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The aim of this paper is to obtain optimal estimates for the first Robin eigenvalue of the anisotropic p -Laplace operator, namely: λ 1 (β , Ω) = min ψ ∈ W 1 , p (Ω) ∖ { 0 } ⁡ ∫ Ω F (∇ ψ) p d x + β ∫ ∂ Ω | ψ | p F (ν Ω) d H N − 1 ∫ Ω | ψ | p d x , where p ∈ ] 1 , + ∞ [ , Ω is a bounded, anisotropic mean convex domain in R N , ν Ω is its Euclidean outward normal, β is a real number, and F is a sufficiently smooth norm on R N. The estimates we found are in terms of the first eigenvalue of a one-dimensional nonlinear problem, which depends on β and on geometrical quantities associated to Ω. More precisely, we prove a lower bound of λ 1 in the case β > 0 , and a upper bound in the case β < 0. As a consequence, we prove, for β > 0 , a lower bound for λ 1 (β , Ω) in terms of the anisotropic inradius of Ω and, for β < 0 , an upper bound of λ 1 (β , Ω) in terms of β. [ABSTRACT FROM AUTHOR]

Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces X and Y. We study the problem of determining the degree of approximation of such operators on a compact subset K X ⊂ X using a finite amount of information. If F : K X → K Y , a well established strategy to approximate F (F) for some F ∈ K X is to encode F (respectively, F (F)) in terms of a finite number d (respectively m) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of m functions on a compact subset of a high dimensional Euclidean space R d , equivalently, the unit sphere S d embedded in R d + 1. The problem is challenging because d , m , as well as the complexity of the approximation on S d are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on S d being O (d 1 / 6). We study different smoothness classes for the operators, and also propose a method for approximation of F (F) using only information in a small neighborhood of F , resulting in an effective reduction in the number of parameters involved. To further mitigate the problem of large number of parameters, we propose prefabricated networks, resulting in a substantially smaller number of effective parameters. The problem is studied in both deterministic and probabilistic settings. [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]

The fractional heat operator (∂ t − Δ x) s and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value representation formulas for functions with respect to (∂ t − Δ x) s and we introduce new nonlocal, nonlinear parabolic operators related to a tug-of-war which accounts for waiting times and space-time couplings. These nonlocal, nonlinear parabolic operators and equations can be seen as nonlocal versions of the evolutionary infinity Laplace operator. [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 The solution of a parabolic problem is expected to reproduce the basic qualitative properties of the original phenomenon, such as nonnegativity/nonpositivity preservation, maximum/minimum principles and maximum norm contractivity, without which the model might lead to unrealistic quantities in conflict with physical reality. This paper presents characterizations of qualitative properties for a general class of nonlinear parabolic systems. This is a continuation of the authors' research, published in a previous paper in this journal on scalar equations. Cooperativity of the systems plays an essential role in most of the results. [ABSTRACT FROM AUTHOR]

This paper presents an integrated framework to construct local-energy solutions to fairly general nonlinear diffusion equations for initial data growing at infinity under suitable assumptions on local-energy estimates for approximate solutions. A delicate issue for constructing local-energy solutions resides in the identification of weak limits of nonlinear terms for approximate solutions in a limiting procedure. Indeed, such an identification process often needs the maximal monotonicity of nonlinear elliptic operators (involved in the doubly-nonlinear equations) as well as uniform estimates for approximate solutions; however, even the monotonicity is violated due to a localization of the equations, which is also necessary to derive local-energy estimates for approximate solutions. In the present paper, such an inconsistency is systematically overcome by reducing the original equation to a localized one, where a (no longer monotone) localized elliptic operator is decomposed into the sum of a maximal monotone operator and a perturbation, and by integrating all the other relevant processes. Furthermore, the general framework developed in the present paper is also applied to the Finsler porous medium and fast diffusion equations , which are variants of the classical PME and FDE and also classified as a doubly-nonlinear equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel projected steepest descent iterative method with frozen derivative. The classical projected steepest descent iterative method involves the computation of derivative of the nonlinear operator at each iterate. The method of this paper requires the computation of derivative of the nonlinear operator only at an initial point. We exhibit the convergence analysis of our method by assuming the conditional stability of the inverse problem on a convex and compact set. Further, by assuming the conditional stability on a nested family of convex and compact subsets, we develop a multi-level method. In order to enhance the accuracy of approximation between neighboring levels, we couple it with the growth of stability constants. This along with a suitable discrepancy criterion ensures that the algorithm proceeds from level to level and terminates within finite steps. Finally, we discuss an inverse problem on which our methods are applicable. [ABSTRACT FROM AUTHOR]

