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Let M n be the space of real matrices of order n. The sign-real spectral radius ξ : M n → R + , introduced in a 1997 paper by S.M. Rump, intervenes for instance in the problem of estimating the componentwise distance to singularity. The function ξ has also a bearing in the analysis of generalized absolute value equations and linear complementarity problems. Although ξ is not a norm, it is at least absolutely homogeneous and continuous. Furthermore, ξ is invariant under transposition, permutation similarity, and a few other linear isomorphisms on M n. A matrix A ∈ M n is called sign-real expansive if ξ (A) ≥ 1. Let Ω n be the set of such matrices. The purpose of this work is to discover new properties of the function ξ and to explore in detail the structure of the set Ω n. [ABSTRACT FROM AUTHOR]

In this paper, we revisit two recently published papers on the iterative approximation of fixed points by Kumam et al. (2019) [17] and Maniu (2020) [19] and reproduce convergence, stability, and data dependency results presented in these papers by removing some strong restrictions imposed on parametric control sequences. We confirm the validity and applicability of our results through various non-trivial numerical examples. We suggest a new method based on the iteration algorithm given by Thakur et al. (2014) [28] to solve the two-point second-order boundary value problems. Furthermore, based on the above mentioned iteration algorithm and S -iteration algorithm, we propose two new gradient type projection algorithms and applied them to supervised learning. In both applications, we present some numerical examples to demonstrate the superiority of the newly introduced methods in terms of convergence, accuracy, and computational time against some earlier methods. [ABSTRACT FROM AUTHOR]

In this present work, we study new algorithms to solve the split equality fixed-point problems for total quasi-asymptotically nonexpansive mappings in Hilbert spaces. In addition, we have established the convergence criteria for the proposed algorithms, at the end, we provided numerical results that justified our theoretical results. The results presented in this paper, provided a unified framework for studying this kind of problem involving different classes of mappings. [ABSTRACT FROM AUTHOR]

• Derive a novel Green's function by perturbation technique with conformal mapping. • Re -derive the Green's function by analytical continuation with nonconformal mapping. • Select proper branch cut to evaluate the complex logarithm functions correctly. • Demonstrate the discontinuity induced by the critical point of nonconformal mapping. • Check the hole boundary condition approximated by the truncated perturbation series. Two sets of Green's functions for anisotropic elastic media containing a polygonal hole with rounded corners are presented. Analyzing non-elliptical holes poses a formidable mathematical challenge for anisotropic materials because existing mapping functions generally fail to meet certain criteria. The nonconformal mapping scheme and the perturbation method with conformal mapping are applied to derive two sets of solutions, respectively. They are solved using the method of analytical continuation. For the nonconformal approach, the hole is mapped onto a unit circle with nonconformal mapping functions, and the solution form is identical to elliptical holes only with a different mapping function. The perturbation solution is expressed in a series form based on the conformal mapping functions and solutions for elliptical holes. Proper adjustments of branch cuts of complex logarithm functions are introduced for both solutions. These adjustments are essential in numerical computation, as they ensure the continuity of an elastic body across branch cuts. Also, the two solutions presented in this paper are compared in-depth. Based upon their pros and cons, a combined use of these two solutions is suggested for application in boundary element method. [Display omitted] [ABSTRACT FROM AUTHOR]

