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In this note, we give affirmative answers to two open problems posed by Agarwal et al. in the paper (J Optim Theory Appl 163(2):482-490, ). [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for efficiently obtaining an approximate solution for constrained nonlinear monotone operator equations. The search direction of the proposed method closely aligns with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) direction, known for its low storage requirement. Notably, the search direction is shown to be sufficiently descent and bounded without using the line search condition. Furthermore, under some standard assumptions, the proposed method converges globally. As an application, the proposed method is applied to solve image restoration problems. The efficiency and robustness of the method in comparison to other methods are tested by numerical experiments using some test problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of equations of viscous compressible and heat-conducting full magnetohydrodynamic (MHD) steady flows in a horizontal layer under the gravitational force and a large temperature gradient across the layer. We assume as boundary conditions, periodic conditions in the horizontal directions, while in the vertical directions, slip-boundary is assumed for the velocity, vertical conditions for the magnetic field, and fixed temperature or fixed heat flux are prescribed for the temperature. The existence of stationary solution in a small neighborhood of a stationary profile close to hydrostatic state is obtained in Sobolev spaces as a fixed point of some nonlinear operator. [ABSTRACT FROM AUTHOR]

Collage Theorem provides a bound for the distance between an element of a given space and a fixed point of a self-map on that space, in terms of the distance between the point and its image. We give in this paper some results of Collage type for Reich mutual contractions in b-metric and strong b-metric spaces. We give upper and lower bounds for this distance, in terms of the constants of the inequality involved in the definition of the contractivity. Reich maps contain the classical Banach contractions as particular cases, as well as the maps of Kannan type, and the results obtained are very general. The middle part of the article is devoted to the invertibility of linear operators. In particular we provide criteria for invertibility of operators acting on quasi-normed spaces. Our aim is the extension of the Casazza-Christensen type conditions for the existence of inverse of a linear map defined on a quasi-Banach space, using different procedures. The results involve either a single map or two operators. The latter case provides a link between the properties of both mappings. The last part of the article is devoted to study the construction of fractal curves in Bochner spaces, initiated by the first author in a previous paper. The objective is the definition of fractal curves valued on Banach spaces and Banach algebras. We provide further results on the fractal convolution of operators, defined in the same reference, considering in this case the nonlinear operators. We prove that some properties of the initial maps are inherited by their convolutions, if some conditions on the elements of the associated iterated function system are satisfied. In the last section of the paper we use the invertibility criteria given before in order to obtain perturbed fractal spanning systems for quasi-normed Bochner spaces composed of Banach-valued integrable maps. These results can be applied to Lebesgue spaces of real valued functions as a particular case. [ABSTRACT FROM AUTHOR]

In this paper, we visualise and analyse the dynamics of fractals (Julia and Mandelbrot sets) for complex polynomials of the form T (z) = z n + m z + r , where n ≥ 2 and m , r ∈ C , by adopting the viscosity approximation type iteration process which is most widely used iterative method for finding fixed points of non-linear operators. We establish a convergence condition in the form of escape criterion which allows to adapt the escape-time algorithm to the considered iteration scheme. We also present some graphical examples of the Mandelbrot and Julia fractals showing the dependency of Julia and Mandelbrot sets on complex polynomials, contraction mappings, and iteration parameters. Moreover, we propose two numerical measures that allow the study of the dependency of the set shape change on the values of the iteration parameters. Using these two measures, we show that the dependency for the considered iteration method is non-linear. [ABSTRACT FROM AUTHOR]

