Search Results (218)

This study proposes a new strongly convergent iterative framework obtained by combining a Krasnosel’skiǐ–Mann type subgradient extragradient process with a hybrid projection strategy and an inertial extrapolation mechanism. The method is applied to address hierarchical fixed-point problems (HFPPs) for nonexpansive and quasi-nonexpansive mappings as well as variational inequality problems (VIPs) involving a pseudomonotone operator in real Hilbert spaces. The proposed scheme employs step sizes that are restricted by the inverse of the Lipschitz constant of the underlying cost operator. Strong convergence of the iterates is achieved under mild hypotheses on the inertial parameter and control sequences. The method is further applied to problems arising in optimization and monotone operator theory. The results show that the proposed framework generalizes and integrates a number of existing approaches while offering improved computational performance. Full article

Certain extensions of $F$-controlled self-mappings in metric spaces to the, as called in this manuscript, ${ F}_{\tau \prime \tau }$ and modified $\ast F_{\tau \prime \tau }$ controlled self-mappings, which are parameterized by two parameters, are addressed. Those parameters govern the properties of local expansivity, asymptotic nonexpansivity, and contractivity properties of the generated sequences. Also, further generalizations to parameterizations by two real sequences of parameters, which are referred to as $F_{{\left\{{\tau }_j^{\prime }\right\}}_{j=0}^{\infty }{\left\{{\tau }_j\right\}}_{j=0}^{\infty }}$-controlled self-mappings, are studied. The main formulated results rely on the asymptotic contractivity and the asymptotic nonexpansivity in metric spaces and some of their relevant properties. In particular, the properties of boundedness of the sequences of distances, as well as those of boundedness of the elements of the sequences themselves, are investigated under asymptotic contractivity or nonexpansivity related to the various types of the above-mentioned $F_{\left(.\right)}$-controlled self-mappings. Also, existence and uniqueness results of fixed points are proved if the metric space is complete, and the resulting Cauchyness properties of sequences and properties of the convergence of such sequences to fixed points are also proved. Finally, two illustrative examples are described if the $F_{\left(.\right)}$-controlled self-mappings are of a cyclic nature when defined using the union of two nonempty closed subsets of the metric space, in the case that those sets intersect, and also in the case when they are disjointed. Full article

In 2024, Kimura proposed the modified shrinking method without assuming the existence of a common fixed point for a family of nonexpansive mappings defined on a complete geodesic space with a nonpositive upper curvature bound. In this paper, we discuss this method for vicinal mappings in an admissible complete geodesic space whose upper curvature bound is an arbitrary real number. Moreover, we investigate the convex minimization problem by using the main result and a resolvent for convex functions. Full article

Building upon classical fixed point theory, the concept of enriched contractions introduces a new class of mappings. For a normed linear space $(X,\parallel ·\parallel )$, a mapping $T:X\to X$ is called an enriched contraction if there exist $b\in [0,\infty )$ and $\theta \in [0,b+1)$ such that $\parallel b(x-y)+Tx-Ty\parallel \le \theta \parallel x-y\parallel ,\forall x,y\in X.$ This class of mappings includes both the well-known Picard–Banach contraction and certain nonexpansive mappings. In this paper, we extend the definition by allowing $b\in \mathbb{R}\backslash \{-1\}$ instead of $b\in [0,\infty )$. This extension enables the condition to cover both contraction and certain nonexpansive mappings. We establish results on the existence and uniqueness of fixed points and present the Krasnosel’skii iteration for approximating such points. An example is provided to demonstrate mapping that meets the extended condition but not the original. Full article

Inertial methods are widely used to accelerate the convergence of iterative algorithms for solving fixed-point problems. However, standard inertial terms often introduce undesirable oscillations, particularly in high-dimensional settings. In this paper, we propose a novel parallel double inertial algorithm with adaptive damping control (D-DIMPMHA) for finding a common fixed point of a finite family of nonexpansive mappings in real Hilbert spaces. By integrating a double inertial step with a self-adaptive damping parameter, the proposed method effectively balances momentum and stability, thereby mitigating numerical oscillations without requiring vanishing inertial conditions. We establish the weak convergence theorem of the generated sequence under suitable control conditions. Furthermore, the practical efficiency of the algorithm is demonstrated through numerical experiments on large-scale convex feasibility problems and image restoration problems. Comparative results indicate that the proposed algorithm achieves superior convergence speed and higher restoration quality compared to existing single inertial methods and FISTA. Full article

In this paper, we present approximation results for a generalized $\alpha$-non-expansive multi-valued mapping using a four-step iteration scheme introduced in the context of a convex metric space. We extend some recent results about generalized $\alpha$-non-expansive multi-valued mappings from the Banach space setting to a convex metric space. Two examples of generalized $\alpha$-non-expansive multi-valued mappings are presented, and it is numerically shown that our iteration scheme enables faster convergence than other well-known schemes in the literature. To demonstrate the application of one of our results, we provide the solution of a non-linear integral equation. Full article

In this paper, we introduced an inertial extragradient algorithm to approximate the common solution of split fixed point, split variational inclusion and split equilibrium problems involving nonexpansive mappings and pseudomonotone Lipschitz-type bifunctions in Hilbert spaces. Moreover, using some assumptions on the control parameters, we prove the strong convergence of the proposed algorithm and then apply our main result to solve the split minimization, split feasibility and split variational inequality problems. We also present some numerical examples to show the effectiveness and applicability of the proposed scheme. We include tables illustrating the number of iterations, the CPU time for convergence, comparisons among different algorithms, and the error analysis. We apply our proposed scheme to solve the image restoration problem as another application of the result presented herein. Full article

