Nonnegative Continuous Function Research Articles

Analysis of the Minimal Traveling Wave Speed via the Methods of Young Measures

In this paper, we consider the equation $u_t = u_{xx} + h(b(x)) (1-u)u, x \in \mathbb{R}$, where $\gamma \mapsto h(\gamma)$ is a nonnegative continuous function and $b(x)$ is a periodic function. In some sense, $b(x)$ represents a controllable parameter and the function $\gamma \mapsto h(\gamma)$ describes how the intrinsic growth rate of $u$ depends on the parameter $b$. It is known that there exists a nonnegative number $c^{*}(h(b))$ such that the traveling wave with average speed $c$ exists if and only if $c \geq c^{*}(h(b))$. We study minimizing problems and maximizing problems of the minimal speed $c^{*}(h(b))$ by varying $b(x)$ under the constraint $\frac{1}{L} \int_{0}^{L} b(x) dx = \alpha$, where $\alpha$ is a given positive constant. We prove the existence of minimizers and maximizers of the minimal speed, but it turns out that the minimizers for certain choices of $h$ do not exist in the class of functions but do exist in the space of Young measures. This means that the minimizing sequence exhi...

A Local Large Deviation Principle for Inhomogeneous Birth-Death Processes

The paper considers a continuous-time birth-death process where the jump rate has an asymptotically polynomial dependence on the process position. We obtain a rough exponential asymptotics for the probability of excursions of a re-scaled process contained within a neighborhood of a given continuous non-negative function.

Singular value inequalities for Hilbert space operators

In this paper we show that if Ai,Bi,Xi are Hilbert space operators such that Xi is compact i = 1,2,..., n and f,g are non-negative continuous functions on [0,?) with f (t)g(t)=t for all t?[0,?), also h is non-negative increasing operator convex function on [0,?), then h(Sj(?n,i=1 ?iAi*Xi*Bi)) ? Sj(?n,i=1 ?ih(Ai* f(|Xi*|)2Ai) ? ?n,i=1 ?ih(Bi* g(|Xi|)2Bi)) for j = 1,2,... and ?n,i=1 ?i = 1. Also, applications of some inequalities are given.

Generalizations of the Aluthge transform of operators

Let A be an operator with the polar decomposition A = U|A|. The Aluthge transform of the operator A, denoted by ?, is defined as ? = |A|1/2U |A|1/2. In this paper, first we generalize the definition of Aluthge transformfor non-negative continuous functions f,g such that f(x)g(x) = x (x ? 0). Then, by using this definition, we get some numerical radius inequalities. Among other inequalities, it is shown that if A is bounded linear operator on a complex Hilbert space H, then h (w(A)) ? 1/4||h(g2 (|A|)) + h(f2(|A|)|| + 1/2h (w(? f,g)), where f,g are non-negative continuous functions such that f(x)g(x) = x (x ? 0), h is a non-negative and non-decreasing convex function on [0,?) and ? f,g = f (|A|)Ug(|A|).

When does Ando–Hiai inequality hold?

Let α be in (0,1) and r>0 and #α stand for the weighted operator geometric mean. We consider the following statement:A,B>0,A#αB≥I⇒Ar#αBr≥I. Ando and Hiai show that if r≥1, then this holds. In the present paper, we first prove the converse of this result, namely, the above statement holds only if r≥1. We next show that for each non-negative continuous function f on [0,∞) with f≤tr and f≠tr, there exist A,B>0 such that A#αB≥I and f(A)#αf(B)≱I. We also try to find a characterization of a continuous function f satisfyingA,B>0,A#αB≥I⇒f(A)#αf(B)≥I.

Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well

In this paper, we consider the following nonlinear Choquard equation driven by fractional Laplacian urn:x-wiley:mma:media:mma4653:mma4653-math-0001 where V(x) is a nonnegative continuous potential function, 0<s<1, N≥2, (N−4s)+<α<N, and λ is a positive parameter. By variational methods, we prove the existence of least energy solution which localizes near the bottom of potential well int(V−1(0)) as λ large enough.

Multiplicity results for a class of quasilinear equations with exponential critical growth

AbstractIn this work, we prove the existence and multiplicity of positive solutions for the following class of quasilinear elliptic equations: urn:x-wiley:0025584X:media:mana201500371:mana201500371-math-0001where is the N‐Laplacian operator, , f is a function with exponential critical growth, μ and ε are positive parameters and A is a nonnegative continuous function verifying some hypotheses. To obtain our results, we combine variational arguments and Lusternik–Schnirelman category theory.

