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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 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]

The goal of the present study is to characterize solutions under a travelling wave formulation to a degenerate Fisher-KPP problem. With the degenerate problem, we refer to the following: a heterogeneous diffusion that is formulated with a high order operator; a non-linear advection and non-Lipstchitz spatially heterogeneous reaction. The paper examines the existence of solutions, uniqueness and travelling wave oscillatory properties (also called instabilities). Such oscillatory behaviour may lead to negative solutions in the proximity of zero. A numerical exploration is provided with the following main finding to declare: the solutions keeps oscillating in the proximity of the null stationary solution due to the high order operator, except if the reaction term is quasi-Lipschitz, in which it is possible to define a region where solutions are positive locally in time. [ABSTRACT FROM AUTHOR]

It is well known that many popular variational, quasi-variational, hemivariational inequalities and variational inclusions involving constraints in a Banach space to convert a fixed point problems for finding the solution of such problems. This paper is to infuse a sequence of penalized problems without constraints and we show under the few reasonable assumptions to the Kuratowski upper limit with respect to the weak topology of the sets of solutions to penalized problems is nonempty. As an application, we explore two complicated partial differential systems of elliptic mixed boundary value problem involving a nonlinear nonhomogeneous differential operator with an obstacle effect, and a nonlinear elastic contact problem in mechanics with unilateral constraints. [ABSTRACT FROM AUTHOR]

The main purpose of the present paper is to introduce and study the notion of neutrosophic b-metric spaces. In this way, we generalize both the notion of neutrosophic metric spaces and fuzzy b-metric spaces. Further, the formulation and proof of neutrosophic b-metric versions of some conventional theorems regarding fixed points via neutrosophic sets are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following non-linear equations in unbounded domains Ω with exterior Dirichlet condition: { (− Δ) p s u (x) = f (u (x)) , x ∈ Ω , u (x) > 0 , x ∈ Ω , u (x) = 0 , x ∈ R n ∖ Ω , where (− Δ) p s is the fractional p -Laplacian defined as (0.1) (− Δ) p s u (x) = C n , s , p P. V. ∫ R n | u (x) − u (y) | p − 2 [ u (x) − u (y) ] | x − y | n + s p d y with 0 < s < 1 and p ≥ 2. We first establish a maximum principle in unbounded domains involving the fractional p -Laplacian by estimating the singular integral in (0.1) along a sequence of approximate maximum points. Then, we obtain the asymptotic behavior of solutions far away from the boundary. Finally, we develop a sliding method for the fractional p -Laplacians and apply it to derive the monotonicity and uniqueness of solutions. There have been similar results for the classical Laplacian [3] and for the fractional Laplacian [39] , which are linear operators. Unfortunately, many approaches there no longer work for the fully non-linear fractional p -Laplacian here. To circumvent these difficulties, we introduce several new ideas, which enable us not only to deal with non-linear non-local equations, but also to remarkably weaken the conditions on f (⋅) and on the domain Ω. We believe that the new methods developed in our paper can be widely applied to many problems in unbounded domains involving non-linear non-local operators. [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 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]

The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space (X , { | ⋅ | p } p ∈ Λ) (X,\{\lvert\,{\cdot}\,\rvert_{p}\}_{p\in\Lambda}) having the 휏-Krein–Šmulian property, where 휏 is a weaker Hausdorff locally convex topology of 푋. The method applied in our study is connected with a family Φ Λ τ \Phi_{\Lambda}^{\tau} -MNC of measures of weak noncompactness and with the concept of 휏-sequential continuity. As a special case, we discuss the existence of solutions for a 2 × 2 2\times 2 block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space C ⁢ (R +) × C ⁢ (R +) C(\mathbb{R}^{+})\times C(\mathbb{R}^{+}) to verify the effectiveness and applicability of our main result. [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]

We consider a mixed variational problem governed by a nonlinear operator and a set of constraints. Existence, uniqueness and convergence results for this problem have already been obtained in the literature. In this current paper we complete these results by proving the well-posedness of the problem, in the sense of Tykhonov. To this end we introduce a family of approximating problems for which we state and prove various equivalence and convergence results. We illustrate these abstract results in the study of a frictionless contact model with elastic materials. The process is assumed to be static and the contact is with unilateral constraints. We derive a weak formulation of the model which is in the form of a mixed variational problem with unknowns being the displacement field and the Lagrange multiplier. Then, we prove various results on the corresponding mixed problem, including its well-posedness in the sense of Tykhonov, under various assumptions on the data. Finally, we provide mechanical interpretation of our results. [ABSTRACT FROM AUTHOR]

