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Abstract: Our aim is to point out an error in the proof of Lemma 2 in the paper “Fixed point theory in symmetric spaces with applications to probabilistic spaces” by T.L. Hicks, B.E. Rhoades [Nonlinear Analysis 36 (1999) 331–344], and to indicate a way to repair it. [Copyright &y& Elsevier]

In this paper, a novel parabolic system involving nonlinear and nonhomogeneous partial differential operator with variable growth structure is introduced for investigating the image denoising and restoration. More precisely, our model is based on regularizing the classical models involving variable exponent operators by considering a nonlinear operator with double phase flux. We begin by investigating theoretically the solvability to the parabolic system under consideration. Under the setting of Musielak-Orlicz spaces, we build a suitable functional framework to study the proposed system. Therefore, we develop the Faedo-Galerkin approach to demonstrate the existence and uniqueness of a weak solution to our model. To illustrate our theoretical results in the context of image noise removal, we present various numerical implementations on some grayscale images. To enrich these simulations, we test the robustness efficiency of the proposed model in the so-called Magnetic Resonance Images (MRI). The obtained numerical results claim that our model is more efficient and robust against noise, in comparison (visually and quantitatively) to some existing state-of-the-art methods. • A novel nonlinear parabolic system is introduced. • The theoretical results are applied to the image denoising and restoration. • Various numerical implementations on some grayscale images are carried out. [ABSTRACT FROM AUTHOR]

In this paper, a data-driven constrained state estimation method is proposed for nonlinear processes. Within the Koopman operator framework, we propose a data-driven model identification procedure for state estimation based on the algorithm of extended dynamic mode decomposition, which seeks an optimal approximation of the Koopman operator for a nonlinear process in a higher-dimensional space that correlates with the original process state-space via a prescribed nonlinear coordinate transformation. By implementing the proposed procedure, a linear state-space model can be established based on historic process data to describe the dynamics of a nonlinear process and the nonlinear dependence of the sensor measurements on process states. Based on the identified Koopman operator, a linear moving horizon estimation (MHE) algorithm that explicitly addresses constraints on the original process states is formulated to efficiently estimate the states in the higher-dimensional space. The states of the treated nonlinear process are recovered based on the state estimates provided by the MHE estimator designed in the higher-dimensional space. Two process examples are utilized to demonstrate the effectiveness and superiority of the proposed framework. • We propose a Koopman-based data-driven modeling method for general nonlinear processes for the state estimation purpose, which can provide infinite- step-ahead state predictions. • We develop a linear moving horizon estimation scheme to handle constrained nonlinear state estimation problems for general nonlinear processes. • We present two case studies, including an experimental study on a water-tank process consisting of four interconnected water tanks. [ABSTRACT FROM AUTHOR]

This paper presents new criteria for the oscillation of all solutions of the third-order nonlinear delay differential equations with noncanonical operators. Our approach to establishing new criteria essentially simplifies and refines the main results obtained in Džurina and Jadlovská (2018) and Grace et al. (2019). Examples illustrating the importance of our results are presented. • This paper presents new criteria for the oscillation of all solutions of the third-order nonlinear delay differential equations with noncanonical operators. We first transform the studied nonlinear noncanonical third-order delay differential equation into semi-canonical type. This approach simplifies the examination of the considered equation. Moreover, our approach to establishing new criteria substantially improves and extends some known results in the relevant literature. [ABSTRACT FROM AUTHOR]

Delay partial differential equations (PDEs) are widely utilized in many fields, such as climate prediction and epidemiology. But observation data in real world is often noisy and discrete. And in order to expand the applications of delay PDEs, we consider numerical Gaussian processes to solve these models. In this paper, numerical Gaussian processes for predicting the latent solution of a type of delay PDEs with multi-delays are investigated, and various delay PDEs are studied, including problems governed by variable-order fractional order operators and nonlinear operators, so as to adapt to the needs of practical applications. Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. And the methodology can clearly quantify the uncertainty of the predicted solution. For complex boundaries controlled by ODEs, we consider mixed boundary conditions of delay PDEs in this paper. And we also apply Runge-Kutta methods to enhance the prediction accuracy of these problems. Finally, we design seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems. • Numerical Gaussian processes for solving delay PDEs models governed by variable-order fractional order operators and nonlinear operators. • Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. • Runge-Kutta methods are applied to enhance the prediction accuracy of these problems. • Seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems. [ABSTRACT FROM AUTHOR]

