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In this paper, using sunny nonexpansive retractions, which are different from the metric projection in Banach spaces, a new type of study regarding the iterative methods in view of two quasi‐nonexpansive nonself mappings is presented. We also give the convergence analysis for the proposed method in the background of uniformly convex Banach spaces. Moreover, we apply our results to find solutions of common zeros of accretive operators, convexly constrained least square problems, and convex minimization problems. Furthermore, we also discuss novel applications of these methods to differential problems, image deblurring, and signal recovering problems. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a new family of iterative methods for approximating the solution of nonlinear systems for non-differentiable operators. The novelty of this family is that it is a m-step generalization of the Steffensen-type method by updating the divided difference operator in the first two steps but not in the following ones. This procedure allows us to increase both the order of convergence and the efficiency index with respect to that obtained in the family that updates divided differences only in the first step. We perform a semilocal convergence study that allows us to fix the convergence domain and uniqueness for real applied problems, where the existence of a solution is not known a priori. After this study, some numerical tests are developed to apply the semilocal convergence theoretical results obtained. Finally, mediating the dynamic planes generated by the different numerical methods that compose the family, we study the symmetry of the basins of attraction generated by each solution, the shape of these basins, and the convergence to each root of a polynomial function. [ABSTRACT FROM AUTHOR]

Multiscale transforms have become a key ingredient in many data processing tasks. With technological development we observe a growing demand for methods to cope with nonlinear data structures such as manifold values. In this paper we propose a multiscale approach for analyzing manifold-valued data using a pyramid transform. The transform uses a unique class of downsampling operators that enable a noninterpolating subdivision schemes as upsampling operators. We describe this construction in detail and present its analytical properties, including stability and coefficient decay. Next, we numerically demonstrate the results and show the application of our method to denoising and anomaly detection. [ABSTRACT FROM AUTHOR]

In this paper, we study the Liouville-type behaviors of the generalized Greiner operators with nonlinear boundary value conditions. We use the technique based upon the method of moving planes. [ABSTRACT FROM AUTHOR]

Let X be a linear space over K ∈ { R , C } , Y be a real or complex Banach space and f : X n → Y . With some fixed a j i , C i 1 ... i n ∈ K ( j ∈ { 1 , ... , n } , i , i k ∈ { 1 , 2 } , k ∈ { 1 , ... , n } ), we study, using the direct and the fixed point methods, the stability and the general stability of the equation f (a 11 x 11 + a 12 x 12 , ... , a n 1 x n 1 + a n 2 x n 2) = ∑ 1 ≤ i 1 , ... , i n ≤ 2 C i 1 ... i n f (x 1 i 1 , ... , x n i n ) , for all x j i j ∈ X ( j ∈ { 1 , ... , n } , i j ∈ { 1 , 2 } ). Our paper generalizes several known results, among others concerning equations with symmetric coefficients, such as the multi-Cauchy equation or the multi-Jensen equation as well as the multi-Cauchy–Jensen equation. We also prove the hyperstability of the above equation in m-normed spaces with m ≥ 2 . [ABSTRACT FROM AUTHOR]