3D mapping is now essential for urban infrastructure resource monitoring, autonomous driving, and the fulfillment of digital earth. LiDAR-based Simultaneous Localization And Mapping (SLAM) technology has been widely studied because of its efficient application in the reconstruction of the 3D environments that benefitted from its sensor characteristic. However, the sparsity of low cost LiDAR (Light Detection And Ranging) sensor provides great challenge for it. Insufficient or poor distribution of the constraints as well as the motion distortion in one scan of point cloud would deteriorate the pose estimation in SLAM. In this paper, a novel 3D mapping method based only on the LiDAR data is proposed to enhance the performance of SLAM-based 3D point clouds map reconstruction. The proposed system consists of two main modules: localization module for rapid ego-motion estimation within the surrounding environment, and mapping module for accurate point clouds fusion. As for the localization, scan-to-map point cloud registration scheme are adopted where structural features are firstly extracted to provide valid correspondences. What's more, an effective method is figured out to balance the cost factors for the optimization function of pose estimation in the process. This further handles the degenerated cases for cloud registration. Focusing on improving the quality of the map, a spline motion model is integrated with non-rigid point cloud fusion to facilitate the continuous and nonrepetitive mapping of the environment. Extensive experiments are carried out for the verification of our system including large-scale outdoor and long-corridor scenes. According to the experimental results, our system has achieved 0.78‰closure error and 27.5 % improvement in the APE (Absolute Pose Errors), outperforming the state-of-the-art LiDAR-based SLAM (e.g. LeGO-LOAM) frameworks in our test sites. To specifically verify the performance of 3D mapping, the precision of 3D points is quantified through comparing with the high-precision point cloud maps collected by MLS (Mobile Laser Scanning) and TLS (Terrestrial Laser Scanning). The RMS of point-to-plane distances have improved 20 % based on the fitted Weibull distribution. In addition, some ablation tests are conducted to reveal the efficacy of different components in our system. [ABSTRACT FROM AUTHOR]

In this paper, we study the simplest and the most standard adaptive finite element method for the second-order nonmonotone quasi-linear elliptic problems with the exact solution u ∈ H 0 1 + α (Ω) , α ≥ 1 2 . The adaptive algorithm is based on the residual-based a posteriori error estimators and Dörfler's marking strategy. We prove the convergence and quasi-optimality of the adaptive finite element method when the initial mesh is sufficiently fine. Numerical experiments are provided to illustrate our findings. [ABSTRACT FROM AUTHOR]

In the context of Hilbert space operators, there is a strong relationship between convex and expansive operators and 2-isometries. In this paper, we investigate the bounded linear operators T on a Hilbert space H which have a 2-isometric lifting S on a Hilbert space K containing H as a closed subspace invariant for S ⁎ S. This last property holds in particular when S | K ⊖ H is an isometry. We relate such 2-isometric liftings S by some convex, concave or expansive liftings of the same type as S. We also examine some power bounded operators with such liftings, as well as an intermediate expansive lifting associated with T on the space H ⊕ ℓ + 2 (H). The latter notion is used to obtain 2-isometric liftings S for T (as above) and to describe the similarity of a quasi-expansive operator to a contraction. [ABSTRACT FROM AUTHOR]

In this paper, we present a new iterative method based on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of common fixed points of a finite family of nonexpansive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative method has strong convergence under some mild conditions imposed on algorithm parameters. The results presented in this paper improve and extend the corresponding results reported by some authors recently. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the mixed Agarwal–O’Regan–Sahu type iterative scheme (Agarwal et al., 2007) for finding a common fixed point of the multi-valued nonexpansive mappings in the setting of hyperbolic spaces . Under suitable conditions, some Δ -convergence theorems of the iterative sequence generated by the proposed scheme to approximate a common fixed point of multi-valued nonexpansive mappings are proved. The results presented in the paper extend and improve some recent results announced in the current literature. [ABSTRACT FROM AUTHOR]

The goal of this paper is to consider damped elastic systems with delay and nonlocal conditions in the framework of ordered Banach spaces. Firstly, we investigate the existence of minimal positive S -asymptotically ω -periodic mild solution for structural damped elastic systems with delay and nonlocal conditions on infinite interval. Secondly, based on monotone iterative technique coupled with fixed point theorem, the existence of minimal positive S -asymptotically ω -periodic mild solution is discussed without assuming the existence of upper and lower solutions. Finally, a concrete problem regarding the vibration equation of simply supported beam is given to illustrate the feasibility of our abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a generalization of Grothendieck's theorem for the space of continuous mappings C λ , μ (X , Y) where Y is a complete uniform space with the uniformity μ endowed with the topology of uniform convergence on the family λ of subsets of X. A new topological game is defined - the Asanov-Velichko game, which makes it possible to single out a class of topological spaces of the Grothendieck type. The developed technique is used to generalize the Grothendieck's theorem for the space of continuous mappings endowed with the set-open topology. [ABSTRACT FROM AUTHOR]