In recent years, multivariate time series prediction has attracted extensive research interests. However, the dynamic changes of the spatial topology and the temporal evolution of multivariate variables bring great challenges to the spatio-temporal time series prediction. In this paper, a novel Dirichlet graph convolution module is introduced to automatically learn the spatio-temporal representation, and we combine graph attention (GAT) and neural differential equation (NDE) based on nonlinear state transition to model spatio-temporal state evolution of nonlinear systems. Specifically, the spatial topology is revealed by the cosine similarity of node embeddings. The use of multi-layer Dirichlet graph convolution aims to enhance the representation ability of the model while suppressing the phenomenon of over-smoothing or over-separation. The GCN and LSTM-based network is used as the nonlinear operator to model the evolution law of the dynamic system, and the GAT updates the strength of the connection. In addition, the Euler trapezoidal integral method is used to model the temporal dynamics and makes medium and long-term prediction in latent space from the perspective of nonlinear state transition. The proposed model can adaptively mine spatial correlations and discover spatio-temporal dynamic evolution patterns through the coupled NDE, which makes the modeling process more interpretable. Experiment results demonstrate the effectiveness of spatio-temporal dynamic discovery on predictive performance. [ABSTRACT FROM AUTHOR]

A modular string averaging procedure (MSA, for short) for a finite number of operators was first introduced by Reich and Zalas in 2016. The MSA concept provides a flexible algorithmic framework for solving various feasibility problems such as common fixed point and convex feasibility problems. In 2001 Bauschke and Combettes introduced the notion of coherence and applied it to proving weak and strong convergence of many iterative methods. In 2019 Barshad, Reich and Zalas proposed a stronger variant of coherence which provides a more convenient sufficient convergence condition for such methods. In this paper we combine the ideas of both modular string averaging and coherence. Focusing on extending the above MSA procedure to an infinite sequence of operators with admissible controls, we establish strong coherence of its output operators. Various applications of these concepts are presented with respect to weak and strong convergence. They also provide important generalizations of known results, where the weak convergence of sequences of operators generated by the MSA procedure with intermittent controls was considered. [ABSTRACT FROM AUTHOR]

Hydraulic brake was widely used for mechanical brake of port crane. The run-state of the brake affects the safety of the crane because of the sudden accident. Because of the complex structure and unsuited to use in real time of traditional assessment model, the condition assessment approach of hydraulic brake was constructed based on state estimation algorithm in this paper. The oil temperature, dynamic friction coefficient, spring stiffness coefficient, brake shoe clearance and contact area were chosen as the state components of the memory matrix based on the analysis of the structure and failure reasons of the brake. Considering the correlation between the state components, the Mahalanobis distance was chosen as the nonlinear operator of algorithm, and the uncertainty factors and random disturbances in state assessment were eliminated by the sliding window residual statistics method. The dynamics simulation of hydraulic brake was constructed for confirming the validity of the approach in this paper. The result shows that it can be accurately judged by the method if the brake is in abnormal state. [ABSTRACT FROM AUTHOR]

This paper presents a new sufficiently descent algorithm for system of nonlinear equations where the underlying operator is pseudomonotone. The conditions imposed on the proposed algorithm to achieve convergence are Lipschitz continuity and pseudomonotonicity which is weaker than monotonicity assumption forced upon many algorithms in this area found in the literature. Numerical experiments on selected test problems taken from the literature validate the efficiency of the new algorithm. Moreover, the new algorithm demonstrates superior performance in comparison with some existing algorithms. Furthermore, the proposed algorithm is applied to reconstruct some disturbed signals. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the optimal risk-sensitive reward for Markov decision processes with compact state space and action space converges to the optimal average reward as the risk-sensitive factor tends to 0. In doing so, a variational formula for the optimal risk-sensitive reward is derived. An extension of the Kreĭn-Rutman Theorem to certain nonlinear operators is involved. Based on these, partially observable Markov decision processes are also investigated. A portfolio optimization problem is presented as an example of an application of the approach, in which a duality-relation between the maximization of risk-sensitive reward and the maximization of upside chance for out-performance over the optimal average reward is established. [ABSTRACT FROM AUTHOR]