In this paper, a novel attempt is made to incorporate the two effective algorithm strategies, where BBO has a strong exploration and Salp Swarm Algorithm (SSA) is used for exploitation of the search space. The proposed algorithm is tested on IEEE CEC 2014 and statistical, convergence graphs are given. The proposed algorithm is also applied to 10 real life problems and compared with its counterpart algorithm. Results obtained by above experiments have demonstrated the outperformance of the hybrid version of BBO over other algorithms. [ABSTRACT FROM AUTHOR]

It is widely acknowledged that neural networks can approximate any continuous (even measurable) functions between finite-dimensional Euclidean spaces to arbitrary accuracy. Recently, the use of neural networks has started emerging in infinite-dimensional settings. Universal approximation theorems of operators guarantee that neural networks can learn mappings between infinite-dimensional spaces. In this paper, we propose a neural network-based method (BasisONet) capable of approximating mappings between function spaces. To reduce the dimension of an infinite-dimensional space, we propose a novel function autoencoder that can compress the function data. Our model can predict the output function at any resolution using the corresponding input data at any resolution once trained. Numerical experiments demonstrate that the performance of our model is competitive with existing methods on the benchmarks, and our model can address the data on a complex geometry with high precision. We further analyze some notable characteristics of our model based on the numerical results. [ABSTRACT FROM AUTHOR]

• A Deep Operator Network (DeepONet) for data-driven transient response prediction. • The Bayesian DeepONet enables a reliable prediction via uncertainty quantification. • The Bayesian DeepONet enables learning even when data is scarce. • The Probabilistic DeepONet enables a reliable prediction at virtually no extra cost This paper proposes a novel data-driven method for the reliable prediction of the power grid's post-fault trajectories, i.e., the power grid's dynamic response after a disturbance or fault. The proposed method is based on the recently proposed concept of Deep Operator Networks (DeepONets). Unlike traditional neural networks that learn to approximate functions, DeepONets are designed to approximate nonlinear operators, i.e., mappings between infinite-dimensional spaces. Under this operator framework, we design a novel and efficient DeepONet that (i) takes as inputs the trajectories collected before and during the fault and (ii) outputs the predicted post-fault trajectories. In addition, we endow our method with the much-needed ability to balance efficiency with reliable/trustworthy predictions via uncertainty quantification. To this end, we propose and compare two novel methods that enable quantifying the predictive uncertainty. First, we propose a Bayesian DeepONet (B-DeepONet) that uses stochastic gradient Hamiltonian Monte-Carlo to sample from the posterior distribution of the DeepONet trainable parameters. Then, we design a Probabilistic DeepONet (Prob-DeepONet) that uses a probabilistic training strategy to enable quantifying uncertainty at virtually no extra computational cost. Finally, we validate the proposed methods' predictive power and uncertainty quantification capability using the New York-New England power grid model. [ABSTRACT FROM AUTHOR]

The paper proposes a new class of nonlinear operators and a dual learning paradigm where optimization jointly concerns both linear convolutional weights and the parameters of these nonlinear operators. The nonlinear class proposed to perform a rich functional representation is composed by functions called rectified parametric sigmoid units. This class is constructed to benefit from the advantages of both sigmoid and rectified linear unit functions, while rejecting their respective drawbacks. Moreover, the analytic form of this new neural class involves scale, shift and shape parameters to obtain a wide range of activation shapes, including the standard rectified linear unit as a limit case. Parameters of this neural transfer class are considered as learnable for the sake of discovering the complex shapes that can contribute to solving machine learning issues. Performance achieved by the joint learning of convolutional and rectified parametric sigmoid learnable parameters are shown to be outstanding in both shallow and deep learning frameworks. This class opens new prospects with respect to machine learning in the sense that main learnable parameters are attached not only to linear transformations, but also to a wide range of nonlinear operators. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid three-term conjugate gradient (CG) method is proposed to solve constrained nonlinear monotone operator equations. The search direction is computed such that it is close to the direction obtained by the memoryless Broyden–Fletcher–Goldferb–Shanno (BFGS) method. Without any condition, the search direction is sufficiently descent and bounded. Moreover, based on some conditions, the search direction satisfy the conjugacy condition without using any line search. The global convergence of the method is established under mild assumptions. Comparison with existing methods is done to test the efficiency of the proposed method through some numerical experiments. Lastly, the applicability of the proposed method is shown. [ABSTRACT FROM AUTHOR]

