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In this paper, we study the tangential touch and C¹ regularity of the free boundary near the fixed boundary of the double obstacle problem for Laplacian and fully nonlinear operator. The main idea to have the properties is regarding the upper obstacle as a solution of the single obstacle problem. Then, in the classification of global solutions of the double problem, it is enough to consider only two cases for the upper obstacle, a/2 xn² or a/2 xn² + bxnx1 for some b ∈ ℝ, b ≠ 0. The second one is a new type of upper obstacle, which does not exist in the study of local regularity of the free boundary of the double problem. Thus, in this paper, a new type of difficulties that come from the second type upper obstacle is mainly studied. [ABSTRACT FROM AUTHOR]

In this paper, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini–Coulomb problem) is unraveled and sharp existence results are proved for the most general two-dimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray–Lions operators, so that a result of Brézis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray–Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the two-dimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini–Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient. [ABSTRACT FROM AUTHOR]

This paper deals with a discrete-time dynamical system generated by a modified susceptible–infected–recovered–dead model (SIRD model; nonlinear operator) in three-dimensional simplex. We introduce a novel approach that incorporates the SIRD model with the quadratic stochastic operator (QSO) that allows for real-time forecasting. The basic reproductive number R 0 is obtained. We describe the set of fixed points of the operator and demonstrate that all fixed points are non-hyperbolic. Further, we study the asymptotical behavior of the trajectories of this system and show that SIRD operators have a regularity property. [ABSTRACT FROM AUTHOR]

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

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

This paper investigates the necessary conditions relating to the existence and uniqueness of solution to impulsive system fractional differential equation with a nonlinear p-Laplacian operator. Our problem is based on two kinds of fractional order derivatives. That is, Atangana–Baleanu–Caputo (ABC) fractional derivative and the Caputo–Fabrizio derivative. To achieve our main aims, we will first convert the proposed impulse system into an integral equation form. Next, we prove the existence and uniqueness of solutions with the help of Leray–Schauder's theory and the Banach contraction principle. We analyze the operator for continuity, boundedness, and equicontinuity. Further, we investigate the stability solution to the proposed impulsive system by using stability techniques. In the last part, we demonstrate the results via an illustrative example for the application of the results. [ABSTRACT FROM AUTHOR]

In this paper, we study the horizontal Newton transformations, which are nonlinear operators related to the natural splitting of the second fundamental form for hypersurfaces in a complex space form. These operators allow to prove the classical Minkowski formulas in the case of real space forms: unlike the real case, the horizontal ones are not divergence-free. Here, we consider the highest order of nonlinearity and we will show how a Minkowski-type formula can be obtained in this case. [ABSTRACT FROM AUTHOR]

We present an operator calculus based on Krawtchouk polynomials, including Krawtchouk transforms and corresponding convolution structure which provides an inherently discrete alternative to Fourier analysis. This approach is well suited for applications such as digital image processing. This paper includes the theoretical aspects and some basic examples. [ABSTRACT FROM AUTHOR]

In this paper, we show how the global bifurcation theory for nonlinear Fredholm operators (Theorem 4.3 of [Shi & Wang, 2009]) and for compact operators (Theorem 1.3 of [Rabinowitz, 1971]) can be used in the study of the nonconstant stationary solutions for a volume-filling chemotaxis model with logistic growth under Neumann boundary conditions. Our results show that infinitely many local branches of nonconstant solutions bifurcate from the positive constant solution (u c , α β u c) at χ = χ ¯ k . Moreover, for each k ≥ 1 , we prove that each Γ k can be extended into a global curve, and the projection of the bifurcation curve Γ k onto the χ -axis contains ( χ ¯ k , ∞). [ABSTRACT FROM AUTHOR]

