Search Results (15)

In this paper, we develop an accelerated three-step iterative scheme for the approximation of fixed points of contraction mappings in Banach spaces, with a particular focus on applications to fractional models. Strong convergence of the proposed iteration is established under standard contraction assumptions, together with stability and data dependence results. A refined rate of convergence analysis shows that the new scheme achieves a smaller effective contraction factor and converges faster than several classical two- and three-step iterative methods, including the Picard, Mann, Ishikawa, and S-iteration processes. The theoretical results are applied to Caputo-type fractional differential equations by reformulating the associated boundary value problems as fixed-point equations. Existence and uniqueness of solutions follow from the Banach contraction principle, while the accelerated convergence of the proposed iteration leads to improved numerical efficiency. Extensive numerical experiments, including fractional differential equations and nonlinear contraction mappings on the real line, are presented to validate the theoretical findings. The results demonstrate that the proposed three-step iteration provides an effective and reliable computational tool for fractional and non-local models. Full article

In this paper, we propose three variants of Mann-type inertial forward–backward iterative methods for approximating the minimum-norm solution of the monotone inclusion problem and the fixed points of nonexpansive mappings. In the first two methods, we compute the Mann-type iteration together with the inertial extrapolation and fixed-point iteration in the initiation of the process, while the last method computes only the Mann-type iteration with inertial extrapolation at the start of the process. We establish the strong convergence results for each method with appropriate assumptions and discuss some applications of the presented methods. Finally, we present numerical examples in both finite- and infinite-dimensional Hilbert spaces to demonstrate their efficiency. A comparative analysis with existing methods is also provided. Full article

Array synthetic aperture radar (SAR) three-dimensional (3D) image reconstruction enables the extraction of target distribution information in 3D space, supporting scattering characteristic analysis and structural interpretation. SAR image reconstruction remains challenging due to issues such as noise contamination and incomplete echo data. By introducing sparse priors such as $L_1$ regularization functions, image quality can be improved to a certain extent and the impact of noise can be reduced. However, in scenarios involving distributed targets, the aforementioned methods often fail to maintain continuous structural features such as edges and contours, thereby limiting their reconstruction performance and adaptability. Recent studies have introduced geometric regularization functions to preserve the structural continuity of targets, yet these lack multi-prior consensus, resulting in limited reconstruction quality and robustness in complex scenarios. To address the above issues, a novel array SAR 3D reconstruction method based on multi-prior collaboration (ASAR-MPC) is proposed in this article. In this method, firstly, each optimization module in 3D reconstruction based on multi-prior is treated as an independent function module, and these modules are reformulated as parallel operations rather than sequential utilization. During the reconstruction process, the solution is constrained within the solution space of the module, ensuring that the SAR image simultaneously satisfies multiple prior conditions and achieves a coordinated balance among different priors. Then, a collaborative equilibrium framework based on Mann iteration is presented to solve the optimization problem of 3D reconstruction, which can ensure convergence to an equilibrium point and achieve the joint optimization of all modules. Finally, a series of simulation and experimental tests are described to validate the proposed method. The experimental results show that under limited echo and noise conditions, the proposed method outperforms existing methods in reconstructing complex target structures. Full article

Forest fires are an important disturbance that affects ecosystem stability and pose a serious threat to the ecosystem. However, the recovery process of forest ecological quality (EQ) after a fire in plateau mountain areas is not well understood. This study utilizes the Google Earth Engine (GEE) and Landsat data to generate difference indices, including NDVI, NBR, EVI, NDMI, NDWI, SAVI, and BSI. After segmentation using the Simple Non-Iterative Clustering (SNIC) method, the data were input into a random forest (RF) model to accurately extract the burned area. A 2005–2020 remote sensing ecological index (RSEI) time series was constructed, and the recovery of post-fire forest EQ was evaluated through Theil–Sen slope estimation, Mann–Kendall (MK) trend test, stability analysis, and integration with topographic information systems. The study shows that (1) from 2006 to 2020, the post-fire forest EQ improved year by year, with an average annual increase rate of 0.014/a. The recovery process exhibited an overall trend of “decline initially-fluctuating increase-stabilization”, indicating that RSEI can be used to evaluate the post-fire forest EQ in complex plateau mountainous regions. (2) Between 2006 and 2020, the EQ of forests exhibited a significant increasing trend spatially, with 84.32% of the areas showing notable growth in RSEI, while 1.80% of the regions experienced a declining trend. (3) The coefficient of variation (CV) of RSEI in the study area was 0.16 during the period 2006–2020, indicating good overall stability in the process of post-fire forest EQ recovery. (4) Fire has a significant impact on the EQ of forests in low-altitude areas, steep slopes, and sun-facing slopes, and recovery is slow. This study offers scientific evidence for monitoring and assessing the recovery of post-fire forest EQ in plateau mountainous regions and can also inform ecological restoration and management efforts in similar areas. Full article