In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order X D s − 1 , q (Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs. [ABSTRACT FROM AUTHOR]

We establish theorems on the existence and compactness of solutions to the σ 2 -Nirenberg problem on the standard sphere S 2. A first significant ingredient, a Liouville type theorem for the associated fully nonlinear Möbius invariant elliptic equations, was established in an earlier paper of ours. Our proof of the existence and compactness results requires a number of additional crucial ingredients which we prove in this paper: A Liouville type theorem for the associated fully nonlinear Möbius invariant degenerate elliptic equations, a priori estimates of first and second order derivatives of solutions to the σ 2 -Nirenberg problem, and a Bôcher type theorem for the associated fully nonlinear Möbius invariant elliptic equations. Given these results, we are able to complete a fine analysis of a sequence of blow-up solutions to the σ 2 -Nirenberg problem. In particular, we prove that there can be at most one blow-up point for such a blow-up sequence of solutions. This, together with a Kazdan-Warner type identity, allows us to prove L ∞ a priori estimates for solutions of the σ 2 -Nirenberg problem under some simple generic hypothesis. The higher derivative estimates then follow from classical estimates of Nirenberg and Schauder. In turn, the existence of solutions to the σ 2 -Nirenberg problem is obtained by an application of the by now standard degree theory for second order fully nonlinear elliptic operators. [ABSTRACT FROM AUTHOR]

In this paper we study quasilinear elliptic equations driven by the so-called double phase operator and with a nonlinear boundary condition. Due to the lack of regularity, we prove the existence of multiple solutions by applying the Nehari manifold method along with truncation and comparison techniques and critical point theory. In addition, we can also determine the sign of the solutions (one positive, one negative, one nodal). Moreover, as a result of independent interest, we prove for a general class of such problems the boundedness of weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonexistence and existence of radial large positive solutions for a class of Schrödinger systems with a nonlinear random operator under certain growth conditions. We also establish the necessary and sufficient conditions for the existence of radial large positive solutions by a successive iteration technique. The results of the paper improve and extend many previous work in this field of research. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis-Stokes system with nonlinear diffusion and rotation: n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ (n S (x , n , c) ⋅ ∇ c) , c t + u ⋅ ∇ c = Δ c − n c , u t + ∇ P = Δ u + n ∇ ϕ + f (x , t) and ∇ ⋅ u = 0 , in a bounded domain Ω ⊂ R 3 , where m > 0 , and ϕ : Ω ¯ → R , f : Ω ¯ × [ 0 , ∞) → R 3 and S : Ω ¯ × [ 0 , ∞) 2 → R 3 × 3 are given sufficiently smooth functions such that f is bounded in Ω × (0 , ∞) and S satisfies | S (x , n , c) | ≤ S 0 (c) (1 + n) − α for all (x , n , c) ∈ Ω ¯ × [ 0 , ∞) 2 with α > 0 and some nondecreasing function S 0 : [ 0 , ∞) → [ 0 , ∞). It is shown that if m + α > 10 9 and m + 5 4 α > 9 8 , then for any reasonably smooth initial data, the corresponding Neumann-Neumann-Dirichlet initial-boundary problem possesses a globally bounded weak solution. This extends the previous global boundedness result for m > 9 8 and α = 0 [43] , and improves that for m ≥ 1 and m + α > 7 6 [34] , or for m + α > 7 6 in the associated fluid-free system [31]. Our proof consists at its core in using, inter alia , the maximal Sobolev regularity theory to elaborately derive some spatio-temporal estimates for the signal and the fluid equations so as to decouple the system. [ABSTRACT FROM AUTHOR]

