In this paper, we introduce and study a modified Mann-type algorithm that combines inertial terms for solving common fixed point problems of two countable families of nonexpansive mappings in Hilbert spaces. Under appropriate assumptions on the sequences of parameters, we establish a strong convergence result for the sequence generated by the proposed method in finding a common fixed point of two countable families of nonexpansive mappings. This method can be applied to solve the monotone inclusion problem. Additionally, we employ a modified Mann-type iterative algorithm to address image restoration problems. Furthermore, we present numerical results across different scenarios to demonstrate the superior efficiency of our algorithm compared to existing algorithms.

In this research paper, we present a novel theoretical technique, referred to as the double Tseng’s algorithm with inertial terms, for finding a common solution to two monotone inclusion problems. Developing the double Tseng’s algorithm in this manner not only comprehensively expands theoretical knowledge in this field but also provides advantages in terms of step-size parameters, which are beneficial for tuning applications and positively impact the numerical results. This new technique can be effectively applied to solve the problem of image deblurring and offers numerical advantages compared to some previously related results. By utilizing certain properties of a Lipschitz monotone operator and a maximally monotone operator, along with the identity associated with the convexity of the quadratic norm in the framework of Hilbert spaces, and by imposing some constraints on the scalar control conditions, we can achieve weak convergence to a common zero point of the sum of two monotone operators. To demonstrate the benefits and advantages of this newly proposed algorithm, we performed numerical experiments to measure the improvement in the signal–to–noise ratio (ISNR) and the structural similarity index measure (SSIM). The results of both numerical experiments (ISNR and SSIM) demonstrate that our new algorithm is more efficient and has a significant advantage over the relevant preceding algorithms.

In this paper, we introduce and study a Halpern inertial method for solving the general perturbed Krasnoselskii-Mann type algorithm in Hilbert space settings, where the underlying mapping is quasi-nonexpansive. We discuss convergence analysis of the method under some mild assumptions on the control sequences. We additionally present a numerical example to demonstrate the effectiveness of the method. Finally, we apply the method in solving image restoration problems. We evaluate the quality of the improvement in the Signal-to-Noise Ratio (ISNR) and the restored images using the Structural Similarity Index Measure (SSIM) metrics. The results of this work generalize and expand upon numerous results found in the literature.

This paper introduces a novel inertial forward–backward–forward algorithm driven by a newly conceptualized moving point projection technique for solving monotone inclusion problems in real Hilbert spaces. By leveraging the properties of a Lipschitz continuous, monotone operator and a maximally monotone operator alongside this innovative projection strategy, we dynamically construct a sequence of nonempty, closed, and convex sets that contain the zeros of the sum of the two operators. This geometric construction ensures that the resulting sequence is well defined and guarantees its weak convergence to a solution. Furthermore, to validate the practical efficacy of the proposed theoretical framework, we evaluate our method on image restoration problems. Numerical experiments measuring the improvement in signal-to-noise ratio (ISNR) and the structural similarity index measure (SSIM) confirm that the proposed algorithm is highly efficient and significantly outperforms existing state-of-the-art methods.

The split convex feasibility problem serves as a unified mathematical framework for solving inverse problems, medical image reconstruction, and signal recovery across multiple Hilbert spaces. Traditional iterative schemes, such as hybrid and shrinking projection algorithms, typically enforce strong convergence by projecting the initial point onto the intersection of successive half-spaces. However, this strategy introduces a severe computational bottleneck due to the necessity of solving constrained quadratic subproblems at every iteration. In this paper, we propose a novel Parallel Moving Point Projection (PMPP) algorithm designed to overcome this limitation. The proposed framework integrates a parallel projection architecture across multiple bounded linear operators with a dynamic moving point projection mechanism, where projections are performed onto the intersection of two adaptively updated half-spaces, which can be computed in explicit closed form. Under mild and standard assumptions on the control parameters, we establish a strong convergence theorem for the generated sequence in real Hilbert spaces without requiring any compact embedding. Furthermore, to validate the theoretical framework and computational efficiency of our approach, we perform extensive numerical experiments on both synthetic benchmark problems in finite-dimensional Euclidean spaces and color image restoration tasks corrupted by motion blur, average blur, and additive Gaussian noise. The experimental results demonstrate that the proposed PMPP algorithm achieves competitive execution times, lower iteration counts in some cases, and higher restoration quality in terms of signal-to-noise ratio (SNR), improvement in signal-to-noise ratio (ISNR), and structural similarity index measure (SSIM) compared to prominent existing parallel hybrid projection methods.