In this paper, we investigate some symmetries and Lie-group transformations of an integrable system by using the symmetry analysis method. It follows that the resulting similarity solutions are obtained by applying the characteristic equations of the symmetries. By applying the software Maple,we work out some exact solutions of the generalized KdV system, including the rational solutions, the periodic solutions, the dark soliton solutions, and so on. Finally, we make use of the self-adjoint operators to investigate the nonlinear self-adjointness and the conservation laws of the generalized KdV integrable system. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of local stable manifolds for a semi-linear differential equation, where the linear part is a Hille–Yosida operator on a Banach space and the nonlinear forcing term f satisfies the φ -Lipschitz conditions, where φ belongs to certain classes of admissible function spaces. The approach being used is the fixed point arguments and the characterization of the exponential dichotomy of evolution equations in admissible spaces of functions defined on the positive half-line. [ABSTRACT FROM AUTHOR]

In this paper, we establish global Sobolev a priori estimates for L p -viscosity solutions of fully nonlinear elliptic equations as follows: F (D 2 u , D u , u , x) = f (x) in  Ω u (x) = φ (x) on  ∂ Ω by considering minimal integrability condition on the data, i.e. f ∈ L p (Ω) , φ ∈ W 2 , p (Ω) for n < p < ∞ and a regular domain Ω ⊂ ℝ n , and relaxed structural assumptions (weaker than convexity) on the governing operator. Our approach makes use of techniques from geometric tangential analysis, which consists in transporting "fine" regularity estimates from a limiting operator, the Recession profile, associated to F to the original operator via compactness methods. We devote special attention to the borderline case, i.e. when f ∈ p − BMO ⊋ L ∞ . In such a scenery, we show that solutions admit BMO type estimates for their second derivatives. [ABSTRACT FROM AUTHOR]

This paper is concerned with a host-macroparasite difference model. By applying the contraction mapping fixed point theorem, we prove the existence of unique almost periodic positive solution. Moreover, we investigate the exponential stability of almost periodic solution by means of Lyapunov functional. [ABSTRACT FROM AUTHOR]

In the present paper we study the deformed harmonic oscillator for the non-Hermitian operator $${\cal H}=\frac{\alpha}{2m}\left(\hat{p}_{1}+\frac{\lambda}{\hbar}\hat q_{2}\right)^{2}+\frac{\beta m\omega^{2}}{2}\left(\hat q_{1}-\frac{\theta}{2\hbar}\hat p_{2}\right)^{2}$$ where λ,θ are real positive parameters, since the parameters α,β,m are for the general case complex. For the case α=1,β=1 and mass m real, we find the eigenfunctions and eigenvalues of energy, the coherent states, the time evolution of the operators $\hat{q}_i$, $\hat{p}_j$ in the Heisenberg picture and the uncertainty relations. In this case the operator ℋ is Hermitian and PT-symmetric. Also for the case m complex α=1,β=1, the operator ℋ is non-Hermitian and no more PT symmetric, but CPT symmetric with real discrete positive spectrum and the CPT symmetry is preserved. In the general case α,β,m complex, for the non-Hermitian operator ℋ, we obtain complex spectrum and for the special values of the complex parameters α,β the spectrum is real discrete and positive and the CPT symmetry is preserved. The general problem of deformed oscillator for non hermitian operators can be applied to the Solid State Physics. [ABSTRACT FROM AUTHOR]

In this paper, we study four-point boundary value problems for a fourth-order ordinary differential equation of the form [formula] with one of the following boundary conditions: [formula] or [formula] Growth conditions on f which guarantee existence of at least three positive solutions for the problems (E)–(B[sub 1]) and (E)–(B[sub 2]) are imposed. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate the dynamics of a non-autonomous density-dependent predator-prey system with Beddington-DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered, such that the studied predator-prey system conforms to the realistically biological environment. We firstly introduce a sufficient condition for the permanence of the system and then use a specific set to obtain a weaker sufficient condition. Afterward, we provide corresponding conditions for the extinction of the system and the existence of boundary periodical solutions, respectively. Further, we get a sufficient condition for global attractiveness of the boundary periodic solution by constructing a Lyapunov function, arriving at the uniqueness of boundary periodic solutions since the uniqueness of boundary periodic solutions can be ensured by global attractiveness. Finally, based on the existence of positive periodic solutions, which can be ensured by the Brouwer fixed-point theorem, we provide a sufficient condition for the uniqueness of positive periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the qualitative analysis of a parabolic-elliptic attraction-repulsion chemotaxis model with logistic source. Applying a fixed point argument, -estimate technique and Moser's iteration, we derive that the model admits a unique global solution provided the initial cell mass satisfying for While for , there are no restrictions on the initial cell mass and the result still holds. [ABSTRACT FROM AUTHOR]