• We propose a 5G NR- multiband for Radio over Fiber link outfitted with an optimized magnitude selective affine (OMSA) method to cover enhanced mobile broadband (eMBB) situations for 2.14 GHz and 10 GHz, correspondingly. • We show experimentally that OMSA further reduces the complexity with better or at least similar performance to that of MSA by using a weighted function as a nonlinear operator to accurately capture the nonlinear behaviour of the system being modelled, which can lead to a reduction in the number of segments and improvement in the accuracy of the model. This means that it will also lead to a reduction in the number of coefficients needed. This helps to make the model more efficient and simpler to use. • The experimental results demonstrate that the OMSA model has lower complexity than the MSA model while achieving a slightly better performance that is assessed in terms of computations, error vector magnitude (EVM) or adjacent channel power ratio (ACPR). The application of analog radio over fiber (A-RoF) systems for 5G new radio (NR) multiband waveforms presents challenges, including nonlinearity introduced by the Mach Zehnder Modulator and the dispersion and attenuation of the optical signal over long fiber lengths. To address these challenges, this paper proposes a new version of the magnitude-selective affine (MSA) model, named the optimized magnitude-selective affine (OMSA) model, which incorporates a power-reliant weighting function to improve performance while reducing complexity via Digital Predistortion (DPD). The OMSA model is tested using 5G NR signals at 10 GHz with 50 MHz and flexible-waveform signals at 2.14 GHz with a 20 MHz bandwidth, transmitted over a 10 km fiber length using a Mach Zehnder Modulator and a 1550 nm optical carrier. The performance of the OMSA model is compared to the MSA and generalized memory polynomial (GMP) models in terms of adjacent channel power ratio, error vector magnitude, and complexity. The results show that the OMSA model outperforms the MSA and GMP models in terms of performance reducing the error vector magnitude to 1.9% as compared to 4% and 3% in case of GMP and MSA respectively. Additionally, in terms of complexity measured via coefficients, OMSA is 168 times more efficient as compared to GMP and offers 1.93 times higher efficiency as compared to MSA while maintaining lower complexity. The experimental demonstration of 5G NR multiband waveforms over A-RoF links using the OMSA model represents a significant step towards the practical implementation of high-performance, low-complexity 5G NR systems over fiber-optic links and provides a promising solution to the challenges associated with DPD linearization for A-RoF systems. [ABSTRACT FROM AUTHOR]

Abstract The paper concerns existence and uniqueness of solutions to a nonlinear parabolic equation with merely integrable data on a Lipschitz bounded domain in R N. Our focal point is to involve the leading, nonlinear part of the operator whose growth is described by anisotropic N -function M inhomogeneous in the space and the time variables. The main goals are proven in absence of Lavrentiev's phenomenon, to ensure which we impose a certain type of balance of interplay between the behavior of M for large | ξ | and small changes of time and space variables. Its instances are log-Hölder continuity of variable exponent (inhomogeneous in time and space) or optimal closeness condition for powers in double phase spaces (possibly changing in time). New delicate approximation-in-time result is proven and applied in the construction of renormalized solutions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a priori C 2 , α estimate up to the boundary for F (D 2 u) = 0 and the regularity of the free boundary of the obstacle problem for fully nonlinear operator under specific conditions for the operator and level sets of the operator. The conditions are variations of conditions for the zero set of the operator in [7]. [ABSTRACT FROM AUTHOR]