When solving many physical and technical problems, a situation arises when only operators (functionals) from the objects under study (signals, images, etc.) are available for observations (measurements). It is required to restore the object from its known operator (functional). In many cases, the correlation (autocorrelation) function acts as an operator. A large number of papers have been devoted to the study of the existence of a solution to the problem of signal reconstruction from its autocorrelation function and the uniqueness of this solution. Since the solution to the problem of restoring a function from its autocorrelation function is not known in an analytical form, the problem of developing approximate methods arises. This is relevant not only in the problems of signal and image recovery, but also in solving the phase problem. From the above, the relevance of the problem of restoring a function (images) from the autocorrelation function follows. The article is devoted to approximate methods for solving this problem. Materials and methods. The construction and justification of the computing scheme is based on a continuous method for solving nonlinear operator equations, based on the theory of stability of solutions to ordinary differential equation systems. The method is stable under perturbations of the parameters of the mathematical model and, when solving nonlinear operator equations, does not require the reversibility of the Gateaux (or Freshet) derivatives of nonlinear operators. Results. In this work, an approximate method of signal reconstruction from its autocorrelation function and calculation of the phase of its spectrum from the reconstructed signal is constructed and substantiated. Conclusions. An approximate method for reconstructing a signal from its autocorrelation function and calculating the phase of its spectrum from the recovered signal is constructed and substantiated. The method does not require additional information about the signal under study. The results of the work can be used in solving a number of problems in optics, crystallography, and biology. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse coefficient problem of identifying both the mass density ρ ⁢ (x) > 0 and flexural rigidity r ⁢ (x) > 0 of a damped Euler–Bernoulli (cantilever) beam governed by the equation ρ ⁢ (x) ⁢ u t ⁢ t + μ ⁢ (x) ⁢ u t + (r ⁢ (x) ⁢ u x ⁢ x ) x ⁢ x = 0 , (x , t) ∈ (0 , ℓ) × (0 , T) , subject to boundary conditions u ⁢ (0 , t) = u x ⁢ (0 , t) = 0 , u x ⁢ x ⁢ (ℓ , t) = 0 , - (r ⁢ (x) ⁢ u x ⁢ x ⁢ (x , t)) x | x = ℓ = g ⁢ (t) , from the available measured boundary deflection ν ⁢ (t) := u ⁢ (ℓ , t) and rotation θ ⁢ (t) := u x ⁢ (ℓ , t) at the free end of the beam. The distinctive feature of the considered inverse coefficient problem is that not one, but two Neumann-to-Dirichlet operators have to be formally defined. The inverse problem is hence formulated as a system of nonlinear Neumann-to-Dirichlet operator equations with the right-hand sides consisting of the measured outputs. As a natural consequence of this approach, a vector-form Tikhonov functional is introduced whose components are squares of the L 2 -norm differences between predicted and measured outputs. We then prove existence of a quasi-solution of the inverse problem and derive explicit gradient formulae for the Fréchet derivatives of both components of the Tikhonov functional. These results are instrumental to any gradient based algorithms for reconstructing the two unknown coefficients of the considered damped Euler–Bernoulli beam. [ABSTRACT FROM AUTHOR]

In this paper, we propose and justify a spline-collocation method with first-order splines for approximate solution of nonlinear hypersingular integral equations of Prandtl's type. We obtained the estimates of the convergence rate and the method error. The constructed computational scheme includes a continuous method for solving nonlinear operator equations, which is stable for perturbations of the coefficients and the right-hand sides of equations. [ABSTRACT FROM AUTHOR]

The present manuscript examines different forms of Initial-Value Problems (IVPs) featuring various types of Ordinary Differential Equations (ODEs) by proposing a proficient modification to the famous standard Adomian decomposition method (ADM). The present paper collected different forms of inverse integral operators and further successfully demonstrated their applicability on dissimilar nonlinear singular and nonsingular ODEs. Furthermore, we surveyed most cases in this very new method, and it was found to have a fast convergence rate and, on the other hand, have high precision whenever exact analytical solutions are reachable. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and the analytic dependence upon domain perturbation of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation. The problem is defined in a pair of sets consisting of a perforated domain and an inclusion whose shape is determined by a suitable diffeomorphism $\phi$. First we analyse the case in which the inclusion is a fixed domain. Then we will perturb the inclusion and study the arising boundary value problem and the dependence of a specific family of solutions upon the perturbation parameter $\phi$. [ABSTRACT FROM AUTHOR]

Kolmogorov-Petrovskii-Piskunov equation can be regarded as a generalized form of the Fitzhugh-Nagumo, Fisher and Huxley equations which have many applications in physics, chemistry and biology. In this paper, two fractional extended versions of the non-linear Kolmogorov-Petrovskii-Piskunov equation are solved by analytical methods. Firstly, a new and more general fractional derivative is defined and some properties of it are given. Secondly, a solution in the form of operator representation of the non-linear Kolmogorov-Petrovskii- Piskunov equation with the defined fractional derivative is obtained. Finally, some exact solutions including kink-soliton solution and other solutions of the non-linear Kolmogorov-Petrovskii-Piskunov equation with Khalil et al.'s fractional derivative and variable coefficients are obtained. It is shown that the fractional- order affects the propagation velocity of the obtained kink-soliton solution. [ABSTRACT FROM AUTHOR]

