Generalized relaxed inertial method with regularization for solving split feasibility problems in real Hilbert spaces

In this paper, we propose a new modified relaxed inertial regularization method for finding a common solution of a generalized split feasibility problem, the zeros of sum of maximal monotone operators, and fixed point problem of two nonlinear mappings in real Hilbert spaces. We prove that the proposed method converges strongly to a minimum-norm solution of the aforementioned problems without using the conventional two cases approach. In addition, we apply our convergence results to the classical variational inequality and equilibrium problems, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

Design of a Hybrid Inertial and Magnetophoretic Microfluidic Device for CTCs Separation from Blood

Circulating tumor cells (CTCs) isolation from a blood sample plays an important role in cancer diagnosis and treatment. Microfluidics offers a great potential for cancer cell separation from the blood. Among the microfluidic-based methods for CTC separation, the inertial method as a passive method and magnetic method as an active method are two efficient well-established methods. Here, we investigated the combination of these two methods to separate CTCs from a blood sample in a single chip. Firstly, numerical simulations were performed to analyze the fluid flow within the proposed channel, and the particle trajectories within the inertial cell separation unit were investigated to determine/predict the particle trajectories within the inertial channel in the presence of fluid dynamic forces. Then, the designed device was fabricated using the soft-lithography technique. Later, the CTCs were conjugated with magnetic nanoparticles and Ep-CAM antibodies to improve the magnetic susceptibility of the cells in the presence of a magnetic field by using neodymium permanent magnets of 0.51 T. A diluted blood sample containing nanoparticle-conjugated CTCs was injected into the device at different flow rates to analyze its performance. It was found that the flow rate of 1000 µL/min resulted in the highest recovery rate and purity of ~95% and ~93% for CTCs, respectively.

Multi-Step Inertial Hybrid and Shrinking Tseng’s Algorithm with Meir–Keeler Contractions for Variational Inclusion Problems

In our paper, we propose two new iterative algorithms with Meir–Keeler contractions that are based on Tseng’s method, the multi-step inertial method, the hybrid projection method, and the shrinking projection method to solve a monotone variational inclusion problem in Hilbert spaces. The strong convergence of the proposed iterative algorithms is proven. Using our results, we can solve convex minimization problems.

Krasnoselski–Mann-type inertial method for solving split generalized mixed equilibrium and hierarchical fixed point problems

In this paper, we present Krasnoselski–Mann-type inertial method for solving split generalized mixed equilibrium and hierarchical fixed point problems for k-strictly pseudocontractive nonself-mappings. We establish that the weak convergence of the proposed accelerated iterative method with inertial terms involves a step size which does not require any prior knowledge of the operator norm under several suitable conditions in Hilbert spaces. Finally, the application to a Nash–Cournot oligopolistic market equilibrium model is discussed, and numerical examples are provided to demonstrate the effectiveness of our iterative method.

An Inertial Method for Split Common Fixed Point Problems in Hilbert Spaces

In this paper, we consider the split common fixed point problem in Hilbert spaces. By using the inertial technique, we propose a new algorithm for solving the problem. Under some mild conditions, we establish two weak convergence theorems of the proposed algorithm. Moreover, the stepsize in our algorithm is independent of the norm of the given linear mapping, which can further improve the performance of the algorithm.

A self adaptive method for solving a class of bilevel variational inequalities with split variational inequality and composed fixed point problem constraints in Hilbert spaces

<p style='text-indent:20px;'>In this work, we propose a new inertial method for solving strongly monotone variational inequality problems over the solution set of a split variational inequality and composed fixed point problem in real Hilbert spaces. Our method uses stepsizes that are generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the operator norm as well as the Lipschitz constant of the operator. In addition, we prove that the proposed method converges strongly to a minimum-norm solution of the problem without using the conventional two cases approach. Furthermore, we present some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature. The results obtained in this paper extend, generalize and improve results in this direction.</p>