This paper deals with improvements of the Trudinger–Moser inequality related to the operator Q V ( u ) : = − Δ n u + V ( x ) | u | n − 2 u , where n ≥ 2 and the potential V : R n → R belongs to a class of nonnegative and continuous functions. Precisely, under suitable assumptions on V we consider the subspace E : = { u ∈ W 1 , n ( R n ) : ∫ R n V ( x ) | u | n d x < ∞ } endowed with the norm ‖ u ‖ : = [ ∫ R n ( | ∇ u | n + V ( x ) | u | n ) d x ] 1 / n and we prove that if ( u k ) is a sequence in E such that ‖ u k ‖ = 1 , u k ⇀ u ≢ 0 in E and 0 < p < p n ( u ) : = β n ( 1 − ‖ u ‖ n ) − 1 / ( n − 1 ) , then ( ⁎ ) sup k ⁡ ∫ R n Ψ ( p | u k | n / ( n − 1 ) ) d x < ∞ , where Ψ ( t ) : = e t − ∑ i = 0 n − 2 t i i ! , β n : = n ω n − 1 1 / ( n − 1 ) and ω n − 1 is the measure of the unit sphere in R n . Furthermore, p n ( u ) is sharp in the sense that there exists a sequence ( u k ) ⊂ E satisfying ‖ u k ‖ = 1 and u k ⇀ u ≢ 0 in E such that the supremum (⁎) is infinite for p ≥ p n ( u ) . As an application of the previous result we prove the following sharp form of the Trudinger–Moser inequality for the subspace E . Considering ℓ ( α ) : = sup { u ∈ E : ‖ u ‖ = 1 } ⁡ ∫ R n Ψ ∘ ν α ( u ) d x , where ν α ( u ) : = β n ( 1 + α ‖ u ‖ n n ) 1 / ( n − 1 ) | u | n / ( n − 1 ) , assuming some conditions of symmetry on V it is established (1) for 0 ≤ α < λ 1 ( V ) we have ℓ ( α ) < ∞ , (2) for α ≥ λ 1 ( V ) , ℓ ( α ) = ∞ and (3) moreover, we prove that for 0 ≤ α < λ 1 ( V ) , an extremal function for ℓ ( α ) exists. Here λ 1 ( V ) denotes the first eigenvalue of Q V ( u ) . [ABSTRACT FROM AUTHOR]

Abstract: The present paper is concerned with the convergence problem of inexact Newton methods. Assuming that the nonlinear operator satisfies the γ-condition, a convergence criterion for inexact Newton methods is established which includes Smale''s type convergence criterion. The concept of an approximate zero for inexact Newton methods is proposed in this paper and the criterion for judging an initial point being an approximate zero is established. Consequently, Smale''s α-theory is generalized to inexact Newton methods. Furthermore, a numerical example is presented to illustrate the applicability of our main results. [Copyright &y& Elsevier]

Let S = U S | S | and T = U T | T | be the polar decompositions of S and T in L ( H ) , respectively. We say that S and T have similar positive parts if | S | and | T | are similar. In this paper, we investigate properties of operators that are preserved under this similarity condition. We describe the form for the polar decomposition of an operator when the operator and its adjoint have similar positive parts. For this case, we also study the existence of invariant subspaces under this assumption. [ABSTRACT FROM AUTHOR]

In this paper we consider the system involving fully nonlinear nonlocal operators: { F α ( u ( x ) ) = C n , α P V ∫ R n G ( u ( x ) − u ( y ) ) | x − y | n + α d y = f ( v ( x ) ) , F β ( v ( x ) ) = C n , β P V ∫ R n G ( v ( x ) − v ( y ) ) | x − y | n + β d y = g ( u ( x ) ) . For carrying on the method of moving planes, a narrow region principle and a decay at infinity are established. Then we prove the radial symmetry and monotonicity for positive solutions to the nonlinear system in the whole space. Furthermore, non-existence of positive solutions to the nonlinear system on a half space is derived. [ABSTRACT FROM AUTHOR]

