Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

On some new generalized sequence spaces

In this paper we define the sequence sets ?_∞(u,Δ²,p), c(u,Δ²,p) and c₀(u,Δ²,p), and give α- and β-duals of these sets. Further we investigate matrix transformations in the spaces and give a characterization of the class (?_∞(u,Δ²,p),?_∞).     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

Homothetic interval orders

We give a characterization of the non-empty binary relations ? on a N^*-set A such that there exist two morphisms of N^*-sets u₁,u₂:A→R₊ verifying u₁?u₂ and x?y⇔u₁(x)>u₂(y). They are called homothetic interval orders. If ? is a homothetic interval order, we also give a representation of ? in terms of one morphism of N^*-sets u:A→R₊ and a map such that x?y⇔σ(x,y)u(x)>u(y). The pairs (u₁,u₂) and (u,σ) are “uniquely” determined by ?, which allows us to recover one from each other. We prove that ? is a semiorder (resp. a weak order) if and only if σ is a constant map (resp. σ=1). If moreover A is endowed with a structure of commutative semigroup, we give a characterization of the homothetic interval orders ? represented by a pair (u,σ) so that u is a morphism of semigroups.     

Optimality conditions for fractional minmax programming

Let V ?H be real separable Hilbert spaces. The abstract wave equation u′' + A(t)u = g(u), where u(t) ?V, A(t) maps V to its dual $\text{V}{}^1$, and g is a nonlinear map from the ball S(R₀) = {u?V: ∥u∥ < R₀} into H, is considered. It is assumed that g is locally Lipschitz in S(R₀) and possibly singular at the boundary. Local existence and continuation theorems are established for the Cauchy problem u(0) = u₀?S(R₀), u′(0) = u₁?H. Global existence is shown for g(u) = εφ(u), where φ has a potential and ε is small. Global nonexistence is shown for g(u) = εφ(u), where φ satisfies an abstract convexity property and ε is large.     

A free boundary problem for p-Laplacian in the plane

We consider the following free boundary problem in an unbounded domain Ω in two dimensions: Δpu=0 in Ω, on J₀, on J₁, where ∂Ω=J₀∪J₁. We prove that if 0<u<1 in Ω, Ji is the graph of a function in and gi is a constant for each i=0,1, then the free boundary ∂Ω must be two parallel straight lines and the solution u must be a linear function. The proof is based on maximum principle.     

Stability and bifurcation of equilibrium solutions of reaction-diffusion equations on an infinite space interval

Under the condition that f(x, y, z, α) and its partial derivatives decay sufficiently fast as $\text{¦x¦ → ∞}$ we will study the (linear) stability and bifurcation of equilibrium solutions of the scalar problem u_t = u_xx + f(x, u, u_x, α), u_x(?∞, t) = u_x(∞, t) = 0 (1) where α is a real bifurcation parameter. After introducing appropriate function spaces X and Y the problem (1) can be rewritten $\text{d}\text{dt}\text{u = G(u, α)}$, (7) where G:X×R → Y is given by G(u, α)(x) = u″(x) + f(x, u(x), u′(x), α). It will be shown, for each (u, α)?X × R, that the Fréchet derivative G_u(u,a): X → Y is not a Fredholm operator. This difficulty is due to the fact that the domain of the space variable x, is infinite and cannot be eliminated by making another choice of X and Y. Since G_u(u, α) is not Fredholm, the hypotheses of most of the general stability and bifurcation results are not satisfied. If (u₀, α₀?S = {(u, α): G(u, α) = 0}, (i.e., (u₀,α₀) is an equilibrium solution of (7)), a necessary condition on the spectrum of G_u(u₀, α₀) for a change in the stability of points in S to occur at G_u(u₀, α₀) will be given. When this condition is met, the principle of exchange of stability which means, in a neighborhood of (u₀, α₀), that adjacent equilibrium solutions for the same α have opposite stability properties in a weakened sense will be established. Also, when G_u or its first order partial derivatives, evaluated at (u₀, α₀), are not too degenerate, the shape of S in a neighborhood of (u₀, α₀) will be described and a strenghtened form of the principle of exchange of stability will be obtained.     

