This paper formulates an intuitionistic non-Archimedean ℒ-fuzzy vector metric space in the context of intuitionism, Archimedeanism, and order structures. In doing so, we derive several essential auxiliary properties, such as left (right)-order continuity and orthogonality. Furthermore, a common fixed point theorem for commuting mappings is established in the space, equipped with a partial order. The method consists of synthesizing the iterative scheme of the contraction mapping principle with the monotone argument.
In this study, we investigate the existence of mild solutions and the exact controllability of Hilfer–Katugampola fractional neutral differential equations involving almost sectorial operators. We first establish the existence of mild solutions and then demonstrate the exact controllability of the proposed system. The existence results are obtained by employing techniques from fractional calculus, semigroup theory, Katugampola fractional integrals and derivatives, and the Laplace transform. Furthermore, a theoretical example is presented to illustrate the obtained results.
We introduce a new class of enriched generalized demimetric mappings and develop an iterative algorithm for finding a common fixed point of these mappings and a common solution of a finite family of monotone inclusion problems in complete CAT(0) spaces. Under suitable assumptions, we establish the strong convergence of the generated sequence to a desired common solution. Numerical experiments demonstrate the applicability of the proposed scheme and show that it achieves a lower average iteration count than the existing methods considered, while satisfying the prescribed residual tolerance. An application to equilibrium problems is also presented, further illustrating the usefulness of the proposed mapping framework for structured nonlinear problems.
Inertial-type iterations were originally introduced to accelerate the weak convergence of iterative methods for various classes of problems. Inspired by these successes, recent studies have extended inertial techniques to Halpern-type schemes to enhance strong convergence performance. In this paper, we establish a structural equivalence between non-inertial and inertial Halpern-type iterations: strong convergence of the error-free scheme guarantees strong convergence of the scheme with error terms. This connection shows that many existing inertial Halpern-type schemes can be unified and derived within a non-inertial framework under weaker assumptions. Furthermore, numerical experiments on split and convex feasibility problems show that inertial terms need not accelerate convergence: in the configurations tested, the non-inertial iteration often attains a smaller error with fewer iterations and less computation time, although the comparison depends on the anchor, the initial points, and the inertial parameter. Finally, we provide a refined perspective on stopping criteria for split feasibility problems, clarifying the applicability of conditions previously discussed in the literature (e.g., Cholamjiak et al. [J. Fixed Point Theory Appl. 20(1):42, 2018]) and proposing a robust alternative for Halpern-type approximations.
In this paper, we prove that the common fixed point set of a given quasi-nonexpansive sequence is identical to the fixed point set of a single quasi-nonexpansive mapping. We also generalize the concept of balanced mappings in Termkaew et al.’s paper (Open Math. 21 (2023)), which was introduced and studied by Hasegawa and Kimura (Linear Nonlinear Anal. 4 (2018)), from a sequence of nonexpansive mappings to a quasi-nonexpansive sequence and investigate its fundamental properties. Furthermore, we can use balanced mapping to prove the above result for a quasi-nonexpansive sequence, and we establish the necessary and sufficient condition to conclude that balanced mapping is a Δ-demiclosed mapping. Moreover, we propose and study one of the convex combinations inspired by Kimura et al. (Fixed Point Theory Appl. 2013:336 (2013)) and obtain the same results as for balanced mappings.