It is a natural expectation that the mathematical models of real-life phenomena have to possess some characteristic qualitative properties of the original process. For parabolic problems the main known qualitative properties are the maximum–minimum principles, nonnegativity–nonpositivity preservation and maximum norm contractivity. These properties have a fundamental relevance concerning the validity of the mathematical or numerical model: without them, the model might produce unphysical quantities that contradict reality. For linear problems with Dirichlet boundary conditions, these properties have been thoroughly investigated and their relations have been characterized. In the present paper, we extend the linear results to nonlinear problems with general boundary conditions. Firstly, we characterize various implications between the qualitative properties. Some of them are given in general, and in certain cases we restrict our study to operators with gradient-dependent principal part or to operators with heat conduction coefficient. Secondly, we give general sufficient conditions to ensure these qualitative properties, both separately and all of them together. The relations are illustrated with several examples. [ABSTRACT FROM AUTHOR]

In this paper, we develop a systematical approach in applying the method of moving planes to study qualitative properties of solutions for nonlocal parabolic equations. We first establish a series of key ingredients in the proofs, such as maximum principles for antisymmetric functions, narrow region principles, maximum principles near infinity, and the Hopf lemma, then we demonstrate how these new tools can be employed to derive the radial symmetry and monotonicity of ancient positive solutions to fractional parabolic equations in the whole space. Our results are in particular applicable to entire solutions, by which we mean solutions defined for all t ∈ R. We believe that the new ideas and methods introduced here can be adapted to study many other nonlocal parabolic problems with more general operators and nonlinear terms. [ABSTRACT FROM AUTHOR]

• Derived a new class of nonlinearly stable flux reconstruction schemes in split form. • Nonlinearly stable flux reconstruction correction functions must be applied on the nonlinear volume terms. • Flux reconstruction in split form with the correction functions only reconstructing the surface is divergent. • The split form must be embedded within the discrete stiffness operator for nonlinear stability. The flux reconstruction (FR) method has gained popularity in the research community as it recovers promising high-order methods through modally filtered correction fields, such as the discontinuous Galerkin method, amongst others, on unstructured grids over complex geometries. Moreover, FR schemes, specifically energy stable FR (ESFR) schemes also known as Vincent-Castonguay-Jameson-Huynh schemes, have proven attractive as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms have recently seen a resurgence in research activity due to their resultant nonlinear (entropy) stability proofs. This paper derives for the first time nonlinearly stable ESFR schemes in split form that enable nonlinear stability proofs for, uncollocated, modal, ESFR split forms with different volume and surface cubature nodes. The critical enabling technology is applying the splitting to the discrete stiffness operator. This naturally leads to appropriate surface and numerical fluxes, enabling both entropy stability and conservation proofs. When these schemes are recast in strong form, they differ from schemes found in the ESFR literature as the ESFR correction functions are incorporated on the volume integral. Furthermore, numerical experiments are conducted for Burgers' equation verifying that the new class of proposed ESFR split forms is nonlinearly stable in contrast to the standard split form ESFR approach. Lastly, the new ESFR split form is shown to obtain the correct orders of accuracy. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of analysis and optimisation of the inerter-based isolators based on a “uni-axial” single-degree-of-freedom isolation system. In the first part, in order to gain an in-depth understanding of inerter from the prospective of vibration, the frequency responses of both parallel-connected and series-connected inerters are analysed. In the second part, three other inerter-based isolators are introduced and the tuning procedures in both the H ∞ optimisation and the H 2 optimisation are proposed in an analytical manner. The achieved H 2 and H ∞ performance of the inerter-based isolators is superior to that achieved by the traditional dynamic vibration absorber (DVA) when the same inertance-to-mass (or mass) ratio is considered. Moreover, the inerter-based isolators have two unique properties, which are more attractive than the traditional DVA: first, the inertance-to-mass ratio of the inerter-based isolators can easily be larger than the mass ratio of the traditional DVA without increasing the physical mass of the whole system; second, there is no need to mount an additional mass on the object to be isolated. [ABSTRACT FROM AUTHOR]