• This paper states some sufficient conditions for the existence of endpoints for some multivalued generalized nonexpansive mappings. • Convergence analysis of a new modified algorithm for approximating common endpoints of two generalized nonexpansive mappings in uniformly convex hyperbolic spaces is studied. • Some numerical examples are given to substantiate the common endpoint results. In this paper, we introduce a new modified iteration process for approximating common endpoint of a multivalued α -nonexpansive mapping and a multivalued mapping satisfying condition (E ′) in uniformly convex hyperbolic spaces. As a by-product, we improve the main results of Panyanak (2018) with a new and faster algorithm for two multivalued mappings. Moreover, we give a numerical example to substantiate our results. Our work is new and holds simultaneously in uniformly convex Banach spaces as well as CAT (0) spaces. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new self adaptive iterative algorithm with an inertial technique for solving split feasibility problems and fixed point problems of relative nonexpansive mappings in p -uniformly convex and uniformly smooth Banach spaces. Under some appropriate assumptions imposed on the parameters and operators, we prove that the iterative sequence generated by our proposed algorithm converges strongly to a common solution of split feasibility problems and fixed point problems. Our algorithm does not require a prior knowledge of the norm of the bounded linear operator as we use a new self adaptive step size. Moreover, we give some numerical examples to show the effectiveness and feasibility of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of monotone α -nonexpansive mappings in an ordered Banach space E with the partial order ≤, which contains monotone nonexpansive mappings as special case. With the help of the Mann iteration, we show some existence theorems of fixed points of monotone α -nonexpansive mappings in uniformly convex ordered Banach space. Also, we prove some weak and strong convergence theorems of the Mann iteration for finding an order fixed point of monotone α -nonexpansive mappings under the condition lim sup n → ∞ β n ( 1 − β n ) > 0 or lim inf n → ∞ β n ( 1 − β n ) > 0 . [ABSTRACT FROM AUTHOR]

Considering that weighted possibilistic fuzzy clustering does not obtain significant performance compared with possibilistic fuzzy clustering, so this paper proposes an enhanced self-adaptive weighted possibilistic fuzzy clustering algorithm. Firstly, the principle of maximum entropy is introduced to weighted possibilistic fuzzy clustering, and the weighted coefficients of fuzzy clustering and possibilistic clustering are subject to regularization entropy constraint and a novel self-learning iterative weighted possibilistic fuzzy clustering is obtained, and its convergence is strictly proved by Zangwill theorem and bordered Hessian matrix. Secondly, a series of clustering validity functions for the proposed algorithm are constructed to determine the optimal number of clusters in the data set. In the end, to enhance the anti-noise robustness of the proposed algorithm, a robust loss function is applied in the adaptive weighted possibilistic fuzzy clustering, and a robust algorithm is obtained for noisy data clustering. Experimental results show that the proposed algorithm outperforms existing possibilistic fuzzy clustering-related algorithms, and the validity functions for the proposed algorithm can accurately determine the optimal number of clusters in the data set, meanwhile, the corresponding robust algorithm effectively enhances the performance of the algorithm in the presence of noise. [ABSTRACT FROM AUTHOR]