In this paper, we initiate the study of a discrete-time dynamical system modeling a trophic network connecting three types of plankton (phytoplankton, zooplankton, mixoplankton) and bacteria. The nonlinear operator V associated with this dynamical system is of type 4-Volterra quadratic stochastic operator with twelve parameters. We give conditions on the parameters under which this operator maps the five-dimensional standard simplex to itself and we find its fixed points. Moreover, we study the limit points of trajectories for this operator. For each situations we give some biological interpretations. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the asymptotic behaviour of a linear combination of operators of the $ p $-Laplacian type in cylinder like domains having some directions going to infinity. The case of a combination of two such operators has been the topic of numerous studies in the recent years. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence of extreme solutions and their properties for a general σ$$ \sigma $$‐Hessian equation involving a nonlinear operator. By introducing a suitable growth condition and developing a iterative technique, some new results on existence and asymptotic estimates of minimum and maximum solutions are derived. Moreover, we also establish the iterative sequences that converge uniformly to the extreme solutions. [ABSTRACT FROM AUTHOR]

Historically, an idea of reaching consensus through repeated averaging was introduced by DeGroot for a structured time-invariant and synchronous environment. Since that time, the consensus which is the most ubiquitous phenomenon of multiagent systems becomes popular in the various scientific fields such as biology, physics, control engineering, and social science. In this paper, we overview the recent development of applications of quadratic stochastic operators on nonlinear consensus problems. We also present some refinement and improvement of the previous results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain necessary and sufficient conditions for the existence of variational principles for a given first-order differential-difference operator equation with a special form of the linear operator Pλ(t) depending on t and the nonlinear operator Q. Under the corresponding conditions the functional is constructed. These conditions are obtained thanks to the well-known criterion of potentiality. Examples show how the inverse problem of the calculus of variations is constructed for given differential-difference operators. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate the detection of quantum states containing fewer than k unentangled particles in multipartite quantum systems. Based on inequalities of nonlinear operators, we derive two families of criteria for detecting N-partite quantum states containing fewer than k unentangled particles. By concrete examples, we point out that both families of criteria can identify some quantum states containing fewer than k unentangled particles that cannot be tested by known criteria. This demonstrates the effectiveness of our criteria. [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]

In this paper, we deal with the nonlinear space-fractional Klein–Gordon–Schrödinger system involving the fractional Laplacian operator of order α for 1 < α ≤ 2 . We propose an accurate numerical method with eneregy-preserving property for solving the well-known system. The problem is discretized in spatial direction by the Fourier spectral method, and in temporal direction by utilizing the fourth-order exponential time-differencing Runge–Kutta technique. We show that the proposed method satisfies both mass and energy conservation. The convergence of this method is proved, and the order of accuracy is obtained, which shows that the order of convergence is near two. Several numerical experiments are tested to validate the accuracy and reliability of the proposed method. The results are presented in tables and figures for the different values of α that show the proposed method is an efficient framework for solving nonlinear space-fractional Klein–Gordon–Schrödinger system. [ABSTRACT FROM AUTHOR]

The blind image deblurring is to find the underlying true image and the blur kernel from a blurred observation. This is a well-known ill-conditional problem in image processing field. To obtain a pleasant deblurred result, additional assumptions and prior knowledge are required. Proposed in this work is a simple and efficient blind image deblurring method which utilizes L1-regularized second-order gradient prior. The inspiration for this work comes from the fact that the absolute values of the second-order gradient elements decrease with motion blur. This change is an essential feature of the motion blur process, and we demonstrate it mathematically in this paper. By enforcing the L1 norm constraint to the term involving second-order gradients and incorporating it into the traditional deblurring framework, an effective optimization scheme is explored. The half-quadratic splitting technique is adopted to handle the non-convex minimum problem. Experimental results illustrate that our algorithm outperforms the state-of-art image deblurring algorithms in both benchmark datasets and ground-truth scenes. Besides, this algorithm is simple since it does not require any heuristic edge selection steps or involves too many nonlinear operators. [ABSTRACT FROM AUTHOR]