Power Normalizations (PN) are useful non-linear operators which tackle feature imbalances in classification problems. We study PNs in the deep learning setup via a novel PN layer pooling feature maps. Our layer combines the feature vectors and their respective spatial locations in the feature maps produced by the last convolutional layer of CNN into a positive definite matrix with second-order statistics to which PN operators are applied, forming so-called Second-order Pooling (SOP). As the main goal of this paper is to study Power Normalizations, we investigate the role and meaning of MaxExp and Gamma, two popular PN functions. To this end, we provide probabilistic interpretations of such element-wise operators and discover surrogates with well-behaved derivatives for end-to-end training. Furthermore, we look at the spectral applicability of MaxExp and Gamma by studying Spectral Power Normalizations (SPN). We show that SPN on the autocorrelation/covariance matrix and the Heat Diffusion Process (HDP) on a graph Laplacian matrix are closely related, thus sharing their properties. Such a finding leads us to the culmination of our work, a fast spectral MaxExp which is a variant of HDP for covariances/autocorrelation matrices. We evaluate our ideas on fine-grained recognition, scene recognition, and material classification, as well as in few-shot learning and graph classification. [ABSTRACT FROM AUTHOR]

This paper considers the approximate controllability for a class of abstract semilinear control systems using a method called surjectivity theorems for nonlinear operator equations under restrictive assumptions. The sufficient conditions for approximate controllability are derived under the natural assumptions on the nonlinear terms, which is an odd homogeneous and the nonlinear inverse considered as a multivalued mapping is bounded. An example is given to illustrate our main results. [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, an iterative method is extended to solve nonlinear two- and three-dimensional mixed Volterra-Fredholm integral equations. We consider a nonlinear operator of these integral equa- tions and then develop the iterative method which was introduced in [J MATH ANAL APPL. 316 (2006) 753-763] to solve them. Conver- gence property of the suggested schemes are proved under some mild assumptions. In both cases, numerical examples are given to compare the performance of the proposed method with some existing methods. [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]

In this paper, we prove the existence of solutions to a nonlinear singular transport equation (ie, transport equation with unbounded collision frequency and unbounded collision operator) with vacuum boundary conditions in bounded spatial domain on Lp‐spaces with 1 ≤ p<+∞. This problem was already considered in6,8,9 under the hypothesis that the collision frequency σ(·) and the collision operator are bounded. In this work, we show that these hypotheses are not necessary; it suffices to assume that σ(·) is locally bounded, and the collision operator is bounded between Xpσ (a weighted space) and Xp (cf Section 2). Although the analysis for p∈(1,+∞) is standard in the sense that it uses the Schauder fixed point theorem, the compactness of the involved operator is not easy to derive. However, the analysis in the case p=1 uses the concept of Dunford‐Pettis operators and a new version of the Darbo fixed point theorem for a measure of weak noncompactness introduced in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we establish the generalized viscosity implicit rules of asymptotically nonexpansive mappings in CAT(0) spaces. The strong convergence theorems of the implicit rules proposed are proved under certain assumptions imposed on the control parameters. The results presented in this paper improve and extend some recent corresponding results announced. [ABSTRACT FROM AUTHOR]

The concern of this paper is to continue the investigation of convergence properties of nonlinear approximation operators, which are defined by Karsli. In details, the paper centers around Urysohn‐type nonlinear counterpart of the Bernstein operators. As a continuation of the study of Karsli, the present paper is devoted to obtain Voronovskaya‐type theorems for the Urysohn‐type nonlinear Bernstein operators. [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]

This paper proposes a novel guidance law based on model predictive control for protecting an aircraft from a homing missile. Considering that in the target‐missile‐defender engagement the attacking missile is pursuing the target aircraft by employing a known linear guidance law, a non‐linear predictive model for the engagement is established. This predictive model allows converting the three‐body engagement to a one‐on‐one engagement. The time‐to‐go for the defender towards the future position of the incoming missile is analytically derived, indicating that the optimal time for the defender to intercept the incoming missile is a fixed‐point of a real‐valued function. As a result, the optimal interception duration can be obtained by a standard fixed‐point iteration method, and thus the interception point can be readily predicted via integrating the predictive model. The guidance command is generated by directly leading the defender to the predicted intercept point. Numerical simulations in different scenarios validate the algorithm to be computationally efficient, and demonstrate that the proposed strategy outperforms previous techniques in terms of miss distance, engagement duration, and the control effort. [ABSTRACT FROM AUTHOR]

In this paper, some common fixed point theorems for pairs of subcompatible and reciprocally continuous mappings or compatible and subsequentially continuous mappings satisfying integral type are obtained. Our results improve several results especially the results of [3], Pathak et al. [13], Djoudi and Aliouche [6], Mbarki [10] and others. [ABSTRACT FROM AUTHOR]