• For the forward process, the clustering algorithm is employed to generate the initial interval type-2 fuzzy rules, and we illustrate the interpretable mathematical process. In addition, a soft version minimum is used as the T-norm operation of IT2TSFNN to realize fuzzy reasoning, and it can be applied to any other fuzzy models to solve the problem of vanishing firing strength due to a lot of input features. • To tune the fuzzy rule parameters, the conjugate gradient method is employed to realize the backpropagation task and the iterative updating formulas are deduced. Using simulation results, the performance enhancement of the conjugate method based fuzzy model is proved. • Convergence analysis of the MIT2TSFNN is presented, which provides the theoretical foundation for the application of IT2 fuzzy model. Weak convergence illustrates that the real data structure can be approximated using IT2 fuzzy model. Strong convergence shows that the conjugate gradient method can help the fuzzy model get the optimal structure. In this paper, to compute the firing strength values of type-2 fuzzy models, a soft version of minimum is presented, which endows the fuzzy model with the ability to solve large dimensional problems. In addition, a conjugate gradient method is borrowed to train the designed interval type-2 Takagi-Sugeno fuzzy model. Compared with the existing gradient-based learning strategy, this scheme can efficiently enhance the fuzzy model performance. Last but not least, convergence analysis for this modified interval type-2 Takagi-Sugeno fuzzy neural network (MIT2TSFNN) is conducted in detail, which proves that the gradient of the error function tends to zero with the iteration increasing (weak convergence) and the sequence of model parameters (weights) convergences to a fixed point (strong convergence). To validate the effectiveness of the proposed MIT2TSFNN and its theoretical results, simulation results of six regression and six classification problems are presented. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iterative algorithm to approximate fixed point of Suzuki’s generalized nonexpansive mappings. We establish some weak and strong convergence theorems in a uniformly convex Banach space. We also provide examples to illustrate the convergence behavior of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E, A be a single valued monotone and Lipschitz continuous mapping of C into E * and T be a single valued relatively nonexpansive mapping of C into itself. In this paper, we consider the composition and the convex combination of T and the gradient projection method for A which Goldstein (1964) proposed and proved the strong convergence to a common element of solutions of the variational inequality problem for A and fixed points of T by the hybrid method in mathematical programming (Haugazeau, 1968). And we get several results which improve the well-known results in a 2-uniformly convex and uniformly smooth Banach space and a Hilbert space. [ABSTRACT FROM AUTHOR]

Recently, Cui and Wang (2014) constructed an algorithm for demicontractive operators that converges weakly, under mild assumptions, to some solution of the split common fixed point problem. In this paper, based on Halpern’s type method (1967), we construct an algorithm for demicontractive operators that produces sequences that always converge strongly to a specific solution of the split common fixed point problem. Particular cases of directed operators and quasi-nonexpansive mappings are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a hybrid viscosity approximation scheme for finding a common element of the solution set for a system of equilibrium problems, the solution set for a system of variational inequality problems and the common fixed point set for an infinite family of nonexpansive mappings in Hilbert spaces. Then we prove the strong convergence of the proposed iterative scheme. Our results improve and extend the results announced by Ceng and Yao (2008) [4]. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we apply methods of proof mining to obtain a highly uniform effective rate of asymptotic regularity for the Ishikawa iteration associated with nonexpansive self-mappings of convex subsets of a class of uniformly convex geodesic spaces. Moreover, we show that these results are guaranteed by a combination of logical metatheorems for classical and semi-intuitionistic systems. [Copyright &y& Elsevier]

The main focus of this paper is on fixed point theory for mapping satisfying local contractive conditions in Banach spaces and in various geodesic spaces. The emphasis is on locally nonexpansive mappings and local contractions. [ABSTRACT FROM AUTHOR]