This paper is concerned with the sensitivity analysis of a class of parameterized fixed-point problems that arise in the context of obstacle-type quasi-variational inequalities. We prove that if the operators in the considered fixed-point equation satisfy a positive superhomogeneity condition, then the maximal and minimal elements of the solution set of the problem depend locally Lipschitz continuously on the involved parameters. We further show that if certain concavity conditions hold, then the maximal solution mapping is Hadamard directionally differentiable and its directional derivatives are precisely the minimal solutions of suitably defined linearized fixed-point equations. Our approach covers the elliptic and the parabolic setting simultaneously and also yields Hadamard directional differentiability results in situations in which the solution set of the fixed-point equation is a continuum and a characterization of directional derivatives via linearized auxiliary problems is provably impossible. To illustrate that our results can be used to study interesting problems arising in practice, we apply them to establish the Hadamard directional differentiability of the solution operator of a nonlinear elliptic quasi-variational inequality, which emerges in impulse control and in which the obstacle mapping is obtained by taking essential infima over certain parts of the underlying domain, and of the solution mapping of a parabolic quasi-variational inequality, which involves boundary controls and in which the state-to-obstacle relationship is described by a PDE. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Newton-iteration scheme based on Galerkin and multi-Galerkin operators to solve non-linear integral equations of the Fredholm-Hammerstein type for both smooth and weakly singular algebraic and logarithmic type kernels. By choosing X n as space of piecewise polynomials subspace of degree ≤ r − 1 , we show that for a smooth kernel, Galerkin and multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders O (h (k + 1) r) and O (h (2 k + 3) r) , respectively, where h is the norm of the partition. For weakly singular kernels, we show that the Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders O (h r + k (1 − α)) , 0 < α < 1 for algebraic kernels and O (h r + k (log ⁡ h) k) for logarithmic kernels. Also the multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders O (h r + (k + 1) (1 − α)) for algebraic kernels and O (h r + k + 1 (log ⁡ h) k + 1) for logarithmic kernels. Numerical results are given for the justification of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the nonlinear single pixel X-ray transform K and study the reconstruction of f from the measurement Kf. Different from the well-known X-ray transform, the transform K is a nonlinear operator and uses a single detector that integrates all rays in the space. We derive stability estimates and an inversion of the linearization at zero. We also consider the case where we integrate along geodesics of a Riemannian metric. Moreover, we conduct several numerical experiments to corroborate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we studied the separability of the non-linear Schrodinger operator of the form S u x = − Δ u x + V x , u u x , where Δ u x = ∑ i = 1 n ∂ 2 u x ∂ x i 2 , with the non-linear matrix potential V x , u . We obtained the sufficient conditions for separability of this operator in the space L 2 R n ℓ and we established the suitable coercive inequalities. [ABSTRACT FROM AUTHOR]

In this paper, the numerical radius of nonlinear operators in Hilbert spaces is studied. First, the relationship between the spectral radius and the numerical radius of nonlinear operators is given. Then, the famous inequality 1 2 ‖ T ‖ ≤ w (T) ≤ ‖ T ‖ and inclusion σ (A - 1 B) ⊆ W (B) ¯ W (A) ¯ of bounded linear operators are generalized to the case of certain nonlinear operators, where w (·) , ‖ · ‖ and σ (·) are the numerical radius, the usual operator norm and the spectrum, respectively. Finally, some numerical radius inequalities for nonlinear operator matrices are obtained. [ABSTRACT FROM AUTHOR]