We present a generalized Ishikawa iterative algorithm with an error term and a variable generalized Ishikawa iterative algorithm. Leveraging the geometric symmetry inherent in uniformly convex Banach spaces, we establish their respective weak convergence theorems for nonexpansive mappings. As applications, we extend several recent results in the literature related to the proximal point algorithm and the split feasibility problem. Consequently, we propose a hyper-generalized proximal point algorithm and a hyper-generalized perturbation CQ algorithm. Our work not only broadens the application scope of these methods but also highlights the foundational role of symmetric space properties in ensuring algorithmic convergence. 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

We introduce and study a new class of noncyclic Chatterjea-type C-nonexpansive mappings in geodesic spaces. We establish a notable existence theorem for best proximity pairs by employing the pivotal geometric property of proximal normal structure within the framework of reflexive Busemann convex spaces. Moreover, we investigate minimal invariant sets associated with these mappings and derive a generalization of the Goebel–Karlovitz lemma. Our main contribution extends this fundamental result to geodesic spaces with property UC, thereby providing a significant generalization of the classical theorem for the case of Chatterjea-type C-nonexpansive mappings. Full article

Maximum power point tracking (MPPT) under partial shading is a nonconvex, rapidly varying control problem that challenges multi-agent policies deployed on photovoltaic modules. We present Fuzzy–MAT3D, a fuzzy-augmented multi-agent TD3 (Twin-Delayed Deep Deterministic Policy Gradient) controller trained under centralized training/decentralized execution (CTDE). On the theory side, we prove that differentiable fuzzy partitions of unity endow the actor–critic maps with global Lipschitz regularity, reduce temporal-difference target variance, enlarge the input-to-state stability (ISS) margin, and yield a global $L_{\infty }$$\gamma$-contraction of fixed-policy evaluation (hence, non-expansive with $\kappa =\gamma <1$). We further state a two-time-scale convergence theorem for CTDE-TD3 with fuzzy features; a PL/last-layer-linear corollary implies point convergence and uniqueness of critics. We bound the projected Bellman residual with the correct contraction factor (for $L^{\infty }$ and $L^2\left(\rho \right)$ under measure invariance) and quantified the negative bias induced by $min\{Q_1,Q_2\}$; an N-agent extension is provided. Empirically, a balanced common-random-numbers design across seven scenarios and 20 seeds, analyzed by ANOVA and CRN-paired tests, shows that Fuzzy–MAT3D attains the highest mean MPPT efficiency (92.0% ± 4.0%), outperforming MAT3D and Multi-Agent Deep Deterministic Policy Gradient controller (MADDPG). Overall, fuzzy regularization yields higher efficiency, suppresses steady-state oscillations, and stabilizes learning dynamics, supporting the use of structured, physics-compatible features in multi-agent MPPT controllers. At the level of PV plants, such gains under partial shading translate into higher effective capacity factors and smoother renewable generation without additional hardware. Full article

This study presents a common fixed-point iteration process that includes two asymptotically nonexpansive self-mappings in a hyperbolic space and their delta convergence. To support our results, we provide an example with a comparison table and sufficient conditions for a modified iteration scheme to have strong convergence to approximate the fixed point. Full article

In this work, we present a Krasnosel’skiǐ–Mann-type subgradient extragradient algorithm to solve variational inequalities and hierarchical fixed-point problems for nonexpansive and quasi-nonexpansive mappings in Hilbert spaces. We establish weak convergence of the generated sequences to a common solution and derive several related results. The algorithm is validated through numerical examples, and several applications are discussed to demonstrate the method’s applicability. The proposed approach extends and unifies existing methods and findings in this field. Full article

In spite of great successes of the inertial step approach (ISA) in various fields, we are investigating the converse inertial step approach (CISA) for the first time. First, the classical Picard iteration for solving nonexpansive mappings converges weakly with CISA integration. Its analysis is based on the newly developed weak quasi-Fejér monotonicity under mild assumptions. We also establish $O(1/k^{\gamma })$ ($\gamma \in (0,1)$) and linear convergence rate under different assumptions. This extends the $O(1/k)$ convergence rate of the Krasnosel’skiĭ–Mann iteration. A generalized version of CISA is then studied. Second, combining CISA with over-relaxed step approach for solving nonexpansive mappings leads to a new algorithm, which not only converges without restrictive assumptions but also allows an inexact calculation in each iteration. Third, with CISA integration, a Backward–Forward splitting algorithm succeeds in accepting a larger step-size, and a Peaceman–Rachford splitting algorithm is guaranteed to converge. Full article

In this work, we propose and study an inertial hybrid projection algorithm to approximate a common solution of a system of unrelated generalized mixed variational-like inequalities and the common fixed points of Bregman asymptotically quasi-nonexpansive mappings in the intermediate sense. We establish a strong convergence theorem for the generated sequence and derive several corollaries. Further, we provide applications of Bregman asymptotically quasi-nonexpansive mappings in the intermediate sense. Numerical examples are provided to demonstrate the effectiveness of the method, and we also present a comparative analysis. Full article