This paper is devoted to approximate methods of solving direct and inverse problems for parabolic equations. An approximate method to solve the initial problem of a multidimensional nonlinear parabolic equation has been proposed. It is based on reducing the initial problem to a nonlinear multidimensional Fredholm intergral equation of the second kind, which is approximated by a system of nonlinear algebraic equatiions using a method of mechanical quadratures. In constructing a computational scheme, the points of local splines have been applied for optimal with respect to order approximation of a functional class that contains the solutions of parabolic equations. For the numerical implementation of the computational scheme, we have used the generalization of a continuous method of solving nonlinear operator equations that is described in the paper. In addition, the inverse problem of a parabolic equation with a fractional order derivative with respect to a time variable has been studied. Approximate methods of determining the fractional order of the time derivative and a coefficient at a spatial derivative have been proposed. [ABSTRACT FROM AUTHOR]

We study a class of nonlinear integral equations with a noncompact Hammerstein– Nemytskii operator on the entire line. Some special cases of such equations have specific applications in various fields of natural science. The combination of a method for constructing invariant cone segments for the corresponding nonlinear monotone operator with methods of the theory of functions of a real variable allows one to prove a constructive theorem on the existence of bounded positive solutions of equations of the class under consideration. The asymptotic behavior of the solution at is studied as well. In particular, we prove that the solution constructed in the paper is an integrable function on the negative half-line and that the difference between the limit at and the solution is integrable on the positive half-line. In one special case, we show that our solution generates a one-parameter family of bounded positive solutions. At the end of the paper, we give specific applied examples of nonlinearities to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we examine the existence and analyze properties of inertial manifolds of a nonlinear semigroup of operators in a Hilbert space. This questions were studied in a general setting that allows generalizing results of the well-known works of K. Foias, J. Sell, and R. Temam. Our reasoning is based on the scheme of proofs of similar assertions proposed earlier by S. Sternberg and F. Hartman for ordinary autonomous differential equations. [ABSTRACT FROM AUTHOR]

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation Au = b. We prove that if A is a quasi-uniformly monotone and hemi-continuous operator, then A−1 is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence of Galerkin approximations to the solution of steady-state electromagnetic p-curl systems. [ABSTRACT FROM AUTHOR]

In this paper we investigate homogenization results for the principal eigenvalue problem associated to 1-homogeneous, uniformly elliptic, second-order operators. Under rather general assumptions, we prove that the principal eigenpair associated to an oscillatory operator converges to the eigenpair associated to the effective one. This includes the case of fully nonlinear operators. Rates of convergence for the eigenvalues are provided for linear and nonlinear problems, under extra regularity/convexity assumptions. Finally, a linear rate of convergence (in terms of the oscillation parameter) of suitably normalized eigenfunctions is obtained for linear problems. [ABSTRACT FROM AUTHOR]

The prior-based blind image deblurring methods have recently achieved good performance. However, many state-of-art algorithms are time-consuming since some nonlinear operators are involved. Presented in this paper is a fast blind image deblurring algorithm which uses the salience map and gradient cepstrum. The inspiration for this work comes from the fact that the extreme values of the salience map of the clear image are more sparse than those of the blurred one. By enforcing the L 0 norm constraint to the terms involving salience map and incorporating them into the traditional deblurring framework, an effective optimization scheme is explored. Furthermore, gradient cepstrum is used to adjust the number of iterations in each scale and determine the size of the initial kernel. Experimental results illustrate that our algorithm outperforms the state-of-art deblurring algorithms in both benchmark datasets and real blur scenes. Besides, this algorithm greatly shortens the running time since it restrains excessive iterations and does not involve any nonlinear operators. [ABSTRACT FROM AUTHOR]

