On the solutions of a higher‐order difference equation in terms of generalized Fibonacci sequences

This paper deals with the solutions, stability character, and asymptotic behavior of the difference equation where and the initial values x_?k,x_{?k + 1},…,x₀ are nonzero real numbers, such that their solutions are associated to Horadam numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

On the solutions of a max‐type difference equation system

In this paper, we study behavior of the solution of the following max‐type difference equation system: where , the parameter A is positive real number, and the initial values x₀,y₀ are positive real numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Qualitative behavior of two systems of higher‐order difference equations

In this paper, we study the qualitative behavior of following two systems of higher‐order difference equations: and where the parameters α,β,γ,α₁,β₁,γ₁,a,b,c,a₁,b₁,andc₁ and the initial conditions x₀, x_?1, ?, x_?k, y₀, y_?1 ,?, y_?k are positive real numbers. More precisely, we study the equilibrium points, local asymptotic stability, instability, global asymptotic stability of equilibrium points, and rate of convergence of positive solutions that converges to the equilibrium point P₀=(0,0) of these systems. Some numerical examples are given to verify our theoretical results. These examples are experimental verification of our theoretical discussions. Copyright © 2015 John Wiley & Sons, Ltd.     

On the periodicity of solutions of max‐type difference equation

This paper is devoted to study the periodic nature of the solution of the following max‐type difference equation: where the initial conditions x_?2,x_?1,x₀ are arbitrary positive real numbers and is a periodic sequence of period two. Copyright © 2014 John Wiley & Sons, Ltd.     

Semistability of two systems of difference equations using centre manifold theory

In this paper, we study the stability of the zero equilibria of the following systems of difference equations: and where a, b, c and d are positive constants and the initial conditions x₀ and y₀ are positive numbers. We study the stability of those systems in the special case when one of the eigenvalues has absolute value equal to 1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiplicity of positive solutions to boundary blow‐up problem with variable exponent and sign‐changing weights

In this paper, we consider the elliptic boundary blow‐up problem where Ω is a bounded smooth domain of are positive continuous functions supported in disjoint subdomains Ω₊,Ω_? of Ω, respectively, a ₊ vanishes on the boundary of satisfies p (x )≥1 in Ω,p (x ) > 1 on ? Ω and , and ε is a parameter. We show that there exists ε ^?>0 such that no positive solutions exist when ε > ε ^?, while a minimal positive solution u _ε exists for every ε ∈(0,ε ^?). Under the additional hypotheses that is a smooth N ? 1‐dimensional manifold and that a ₊,a _? have a convenient decay near Γ, we show that a second positive solution v _ε exists for every ε ∈(0,ε ^?) if , where N ^?=(N + 2)/(N ? 2) if N > 2 and if N = 2. Our results extend that of Jorge Garcá‐Melián in 2011, where the case that p > 1 is a constant and a ₊>0 on ? Ω is considered. Copyright © 2016 John Wiley & Sons, Ltd.     

On the uniform convergence of the Fourier series for one spectral problem with a spectral parameter in a boundary condition

In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problem where λ is a spectral parameter, q(x) is a real‐valued continuous function on the interval [0,1], and a₁,b₀,b₁,c₁,d₀, and d₁ are real constants that satisfy the conditions Copyright © 2015 John Wiley & Sons, Ltd.     

On singular quasilinear Schrödinger equations with critical exponents

We consider a class of singular quasilinear Schrödinger equations of the form where are given functions, N ?3,λ is a positive constant, . By using variational methods together with concentration–compactness principle, we prove the existence of positive solutions of the aforementioned equations under suitable conditions on V (x ) and K (x ). Copyright © 2017 John Wiley & Sons, Ltd.     

Global existence and boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant and logistic source

This paper deals with the following chemotaxis system: under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Here, δ and χ are some positive constants and f is a smooth function that satisfies with some constants a ?0,b  > 0, and γ  > 1. We prove that the classical solutions to the preceding system are global and bounded provided that Copyright © 2016 John Wiley & Sons, Ltd.     

Boundedness in a quasilinear chemotaxis‐haptotaxis system with logistic source

This paper studies the chemotaxis‐haptotaxis system with nonlinear diffusion subject to the homogeneous Neumann boundary conditions and suitable initial conditions, where χ , ξ and μ are positive constants, and (n ?2) is a bounded and smooth domain. Here, we assume that D (u )?c _D u ^{m  ? 1} for all u  > 0 with some c _D  > 0 and m ?1. For the case of non‐degenerate diffusion, if μ  > μ ^?, where it is proved that the system possesses global classical solutions which are uniformly‐in‐time bounded. In the case of degenerate diffusion, we show that the system admits a global bounded weak solution under the same assumptions. Copyright © 2016 John Wiley & Sons, Ltd.     

Solution form of a higher‐order system of difference equations and dynamical behavior of its special case

The solution form of the system of nonlinear difference equations where the coefficients a ,b ,α ,β and the initial values x  _? i ,y  _? i ,i ∈{0,1,…,k } are non‐zero real numbers, is obtained. Furthermore, the behavior of solutions of the aforementioned system when p  = 1 is examined. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamics of a three‐dimensional systems of rational difference equations

We investigate in this paper the solutions and the periodicity of the following rational systems of difference equations of three‐dimensional with initial conditions x_?2,x_?1,x₀,y_?2,y_?1,y₀,z_?2,z_?1andz₀ are nonzero real numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

Global well‐posedness and scattering of a generalized nonlinear fourth‐order wave equation

In this paper, we study the global well‐posedness and scattering theory of the solution to the Cauchy problem of a generalized fourth‐order wave equation where if d ?4, and if d ?5. The main strategy we use in this paper is concentration‐compactness argument, which was first introduced by Kenig and Merle to handle the scattering problem vector so as to control the momentum. Copyright © 2017 John Wiley & Sons, Ltd.     