We extend some of main results of Abasov et al., 2016, Fotiy et al., (2020), Mykhaylyuk et al., (2015), Pliev and Fang, (2017), Pliev and Popov, (2014) to the setting of orthogonally additive operators on lattice-normed spaces. We introduce a new class of C -complete lattice-normed spaces which strictly includes a class of Banach–Kantorovich spaces. The first main result of the paper asserts that every laterally-to-norm continuous C -compact orthogonally additive operator T : X → Y from an atomless C -complete lattice-normed space X to a Banach space Y is narrow. As a non expecting consequence of the first main result we obtain necessary and sufficient conditions for a nonlinear superposition operator T N : E (X) → E (X) to be C -compact. We also show that the sum of two orthogonally additive operators G and T , where G is narrow and T is laterally-to-norm continuous and C -compact, is a narrow operator. In the last part of the article we investigate dominated orthogonally additive operators. In particular we show the narrowness of dominated operators taking values in a Banach sequence space Y or in a separable symmetric operator space C E. [ABSTRACT FROM AUTHOR]

This paper is devoted to numerical analysis of a new class of elliptic variational–hemivariational inequalities in the study of a family of contact problems for elastic ideally locking materials. The contact is described by the Signorini unilateral contact condition and the friction is modeled by a nonmonotone multivalued subdifferential relation allowing slip dependence. The problem involves a nonlinear elasticity operator, the subdifferential of the indicator function of a convex set for the locking constraints and a nonconvex locally Lipschitz friction potential. Solution existence and uniqueness result on the inequality can be found in Migórski and Ogorzaly (2017). In this paper, we introduce and analyze a finite element method to solve the variational–hemivariational inequality. We derive a Céa type inequality that serves as a starting point of error estimation. Numerical results are reported, showing the performance of the numerical method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we offer a method aiming to minimize the role of distance metric used in clustering. It is well known that distance metrics used in clustering algorithms heavily influence the end results and also make the algorithms sensitive to imbalanced attribute/feature scales. To solve these problems, a new clustering algorithm using a per-attribute/feature ranking operating mechanism is proposed in this paper. Ranking is a rarely used discrete, nonlinear operator by other clustering algorithms. However, it also has unique advantages over the dominantly used continuous operators. The proposed algorithm is based on the ranks of the data samples in terms of their spatial separation and is able to provide a more objective clustering result compared with the alternative approaches. Numerical examples on benchmark datasets prove the validity and effectiveness of the proposed concept and principles. Highlights • A new method is proposed to minimize the role of distance metric used in clustering. • This method employs the rarely-used ranking operator as the "core". • It is insensitive to the type of distance metric that is used for clustering. • It is insensitive to the imbalanced attribute scales. • It is able to provide a more objective clustering result compared with the alternatives. [ABSTRACT FROM AUTHOR]