Recent advances in modeling large-scale, complex physical systems have shifted research focuses towards data-driven techniques. However, generating datasets by simulating complex systems can require significant computational resources. Similarly, acquiring experimental datasets can prove difficult. For these systems, often computationally inexpensive, but in general inaccurate models, known as the low-fidelity models, are available. In this paper, we propose a bi-fidelity modeling approach for complex physical systems, where we model the discrepancy between the true system's response and a low-fidelity response in the presence of a small training dataset from the true system's response using a deep operator network, a neural network architecture suitable for approximating nonlinear operators. We apply the approach to systems that have parametric uncertainty and are partially unknown. Three numerical examples are used to show the efficacy of the proposed approach to model uncertain and partially unknown physical systems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear systems involving fully nonlinear nonlocal operators { F α (u (x)) = v p (x) + k 1 (x) u r (x) , x ∈ ℝ N , G β (v (x)) = u q (x) + k 2 (x) v s (x) , x ∈ ℝ N and { F α (u (x)) = v p (x) | x | a + u r (x) | x | b , x ∈ ℝ N \ { 0 } , G β (v (x)) = u q (x) | x | c + v s (x) | x | d , x ∈ ℝ N \ { 0 } , where ki(x) ≥ 0, i = 1, 2, 0 < α, β < 2, p, q, r, s > 1, a, b, c, d > 0. By proving a narrow region principle and other key ingredients for the above systems and extending the direct method of moving planes for the fractional p-Laplacian, we derive the radial symmetry of positive solutions about the origin. During these processes, we estimate the local lower bound of the solutions by constructing sub-solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the symmetry properties of positive solutions for nonlocal parabolic equations in the whole space. We obtain various asymptotic maximum principles for carrying out the asymptotic method of moving planes. With the help of these results, we show that if the equation converges to a symmetric one, then the solutions will converge to radially symmetric functions. The methods and techniques used here can be easily applied to study a variety of nonlocal parabolic equations with more general operators and nonlinear terms. [ABSTRACT FROM AUTHOR]

This paper focuses on the problem of convex constraint nonlinear equations involving monotone operators in Euclidean space. A Fletcher and Reeves type derivative-free conjugate gradient method is proposed. The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems. [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]

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, 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]

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]

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 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]

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]

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]

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]

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 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]

This paper is devoted to the qualitative properties of discretized parabolic operators, such as nonnegativity and nonpositivity preservation, maximum/minimum principles and maximum norm contractivity. In the linear case, earlier papers of the authors (Faragó and Horváth in SIAM Sci Comput 28:2313–2336, 2006, IMA J Numer Anal 29:606–631, 2009) have established the connections between the above qualitative properties and have given sufficient conditions for their validity. The present paper extends the above results to nonlinear discretized parabolic operators, also motivated by the authors' recent paper (Faragó and Horváth in J Math Anal Appl 448:473–497, 2017), which has given related results on the continuous PDE level. A systematic study is presented, ranging from general discrete mesh operators to proper finite element applications. [ABSTRACT FROM AUTHOR]

In this paper, we establish the Caccioppoli estimates for the nonlinear differential equation − D ‾ (| D v | p − 2 D v) = λ | v | p − 2 v , 1 < p < ∞ , where D is the Dirac operator. Moreover, we obtain general weighted versions of the Caccioppoli-type inequalities for the Dirac operators. [ABSTRACT FROM AUTHOR]

As a well-known ill-conditional problem in the image processing field, image deblurring has become a hot topic recently. The prior-based blind image deblurring methods have recently shown promising effectiveness. A lot of advanced algorithms such as dark channel prior, bright channel prior, and local maximum gradient prior are time-consuming since nonlinear operators are involved. Presented in this paper is a fast blind image deblurring algorithm which uses the simplified extreme channel prior (SECP) and gradient cepstrum. The inspiration for this work comes from the fact that the simplified bright channel prior (SBCP) of the clear image has fewer non-one elements than the blurred one. We propose a novel SECP based on the proposed SBCP and the simplified dark channel prior (SDCP). By enforcing the L 0 norm constraint to the terms involving SECP and incorporating them into the traditional deblurring framework, an effective optimization scheme is explored. Furthermore, gradient cepstrum is used to determine the size of the initial kernel and restrain excessive iterations in each scale. Experimental results illustrate that our algorithm outperforms the state-of-the-art deblurring algorithms in terms of computational efficiency and deblurring effect on both benchmark datasets and real-world blur scenes. [ABSTRACT FROM AUTHOR]