Max‐type system of difference equations with positive two‐periodic sequences

This paper deals with the form and the periodicity of the solutions of the max‐type system of difference equations where , and are positive two‐periodic sequences and initial values x₀, x _{? 1}, y₀, y _{? 1} ∈ (0, + ∞ ). Copyright © 2014 John Wiley & Sons, Ltd.     

On a class of coupled Schrödinger systems with critical Sobolev exponent growth

Consider the following two critical nonlinear Schrödinger systems: (0.1) (0.2) where is a smooth bounded domain, N≥3,?λ(Ω) < λ₁,λ₂<0,μ₁,μ₂>0,α,β≥1 with α + β = 2^?,γ ≠ 0,λ(Ω) is the first eigenvalue of ?Δ with the Dirichlet boundary condition and For N = 3,λ₁=λ₂,γ > 0 small, we obtain the existence of positive least energy solution of 0.1 and 0.2 . For N≥5,γ > 0, the existence of positive least energy solution of 0.2 is established. For N≥5,γ ≠ 0, we prove that 0.1 possesses a positive least energy solution. The limit behavior of the positive least energy solutions when γ→?∞ and phase separation for 0.1 are also considered. Copyright © 2015 John Wiley & Sons, Ltd.     

Characterizations of Sobolev spaces associated to operators satisfying off‐diagonal estimates on balls

Let be a metric measure space of homogeneous type and L be a one‐to‐one operator of type ω on for ω ∈[0, π /2). In this article, under the assumptions that L has a bounded H _∞ ‐functional calculus on and satisfies (p _L , q _L ) off‐diagonal estimates on balls, where p _L ∈[1, 2) and q _L ∈(2, ∞ ], the authors establish a characterization of the Sobolev space , defined via L ^{α /2}, of order α ∈(0, 2] for p ∈(p _L , q _L ) by means of a quadratic function S _{α , L} . As an application, the authors show that for the degenerate elliptic operator L _w : =? w  ^? 1div(A ?) and the Schrödinger type operator with a ∈(0, ∞ ) on the weighted Euclidean space with A being real symmetric, if n ?3, with q ∈[1, 2], , p ∈(1, ∞ ) and with , then, for all , , where the implicit equivalent positive constants are independent of f , denotes the class of Muckenhoupt weights, the reverse Hölder class, and D (L _w ) and the domains of L _w and , respectively. Copyright © 2016 John Wiley & Sons, Ltd.     

Oscillation criteria for higher‐order nonlinear dynamic equations with Laplacians and a deviating argument on time scales

In this paper, we study the n th‐order nonlinear dynamic equation with Laplacians and a deviating argument on an above‐unbounded time scale, where n ?2, New oscillation criteria are established for the cases when n is even and odd and when α  > γ ,α  = γ , and α  < γ , respectively, with α  = α ₁?α _{n  ? 1}. Copyright © 2017 John Wiley & Sons, Ltd.     

Maxwell meets Korn: A new coercive inequality for tensor fields in with square‐integrable exterior derivative

For a bounded domain with connected Lipschitz boundary, we prove the existence of some c > 0, such that holds for all square‐integrable tensor fields , having square‐integrable generalized “rotation” tensor fields and vanishing tangential trace on ?Ω, where both operations are to be understood row‐wise. Here, in each row, the operator curl is the vector analytical reincarnation of the exterior derivative d in . For compatible tensor fields T, that is, T = ? v, the latter estimate reduces to a non‐standard variant of Korn's first inequality in , namely for all vector fields , for which ? v_n,n = 1, … ,N, are normal at ?Ω. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence and concentration of bound states for a Kirchhoff type problem with potentials vanishing or unbounded at infinity

In this paper, we study the existence and concentration behavior of positive solutions for the following Kirchhoff type equation: where ɛ is a positive parameter, a and b are positive constants, and 3<p<5. Let denotes the ground energy function associated with , , where is regard as a parameter. Suppose that the potential V(x) decays to zero at infinity like |x|^−α with 0<α≤2, we prove the existence of positive solutions u_ɛ belonging to for vanishing or unbounded K(x) when ɛ > 0 small. Furthermore, we show that the solution u_ɛ concentrates at the minimum points of as ɛ→0⁺.     

Three rapidly convergent parareal solvers with application to time‐dependent PDEs with fractional Laplacian

Time‐dependent PDEs with fractional Laplacian ( ? Δ)^α play a fundamental role in many fields and approximating ( ? Δ)^α usually leads to ODEs' system like u ^′ (t ) + A u (t ) =  g (t ) with A  = Q ^α , where is a sparse symmetric positive definite matrix and α  > 0 denotes the fractional order. The parareal algorithm is an ideal solver for this kind of problems, which is iterative and is characterized by two propagators and . The propagators and are respectively associated with large step size ΔT and small step size Δt , where ΔT  = J Δt and J ?2 is an integer. If we fix the ‐propagator to the Implicit‐Euler method and choose for some proper Runge–Kutta (RK) methods, such as the second‐order and third‐order singly diagonally implicit RK methods, previous studies show that the convergence factors of the corresponding parareal solvers can satisfy and , where σ (A ) is the spectrum of the matrix A . In this paper, we show that by choosing these two RK methods as the ‐propagator, the convergence factors can reach , provided the one‐stage complex Rosenbrock method is used as the ‐propagator. If we choose for both and , the complex Rosenbrock method, we show that the convergence factor of the resulting parareal solver can also reach . Numerical results are given to support our theoretical conclusions. Copyright © 2017 John Wiley & Sons, Ltd.