Search

This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized multiple-set split feasibility problem including the common fixed-point problem for a finite family of generalized demimetric mappings and the monotone inclusion problem in 2-uniformly convex and uniformly smooth Banach spaces. We propose an inertial Halpern-type iterative algorithm for obtaining a solution of the problem and derive a strong convergence theorem for the algorithm. Then, we apply our convergence results to the convex minimization problem, the variational inequality problem, the multiple-set split feasibility problem and the split common null-point problem in Banach spaces. [ABSTRACT FROM AUTHOR]

This is the first paper dealing with the study of minimum and maximum principle sufficiency properties for nonsmooth variational inequalities by using gap functions in the setting of Hadamard manifolds. We also provide some characterizations of these two sufficiency properties. We conclude the paper with a discussion of the error bounds for nonsmooth variational inequalities in the setting of Hadamard manifolds. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a class of generalized nonexpansive mappings called generalized nearly asymptotically nonexpansive mappings and to show that it properly includes the class of nearly asymptotically nonexpansive mappings. We investigate the problem of approximating a common element of the set of solutions of a system of generalized nonlinear variational-like inclusions involving P-η-accretive mappings and of the set of fixed points of a generalized nearly asymptotically nonexpansive mapping. To this end, we suggest a new iterative algorithm with mixed errors. As an application of the obtained equivalence, we prove the strong convergence and stability of the sequence generated by the proposed iterative algorithm to a common point of the two sets mentioned above. [ABSTRACT FROM AUTHOR]

This paper aims at proposing an iterative algorithm for finding an element in the intersection of the solutions set of a system of variational inclusions and the fixed-points set of a total uniformly L-Lipschitzian mapping. Applying the concepts of graph convergence and the resolvent operator associated with an Ĥ-accretive mapping, a new equivalence relationship between graph convergence and resolvent-operator convergence of a sequence of Ĥ-accretive mappings is established. As an application of the obtained equivalence relationship, the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the above two sets is proved under some suitable hypotheses imposed on the parameters and mappings. At the same time, the notion of H (⋅ , ⋅) -accretive mapping that appeared in the literature, where H (⋅ , ⋅) is an α, β-generalized accretive mapping, is also investigated and analyzed. We show that the notions H (⋅ , ⋅) -accretive and Ĥ-accretive operators are actually the same, and point out some comments on the results concerning them that are available in the literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and study the mixed set-valued vector inverse quasi-variational inequality problems (MSVIQVIPs) and to obtain error bounds for this kind of MSVIQVIP in terms of the residual gap function, the regularized gap function, and the D-gap function. These bounds provide effective estimated distances between an arbitrary feasible point and the solution set of mixed set-valued vector inverse quasi-variational inequality problem. The results presented in the paper improve and generalize some recent results. [ABSTRACT FROM AUTHOR]

In this paper, a new form of the symmetric vector equilibrium problem is introduced and, by mixing properties of the nonlinear scalarization mapping and the maximal element lemma, an existence theorem for it is established. We show that Ky Fan's lemma, as a usual technique for proving the existence results for equilibrium problems, implies the maximal element lemma, while it is useless for proving the main theorem of this paper. Our results can be viewed as an extension and improvement of the main results obtained by Farajzadeh (Filomat 29(9):2097-2105, 2015) and some corresponding results that appeared in this area by relaxing the lower semicontinuity, quasiconvexity on the mappings and being nontrivial of the dual cones. Finally, some examples are given to support the main results. [ABSTRACT FROM AUTHOR]

In this paper, weak sharp solutions are investigated for a variational-type inequality governed by (ρ , b , d) -convex path-independent curvilinear integral functional. Moreover, an equivalence between the minimum principle sufficiency property and the weak sharpness property of the solution set associated with the considered variational-type inequality is established. [ABSTRACT FROM AUTHOR]

In this paper, we propose a parallel descent LQP alternating direction method for solving structured variational inequality with three separable operators. The $O(1/t)$ convergence rate for this method is studied. We also present some numerical examples to illustrate the efficiency of the proposed method. The results presented in this paper extend and improve some well-known results in the literature. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the introduction a new class of equilibrium problems named generalized multivalued equilibrium-like problems which includes the classes of hemiequilibrium problems, equilibrium-like problems, equilibrium problems, hemivariational inequalities, and variational inequalities as special cases. By utilizing the auxiliary principle technique, some new predictor-corrector iterative algorithms for solving them are suggested and analyzed. The convergence analysis of the proposed iterative methods requires either partially relaxed monotonicity or jointly pseudomonotonicity of the bifunctions involved in generalized multivalued equilibrium-like problem. Results obtained in this paper include several new and known results as special cases. [ABSTRACT FROM AUTHOR]

Most mathematical models arising in stationary filtration processes as well as in the theory of soft shells can be described by single-valued or generalized multivalued pseudomonotone mixed variational inequalities with proper convex nondifferentiable functionals. Therefore, for finding the minimum norm solution of such inequalities, the current paper attempts to introduce a modified two-layer iteration via a boundary point approach and to prove its strong convergence. The results here improve and extend the corresponding recent results announced by Badriev, Zadvornov and Saddeek (Differ. Equ. 37:934-942, 2001). [ABSTRACT FROM AUTHOR]