This paper focuses on the following degenerate parabolic equation u t − d i v (σ (x) | ∇ u | p − 2 ∇ u) + f (x , u) = g i n Ω × R + , u = 0 o n ∂ Ω × R + , u (x , 0) = u 0 (x) i n Ω , where Ω is a smooth bounded domain in R N , (N ≥ 2) , 1 < p < N , u 0 , g ∈ L 1 (Ω) , σ (x) is positive almost everywhere and satisfies proper degenerate conditions. The existence and uniqueness result is proved in the framework of entropy solutions. For the long-time behavior, we prove the existence of a global attractor in L q (Ω) by using some delicate estimates on the solution, which are derived by taking advantage of both the leading operator and the zero-order nonlinear term. The aforementioned results improve some previous results in the literature in several aspects. [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]

This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors' recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search. [ABSTRACT FROM AUTHOR]

In this paper, we establish new Lyapunov-type inequalities for (n + 1)st order nonlinear differential equation including p-relativistic operator and q -prescribed curvature operator under the antiperiodic boundary conditions. [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]

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]

In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully nonlinear operators, such as Pucci's extremal operators, being new even for scalar equations. [ABSTRACT FROM AUTHOR]

The aim of the paper is to discuss data dependence, existence of fixed points, strict fixed points, and well posedness of some multivalued generalized contractions in the setting of complete metric spaces. Using auxiliary functions, we introduce Wardowski type multivalued nonlinear operators that satisfy a novel class of contractive requirements. Furthermore, the existence and data dependence findings for these multivalued operators are obtained. A nontrivial example is also provided to support the results. The results generalize, improve, and extend existing results in the literature. [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]

In this paper, the optimal dividend problem in a discrete-time risk model with interest is discussed. Assume that the premium received per unit time is a positive real-valued random variable, and the sequence of premiums is a Markov chain owing to the environmental effects. In arbitrary unit time whether a claim occurs or not is related to the premium received in the corresponding period. Under the constraint of a given survival probability, the optimal control strategy for dividends paid periodically to the shareholders is considered. We provide some properties and an algorithm for the optimal control strategy by structuring a non-linear operator and applying the fixed point theorem. Numerical examples are presented to illustrate the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the higher-order multi-point fractional boundary value problem. We establish the criteria for the existence of at least one and three positive solutions for higher order nonlinear m-point fractional boundary value problem by using the Krasnosel'skii fixed point theorem and the Legget-Williams fixed point theorem, respectively. [ABSTRACT FROM AUTHOR]

In this paper, by considering theWardowski's technique, we present fixed point results for multivalued mapping on a space with two metrics. Also, taking into account β-admissibility of a multivalued mapping, we provide some more general results. [ABSTRACT FROM AUTHOR]

The goals of the present paper are to introduce truncated Lupaş type operators of max-product kind and give an estimation for the degree of approximation with respect to first modulus of continuity function. We prove that this estimate can not be improved; on the other hand, for some subclasses of functions, better degree of approximation is obtained. We also showed the piecewise convexity of the constructed operators on the interval [0, 1]. [ABSTRACT FROM AUTHOR]

The ubiquity of semilinear parabolic equations is clear from their numerous applications ranging from physics and biology to materials and social sciences. In this paper, we consider a practically desirable property for a class of semilinear parabolic equations of the abstract form ut = u + f[u], with a linear dissipative operator and f a nonlinear operator in space, namely, a time-invariant maximum bound principle, in the sense that the timedependent solution u preserves for all time a uniform pointwise bound in absolute value imposed by its initial and boundary conditions. We first study an analytical framework for sufficient conditions on and f that lead to such a maximum bound principle for the time-continuous dynamic system of infinite or finite dimensions. Then we utilize a suitable exponential time-differencing approach with a properly chosen generator of the contraction semigroup to develop first- and second-order accurate temporal discretization schemes that satisfy the maximum bound principle unconditionally in the time-discrete setting. Error estimates of the proposed schemes are derived along with their energy stability. Extensions to vector- and matrix-valued systems are also discussed. We demonstrate that the abstract framework and analysis techniques developed here offer an effective and unified approach to studying the maximum bound principle of the abstract evolution equation that covers a wide variety of well-known models and their numerical discretization schemes. Some numerical experiments are also carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Runaway electron modelling efforts are motivated by the risk these energetic particles pose to large fusion devices. The sophisticated kinetic models can capture most features of the runaway electron generation but have high computational costs, which can be avoided by using computationally cheaper reduced kinetic codes. This paper compares the reduced kinetic and kinetic models to determine when the former solvers, based on analytical calculations assuming quasi-stationarity, can be used. The Dreicer generation rate is calculated by two different solvers in parallel in a workflow developed in the European integrated modelling framework, and this is complemented by calculations of a third code that is not yet integrated into the framework. Runaway Fluid, a reduced kinetic code, NORSE, a kinetic code using non-linear collision operator, and DREAM, a linearized Fokker–Planck solver, are used to investigate the effect of a dynamic change in the electric field for different plasma scenarios spanning across the whole tokamak-relevant range. We find that on time scales shorter than or comparable to the electron–electron collision time at the critical velocity for runaway electron generation, kinetic effects not captured by reduced kinetic models play an important role. This characteristic time scale is easy to calculate and can reliably be used to determine whether there is a need for kinetic modelling or cheaper reduced kinetic codes are expected to deliver sufficiently accurate results. This criterion can be automated, and thus it can be of great benefit for the comprehensive self-consistent modelling frameworks that are attempting to simulate complex events such as tokamak start-up or disruptions. [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]

The maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations, in the sense that the time-dependent solution of the equation with appropriate initial and boundary conditions and nonlinear operator preserves for all time a uniform pointwise bound in absolute value. It has been a challenging problem to design unconditionally MBPpreserving high-order accurate time-stepping schemes for these equations. In this paper, we combine the integrating factor Runge--Kutta (IFRK) method with the linear stabilization technique to develop a stabilized IFRK (sIFRK) method, and we successfully derive sufficient conditions for the proposed method to preserve the MBP unconditionally in the discrete setting. We then elaborate some sIFRK schemes with up to the third-order accuracy, which are proven to be unconditionally MBP-preserving by verifying these conditions. In addition, it is shown that many classic strong stability-preserving sIFRK schemes do not satisfy these conditions except the first-order one. Extensive numerical experiments are also carried out to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

We consider synchronization of the all-to-all power grid model with inertia and nonlinear terms of the first- and second-order derivatives. This model was derived from the energy conservation law in [G. Filatrella, A. H. Nielsen, and N. F. Pedersen, Eur. Phys. J. B, 61 (2008), pp. 485-491]. Since the model contains a nonlinear differential operator, we cannot use a Lyapunov functional and an a priori estimate method directly. In this paper, we provide a new framework to obtain complete phase and frequency synchronization of the power grid model. We assume that the coupling strength and the moment of inertia are positive and the initial data are near the synchronization state. After applying an appropriate transform, we derive a system of differential equations whose coefficient matrix is negative definite. The initial data assumption and the negative definite coefficient matrix lead to the global existence and synchronization of the power grid model with a nonlinear differential operator. For both identical and nonidentical cases, the exponential decay for the frequency difference is determined and the square of frequencies converges to the average of the normalized energy feeding rates. [ABSTRACT FROM AUTHOR]

In this paper, we establish the generalized viscosity implicit rules of asymptotically nonexpansive mappings in CAT(0) spaces. The strong convergence theorems of the implicit rules proposed are proved under certain assumptions imposed on the control parameters. The results presented in this paper improve and extend some recent corresponding results announced. [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]

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]

Highlights • Nonlinear operators are used to smooth a hysteretic system with rate-independent hysteresis property. • It is applied the harmonic probing method on the equivalent system that carries the nonlinear smooth operator. • Analytical expressions to the higher-order frequency response functions are derived to evaluate a hysteretic system. • The analytical approach proposed can reproduce the hysteresis loop of a benchmark involving a Bouc-Wen oscillator for narrow-banded input conditions. Abstract This paper proposes a new way to approach hysteresis with the rate-independent property through an analytical response for the non-smooth hysteresis loop using frequency response approximations. The method consists of rewriting the loading and unloading loop using smooth operators and after applying a harmonic probing in the equivalent system to obtain the higher-order frequency response functions computed by Volterra series. The novelty of this paper lies on predicting analytically, through closed-form equations of the Volterra kernels, the output and the hysteresis loop from a non-smooth system. To illustrate the applicability of the proposed approach, a challenging benchmark with hysteretic damping, described by the Bouc-Wen model, is simulated through a numerical integration scheme. The hysteresis loops, as well as the outputs, are compared to the analytical approach proposed here. The results show that the Volterra model is able to predict the hysteretic outputs when the excitation amplitude is weak and the hysteresis draws a single loop in the restoring force × displacement plane. The higher-order FRFs are given as a function of the model parameters. This framework could turn into an alternative tool to perform nonlinear modal analysis on a hysteretic system. [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]