This paper aims to provide a comprehensive analysis of the advancements made in understanding Iterative Fixed-Point Schemes, which builds upon the concept of digital contraction mappings. Additionally, we introduce the notion of an Iterative Fixed-Point Schemes in digital metric spaces. In this study, we extend the idea of Iteration process Mann, Ishikawa, Agarwal, and Thakur based on the ϝ-Stable Iterative Scheme in digital metric space. We also design some fractal images, which frame the compression of Fixed-Point Iterative Schemes and contractive mappings. Furthermore, we present a concrete example that exemplifies the motivation behind our investigations. Moreover, we provide an application of the proposed Fractal image and Sierpinski triangle that compress the works by storing images as a collection of digital contractions, which addresses the issue of storing images with less storage memory in this paper. Full article

This paper is dedicated to the advancement of fixed-point results for multi-valued asymptotically non-expansive maps regarding convergence criteria in complete uniformly convex hyperbolic metric spaces that are endowed with a graph. The famous fixed-point theorems of Goebel and Kirk, Khamsi and Khan, along with other recent results in the literature can be obtained as corollaries of these main results. A nice graph and an interesting example are also provided in support of the hypothesis of the main results. Full article

In this work, an algorithm is introduced for the problem of finding a common fixed point of a finite family of G-nonexpansive mappings in a real Hilbert space endowed with a directed graph G. This algorithm is a modified parallel algorithm inspired by the inertial method and the Mann iteration process. Moreover, both weak and strong convergence theorems are provided for the algorithm. Furthermore, an application of the algorithm to a signal recovery problem with multiple blurring filters is presented. Consequently, the numerical experiment shows better results compared with the previous algorithm. Full article

In a real Hilbert space, let the CFPP, VIP, and HFPP denote the common fixed-point problem of countable nonexpansive operators and asymptotically nonexpansive operator, variational inequality problem, and hierarchical fixed point problem, respectively. With the help of the Mann iteration method, a subgradient extragradient approach with a linear-search process, and a hybrid deepest-descent technique, we construct two modified Mann-type subgradient extragradient rules with a linear-search process for finding a common solution of the CFPP and VIP. Under suitable assumptions, we demonstrate the strong convergence of the suggested rules to a common solution of the CFPP and VIP, which is only a solution of a certain HFPP. Full article

We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include combinations of the Mann iteration method, subgradient extra gradient method with the line-search process, and viscosity approximation method. Under suitable assumptions, we obtain the strong convergence results of sequence of iterates generated by our methods for a solution to HVI with the CFPP constraint. Full article

In this work, we introduce the notion of cascading non-expansive mappings in the setting of $\mathrm{CAT}\left(0\right)$ spaces. This family of mappings properly contains the non-expansive maps, but it differs from other generalizations of this class of maps. Considering the concept of $\Delta$-convergence in metric spaces, we prove a principle of demiclosedness for this type of mappings and a $\Delta$-convergence theorem for a Mann iteration process defined using cascading operators. Full article

The purpose of this paper is to introduce two different kinds of iterative algorithms with inertial effects for solving variational inequalities. The iterative processes are based on the extragradient method, the Mann-type method and the viscosity method. Convergence theorems of strong convergence are established in Hilbert spaces under mild assumption that the associated mapping is Lipschitz continuous, pseudo-monotone and sequentially weakly continuous. Numerical experiments are performed to illustrate the behaviors of our proposed methods, as well as comparing them with the existing one in literature. Full article

Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants. Full article

In a real Hilbert space, we denote CFPP and VIP as common fixed point problem of finitely many strict pseudocontractions and a variational inequality problem for Lipschitzian, pseudomonotone operator, respectively. This paper is devoted to explore how to find a common solution of the CFPP and VIP. To this end, we propose Mann viscosity algorithms with line-search process by virtue of subgradient extragradient techniques. The designed algorithms fully assimilate Mann approximation approach, viscosity iteration algorithm and inertial subgradient extragradient technique with line-search process. Under suitable assumptions, it is proven that the sequences generated by the designed algorithms converge strongly to a common solution of the CFPP and VIP, which is the unique solution to a hierarchical variational inequality (HVI). Full article

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI). Full article

In this work, we extend the fundamental results of Schu to the class of monotone asymptotically nonexpansive mappings in modular function spaces. In particular, we study the behavior of the Fibonacci–Mann iteration process, introduced recently by Alfuraidan and Khamsi, defined by $$x_{n+1}=t_n{T}^{\varphi \left(n\right)}\left(x_n\right)+(1-t_n)x_n,$$ for $n\in \mathbb{N}$ , when T is a monotone asymptotically nonexpansive self-mapping. Full article