Three solutions for a fractional Schrödinger equation with vanishing potentials

In this paper, we study the following fractional Schrödinger equation ( − Δ ) s u + V ( x ) u = K ( x ) f ( u ) + λ W ( x ) | u | p − 2 u , x ∈ R N , where λ > 0 is a parameter, ( − Δ ) s denotes the fractional Laplacian of order s ∈ ( 0 , 1 ) , N > 2 s , W ∈ L 2 2 − p ( R N , R + ) , 1 < p < 2 , V , K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Under some mild assumptions, we prove that the above equation has three solutions.

Multiplicity results for a class of singular elliptic equation involving sublinear Neumann boundary condition in $$\mathbb {R}^2$$ R 2

Let $$\Omega \subset {\mathbb R}^2$$ be a bounded domain with boundary of class $$C^2$$ . The purpose of this paper is to study the existence of positive solution for the following inhomogeneous singular Neumann problem: $$\begin{aligned} (P_{\mu ,\lambda })\qquad \left\{ \begin{array}{ll} &{} - \Delta u+u = \mu u^{-\delta }+h(x,u)e^{u^\alpha }, \quad u>0 \quad \text {in } \Omega , \\ &{}\frac{\partial u}{\partial \nu }= \lambda \psi u^q \quad \text {on }\partial \Omega . \end{array}\right. \end{aligned}$$ where $$\mu ,\lambda >0,$$ $$0<\delta <3,$$ $$1\le \alpha \le 2,$$ $$0\le q<1,$$ and $$\psi $$ is a non-negative Holder continuous function on $$\overline{\Omega }.$$ Here, h(x, u) is a $$C^{1}(\overline{\Omega }\times \mathbb {R})$$ having superlinear growth at infinity. Using variational methods, we show that there exists a region $$\mathcal {R}\subset \{(\mu ,\lambda ):\;\mu ,\lambda >0\}$$ bounded by the graph of a map $$\Lambda $$ , such that $$(P_{\mu ,\lambda })$$ admits at least two solutions for all $$(\mu ,\lambda ) \in \mathcal {R},$$ at least one solution for $$(\mu ,\lambda )\in \partial \mathcal {R}$$ and no solution for all $$(\mu ,\lambda )$$ outside $$\overline{\mathcal {R}}.$$

Universal inequalities for eigenvalues of a class of operators on Riemannian manifold

In this paper, we investigate the boundary value problem of the following operator $$\left\{\begin{array}{l}\Delta^{2}u-a\rm div\it A\nabla{u}+Vu=\rho\lambda{u} \hbox{in} \rm\Omega, u|_{\partial \Omega}=\frac{\partial u}{\partial v}|_{\partial \Omega}=0, \end{array}\right.$$ where $\Omega$ is a bounded domain in an $n$-dimensional complete Riemannian manifold $M^n$, $A$ is a positive semidefinite symmetric (1,1)-tensor on $M^n$, $V$ is a non-negative continuous function on $\Omega$, $v$ denotes the outwards unit normal vector field of $\partial \Omega$ and $\rho$ is a weight function which is positive and continuous on $\Omega$. By the Rayleigh-Ritz inequality, we obtain universal inequalities for the eigenvalues of these operators on bounded domain of complete manifolds isometrically immersed in a Euclidean space, and of complete manifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below, a class of warped product manifolds, the product of Euclidean spaces with any complete manifold and manifolds admitting eigenmaps to a sphere.

THE SHAPE OF THE ONE-DIMENSIONAL PHYLOGENETIC LIKELIHOOD FUNCTION.

By fixing all parameters in a phylogenetic likelihood model except for one branch length, one obtains a one-dimensional likelihood function. In this work, we introduce a mathematical framework to characterize the shapes of such one-dimensional phylogenetic likelihood functions. This framework is based on analyses of algebraic structures on the space of all frequency patterns with respect to a polynomial representation of thspace likelihood functions. Using this framework, we provide conditions under which the one-dimensional phylogenetic likelihood functions are guaranteed to have at most one stationary point, and this point is the maximum likelihood branch length. These conditions are satisfied by common simple models including all binary models, the Jukes-Cantor model and the Felsenstein 1981 model. We then prove that for the simplest model that does not satisfy our conditions, namely, the Kimura 2-parameter model, the one-dimensional likelihood functions may have multiple stationary points. As a proof of concept, we construct a non-degenerate example in which the phylogenetic likelihood function has two local maxima and a local minimum. To construct such examples, we derive a general method of constructing a tree and sequence data with a specified frequency pattern at the root. We then extend the result to prove that the space of all rescaled and translated one-dimensional phylogenetic likelihood functions under the Kimura 2-parameter model is dense in the space of all non-negative continuous functions on [0, ∞) with finite limits. These results indicate that one-dimensional likelihood functions under advanced evolutionary models can be more complex than it is typically assumed by phylogenetic inference algorithms; however, these complexities can be effectively captured by the Kimura 2-parameter model.