Abstract In this paper, some new p -stability criteria and boundedness results of reaction–diffusion BAM neural networks are derived by way of fixed point theorem, Laplacian semigroup theory and L ∞-estimate technique, which are novel against those of the previous related literature. Since the T–S fuzzy impulsive reaction–diffusion neural networks was investigated by previous literature, the main difficulty of this paper is to find out a novel method to give simpler conclusions than existing results. Finally, a numerical example is presented to illustrate the effectiveness and feasibility of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper we introduce a novel methodology to face the problem of finding, for every fuzzy set of the real line, a fuzzy number which can be considered as an approximation of the first one in some reasonable sense. This methodology depends on a wide variety of initial parameters that each researcher may set depending on his/her own interests. The main objective of this new methodology is to ensure that many of the techniques that are currently available for fuzzy numbers can also be extended to the setting of fuzzy sets of the real line which are, in many ways, much more enriching. To do this, we carry out a study of the families of nested sets that can determine fuzzy numbers through their level sets. Next, we describe some of the main properties that this approximation methodology verifies and we show some examples to illustrate how the initial parameters influence the result of the approximation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a dual-mixed formulation for stationary viscoplastic flows with yield, such as the Bingham or the Herschel-Bulkley flow. The approach is based on a Huber regularization of the viscosity term and a two-fold saddle point nonlinear operator equation for the resulting weak formulation. We provide the uniqueness of solutions for the continuous formulation and propose a discrete scheme based on Arnold-Falk-Winther finite elements. The discretization scheme yields a system of slantly differentiable nonlinear equations, for which a semismooth Newton algorithm is proposed and implemented. Local superlinear convergence of the method is also proved. Finally, we perform several numerical experiments in two and three dimensions to investigate the behavior and efficiency of the method. [ABSTRACT FROM AUTHOR]

Based on the delta operator approach, a new method of actuator fault estimation for a two-stage chemical reactor system is presented with time-delay and outside disturbances. Firstly, a nonlinear delta operator mathematical model is established to describe the two-stage chemical reactor systems by using the mechanism analysis method. In order to estimate the actuator fault, the proportional–integral observer (PIO) is employed as the state estimator. Meanwhile, a novel delta operator-based actuator fault estimation algorithm is proposed. Then, by means of the Lyapunov–Krasovskii functional technique, sufficient conditions are derived to guarantee that the error system is asymptotically stable with a predefined H ∞ performance index. Next, the desired PIO can be designed. Finally, simulation results are given to illustrate the effectiveness of the proposed methods. • A novel proportional and integral fault estimation algorithm is firstly proposed. • This method connects continuous-time systems and discrete-time systems. • A practical simulation example is given to verify the method. • The system model in this paper is more universal and practical. [ABSTRACT FROM AUTHOR]

This paper presents the numerical analysis for a variant of the nonlinear multiscale Dynamic Diffusion (DD) method for the advection-diffusion-reaction equation initially proposed by Arruda et al. [1] and recently studied by Valli et al. [2]. The new DD method, based on a two-scale approach, introduces locally and dynamically an extra stability through a nonlinear operator acting in all scales of the discretization. We prove existence of discrete solutions, stability, and a priori error estimates. We theoretically show that the new DD method has convergence rate of O (h 1 / 2) in the energy norm, and numerical experiments have led to optimal convergence rates in the L 2 (Ω) , H 1 (Ω) , and energy norms. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution for a class of nonlinear weakly singular Volterra integral equation of the first kind. The existence and uniqueness issue of this nonlinear Volterra integral equations is studied completely. An hp -version collocation method in conjunction with Jacobi polynomials is introduced so as an appropriate numerical solution to be found. We analyze it properly and find an error estimation in L 2 -norm. The efficiency of the method is illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

In numerical analysis of frame structures, modeling of the connection between structural members often requires the introduction of rigid end offsets to correctly describe the stiffness of the joints. This is typical of beam-to-column connections in civil constructions but is also common in lattice materials, where the element overlapping at the joints locally increases the stiffness of the nodes. However, when nonlinear geometric effects are included, correct simulation of such phenomena is challenging. This paper presents a geometrically nonlinear three-dimensional force-based beam finite element that efficiently accounts for the effects of rigid joint offsets. The model imposes the element equilibrium in the local reference system referring to the element deformed configuration, considering the von Kármán nonlinear terms, and exploits a corotational formulation to account for rigid large displacements and rotations of the beam. Two alternative approaches are used to introduce the rigid offsets. The first derives from an existing proposal where kinematic and static nonlinear operators are adopted to describe the behavior of the rigid portions at the element ends. The second is a novel approach, easier to implement and use, that involves modifying the integration, along the element axis, of the cross-section strains and flexibility. This can be easily defined for beam models under linear geometry assumption but requires particular attention for the analysis of frame structures under large displacements and strains. Both approaches are validated through numerical examples, including comparison with other numerical methods and experimental results available in the literature. • Corotational force-based 3D Timoshenko beam model with large deformations. • Introduction of the end rigid offset contribution into the corotational formulation. • Alternative simpler approach to model end rigid offsets under nonlinear geometry. • Simulation of experimentally tested lattice structures under nonlinear response. [ABSTRACT FROM AUTHOR]