Stability of solutions bifurcating from multiple eigenvalues

Let X and Y be real Banach spaces and G:X × $\text{R}$ be a twice continuously differentiate function which is not necessarily linear. Suppose G(u₀, α₀) = 0 and the dimension of the null space of G_u(u₀, α₀) is m, where 1 ? m < ∞. Usually, S = {(u, α):G(u, α) = 0}, in a neighborhood of (u₀, α₀), consists of a finite number of curves emanating from (u₀, α₀). We will determine the stability of points, (u, α), in S (i.e., the maximum of the real parts of the spectrum of G_u(u, α) for each (u, α) ∈ S) using a general perturbation theorem of Kato. Our results contain as a special case the stability theorems of Crandall and Rabinowitz for the case m = 1. We will also tie our stability theorems together with some bifurcation results of Decker and Keller. Finally we apply our results to systems of reaction diffusion equations.     

Locally Lipschitz solutions of a single conservation law in several space variables

We consider the Cauchy problem for a single conservation law in several space variables. Letting u(x, t) denote the solution with initial data u₀, we state necessary and sufficient conditions on u₀ so that u(x, t) is locally Lipschitz continuous in the half space {t > 0}. These conditions allow for the preservation of smoothness of u₀ as well as for the smooth resolution of discontinuities in u₀. One consequence of our result is that u(x, t) cannot be locally Lipschitz unless u₀ has locally bounded variation. Another is that solutions which are bounded and locally Lipschitz continuous in {t > 0} automatically have boundary values u₀ at t = 0 in the sense that u(·, t) → u₀ in L_loc¹. Finally, we give an elementary proof that locally Lipschitz solutions satisfy Kruzkov's uniqueness condition.     

Asymptotic behaviors of the solution to an initial-boundary value problem for scalar viscous conservation laws

This paper is concerned with the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws u_t + f(u)_x = u_xx on [0, 1], with the boundary condition u(0, t) = u₋(t) → u₋, u(1, t) = u₊(t) → u₊, as t → +∞ and the initial data u(x,0) = u₀(x) satisfying u₀(0) = u₋(0), u₀(1) = u₊(1), where u± are given constants, u₋ ≠ u₊ and f is a given function satisfying f″(u) > 0 for u under consideration. By means of an elementary energy estimates method, both the global existence and the asymptotic behavior are obtained. When u₋ 〈 u₊, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for u₋ > u₊, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, that is, |u₋−u₊| is small. Moreover, when u_±(t) ≡ u_±, t ≥ 0, exponential decay rates are both obtained.     

Quenching for nonlinear degenerate parabolic problems

For the problem given by u_τ=(ξ^ru^mu_ξ)_ξ/ξ^r+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u₀(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that lim_u→c⁻f(u)=∞ for some positive constant c, and u₀(ξ) is a given function satisfying u₀(0)=0=u₀(a), this paper studies quenching of the solution u.     

Resolving Extensions of Finitely Presented Systems

In this paper we extend certain central results of zero dimensional systems to higher dimensions. The first main result shows that if (Y,f) is a finitely presented system, then there exists a Smale space (X,F) and a u-resolving factor map π ₊:X→Y. If the finitely presented system is transitive, then we show there is a canonical minimal u-resolving Smale space extension. Additionally, we show that any finite-to-one factor map between transitive finitely presented systems lifts through u-resolving maps to an s-resolving map.     

On a class of inverse optimal control problems

Four distinct, though closely related, inverse optimal control problems are considered. Given a closed, convex setU in a real Hilbert spaceX and an elementu ₀ inU, it is desired to find all functionals of the form (u,Ru) such that (i)R is a self-adjoint positive operator and (u,Ru) is minimized over the setU at the pointu ₀, (ii)R is self-adjoint, positive definite and (u,Ru) is minimized overU atu ₀, (iv)R is self-adjoint, positive definite and (u,Ru) is uniquely minimized overU atu ₀. The interrelationships among the sets of solutions of these problems are pointed out. Necessary and sufficient conditions which explicitly characterize the solutions to each of these problems are derived. The question of existence of a solution (namely, Given a particular setU and a particular elementu ₀, under what conditions does there exist an operatorR having certain required properties?) is discussed. The results derived are illustrated by an example.     