Advances in microtechnology have enabled an exponential increase in the number of neurons that can be simultaneously recorded. To meet high-channel count and implantability demands, emerging applications require new methods for local real-time processing to reduce the data to transmit. Nonlinear energy operators are widely used to distinguish neural spikes from background noise featuring a good tradeoff between hardware resources and accuracy. However, they require an additional smoothing filter, which affects both area occupation and power dissipation. In this paper, we investigate a spike detector, based on a series of two nonlinear energy operators, and a simple and adaptive threshold, based on a three-point median operator. We show that our proposal provides good accuracy compared to other energy-based detectors on a synthetic dataset at different noise levels. Based on the proposed technique, a 1024-channel neural signal processor was designed in a 28 nm TSMC CMOS process by using latch-based static random-access memory (SRAM), demonstrating a total power consumption of 1.4 μW/ch and a silicon area occupation of 230 μm2/ch. These features, together with a comparison with the state of the art, demonstrate that our proposal constitutes an alternative for the development of next-generation multichannel neural interfaces. [ABSTRACT FROM AUTHOR]

The Deep Operator Network (DeepONet) framework is a different class of neural network architecture that one trains to learn nonlinear operators, i.e., mappings between infinite-dimensional spaces. Traditionally, DeepONets are trained using a centralized strategy that requires transferring the training data to a centralized location. Such a strategy, however, limits our ability to secure data privacy or use high-performance distributed/parallel computing platforms. To alleviate such limitations, in this paper, we study the federated training of DeepONets for the first time. That is, we develop a framework, which we refer to as Fed-DeepONet, that allows multiple clients to train DeepONets collaboratively under the coordination of a centralized server. To achieve Fed-DeepONets, we propose an efficient stochastic gradient-based algorithm that enables the distributed optimization of the DeepONet parameters by averaging first-order estimates of the DeepONet loss gradient. Then, to accelerate the training convergence of Fed-DeepONets, we propose a moment-enhanced (i.e., adaptive) stochastic gradient-based strategy. Finally, we verify the performance of Fed-DeepONet by learning, for different configurations of the number of clients and fractions of available clients, (i) the solution operator of a gravity pendulum and (ii) the dynamic response of a parametric library of pendulums. [ABSTRACT FROM AUTHOR]

In this paper, we employ the well-known Krasnoselskii fixed point theorem to study the existence and n-multiplicity of positive periodic solutions for the periodic impulsive functional differential equations with two parameters. The form including an impulsive term of the equations in this paper is rather general and incorporates as special cases various problems which have been studied extensively in the literature. Easily verifiable sufficient criteria are obtained for the existence and n-multiplicity of positive periodic solutions of the impulsive functional differential equations. [ABSTRACT FROM AUTHOR]

In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method. [ABSTRACT FROM AUTHOR]

Let u0 = F(u; t); u(0) = u0; (1), u 2 H, H is a Hilbert space, F(u; t) is a nonlinear operator in H. If F(u; t) = A(t)u + B(u; t), where A(t) is a linear operator, B(u; t) = O(kuk2) for kuk ! 0, then problem (1) has a solution u = 0 if u0 = 0. If ku0k is small then the stability problem is: will the solution to (1) exist for all t > 0 and be small for all t > 0, A.M. Lyapunov gave in 1892 sufficient conditions for this to happen. In our paper a new technical tool is given for answering the above question. This tool (a nonlinear inequality) allows one to give old and new results on Lyapunov stability. One of such results is proved in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of the existence and the uniqueness of a recursive utility function defined on intertemporal lotteries. The purpose of this paper is to provide the results of the existence and the uniqueness of a recursive utility function. The utility function is obtained as the limit of iterations on a nonlinear operator and is independent on initial starting points, with iterations converging at an exponential rate. We also find the maximum utility and an optimal strategy by means of iterations of the Bellman operator. [ABSTRACT FROM AUTHOR]

In this paper we extend the result of Geraghty([4],[5]) about φ- contractions, starting from the papers of O.Popescu( [7]) respectively A.Fulga, A. Proca ([2]). We introduce a new notation and we establish a fixed point theorem for such mapping in a complete metric space. [ABSTRACT FROM AUTHOR]