This paper establishes an axiomatic definition of inclusion measures between Atanassov's intuitionistic fuzzy (A-IF for short) sets. Some kinds of A-IF inclusion measures are constructed by different A-IF operators especially by A-IF implicator, and some new methods for measuring the degree of similarity between A-IF sets are proposed. Moreover, the similarity measure obtained from an A-IF inclusion measure satisfies properties of normal similarity measure. We then define a compatibility measure by a predicates logical idea and construct several functions to measure compatibility for an intuitionistic t-norm. [ABSTRACT FROM AUTHOR]

In this work, we build a new numerical method to approximate the solution of Volterra's nonlinear integro-differential equation. This method needs fewer conditions to converge, compared to the direct Nytröm method. Numerical tests show its efficiency. This new method is more practical and compatible with the Volterra nonlinear integro-differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we find large classes of positive functions, others than those in [1], having even a Jackson-type estimate, ω1(f;1/n), in approximation by the nonlinear max-product Bernstein operator. The uniform estimate of the order O[nω1(f;1/n)2 + ω1(f;1/n)] is achieved, while near to the endpoints 0 and 1, the better pointwise estimate of the order $\omega_{1}(f, \sqrt{x(1-x)/n})$ is obtained. Finally, we prove that besides the preservation of quasi-convexity found in [1], the nonlinear max-product Bernstein operator preserves the quasi-concavity too. [ABSTRACT FROM AUTHOR]

The main scope of this paper is to obtain Aleksandrov–Bakelman–Pucci estimates (ABP estimates) for viscosity solutions of singular fully nonlinear operator, which includes the p-Laplacian operator, p > 1. [ABSTRACT FROM AUTHOR]

Let A be a semiprime associative algebra with an involution ∗ over a field of characteristic not 2, let KA be the Lie algebra of all skew elements of A, and let ZKA denote the annihilator of KA. The aim of this paper is to prove that if Q is a ∗-subalgebra of Qs(A) (the Martindale symmetric algebra of quotients of A) containing A, then KQ/ZKQ is a Lie algebra of quotients of KA/ZKA. Similarly, [KQ, KQ]/Z[KQ,KQ] is a Lie algebra of quotients of [KA,KA]/Z[KA,KA]. [ABSTRACT FROM AUTHOR]

In this paper, a predictive control method is suggested for chaos control in continuous-time systems. This method combines the delayed feedback control of high order continuous-time chaotic systems with the prediction-based method of discrete-time chaotic systems. Moreover, we give necessary and sufficient conditions for exponential stabilization of unstable fixed points by the proposed method. Both control performance and system sensitivity to initial conditions of this approach are demonstrated by numerical simulations. [ABSTRACT FROM AUTHOR]

One purpose of this paper is to document the fact that, in dynamical systems described by ordinary differential equations, the trajectories can be organized not only around fixed points (steady states), but also around lines. In 2D, these lines are the nullclines themselves, in 3D, the intersections of the nullclines two by two, etc. We precise the concepts of "partial steady states" (i.e. steady states in a subsystem that consists of sections of phase space by planes normal to one of the axes) and of "partial multistationarity" (multistationarity in such a subsystem). Steady states, nullclines or their intersections are revisited in terms of circuits, defined from nonzero elements of the Jacobian matrix. It is shown how the mere examination of the Jacobian matrix and the sign patterns of its circuits can help interpreting (and often predicting) aspects of the dynamics of systems. The results reinforce the idea that chaotic dynamics requires both a positive circuit, to provide (if only partial) multistationarity, and a negative circuit, to provide sustained oscillations. As shown elsewhere, a single circuit may suffice if it is ambiguous (i.e. positive or negative depending on the location in phase space). The description in terms of circuits is by no means exclusive of the classical description. In many cases, a fruitful approach involves repeated feedback between the two viewpoints. [ABSTRACT FROM AUTHOR]