A novel algorithm for enhancing the details of the blurred infrared images based on frequency extrapolation has been raised in this paper. Unlike other researchers’ work, this algorithm mainly focuses on how to predict the higher frequency information based on the Laplacian pyramid separation of the blurred image. This algorithm uses the first level of the high frequency component of the pyramid of the blurred image to reverse-generate a higher, non-existing frequency component, and adds back to the histogram equalized input blurred image. A simple nonlinear operator is used to analyze the extracted first level high frequency component of the pyramid. Two critical parameters are participated in the calculation known as the clipping parameter C and the scaling parameter S . The detailed analysis of how these two parameters work during the procedure is figure demonstrated in this paper. The blurred image will become clear, and the detail will be enhanced due to the added higher frequency information. This algorithm has the advantages of computational simplicity and great performance, and it can definitely be deployed in the real-time industrial applications. We have done lots of experiments and gave illustrations of the algorithm’s performance in this paper to convince its effectiveness. [ABSTRACT FROM AUTHOR]

This paper presents a new generalized Nicholson’s blowflies model with a linear harvesting term and mixed delays. The main purpose of this paper is to study the existence and the attractivity of the pseudo almost periodic solutions, which are more general and complicated than periodic and almost periodic solutions. Under suitable assumptions, and by using fixed point theorem, sufficient conditions are given to study the pseudo almost periodic solution for the considered model. Moreover, an illustrative example is given to demonstrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we present a collocation method for nonlinear Volterra integral equation of the first kind. This method benefits from the idea of hp -version projection methods. We provide an approximation based on the Legendre polynomial interpolation. The convergence of the proposed method is completely studied and an error estimate under the L 2 -norm is provided. Finally, several numerical experiments are presented in order to verify the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the first eigenvalue of a nonlinear elliptic system involving p -Laplacian as the differential operator. The principal eigenvalue of the system and the corresponding eigenfunction are investigated both analytically and numerically. An alternative proof to show the simplicity of the first eigenvalue is given. In addition, an upper and lower bounds of the first eigenvalue are provided. Then, a numerical algorithm is developed to approximate the principal eigenvalue. This algorithm generates a decreasing sequence of positive numbers and various examples numerically indicate its convergence. Further, the algorithm is generalized to a class of gradient quasilinear elliptic systems. [ABSTRACT FROM AUTHOR]

In this paper an improved version of a verification algorithm for solving nonlinear systems of equations based on Krawczyk operator is presented. Compared with the original algorithm, the improved verification algorithm not only saves computing time, but also computes a narrower (or at least the same) inclusion of the solution to nonlinear systems of equations for certain classes of problems. Numerical results demonstrate the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper we obtain symmetry and monotonicity of positive solutions for the systems involving fully nonlinear nonlocal operators in a domain (bounded or unbounded) in R n via using a direct method of moving planes. This extends the results in Wang and Niu (2017) and also is the first result of the symmetry for a fully nonlinear nonlocal system containing gradient terms with different order. [ABSTRACT FROM AUTHOR]