In a recent paper [Pata, V., A fixed point theorem in metric spaces, J. Fixed Point Theory Appl., 10 (2011), No. 2, 299–305], the author stated and proved a fixed point theorem that is intended to generalize the well known Banach's contraction mapping principle. In this note we show that the main result in the above paper does not hold at least in two extremal cases for the parameter #949; involved in the contraction condition used there. We also present some illustrative examples and related results. [ABSTRACT FROM AUTHOR]

This paper focuses on second-order differential equations involving causal operators with nonlinear two-point boundary conditions. By applying the monotone iterative technique in the presence of upper and lower solutions, with a new comparison theorem, we obtain the existence of extremal solutions. This is an extension of classical theory of second-order differential equations. Finally, we present two examples to show the usefulness of our results. [ABSTRACT FROM AUTHOR]

This paper provides iterative solutions, via some variants of the extragradient approximants, associated with the pseudomonotone equilibrium problem (EP) and the fixed point problem (FPP) for a finite family of η-demimetric operators in Hilbert spaces. The classical extragradient algorithm is embedded with the inertial extrapolation technique, the parallel hybrid projection technique and the Halpern iterative methods for the variants. The analysis of the approximants is performed under suitable set of constraints and supported with an appropriate numerical experiment for the viability of the approximants. [ABSTRACT FROM AUTHOR]

In this paper, a conformable fractional time derivative of order α ∈ (0 , 1 ] is considered in view of the Lax-pair of nonlinear operators to derive a fractional nonlinear evolution system of partial differential equations, called the Fractional-Six-Wave-Interaction-Equations, which is derived in terms of one temporal plus one and two spatial dimensions. Further, an ansatz consisting of linear combinations of hyperbolic functions with complex coefficients is utilized to obtain an infinite set of exact soliton solutions for this system. Certain numerical examples are introduced to show the effectiveness of the ansatz method in obtaining exact solutions for similar systems of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this paper, we address 3D inverse Cauchy issues of highly nonlinear elliptic equations in large cuboids by utilizing the new 3D homogenization functions of different orders to adapt all the specified boundary data. We also add the average classification as an approximate solution to the nonlinear operator part, without requiring to cope with nonlinear equations to resolve the weighting coefficients because these constructions are owned many conditions about the true solution. The unknown boundary conditions and the result can be retrieved straightway by coping with a small-scale linear system when the outcome is described by a new 3D homogenization function, which is right to find the numerical solutions with the errors smaller than the level of noise being put on the over-specified Neumann conditions on the bottom of the cuboid. Besides, note that the new homogenization functions method (HFM) does not require dealing with the regularization and highly nonlinear equations. The robustness and accuracy of theHFMare verified by comparing the recovered results of several numerical experiments to the exact solutions in the entire region, even though a very large level of noise 50% is imposed on the over specified Neumann conditions. The numerical errors of our scheme are in the order of O(10-1)-O(10-4). [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and to investigate the basic properties of qconvex, q-affine and q-concave sequences and to establish their surprising connection to Chebyshev polynomials of the first and of the second kind. One of the main results shows that qconcave sequences are the pointwise minima of q-affine sequences. As an application, we consider a nonlinear selfmap of the n-dimensional space and prove that it has a unique fixed point. For the proof of this result, we introduce a new norm on the space in terms of a q-concave sequence and show that the nonlinear operator becomes a contraction with respect to this norm, and hence, the Banach Fixed Point theorem can be applied. [ABSTRACT FROM AUTHOR]

In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation x ˙ (t) = f (t , x (t) , x [ 2 ] (t) ,... , x [ n ] (t)). x ˙ (t) = f (t , x (t) , x [ 2 ] (t) ,... , x [ n ] (t)). As n = 2 n = 2 , this equation can be regarded as a mixed-type functional differential equation with state-dependence x ˙ (t) = f (t , x (t) , x (T (t , x (t)))) x ˙ (t) = f (t , x (t) , x (T (t , x (t)))) of a special form but, being a nonlinear operator, n n -th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively. [ABSTRACT FROM AUTHOR]

Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and nonexpansive fixed point iterations with a new tool called the scaled relative graph. The SRG provides a correspondence between nonlinear operators and subsets of the 2D plane. Under this framework, a geometric argument in the 2D plane becomes a rigorous proof of convergence. [ABSTRACT FROM AUTHOR]

Although data-driven models, especially deep learning, have achieved astonishing results on many prediction tasks for nonlinear sequences, challenges still remain in finding an appropriate way to embed prior knowledge of physical dynamics in these models. In this work, we introduce a convex relaxation approach for learning robust Koopman operators of nonlinear dynamical systems, which are intended to construct approximate space spanned by eigenfunctions of the Koopman operator. Different from the classical dynamic mode decomposition, we use the layer weights of neural networks as eigenfunctions of the Koopman operator, providing intrinsic coordinates that globally linearize the dynamics. We find that the approximation of space can be regarded as an orthogonal Procrustes problem on the Stiefel manifold, which is highly sensitive to noise. The key contribution of this paper is to demonstrate that strict orthogonal constraint can be replaced by its convex relaxation, and the performance of the model can be improved without increasing the complexity when dealing with both clean and noisy data. After that, the overall model can be optimized via backpropagation in an end-to-end manner. The comparisons of the proposed method against several state-of-the-art competitors are shown on nonlinear oscillators and the lid-driven cavity flow. [ABSTRACT FROM AUTHOR]

Ambartsumian equation and its generalizations are some of the main integral equations of astrophysics, which have found wide application in many areas of physics and technology. An analytical solution to this equation is currently unknown, and the development of approximate methods is urgent. To solve the Ambartsumian equation, several iterative methods are proposed that are used in solving practical problems. Methods of collocations and mechanical quadratures have also been constructed and substantiated under rather severe conditions. It is of considerable interest to construct an iterative method adapted to the coefficients and kernels of the equation. This paper is devoted to the construction of such method. The construction of the iterative method is based on a continuous method for solving nonlinear operator equations. The method is based on the Lyapunov stability theory and is stable against perturbation of the initial conditions, coefficients, and kernels of the equations being solved. An additional advantage of the continuous method for solving nonlinear operator equations is that its implementation does not require the reversibility of the Gateaux derivative of the nonlinear operator. An iterative method for solving the Ambartsumian equation is constructed and substantiated. Model examples were solved to illustrate the effectiveness of the method. Equations generalizing the classical Ambartsumian equation are considered. To solve them, computational schemes of collocation and mechanical quadrature methods are constructed, which are implemented by a continuous method for solving nonlinear operator equations. [ABSTRACT FROM AUTHOR]

In this paper using new fixed point results of the author, we establish a variety of existence results for equilibria for abstract economies. [ABSTRACT FROM AUTHOR]

To describe the dynamics of a size-structured population and its unstructured resource, we formulate bookkeeping equations in two different ways. The first, called the PDE formulation, is rather standard. It employs a first-order partial differential equation, with a non-local boundary condition, for the size-density of the consumer, coupled to an ordinary differential equation for the resource concentration. The second is called the DELAY formulation and employs a renewal equation for the population level birth rate of the consumer, coupled to a delay differential equation for the (history of the) resource concentration. With each of the two formulations we associate a constructively defined semigroup of nonlinear solution operators. The two semigroups are intertwined by a non-invertible operator. In this paper, we delineate in what sense the two semigroups are equivalent. In particular, we (i) identify conditions on both the model ingredients and the choice of state space that guarantee that the intertwining operator is surjective, (ii) focus on large time behavior and (iii) consider full orbits, i.e. orbits defined for time running from − ∞ to + ∞. Conceptually, the PDE formulation is by far the most natural one. It has, however, the technical drawback that the solution operators are not differentiable, precluding rigorous linearization. (The underlying reason for the lack of differentiability is exactly the same as in the case of state-dependent delay equations: we need to differentiate with respect to a quantity that appears as argument of a function that may not be differentiable.) For the delay formulation, one can (under certain conditions concerning the model ingredients) prove the differentiability of the solution operators and establish the Principle of Linearized Stability. Next, the equivalence of the two formulations yields a rather indirect proof of this principle for the PDE formulation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a concept of nonuniform exponential dichotomy through a certain class of strongly continuous evolution operators defined with the aid of a particular family of projections acting on the state space. This class easily emphasizes the fact that, in the case of uniform exponential dichotomy, the uniform exponential growth is essential in order to prove the boundedness of the dichotomic family of projections. The main result of the paper is the extension of the boundedness result in the nonuniform setting. [ABSTRACT FROM AUTHOR]