Multiple solutions for the fourth-order elliptic equation with vanishing potential

This paper is concerned with the following fourth-order elliptic equation Δ 2 u − Δ u + V ( x ) u = K ( x ) f ( u ) + μ ξ ( x ) | u | p − 2 u , x ∈ R N , u ∈ H 2 ( R N ) , where Δ 2 ≔ Δ ( Δ ) is the biharmonic operator, N ≥ 5 , V , K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. By working in weighted Sobolev spaces and using a variational method, we prove that the above equation has two nontrivial solutions.

On distribution of local dimensions of doubling measures on Euclidean space

We prove that for every nonnegative, increasing and right continuous function g on [0,n] with g(n)=1 and g(ϵ)=0 for some ϵ∈(0,n) there exists a doubling probability measure μ on [0,1]n such that the distribution Gμ of the lower local dimension of μ is g.

Some inequalities for operator (p, h)-convex functions

Let p be a positive number and h a function on satisfying for any . A non-negative continuous function f on is said to be operator (p, h)-convex ifholds for all positive semidefinite matrices A, B of order n with spectra in K, and for any . In this paper, we study properties of operator (p, h)-convex functions and prove the Jensen, Hansen–Pedersen type inequalities for them. We also give some equivalent conditions for a function to become an operator (p, h)-convex. In applications, we obtain Choi–Davis–Jensen type inequality for operator (p, h)-convex functions and a relation between operator (p, h)-convex functions with operator monotone functions.

Erratum to: Isomorphisms of Lattices of Subalgebras of Semirings of Continuous Nonnegative Functions with the Max-Plus

Erratum to: Isomorphisms of Lattices of Subalgebras of Semirings of Continuous Nonnegative Functions with the Max-Plus

Lattices of Subalgebras of Semirings of Continuous Nonnegative Functions with the Max-Plus

Isomorphisms φ of semirings C∨(X) of continuous nonnegative functions over an arbitrary Hewitt space X with the condition φ(ℝ+) = ℝ+ are characterized in this work. It is proved that any isomorphism of lattices of all subalgebras of semirings C∨ (X) and C∨ (Y) is induced by a unique isomorphism of semirings excepting the case of one- and two-point Tychonovization of spaces.

Metric Character for the Sub-Hamilton-Jacobi Obstacle Equation

For a given nonnegative continuous function $\mathfrak{g},$ we establish a new explicit formula of the distance related to the sub--Hamilton--Jacobi obstacle equation : $u\geq \mathfrak{g}$ and $H(x,\nabla u)=0$ in the set $[u>\mathfrak{g}].$ We introduce a new inf-sup integral formula involving the trajectories joining two given points and the obstacle $\mathfrak{g}. $ This defines a new length quasi-metric ${\mathcal I}_{\mathfrak{g}} $ in ${I\!\!R}^N$ which handles the obstacle and enters in a representation formula for a viscosity solution of the sub-Hamilton-Jacobi obstacle equation.

Global Attractor of Thermoelastic Coupled Beam Equations with Structural Damping

In this paper, we study the existence of a global attractor for a class ofn-dimension thermoelastic coupled beam equations with structural dampingutt+Δ2u+Δ2ut-[σ(∫Ω‍(∇u)2dx)+ϕ(∫Ω‍∇u∇utdx)]Δu+f1(u)+g(ut)+νΔθ=q(x), in Ω×R+, andθt-Δθ+f2(θ)-νΔut=0. HereΩis a bounded domain ofRN, andσ(·)andϕ(·)are both continuous nonnegative nonlinear real functions andqis a static load. The source termsf1(u)andf2(θ)and nonlinear external dampingg(ut)are essentially|u|ρu,|θ|ϱθ, and|ut|rutrespectively.

Large solutions for non-divergence structure equations with singular lower order terms

In this paper we investigate infinite boundary value problems associated with the semi-linear PDE Lu=k(x)f(u) on a bounded smooth domain Ω⊂Rn, where L is a non-divergence structure, uniformly elliptic operator with singular lower order terms. The weight k is a continuous non-negative function and f is a continuous nondecreasing function that satisfies the Keller–Osserman condition. We study a sufficient condition on k that ensures existence of a large solution u. In case the lower order terms of L are bounded, under further assumptions on f and k we establish asymptotic bounds of solutions u near the boundary ∂Ω and, as a consequence a uniqueness result.

Perron's method for the porous medium equation

This work extends Perron’s method for the porous medium equation in the slow diffusion case. The main result shows that nonnegative continuous boundary functions are resolutive in a general cylindrical domain.