In the paper, we shall prove the results on the existence of fixed points of mapping defined on generalized metric space satisfying a nonlinear contraction condition, which is a generalization of Diaz and Margolis theorem (see [A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968) 305-309]). We also present local fixed point theorems both in generalized and ordinary metric spaces. Our results are generalizations of Banach fixed point theorem and many other results. [ABSTRACT FROM AUTHOR]

We obtain a Liouville-type theorem for cylindrical viscosity solutions of fully nonlinear CR invariant equations on the Heisenberg group. As a by-product, we also prove a comparison principle with finite singularities for viscosity solutions to more general fully nonlinear operators on the Heisenberg group. [ABSTRACT FROM AUTHOR]

We follow the guiding line offered by canonical operators on the full Fock space, in order to identify what kind of cumulant functionals should be considered for the concept of bi-free independence introduced in the recent work of Voiculescu. By following this guiding line we arrive to consider, for a general noncommutative probability space (풜, φ), a family of "(ℓ, r )-cumulant functionals" which enlarges the family of free cumulant functionals of the space. In the motivating case of canonical operators on the full Fock space we find a simple formula for a relevant family of (ℓ, r )-cumulants of a (2d)-tuple (A1,...,Ad, B1,...,Bd), with A1,...,Ad canonical operators on the left and B1,...,Bd canonical operators on the right. This extends a known one-sided formula for free cumulants of A1,...,Ad, which establishes a basic operator model for the R-transform of free probability. [ABSTRACT FROM AUTHOR]

We investigate the appropriate and sufficient conditions for the existence and uniqueness of a solution for a coupled system of Atangana–Baleanu fractional equations with a p-Laplacian operator. We also study the HU-stability of the solution by using the Atangana–Baleanu–Caputo (ABC) derivative. To achieve these goals, we convert the coupled system of Atangana–Baleanu fractional equations into an integral equation form with the help of Green functions. The existence of the solution is proven by using topological degree theory and Banach's fixed point theorem, with which we analyze the solution's continuity, equicontinuity and boundedness. Then, we use Arzela–Ascolli theory to ensure that the solution is completely continuous. Uniqueness is established using the Banach contraction principle. We also investigate several adequate conditions for HU-stability and generalized HU-stability of the solution. An illustrative example is presented to verify our results. [ABSTRACT FROM AUTHOR]

In this paper we prove that for every n ≥ 3, there exists a permutation 3-group Pn such that Pn has five orbits, each of size at least 32n-1, and Pn has no fixed-point-free element. In particular, this gives, for the prime 3, an answer to a conjecture in [2] and [4]. [ABSTRACT FROM AUTHOR]

This work investigates maps T of the plane with points p ∈ ℝ2 where T is undefined. Through the study of certain families with a singularity at the origin two very different dynamics will be illustrated. The first example will show that a singularity has no apparent effect on forward mapping. In this example there exists also an attractive fixed point. The second example will illustrate a singularity that exhibits attracting fixed point-like behavior. [ABSTRACT FROM AUTHOR]

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood-Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the K-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on Td, Rd, [-1,1], nonlinear wavelet approximation, etc. [ABSTRACT FROM AUTHOR]

We consider the Leslie's prey–predator model with discrete time. This model is given by a nonlinear evolution operator depending on five parameters. We show that this operator has two fixed points and define type of each fixed point depending on the parameters. Finding two invariant sets of the evolution operator, we study the dynamical systems generated by the operator on each invariant set. Depending on the parameters, we classify the dynamics between a predator and a prey of the Leslie's model. [ABSTRACT FROM AUTHOR]