A nonlinear Markov chain is a discrete time stochastic process whose transitions may depend on both the current state and the current distribution of the process. The nonlinear Markov chain over a finite state space can be identified by a continuous mapping (the so-called nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex) of the finite state space and by a family of transition matrices depending on occupation probability distributions of states. In this paper, we introduce a notion of Dobrushin's ergodicity coefficients for stochastic hypermatrices and provide a criterion for the contraction nonlinear Markov operator by means of Dobrushin's ergodicity coefficients. We also introduce a notion of p − majorizing nonlinear Markov operators associated with stochastic hypermatrices and provide a criterion for strong ergodicity of such kind of operator. We show that the p − majorizing nonlinear Markov operators associated with scrambling , Sarymsakov , and Wolfowitz stochastic hypermatrices are strongly ergodic. These classes of p − majorizing nonlinear Markov operators assure an existence of a residual set of strongly ergodic nonlinear Markov operators which are not contractions. Some supporting examples are also provided. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded open set of R n , n ≥ 2. In this paper we mainly study some properties of the second Dirichlet eigenvalue λ 2 (p , Ω) of the anisotropic p -Laplacian − Q p u : = − div (F p − 1 (∇ u) F ξ (∇ u)) , where F is a suitable smooth norm of R n and p ∈ ] 1 , + ∞ [. We provide a lower bound of λ 2 (p , Ω) among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as p → + ∞. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to investigate some subordination implications for a class of nonlinear integral operators associated with multivalent functions in the open unit disk. Some known results associated with a class of nonlinear averaging integral operators are also pointed out as special cases of the main findings which are presented here. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of W 2 , p (n < p < ∞) solutions of a double obstacle problem. Moreover, we show the optimal regularity of the solution and the local C 1 regularity of the free boundary under a thickness assumption at the free boundary point on the intersection of two free boundaries. In the study of the regularity of the free boundary, we deal with a general problem, the no-sign reduced double obstacle problem with an upper obstacle ψ , F (D 2 u , x) = f χ Ω (u) ∩ { u < ψ } + F (D 2 ψ , x) χ Ω (u) ∩ { u = ψ } , u ≤ ψ in B 1 , where Ω (u) = B 1 ∖ { u = 0 } ∩ { ∇ u = 0 } . [ABSTRACT FROM AUTHOR]

Teaching Learning Algorithm (TLA) is a recently developed nature-inspired optimization technique applicable to complex optimization problems. This paper proposes an improved TLA version using the Laplacian operator of the Genetic Algorithm (GA), named LX-TLA. The proposed algorithm is tested on benchmark optimization problems, including unimodal and multimodal problems. The numerical results are obtained in the form of objective function values, and a t-test is applied to compare the performance of LX-TLA and basic TLA. Convergence plots are given to provide insight into the convergence behavior of LX-TLA. The results reveal that proposed algorithm provides effective and efficient performance in solving benchmark test functions. The proposed algorithm is also applied to engineering design problems, such as Tuned Mass Damper (TMD), truss structure, welded beam, tension string, and pressure vessel. The results obtained using LX-TLA are compared with other nature-inspired optimization algorithms. The results demonstrate that the proposed algorithm is a robust and effective tool for solving complex optimization problems. • Proposed an improved Teaching Learning Algorithm using Laplacian Operator of Genetic algorithm. • Tested on benchmarks functions of varying complexity. • Applied to real life problems in civil and mechanical engineering problems. [ABSTRACT FROM AUTHOR]

• A clean ECG data is modeled as a combination of different components. • The components are disjoint in the time domain, and they have overlapped spectral coefficients. • A segmentation procedure is developed to segment the measured data. • Ideal filters are designed to filter each segment. • The proposed algorithm outperforms many of the existing techniques. A new algorithm for denoising ECG data contaminated by wideband noise is proposed in this paper. In the proposed algorithm, a clean ECG data is modeled as a combination of different components. The components have the characteristics that they are disjoint in the time domain, their spectral coefficients overlap in the frequency domain, and they have different bandwidths. Based on this model, a successive local filtering approach is suggested in this paper to remove wideband noise from a recorded ECG data. In the proposed algorithm, a segmentation procedure is first developed to segment the recorded ECG data such that each segment approximately contains one dominant component. The denoised ECG signal is then constructed by successively denoising the constructed segments using ideal filters. The ideal filters are designed in the frequency domain by minimizing a penalized least-squares objective function, where the weighted ℓ 0 -norm is utilized as the penalty term to encourage the on-off group-sparsity of the ideal filters. In the proposed algorithm, the BW of each ideal filter is automatically adjusted to the BW of the dominant component in the analyzed segment. Simulation results on simulated and real ECG data show that the proposed algorithm can be successfully utilized to denoise ECG data contaminated by wideband noise. In addition, the proposed algorithm is also shown to produce significantly improved results compared to some existing ECG denoising techniques. [ABSTRACT FROM AUTHOR]