Short cycles in repeated exponentiation modulo a prime

Given a prime p, we consider the dynamical system generated by repeated exponentiations modulo p, that is, by the map \({u \mapsto f_g(u)}\), where f _g (u) ≡ g ^u (mod p) and 0 ≤ f _g (u) ≤ p ? 1. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system.     

Characterizing sequences for precompact group topologies

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact  ) if there is a sequence u=(_un)$\mathbf{u}=(u_n)$ in G such that τ is the finest precompact group topology on G   making u=(_un)$\mathbf{u}=(u_n)$ converge to zero. It is proved that a metrizable precompact abelian group (G,τ)$(G,\tau )$ is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ   and the groups (G,τ)$(G,\tau )$ and (G,η)$(G,\eta )$ have the same Pontryagin dual groups (in other words, (G,τ)$(G,\tau )$ is not a Mackey group in the class of maximally almost periodic groups).     

Weak limit and blowup of approximate solutions to H-systems

Let be a continuous function such that H(p)→H₀∈R as |p|→+∞. Fixing a domain Ω in R² we study the behaviour of a sequence (un) of approximate solutions to the H-system Δu=2H(u)ux∧uy in Ω. Assuming that supp∈R³|(H(p)−H₀)p|<1, we show that the weak limit of the sequence (un) solves the H-system and un→u strongly in H¹ apart from a countable set S made by isolated points. Moreover, if in addition H(p)=H₀+o(1/|p|) as |p|→+∞, then in correspondence of each point of S we prove that the sequence (un) blows either an H-bubble or an H₀-sphere.     

Globally smooth solutions of quasilinear hyperbolic systems in diagonal form

We prove the existence of a large class of globally smooth solutions of the Cauchy problem for the system of n equations u_t + Λ(x, t, u)u_x = 0, where Λ is a diagonal matrix. We show that, under certain monotonicity conditions on both Λ and the initial data u₀, the solution u will be locally Lipschitz continuous at positive times. Since u₀ is a function of locally bounded variation, our result thus provides both for the smoothing of discontinuities in u₀ as well as for the global preservation of smoothness. The global existence results from an a priori estimate of $\text{?u}\text{?x}$, which we obtain by a device which enables us to effectively uncouple the system of equations for $\text{?u}\text{?x}$. Finally, we prove a partial converse which demonstrates that our hypotheses are not overly restrictive.     

Nonlinear parabolic equations with p-growth and unbounded data

The purpose of this paper is to prove the existence of a solution for a nonlinear parabolic equation in the form u_t - div(a(t, x, u, Du)) = H(t, x, u, Du) - div(g(t, x)) in Q_T =]0,T[×Ω, Ω ⊂ R^N, with an initial condition u(0) = u₀, where u₀ is not bounded, |H(t,x, u, ξ)⩽ β|ξ|^p + f(t,x) + βe^λ₁^|u|f, |g|^p/(p-1) ∈ L^r(Q_T) for some r = r{N) ⩾ 1, and - div(a(t,x,u, Du)) is the usual Leray-Lions operator.     

An efficient algorithm for the multivariable Adomian polynomials

In this article the sum of the series of multivariable Adomian polynomials is demonstrated to be identical to a rearrangement of the multivariable Taylor expansion of an analytic function of the decomposition series of solutions u₁, u₂, … , u_m about the initial solution components u_1,0, u_2,0, … , u_m,0; of course the multivariable Adomian polynomials were developed and are eminently practical for the solution of coupled nonlinear differential equations. The index matrices and their simplified forms of the multivariable Adomian polynomials are introduced. We obtain the recurrence relations for the simplified index matrices, which provide a convenient algorithm for rapid generation of the multivariable Adomian polynomials. Another alternative algorithm for term recurrence is established. In these algorithms recurrence processes do not require complicated operations such as parametrization, expanding and regrouping, derivatives, etc. as practiced in prior art. The MATHEMATICA program generating the Adomian polynomials based on the algorithm in this article is designed.     

Existence of Semilinear Differential Equations with Nonlocal Initial Conditions

In this paper we give the existence of mild solutions for semilinear Cauchy problems u′（t） = Au（t） ＋f（t, u（t））, t ∈ I, a.e. with nonlocal initial condition u（O） = g（u） ＋uo when the map g loses compactness in Banach spaces.