A remarkable result by S. Artstein-Avidan and V. Milman states that, up to pre-composition with affine operators, addition of affine functionals, and multiplication by positive scalars, the only fully order preserving mapping acting on the class of lower semicontinuous proper convex functions defined on R n is the identity operator, and the only fully order reversing one acting on the same set is the Fenchel conjugation. Here fully order preserving (reversing) mappings are understood to be those which preserve (reverse) the pointwise order among convex functions, are invertible, and such that their inverses also preserve (reverse) such order. In this paper we establish a suitable extension of these results to order preserving and order reversing operators acting on the class of lower semicontinuous proper convex functions defined on arbitrary infinite dimensional Banach spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we make it mathematically rigorous the formulation of the following quantum Schrödinger–Langevin nonlinear operator for the wavefunction in bounded domains via its mild interpretation. The a priori ambiguity caused by the presence of the multi-valued potential , proportional to the argument of the complex-valued wavefunction is circumvented by subtracting its positional expectation value, as motivated in the original derivation (Kostin, 1972 ). The problem to be solved in order to find is mostly deduced from the modulus-argument decomposition of ψ and dealt with much like in Guerrero et al. (2010) . Here ℏ is the (reduced) Planck constant, m is the particle mass, λ is a friction coefficient, is the local probability density, denotes the electric current density, and is a general operator (eventually nonlinear) that only depends upon the macroscopic observables and . In this framework, we show local well-posedness of the initial-boundary value problem associated with the Schrödinger–Langevin operator in bounded domains. In particular, all of our results apply to the analysis of the well-known Kostin equation derived in Kostin (1972) and of the Schrödinger–Langevin equation with Poisson coupling and enthalpy dependence (Jüngel et al., 2002 ). [Copyright &y& Elsevier]

Abstract: In this paper we study one-phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials, with . We establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the ε-parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transition, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, , we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed. [Copyright &y& Elsevier]

Abstract: In the present paper we study nonlinear dynamics of quantum quadratic operators (q.q.o.) acting on the algebra of matrices . First, we describe q.q.o. with Haar state as well as quadratic operators with the Kadison–Schwarz property. By means of such a description we provide an example of q.q.o. which is not the Kadison–Schwarz operator. Then we study stability of dynamics of q.q.o. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the existence of positive solutions for a second-order differential system by using the fixed point theorem of cone expansion and compression. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we are interested in the existence of a principal eigenfunction of a nonlocal operator which appears in the description of various phenomena ranging from population dynamics to micro-magnetism. More precisely, we study the following eigenvalue problem: where is an open connected set, J a non-negative kernel and g a positive function. First, we establish a criterion for the existence of a principal eigenpair . We also explore the relation between the sign of the largest element of the spectrum with a strong maximum property satisfied by the operator. As an application of these results we construct and characterise the solutions of some nonlinear nonlocal reaction diffusion equations. [Copyright &y& Elsevier]

Abstract: For singular elliptic fully-nonlinear operators we prove that the eigenfunctions corresponding to the principal eigenvalues in bounded domains are simple. The proof uses in particular a regularity result obtained in the first part of the paper which is of independent interest. [Copyright &y& Elsevier]

Abstract: A version of the Dynamical Systems Method (DSM) for solving ill-posed nonlinear equations with monotone operators F in a Hilbert space is studied in this paper under less restrictive assumptions on the nonlinear operators F than the assumptions used earlier. A new method of proof of the basic results is used. An a posteriori stopping rule, based on a discrepancy-type principle, is proposed and justified mathematically under weaker assumptions on the nonlinear operator F, than the assumptions used earlier. [Copyright &y& Elsevier]

Abstract: This paper considers existence of multiple positive fixed points for some nonlinear operators, a particular case of the operators is sum of an e-concave operator and an e-convex operator. Then we apply the results to nonlinear integral equations. [Copyright &y& Elsevier]