In the last decades, wavelets have become a dominant tool for having applications in almost all the areas of engineering and science such as numerical simulation of partial differential equations (PDEs). The performance of the conventional numerical methods has been found to involve some difficulty to observe fast convergence in low computational time. To overcome this difficulty, we presented wavelet-based full approximation scheme (WFAS) for the numerical solution of some class of nonlinear PDEs using Daubechies wavelet intergrid operators. The numerical results obtained by this scheme are compared with the exact solution to reveal the accuracy and also speed up convergence in lesser computational time as compared with the existing schemes. Some test problems are presented to show the applicability and attractiveness of WFAS. [ABSTRACT FROM AUTHOR]

We focus on the structure of the solution set for the nonlinear equation u = L (λ) u + H (λ , u) , where L (⋅) and H (λ , u) are continuous operators. Under certain hypotheses on L (⋅) and H (⋅ , ⋅) , unilateral global bifurcations for eigenvalue problems are presented. Some applications are illustrated for nonlinear ordinary and partial differential equations. In particular, the existence and multiplicity of one-sign solutions for Monge–Ampère equation is discussed. [ABSTRACT FROM AUTHOR]

We prove interior Hessian estimates in the setting of weighted Orlicz spaces for viscosity solutions of fully nonlinear, uniformly elliptic equations F (D 2 u , x) = f (x) in  B 1 under asymptotic assumptions on the nonlinear operator F. The results are further extended to fully nonlinear, asymptotically elliptic equations. [ABSTRACT FROM AUTHOR]

Boundary value problems of second-order singular differential equations with nonlinear operator Φ on whole lines are discussed. By applying the nonlinear alternative of Leray-Schauder-type fixed point theorem, some existence results of solutions for integral boundary value problems of differential equations on whole lines are established. The emphasis is put on the nonlinear operator [Φ(ρ(t)x′(t))]′ involved with the nonnegative function ρ that may satisfy ρ(0) = 0, the strictly increasing sup-multiplicative-like function Φ and the differential equations are defined on the whole line. Three examples and a remark are presented to illustrate the main theorems. [ABSTRACT FROM AUTHOR]

We study the attractor flow and near horizon geometry of two-charge small black holes in heterotic string theory. The Hessian of Sen's entropy function with respect to the moduli fields has standard attractor properties and shows the interesting factorization at the attractor fixed points. We notice that the stability conditions are preserved under arbitrary α′-corrections to the black hole solutions. [ABSTRACT FROM AUTHOR]

For a large class of noncoercive operator inclusions, including those generated by maps of Sk type, we obtain a general theorem on the existence of solutions. We apply this result to a particular example. This theorem is proved using the method of Faedo-Galerkin approximations. [ABSTRACT FROM AUTHOR]

Having performed discrete stationary wavelet transform (DSWT) to an infrared image and employing generalized cross validation (GCV), noise is reduced in high frequency sub-bands. These sub-bands are at finer resolution levels and the local contrast is enhanced by combining the de-noising method with non-linear operator in these sub-bands which are at coarser resolutionn levels. In order to enhance global contrast for an infrared image, the low frequency sub-band image is also enhanced employing non-incomplete Beta transform (IBT), simulated annealing algorithm (SA) and wavelet neural network (WNN). IBT is used to obtain non-linear gray transform curve. Transform parameters are determined by SA so as to obtain optimal non-linear gray transform parameters. Contrast type of an original image is determined by a new criterion. Gray transform parameters space is determined respectively according to different contrast types. A kind of WNN is proposed to approximate the IBT in the whole low frequency sub-band image. The quality of enhanced image is evaluated by an overall cost criterion. Experimental results show that the new algorithm can greatly improve the global and local contrast for an infrared image while efficiently reducing the correlative noise (CN), gauss white noise (GWN) and multiplicative noise (MN) in the infrared image. [ABSTRACT FROM AUTHOR]

Our aim is to show that it is impossible to find a bound for the power of the first fixed point of the aleph function. [ABSTRACT FROM AUTHOR]

This note presents closed-form formulas for determining the critical points of general n-dimensional differential equations. The formulas do not require commutating the eigenvalues of the Jacobian of a system. Based on the Hurwitz criterion, explicit necessary and sufficient conditions are obtained. Particular attention is focused on Hopf and double Hopf bifurcations. A model of induction machine is presented to show the application of the results. [ABSTRACT FROM AUTHOR]