Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and a semi-linear integral equation . We prove the existence of stable manifolds of solutions to this equation in the case that (U(t,s))_t?s?0 has an exponential dichotomy and the nonlinear forcing term f(t,x) satisfies the non-uniform Lipschitz conditions: ‖f(t,x₁)−f(t,x₂)‖?φ(t)‖x₁−x₂‖ for φ being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory.     

Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and an integral equation . We characterize the exponential dichotomy of the evolution family through solvability of this integral equation in admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory. We then apply our results to study the robustness of the exponential dichotomy of evolution families on a half-line under small perturbations.     

Uniform asymptotic stability for perturbed neutral delay differential equations

One-dimensional perturbed neutral delay differential equations of the form (x(t)−P(t,x(t−τ)))′=f(t,x_t)+g(t,x_t) are considered assuming that f satisfies −v(t)M(φ)?f(t,φ)?v(t)M(−φ), where M(φ)=max{0,max_s∈[−r,0]φ(s)}. A typical result is the following: if ‖g(t,φ)‖?w(t)‖φ‖ and , then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation (x(t)−P(t,x(t−τ)))′=f(t,x_t) is uniformly asymptotically stable. Some known results associated with this equation are extended and improved.     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

Selections of bounded variation under the excess restrictions

Let X be a metric space with metric d, c(X) denote the family of all nonempty compact subsets of X and, given F,G∈c(X), let e(F,G)=supx∈Finfy∈Gd(x,y) be the Hausdorff excess of F over G. The excess variation of a multifunction , which generalizes the ordinary variation V of single-valued functions, is defined by where the supremum is taken over all partitions of the interval [a,b]. The main result of the paper is the following selection theorem: If,V₊(F,[a,b])<∞,t₀∈[a,b]andx₀∈F(t₀), then there exists a single-valued functionof bounded variation such thatf(t)∈F(t)for allt∈[a,b],f(t₀)=x₀,V(f,[a,t₀))?V₊(F,[a,t₀))andV(f,[t₀,b])?V₊(F,[t₀,b]). We exhibit examples showing that the conclusions in this theorem are sharp, and that it produces new selections of bounded variation as compared with [V.V. Chistyakov, Selections of bounded variation, J. Appl. Anal. 10 (1) (2004) 1-82]. In contrast to this, a multifunction F satisfying e(F(s),F(t))?C(t−s) for some constant C?0 and all s,t∈[a,b] with s?t (Lipschitz continuity with respect to e(⋅,⋅)) admits a Lipschitz selection with a Lipschitz constant not exceeding C if t₀=a and may have only discontinuous selections of bounded variation if a<t₀?b. The same situation holds for continuous selections of when it is excess continuous in the sense that e(F(s),F(t))→0 as s→t−0 for all t∈(a,b] and e(F(t),F(s))→0 as s→t+0 for all t∈[a,b) simultaneously.     

A maximum principle for evolution Hamilton-Jacobi equations on Riemannian manifolds

We establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form ut+F(t,dxu)=0, u(0,x)=u₀(x), where is a bounded uniformly continuous function, M is a Riemannian manifold, and . This yields uniqueness of the viscosity solutions of such Hamilton-Jacobi equations.     

Nonresonance for one-dimensional p-Laplacian with regular restoring

In this paper, the existence of periodic solutions of the equation (φ_p(x′))′+f(t,x)=0 is proved, where has some nonlinear growth near x=±∞, and the nonresonance condition (S_P), which can be explained using the periodic eigenvalues concerning p-Laplacian, is satisfied.     

Flow invariance for solutions to nonlinear nonautonomous partial differential delay equations

We investigate the problem of existence and flow invariance of mild solutions to nonautonomous partial differential delay equations , t?s, us=φ, where B(t) is a family of nonlinear multivalued, α-accretive operators with D(B(t)) possibly depending on t, and the operators F(t,.) being defined—and Lipschitz continuous—possibly only on “thin” subsets of the initial history space E. The results are applied to population dynamics models. We also study the asymptotic behavior of solutions to this equation. Our analysis will be based on the evolution operator associated to the equation in the initial history space E.     

Invariance of quasi-arithmetic means with function weights

Let I⊂R be a non-trivial interval and let . We present some results concerning the following functional equation, generalizing the Matkowski-Sutô equation,

λ(x,y)φ⁻¹(μ(x,y)φ(x)+(1−μ(x,y))φ(y))+(1−λ(x,y))ψ⁻¹(ν(x,y)ψ(x)+(1−ν(x,y))ψ(y))=λ(x,y)x+(1−λ(x,y))y,     

On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary

We study the differentiability of very weak solutions v∈L¹(Ω) of ₀(v,L^?φ)=₀(f,φ) for all vanishing at the boundary whenever f is in L¹(Ω,δ), with δ=dist(x,∂Ω), and L^* is a linear second order elliptic operator with variable coefficients. We show that our results are optimal. We use symmetrization techniques to derive the regularity in Lorentz spaces or to consider the radial solution associated to the increasing radial rearrangement function of f.     

Harmonic measures for a point may form a square

Let X be a Green domain in Rd, d?2, x∈X, and let Mx(P(X)) denote the compact convex set of all representing measures for x. Recently it has been proven that the set of harmonic measures , U open in X, x∈U, which is contained in the set of extreme points of Mx(P(X)), is dense in Mx(P(X)). In this paper, it is shown that Mx(P(X)) is not a simplex (and hence not a Poulsen simplex). This is achieved by constructing open neighborhoods U₀, U₁, U₂, U₃ of x such that the harmonic measures are pairwise different and . In fact, these measures form a square with respect to a natural L²-structure. Since the construction is mainly based on having certain symmetries, it can be carried out just as well for Riesz potentials, the Heisenberg group (or any stratified Lie algebra), and the heat equation (or more general parabolic situations).     

Non-autonomous scalar discontinuous ordinary differential equation

In this paper we prove that the problem x^′(t)=f(x(t))+h(t), x(0)=0, where f and h are non-negative, f is finite a.e. and , h are Lebesgue integrable, has an absolutely continuous solution.     

Blow-up and stability of semilinear PDEs with gamma generators

We investigate finite-time blow-up and stability of semilinear partial differential equations of the form , w₀(x)=φ(x)?0, x∈R₊, where Γ is the generator of the standard gamma process and ν>0, σ∈R, β>0 are constants. We show that any initial value satisfying c₁x−a₁?φ(x), x>x₀, for some positive constants x₀, c₁, a₁, yields a non-global solution if a₁β<1+σ. If , where x₀,c₂,a₂>0, and a₂β>1+σ, then the solution wt is global and satisfies , for some constant C>0. This complements the results previously obtained in [M. Birkner et al., Proc. Amer. Math. Soc. 130 (2002) 2431; M. Guedda, M. Kirane, Bull. Belg. Math. Soc. Simon Stevin 6 (1999) 491; S. Sugitani, Osaka J. Math. 12 (1975) 45] for symmetric α-stable generators. Systems of semilinear PDEs with gamma generators are also considered.