In this paper we consider the following nonlinear nonlocal Choquard equation F α u (x) + ω u (x) = C n , 2 s | x | 2 s - n ∗ u q (x) u r (x) , x ∈ R n , where 0 < s < 1 , 0 < α < 2 , F α is the fully nonlinear nonlocal operator: F α (u (x)) = C n , α P. V. ∫ R n F (u (x) - u (y)) x - y n + α d y. The positive solution to nonlinear nonlocal Choquard equation is shown to be symmetric and monotone by using the moving plane method which has been introduced by Chen, Li and Li in 2015. We first turn single equation into equivalent system of equations. Then the key ingredients are to obtain the "narrow region principle" and "decay at infinity" for the corresponding problems. We also get radial symmetry results of positive solution for the Schrödinger-Maxwell nonlocal equation. Similar ideas can be easily applied to various nonlocal problems with more general nonlinearities. [ABSTRACT FROM AUTHOR]

The present paper represents a continuation of Migórski et al. (J Elast 127:151–178, 2017). There, the analysis of a new class of elliptic variational–hemivariational inequalities in reflexive Banach spaces, including existence and convergence results, was provided. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. In the current paper we complete this study with new results, including a convergence result with respect the set of constraints. Then we formulate two optimal control problems for which we prove the existence of optimal pairs, together with some convergence results. Finally, we exemplify our results in the study of a one-dimensional mathematical model which describes the equilibrium of an elastic rod in unilateral contact with a foundation, under the action of a body force. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence, regularity, and approximation of the solution for a class of nonlinear fractional differential equations. In order to do this, suitable variational formulations are defined for nonlinear boundary value problems with Riemann-Liouville and Caputo fractional derivatives together with the homogeneous Dirichlet condition. We investigate the well-posedness and also the regularity of the corresponding weak solutions. Then, we develop a Galerkin finite element approach for the numerical approximation of the weak formulations and drive a priori error estimates and prove the stability of the schemes. Finally, some numerical experiments are provided to demonstrate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

By using the classical fixed point theorem for operators on a cone, in this paper, some results of single and multiple positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained. It is worth noticing that the nonlinearity f and the pulse function in this paper are not positive. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear cancer invasion mathematical model with proliferation, growth and haptotaxis effects. We obtain lower bounds for the finite-time blow-up of solutions of the considered system with nonlinear diffusion operator when blow-up occurs. We have assumed both the Dirichlet and Neumann boundary conditions in R n , n ≥ 2 to attain the desire result. [ABSTRACT FROM AUTHOR]

Nonlinear grey Bernoulli model, abbreviated as NGBM model, has been validly used in real applications due to its high accuracy in nonlinear time series forecasting. However, there remain technical challenges to explain the mechanism of the accumulative sum operator in nonlinear grey modelling process and estimate structural parameters independent from the initial values. This paper aims to reconstruct the modelling process of the NGBM model so as to explain the modelling mechanism better by utilizing the integral matching approach, which consists of an integral formula and the numerical discretization-based least squares. First, the integral formula is employed to investigate the accumulative sum operator and further reconstruct the NGBM model to a generalized form, referred as to INGBM model. Then, a novel parameter estimation strategy, estimating structure parameters and initial values simultaneously, is developed by utilizing the numerical discretization-based least squares approach. Next, Monte Carlo simulation studies are designed to evaluate the finite sample performance of both models. Comparisons show that the INGBM model outperforms to the original one in terms of parameter estimation accuracy, forecasting accuracy and robustness to noise. Finally, we apply the INGBM model at a coal consumption in China study to further illustrate the usefulness of this model. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and approximate controllability for a class of fractional nonlocal control systems governed by abstract fractional evolution equations with multiple delays. Under some weaker assumptions, the existence as well as the approximate controllability is established by using fixed point theory. An example is given to illustrate the applicability of the abstract results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of fractional boundary value problems with the derivative term and nonlinear operator term. By establishing new mixed monotone fixed point theorems, we prove these problems to have a unique solution, and we construct the corresponding iterative sequences to approximate the unique solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonhomogeneous multipoint boundary-value problem (BVP) of a second-order differential equation with one-dimensional p-Laplacian. Using multiple fixed-point theorems, new sufficient conditions to guarantee the existence of at least three solutions of this BVP are established. An example is presented to illustrate the main results. The first emphasis of this paper is to show that the approach to get three positive solutions of a BVP by using multiple fixed-point theorems can be extended to treat nonhomogeneous BVPs. The second emphasis is put on the nonlinear term f involved with the first-order delta operator. [ABSTRACT FROM AUTHOR]

