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In this paper we propose and analyze three parallel hybrid extragradient methods for finding a common element of the set of solutions of equilibrium problems involving pseudomonotone bifunctions and the set of fixed points of nonexpansive mappings in a real Hilbert space. Based on parallel computation we can reduce the overall computational effort under widely used conditions on the bifunctions and the nonexpansive mappings. A simple numerical example is given to illustrate the proposed parallel algorithms.

In this paper, we introduce an explicit iterative algorithm for solving a (pseudo) monotone variational inequality under Lipschitz condition in a Hilbert space. The algorithm is constructed around some projections incorporated by inertial terms. It uses variable stepsizes which are generated at each iteration by some simple computations. Furthermore, it can be easily implemented without the prior knowledge of the Lipschitz constant of the operator. Theorems of weak convergence are established under mild conditions, and some numerical results are reported for the purpose of comparison with other algorithms. The obtained results in this paper extend some related works in the literature.

Optimal positioning of the acetabular component is crucial in total hip replacement surgery to minimize postoperative dislocation rates. The transverse acetabular ligament (TAL) has been proposed as a useful anatomical landmark for cup orientation. This prospective cohort study included 122 patients who underwent total hip replacement at E Hospital, Hanoi, between January 2021 and December 2022. Orientation angles of the TAL and acetabulum were assessed preoperatively using CT with multiplanar reconstruction and MRI arthrography, and intraoperatively with a handmade protractor developed at our institution. The mean anteversion of TAL on CT and MRI was approximately 10°, with a mean inclination of 45–46°. Intraoperatively, the mean TAL anteversion was 10.2° and the acetabular anteversion was 12.0°, while the mean TAL inclination was 44.9° and the acetabular inclination 41.9°. These findings demonstrate significant correlations between TAL orientation and acetabular alignment across imaging and intraoperative measurements. TAL is a readily identifiable landmark, and its use can facilitate accurate, patient-specific acetabular cup positioning within the safe zone, thereby enhancing surgical outcomes.

The paper considers the problem of finding a common solution of a pseudomonotone and Lipschitz-type equilibrium problem and a fixed point problem for a quasi nonexpansive mapping in a Hilbert space. A new hybrid algorithm is introduced for approximating a solution of this problem. The presented algorithm can be considered as a combination of the extragradient method (two-step proximal-like method) and a modified version of the normal Mann iteration. It is well known that the normal Mann iteration has the weak convergence, but in this paper we has obtained the strong convergence of the new algorithm under some mild conditions on parameters. Several numerical experiments are reported to illustrate the convergence of the algorithm and also to show the advantages of it over existing methods. First Published Online: 21 Nov 2018

In this paper, we propose two novel parallel hybrid methods for finding a common element of the set of solutions of a finite family of generalized equilibrium problems for monotone bifunctions f i i = 1 N and α -inverse strongly monotone operators A i i = 1 N and the set of common fixed points of a finite family of (asymptotically) κ -strictly pseudocontractive mappings S j j = 1 M in Hilbert spaces. The strong convergence theorems are established under the standard assumptions imposed on equilibrium bifunctions and operators. Some numerical examples are presented to illustrate the efficiency of the proposed parallel methods.

The purpose of this work is to present a buckling analysis of functionally graded (FG) microplates by combining nonlocal strain gradient theory (NSGT) with the higher-order shear deformation plate theories (HSDPTs). The microplate is assumed to be composed of a combination of ceramic and metal materials, and the material properties are assumed to vary continuously in the thickness direction based on a simple power law. The equilibrium equations and the boundary conditions are derived using the principle of minimum potential energy. Analytical solutions are determined for the critical buckling loads of the rectangular FG microplates with different boundary conditions. The results obtained have been verified by comparison with existing findings in the literature. Furthermore, some numerical illustrations are provided to investigate the effects of nonlocal parameters, the material length scale parameter, shear deformation, aspect ratios, the power-law index, and the surface energy on the buckling response of the rectangular FG microplates.

In this paper, post-buckling and free nonlinear vibration of microbeams resting on nonlinear elastic foundation subjected to axial force are investigated. The equations of motion of microbeams are derived by using the modified couple stress theory. Using Galerkin’s method, the equation of motion of microbeams is reduced to the nonlinear ordinary differential equation. By using the equivalent linearization in which the averaging value is calculated in a new way called the weighted averaging value, approximate analytical expressions for the nonlinear frequency of microbeams with pinned–pinned and clamped–clamped end conditions are obtained in closed-forms. Comparisons with previous solutions are showed accuracy of the present solutions. Effects of the material length scale parameter and the axial compressive force on the frequency ratios of microbeams; and effect of the material length scale parameter on the buckling load ratios of microbeams are investigated in this paper.

The paper concerns with an inertial-like algorithm for approximating solutions of equilibrium problems in Hilbert spaces. The algorithm is a combination around the relaxed proximal point method, inertial effect and the Krasnoselski–Mann iteration. The using of the proximal point method with relaxations has allowed us a more flexibility in practical computations. The inertial extrapolation term incorporated in the resulting algorithm is intended to speed up convergence properties. The main convergence result is established under mild conditions imposed on bifunctions and control parameters. Several numerical examples are implemented to support the established convergence result and also to show the computational advantage of our proposed algorithm over other well known algorithms.

The paper concerns with a new iterative method for solving a monotone variational inclusion problem in a Hilbert space. The method is of the proximal contraction type incorporated with the regularization technique. Under the prediction stepsize conditions, we establish the strong convergence of the iterative sequences generated by the method to a particular solution of the problem satisfying a variational inequality problem. Finally, we give some numerical examples to illustrate the behavior of the new method in comparison with existing ones.

The article introduces a new algorithm for solving a class of equilibrium problems involving strongly pseudomonotone bifunctions with a Lipschitz-type condition. We describe how to incorporate the proximal-like regularized technique with inertial effects. The main novelty of the algorithm is that it can be done without previously knowing the information on the strongly pseudomonotone and Lipschitz-type constants of cost bifunction. A reasonable explain for this is that the algorithm uses a sequence of stepsizes which is diminishing and non-summable. Theorem of strong convergence is proved. In the case, when the information on the modulus of strong pseudomonotonicity and Lipschitz-type constant is known, the rate of linear convergence of the algorithm has been established. Several of experiments are performed to illustrate the numerical behavior of the algorithm and also compare it with other algorithms.