The property that a nonlinear operator on a Banach space preserves an order relation, is subhomogeneous or order concave w.r.t. an order cone has profound consequences. In Nonlinear Analysis it allows to solve related equations by means of suitable fixed point or monotone iteration techniques. In Dynamical Systems the possible long term behavior of associate integrodifference equations is drastically simplified. This paper contains sufficient conditions for vector-valued Urysohn integral operators to be monotone, subhomogeneous or concave. It also provides conditions guaranteeing that these properties are preserved under spatial discretization of particularly Nyström type. This fact is crucial for numerical schemes to converge, or for simulations to reproduce the actual behavior and asymptotics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce fully implementable, adaptive Euler–Maruyama schemes for McKean–Vlasov stochastic differential equations (SDEs) assuming only a standard monotonicity condition on the drift and diffusion coefficients but no global Lipschitz continuity in the state variable for either, while global Lipschitz continuity is required for the measure component. We prove moment stability of the discretised processes and a strong convergence rate of 1 / 2. Several numerical examples, centred around a mean-field model for FitzHugh–Nagumo neurons, illustrate that the standard uniform scheme fails and that the adaptive approach shows in most cases superior performance to tamed approximation schemes. In addition, we introduce and analyse an adaptive Milstein scheme for a certain sub-class of McKean–Vlasov SDEs with linear measure-dependence of the drift. [ABSTRACT FROM AUTHOR]

In this paper, the initial–boundary value problem of the coupled inhomogeneous incompressible Navier–Stokes equations and Vlasov–Boltzmann equation for the moderately thick spray is considered in three-dimensional space. The global existence of weak solutions is established by an approximation scheme, a fixed point argument and the weak convergence method. [ABSTRACT FROM AUTHOR]

Expansion of mudstone through absorbing water can increase local stress in information, which is one of the main reasons for casing damage in mudstone. To establish a reasonable mechanical model of local stress increase is the key problem for casing damage in locally expansive mudstone. Many models of uniform stress or linear stress are established to simulate the local stress increase. However, the accuracy and scientificity for those models are insufficient. In this paper, a study on casing strength in locally expansive mudstone is carried out based on theoretical analysis and numerical calculation. Firstly, a mechanical model of casings under parabolic-like loads is established to simulate the local stress increase. Then, an analytical solution of casings is obtained based on the stress function method. Next, a finite element calculation of casings under the same boundary condition is carried out. In the end, the comparisons between analytical solution and finite element method (FEM) are conducted. The results reveal that the theoretical solutions are in good agreement with the FEM. In addition, a novel predicting method of the casing stress in locally expansive mudstone is presented. Finally, our theoretical analysis can be the standard for numerical and approximate calculations. It is helpful to explore the casing damage mechanism in mudstone layer and reduce the casing collapse in complex underground environment. • The action of local swelling mudstone on casing is simplified to a parabolic loading. • An analytical solution of the casing under local swelling mudstone is obtained. • The stress distributions of casings under parabolic loads are analyzed theoretically. [ABSTRACT FROM AUTHOR]

In this paper we study a simplified transient energy-transport model in semiconductors with a general conductivity and the Dirichlet boundary conditions on an interval. By using a new iterative scheme, we prove the global existence and uniqueness of strong solutions provided that the variation of the temperature is small. Also, the existence and stability of stationary solutions are proved if the temperature is large. [ABSTRACT FROM AUTHOR]

In this paper, in order to solve the multiple-sets split feasibility problem in Hilbert spaces, we introduce the regularization methods of Lavrentiev's and Bruck–Bakushinskii's type, where the iterative parameter is chosen independently of the operator norm. We also consider some particular cases of the introduced methods. Applications and numerical experiments are given for illustration and comparison. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a large class of mappings, called enriched Kannan mappings , that includes all Kannan mappings and some nonexpansive mappings. We study the set of fixed points and prove a convergence theorem for the Krasnoselskij iteration used to approximate fixed points of enriched Kannan mappings in Banach spaces. We further extend these mappings to the class of enriched Bianchini mappings. Examples to illustrate the effectiveness of our results are given. As applications of our main fixed point theorems, we present two Krasnoselskij projection type algorithms for solving split feasibility problems and variational inequality problems in the class of enriched Kannan mappings and enriched Bianchini mappings, respectively. [ABSTRACT FROM AUTHOR]