In this paper we investigate the following nonlocal problem with singular term and critical Hardy-Sobolev exponent (P) (− Δ) s u = λ u γ + | u | 2 α ∗ − 2 u | x | α in Ω , u > --> 0 in Ω , u = 0 in R N ∖ Ω , $$\begin{array}{} ({\rm P}) \left\{ \begin{array}{ll} (-\Delta)^s u = \displaystyle{\frac{\lambda}{u^\gamma}+\frac{|u|^{2_\alpha^*-2}u}{|x|^\alpha}} \ \ \text{ in } \ \ \Omega, \\ u >0 \ \ \text{ in } \ \ \Omega, \quad u = 0 \ \ \text{ in } \ \ \mathbb{R}^{N}\setminus \Omega, \end{array} \right. \end{array}$$ where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < α < 2s < N, 0 < γ < 1 < 2 < 2 s ∗ $\begin{array}{} \displaystyle 2_s^* \end{array}$ , where 2 s ∗ = 2 N N − 2 s and 2 α ∗ = 2 (N − α) N − 2 s $\begin{array}{} \displaystyle 2_s^* = \frac{2N}{N-2s} ~\text{and}~ 2_\alpha^* = \frac{2(N-\alpha)}{N-2s} \end{array}$ are the fractional critical Sobolev and Hardy Sobolev exponents respectively. The fractional Laplacian (–Δ)s with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by (− Δ) s u (x) = − 1 2 ∫ R N u (x + y) + u (x − y) − 2 u (x) | y | N + 2 s d y , for all x ∈ R N. $$\begin{array}{} \displaystyle (-\Delta)^s u(x)=-\frac{1}{2} \displaystyle\int_{\mathbb{R}^N} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{N+2s}}{\rm d }y, \;\; \text{ for all }\, x \in \mathbb{R}^N. \end{array}$$ By combining variational and approximation methods, we provide the existence of two positive solutions to the problem (P). [ABSTRACT FROM AUTHOR]

Contraction coefficients are distribution dependent constants that are used to sharpen standard data processing inequalities for f-divergences (or relative f-entropies) and produce so-called "strong" data processing inequalities. For any bivariate joint distribution, i.e., any probability vector and stochastic matrix pair, it is known that contraction coefficients for f-divergences are upper bounded by unity and lower bounded by the contraction coefficient for χ2-divergence. In this paper, we elucidate that the upper bound is achieved when the joint distribution is decomposable, and the lower bound can be achieved by driving the input f-divergences of the contraction coefficients to zero. Then, we establish a linear upper bound on the contraction coefficients of joint distributions for a certain class of f-divergences using the contraction coefficient for χ2-divergence, and refine this upper bound for the salient special case of Kullback-Leibler (KL) divergence. Furthermore, we present an alternative proof of the fact that the contraction coefficients for KL and χ2-divergences are equal for bivariate Gaussian distributions (where the former coefficient may impose a bounded second moment constraint). Finally, we generalize the well-known result that contraction coefficients of stochastic matrices (after extremizing over all possible probability vectors) for all nonlinear operator convex f-divergences are equal. In particular, we prove that the so-called "less noisy" preorder over stochastic matrices can be equivalently characterized by any nonlinear operator convex f-divergence. As an application of this characterization, we also derive a generalization of Samorodnitsky's strong data processing inequality. [ABSTRACT FROM AUTHOR]

The object of this paper was to study two majorization results for the subclasses S s ∗ and S c ∗ of starlike functions, which are, respectively, associated with the sine and cosine functions, without acting upon any linear or nonlinear operators to the above function classes. [ABSTRACT FROM AUTHOR]