This paper deals with the mathematical analysis of a class of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator. We are concerned both with the coercive and the noncoercive (and nonresonant) cases, which are in relationship with two associated Rayleigh quotients. The proof combines critical point theory arguments and the dual variational principle. The arguments developed in this paper can be extended to other classes of nonlinear eigenvalue problems with nonstandard growth. [ABSTRACT FROM AUTHOR]

In this paper, we consider the minimization of a Tikhonov functional with an ℓ1 penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to transform the Tikhonov functional into one with an ℓ2 penalty term but a nonlinear operator. The transformed problem can then be analyzed and minimized using standard methods. However, by the nature of this transform, the resulting functional is only once continuously differentiable, which prohibits the use of second order methods. Hence, in this paper, we propose a different transformation, which leads to a twice differentiable functional that can now be minimized using efficient second order methods like Newton's method. We provide a convergence analysis of our proposed scheme, as well as a number of numerical results showing the usefulness of our proposed approach. [ABSTRACT FROM AUTHOR]

In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional p (·) -Laplacian with variable exponents, which is a fractional version of the nonhomogeneous p (·) -Laplace operator. The paper is divided into two parts. In the first part, under suitable conditions on the nonlinearity f, we analyze the problem (P 1) in a bounded domain Ω of R N and we establish the well-posedness of solutions by using techniques of monotone operators. We also study the large-time behaviour and extinction of solutions and we prove that the fractional p (·) -Laplacian operator generates a (nonlinear) submarkovian semigroup on L 2 (Ω). In the second part of the paper we establish the existence of global attractors for problem (P 2) under certain conditions in the potential V. Our results are new in the literature, both for the case of variable exponents and for the fractional p-laplacian case with constant exponent. [ABSTRACT FROM AUTHOR]

The current task scheduling is separated from the virtual machine (VM) allocation, which, to some extent, wastes resources or degrades application performance. The scheduling algorithm influences the demand of VMs in terms of service-level agreement, while the number of VMs determines the performance of task scheduling. Workload plays an indispensable role in both dynamic VM allocation and task scheduling. To address this problem, we coordinate task scheduling and VM allocation based on workload characteristics. Workload is empirically time-varying and stochastic. We demonstrate that the acquired workload data set has Markov property which can be modeled as a Markov chain. Then, three workload characteristic operators are extracted: persistence, recurrence and entropy, which quantify the relative stability, burstiness, and unpredictability of the workload, respectively. Experiments indicate that the persistence and recurrence of workloads has a direct bearing on the average response time and resource utilization of the system. A nonlinear model between the load characteristic operators and the number of VMs is established. In order to test the performance of the collaborative framework, we design a scheduling algorithm based on genetic algorithm (GA), which takes the estimated number of VMs as input and the task completion time as the optimization target. Simulation experiments have been performed on the CloudSim platform, testifying that the estimated average absolute VMs error is only 2.6%. The GA-based task scheduling algorithm could improve resource utilization and reduce task completion time compared with the first come first serve and greedy algorithm. The proposed coordination mechanism in this paper has proved able to find the optimal match and reduce the resource cost by utilizing the interaction between VM allocation and task scheduling. [ABSTRACT FROM AUTHOR]