Abstract: We study the fully nonlinear elliptic equation in a smooth bounded domain Ω, under the assumption that the nonlinearity F is uniformly elliptic and positively homogeneous. Recently, it has been shown that such operators have two principal “half” eigenvalues, and that the corresponding Dirichlet problem possesses solutions, if both of the principal eigenvalues are positive. In this paper, we prove the existence of solutions of the Dirichlet problem if both principal eigenvalues are negative, provided the “second” eigenvalue is positive, and generalize the anti-maximum principle of Clément and Peletier [P. Clément, L.A. Peletier, An anti-maximum principle for second-order elliptic operators, J. Differential Equations 34 (2) (1979) 218–229] to homogeneous, fully nonlinear operators. [Copyright &y& Elsevier]

In this paper, we consider the following problem { (− Δ) p s u (x) = f (u) , u (x) > 0 in Ω , u (x) = 0 on R n ∖ Ω , where (− Δ) p s is the fractional p -Laplacian with p ≥ 2 and Ω : = { x = (x ′ , x n) | x n > φ (x ′) } is an unbounded domain. In the case φ ≡ 0 , it reduces to the upper half space. Without assuming any asymptotic behavior of u near infinity, we first develop narrow region principles in unbounded domains, then using the method of moving planes, we establish the monotonicity of the positive solutions. In most previous literature, to apply the method of moving planes on unbounded domains, one usually needed to assume that the solution tends to zero in certain rate near infinity; or make a Kelvin transform, or divide the solution by a function, so that the new solution possesses such an asymptotic decay. Here we introduce a new idea, estimating the singular integral defining (− Δ) p s along a sequence of auxiliary functions at their maximum points. This way, we only require the solutions be bounded. We believe that this new method will become a useful tool in investigating qualitative properties of solutions for equations involving nonlinear nonlocal operators. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we provide a formulation of initial value problems for (explicit and implicit) difference equations in terms of abstract equations in sequence spaces. They will be solved using appropriate fixed point theorems and we obtain quantitative attractivity properties. [Copyright &y& Elsevier]

Abstract: In this paper, the competitor–competitor–mutualist three-species Lotka–Volterra model is discussed. Firstly, by Schauder fixed point theory, the coexistence state of the strongly coupled system is given. Applying the method of upper and lower solutions and its associated monotone iterations, the true solutions are constructed. Our results show that this system possesses at least one coexistence state if cross-diffusions and cross-reactions are weak. Secondly, the existence and asymptotic behavior of T-periodic solutions for the periodic reaction–diffusion system under homogeneous Dirichlet boundary conditions are investigated. Sufficient conditions which guarantee the existence of T-periodic solution are also obtained. [Copyright &y& Elsevier]

Abstract: Enormous progress has been made in the construction and analysis of adaptive wavelet methods in the recent years. Cohen, Dahmen, and DeVore showed that such methods converge for a wide class of operator equations, both linear and nonlinear. Moreover, they showed that the rate of convergence is asymptotically optimal and that the methods are asymptotically optimally efficient. So far, these methods are based upon biorthogonal wavelets with compactly supported primal and dual functions. Semiorthogonal spline wavelets offer some quantitative advantages, namely small supports and good conditioning of the bases. On the other hand, the corresponding dual functions are globally supported so that they are ruled out for existing wavelet methods for nonlinear variational problems. In this paper, we focus on a core ingredient of adaptive wavelet methods for nonlinear problems, namely the adaptive evaluation of nonlinear functions. We present an efficient adaptive method for approximately evaluating nonlinear functions of wavelet expansions using semiorthogonal spline wavelets. This is achieved by modifying and extending a method for compactly supported biorthogonal wavelets by Dahmen, Schneider, and Xu. In order to do so, we introduce a new adaptive quasi-interpolation scheme, a corresponding prediction and a new decomposition. We give a complete analysis including an investigation of the complexity. [Copyright &y& Elsevier]