In this paper, inspired by Hussain et al. (Fixed Point Theory Appl. 2015:17, ), we study a modified Mann method to approximate strongly fixed points of strict pseudo-contractive mappings. In (Hussain et al. in Fixed Point Theory Appl. 2015:17, ) it is shown that the same algorithm converges strongly to a fixed point of a nonexpansive mapping under suitable hypotheses on the coefficients. Here the assumptions on the coefficients are different, as well as the techniques of the proof. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of gap functions of random generalized variational inequality problems in a fuzzy environment. Further by using the residual vector we compute error bounds for random generalized variational inequality problems and we generalize the regularized gap functions proposed by Fukushima. Furthermore we study various properties of the generalized regularized gap functions in random fuzzy mappings and derive the global error bounds for random generalized variational inequality problems. Our results are new and generalize a number of known results to generalized variational inequality problems with fuzzy mappings. [ABSTRACT FROM AUTHOR]

The present paper deals with the common solution method for finding a fixed point of a nonexpansive mapping and a solution of split hierarchical Minty variational inequality problems. We discuss the weak convergence of the sequences generated by the proposed method to a common solution of a fixed point problem and a split hierarchical Minty variational inequality problem. An example is presented to illustrate the proposed algorithm and result. [ABSTRACT FROM AUTHOR]

In this paper, we propose a descent direction method for solving variational inequalities. A new iterate is obtained by searching the optimal step size along a new descent direction which is obtained by the linear combination of two descent directions. Under suitable conditions, the global convergence of the proposed method is studied. Two numerical experiments are presented to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two general iterative methods (one implicit method and one explicit method) for finding a solution of a general system of variational inequalities (GSVI) with the constraints of finitely many generalized mixed equilibrium problems and a fixed point problem of a continuous pseudocontractive mapping in a Hilbert space. Then we establish strong convergence of the proposed implicit and explicit iterative methods to a solution of the GSVI with the above constraints, which is the unique solution of a certain variational inequality. The results presented in this paper improve, extend, and develop the corresponding results in the earlier and recent literature. [ABSTRACT FROM AUTHOR]

In this paper, we provide the connectedness of the sets of weak efficient solutions, Henig efficient solutions and Benson proper efficient solutions for set-valued vector equilibrium problems under improvement sets. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified Krasnoselski–Mann type iterative method for capturing a common solution of a split mixed equilibrium problem and a hierarchical fixed point problem of a finite collection of k-strictly pseudocontractive nonself-mappings. Many of the algorithms for solving the split mixed equilibrium problem involve a step size which depends on the norm of a bounded linear operator. Since the computation of the operator norm is very difficult, we formulate our iterative algorithm in such a way that the implementation of the proposed algorithm does not require any prior knowledge of operator norm. Weak convergence results are established under mild conditions. We also establish strong convergence results for a certain class of hierarchical fixed point and split equilibrium problem. Our results generalize some important results in the recent literature. [ABSTRACT FROM AUTHOR]

In the paper, we consider the following hemivariational inequality problem involving the fractional Laplacian: { (− Δ) s u + λ u ∈ α (x) ∂ F (x , u) x ∈ Ω , u = 0 x ∈ R N ∖ Ω , where Ω is a bounded smooth domain in R N with N ≥ 3 , (− Δ) s is the fractional Laplacian with s ∈ (0 , 1) , λ > 0 is a parameter, α (x) : Ω → R is a measurable function, F (x , u) : Ω × R → R is a nonsmooth potential, and ∂ F (x , u) is the generalized gradient of F (x , ⋅) at u ∈ R . Under some appropriate assumptions, we obtain the existence of a nontrivial solution of this hemivariational inequality problem. Moreover, when F is autonomous, we obtain the existence of infinitely many solutions of this problem when the nonsmooth potentials F have suitable oscillating behavior in any neighborhood of the origin (respectively the infinity) and discuss the properties of the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new viscosity-type iteration process for approximating a common solution of a finite family of split variational inclusion problem and fixed point problem. We prove that the proposed algorithm converges strongly to a common solution of a finite family of split variational inclusion problems and fixed point problem for a finite family of type-one demicontractive mappings between a Hilbert space and a Banach space. Furthermore, we applied our results to study a finite family of split convex minimization problems, and also considered a numerical experiment of our results to further illustrate its applicability. Our results extend and improve the results of Byrne et al. (J. Nonlinear Convex Anal. 13:759–775, 2012), Kazmi and Rizvi (Optim. Lett. 8(3):1113–1124, 2014), Moudafi (J. Optim. Theory Appl. 150:275–283, 2011), Shehu and Ogbuisi (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 110(2):503–518, 2016), Takahashi and Yao (Fixed Point Theory Appl. 2015:87, 2015), Chidume and Ezeora (Fixed Point Theory Appl. 2014:111, 2014), and a host of other important results in this direction. [ABSTRACT FROM AUTHOR]

The goal of this paper is to discuss iteration complexity of two projection type methods for solving generalized complementarity problems and establish their worst-case sub-linear convergence rates measured by the iteration complexity in both the ergodic and the nonergodic sense. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation problem of solutions for the semistrictly quasi-monotone variational inequalities in infinite-dimensional Hilbert spaces. We prove that the iterative sequence generated by the algorithm for solving the semistrictly quasi-monotone variational inequalities converges weakly to a solution. [ABSTRACT FROM AUTHOR]