate way. Without assuming additional assumptions such as the existence of sub- and supersolutions, we are able not only to prove the existence of solutions, but also show the compactness of the set of all solutions in V. Finally, an extension of the theory is established, which allows us to deal with noncoercive bilateral variational-hemivariational inequalities in ℝN. The proof of our main existence result is based on a modified penalty approach. [ABSTRACT FROM AUTHOR]

In this paper, we study the variational inequality over the solution set of the split common solution problem with multiple output sets for monotone operator equations in real Hilbert spaces. We propose a new approach for solving this problem without depending on the norm of the transfer mappings. Some of its applications for solving the variational inequality over the solution set of the split common fixed point problem with multiple output sets and related problems are also mentioned. Finally, we give numerical examples to illustrate the performance of the proposed algorithm and compare it with several existing previous algorithms. [ABSTRACT FROM AUTHOR]

We present a refined, simplified, and unified proof of Aristarchus' inequality, without resorting to trigonometry, calculus or extended geometric arguments. [ABSTRACT FROM AUTHOR]

We consider an elliptic variational inequality with unilateral constraints in a Hilbert space X which, under appropriate assumptions on the data, has a unique solution u. We formulate a convergence criterion to the solution u, i.e. we provide necessary and sufficient conditions on a sequence $ \{u_n\}\subset X $ { u n } ⊂ X which guarantee the convergence $ u_n\to u $ u n → u in the space X. Then we illustrate the use of this criterion to recover well-known convergence results and well-posedness results in the sense of Tykhonov and Levitin–Polyak. We also provide two applications of our results, in the study of a heat transfer problem and an elastic frictionless contact problem, respectively. [ABSTRACT FROM AUTHOR]

In this paper, an inertial subgradient extragradient algorithm is proposed to solve the pseudomonotone variational inequality problems in Banach space. This iterative scheme employs a new line-search rule. Strong convergence theorems for the proposed algorithms are established under the assumptions that the operators are non-Lipschitz continuous. Furthermore, several numerical experiments are given to show that our method has better convergence performance than the known ones in the literatures. [ABSTRACT FROM AUTHOR]