Abstract: In this paper, by using the fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result of bounded positive solutions of semilinear elliptic equations in exterior domain of , . [Copyright &y& Elsevier]

Abstract: In this paper, by using the fixed point theory, under quite general condition on the nonlinear term, we obtain an existence result concerning bounded continuous nonoscillatory solutions of a second-order nonlinear difference equation with continuous variable. [Copyright &y& Elsevier]

Abstract: In this paper, by using the fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result of bounded positive solutions of Schrödinger equations in two-dimensional exterior domains. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of positive solutions for the nonlinear four-point singular boundary value problem for higher-order with p-Laplacian operator. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear singular boundary value problem with p-Laplacian operator are obtained. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to introduce a generalized -concave condition, and by using Himmelberg''s fixed point theorem, to prove a new existence theorem of Nash equilibrium in non-compact generalized game with -concavity. As applications, we shall prove a minimax theorem in non-compact settings and prove a minimax inequality in compact settings. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a generalized iterative process with errors to approximate the common fixed points of two asymptotically quasi-nonexpansive mappings. A convergence theorem has been obtained which generalizes a known result. This theorem has then been used to prove another convergence theorem which, in turn, generalizes a number of results. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the density of extremal points appeared in Ekeland''s variational principle. By introducing radial intersections of sets, we give a very general result on the density of extremal points in the framework of locally convex spaces. This solves a problem proposed by G. Isac in 1997. From the general result we deduce several convenient criterions for judging the density of extremal points, which extend and improve a result of F. Cammaroto and A. Chinni. Using the equivalence between Ekeland''s variational principle and Caristi''s fixed point theorem, we obtain some density results on Caristi''s fixed points. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss Moudafi''s viscosity approximations with Meir–Keeler contractions. We first present very simple proofs of Xu''s theorems concerning Moudafi''s approximations. We next prove that Browder''s and Halpern''s type convergence theorems imply Moudafi''s viscosity approximations. Using them, we finally state several deduced theorems. [Copyright &y& Elsevier]

Abstract: In this paper we discuss some existence results and the application of quasilinearization methods, developed so far for differential equations, to the solution of the abstract problem in a Hilbert space . Under fairly general assumptions on , F and , we show that this problem has a solution that can be obtained as the limit of a quadratically convergent nondecreasing sequence of approximate solutions. If the assumptions are strengthened, we show that the abstract problem has a solution which is quadratically bracketed between two monotone sequences of approximate solutions of certain related linear equations. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the concept of τ-function which generalizes the concept of w-distance studied in the literature. We establish a generalized Ekeland''s variational principle in the setting of lower semicontinuous from above and τ-functions. As applications of our Ekeland''s variational principle, we derive generalized Caristi''s (common) fixed point theorems, a generalized Takahashi''s nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces. We also prove that these theorems also imply our Ekeland''s variational principle. [Copyright &y& Elsevier]

Abstract: In this paper, we proved a common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type and a property (E.A) introduced in [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181–188]. Our theorem generalizes Theorem 2.2 of [M. Aamri, D. El Moutawakil, Common fixed points under contractive conditions in symmetric spaces, Appl. Math. E-Notes 3 (2003) 156–162] and Theorem 2 of [M. Aamri, D. El Moutawakil, Some new common fixed point theorems under strict contractive conditions, J. Math. Anal. Appl. 270 (2002) 181–188]. [Copyright &y& Elsevier]

Abstract: In this paper, we establish sufficient conditions to guarantee the existence of at least one positive solution, a unique positive solution, and multiple positive solutions for the Sturm–Liouville boundary value problem on the half-line. By using an effective operator, the fixed point theorems in cone, especially Krasnoselskii fixed point theorem, can be applied to such systems and then existence criteria are established. The interesting point of the results is that the nonlinear term f can be sign-changing. [Copyright &y& Elsevier]