Abstract In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization. It is shown that these algorithms are as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size H and the fine grid size h satisfy H = O (h r − 1 r ). Especially for PIDEs with nonlinear memory defined by a lower order nonlinear operator, our two-grid algorithm can save significant storage and computing time. Numerical experiments are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract Risk-averse model predictive control (MPC) offers a control framework that allows one to account for ambiguity in the knowledge of the underlying probability distribution and unifies stochastic and worst-case MPC. In this paper we study risk-averse MPC problems for constrained nonlinear Markovian switching systems using generic cost functions, and derive Lyapunov-type risk-averse stability conditions by leveraging the properties of risk-averse dynamic programming operators. We propose a controller design procedure to design risk-averse stabilizing terminal conditions for constrained nonlinear Markovian switching systems. Lastly, we cast the resulting risk-averse optimal control problem in a favorable form which can be solved efficiently and thus deems risk-averse MPC suitable for applications. [ABSTRACT FROM AUTHOR]

Abstract: Let be a nonempty, closed and convex subset of a real Hilbert space . Let , be a finite family of generalized asymptotically nonexpansive mappings. It is our purpose, in this paper to prove strong convergence of Mann’s type method to a common fixed point of provided that the interior of common fixed points is nonempty. No compactness assumption is imposed either on or on . As a consequence, it is proved that Mann’s method converges for a fixed point of nonexpansive mapping provided that interior of . The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings. [Copyright &y& Elsevier]

Abstract: Kirk and Xu introduced the concept of asymptotic pointwise contractions. In this paper, we investigate these kinds of mappings in modular spaces. Moreover, the fixed point theorem for asymptotic pointwise nonexpansive mappings in modular spaces is also studied. The results of this paper improve and extend the results of Razani et al. [A. Razani, E. Nabizadeh, M. Beyg Mohamadi, S. Homaei Pour, Fixed points of nonlinear and asymptotic contractions in the modular spaces, Abstract and Applied Analysis, 2007 (2007)], and Kirk and Xu [W. A. Kirk, Hong-Kun Xu, Asymptotic pointwise contractions, Nonlinear Analysis 69 (2008) 4706–4712] to modular spaces. [Copyright &y& Elsevier]

Abstract: This paper deals with the second order nonlinear neutral delay difference equation By using Krasnoselskii’s fixed point theorem and some new techniques, we obtain the existence results of uncountably many bounded positive solutions for the above equation. Examples that cannot be solved by known results are given to illustrate the results presented in this paper. [Copyright &y& Elsevier]

Abstract: The fixed point theory for cone metric spaces, which was introduced in 2007 by Huang and Zhang in the paper [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive maps, J. Math. Anal. Appl. 332 (2007) 1467–1475] has recently become a subject of interest for many authors. Cone metric spaces are generalizations of metric spaces where the metric is replaced by the mapping , where , and is a real Banach space. In the present paper for a cone metric space and for the family of subsets of we establish a new cone metric . Next, we introduce the concept of set-valued contraction of Nadler type and prove a fixed point theorem. Examples are provided. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce and study a new class of nonlinear random -resolvent operator equations with random relaxed cocoercive operators in Hilbert spaces. By using Chang’s lemma, Theorem 3.1 of Liu and Li, and the resolvent operator technique for -monotone operators due to Lan, we also prove the existence theorems of the solutions and convergence theorems of the new generalized random iterative procedures with errors for this nonlinear random -resolvent operator equations involving non-monotone random set-valued operators in Hilbert spaces. The results presented in this paper improve and generalize some known corresponding results in the literature. [Copyright &y& Elsevier]