Nondecreasing Continuous Function Research Articles

Elliptic equations with vertical asymptotes in the nonlinear term

We study the existence of solutions of the nonlinear problem {fx349-1} where μ is a bounded measure andg is a continuous nondecreasing function such thatg(0)=0. In this paper, we assume that the nonlinearityg satisfies {fx349-2} Problem (0.1) need not have a solution for every measure μ. We prove that, given μ, there exists a “closest” measure μ* for which (0.1) can be solved. We also explain how assumption (0.2) makes problem (0.1) different from the case whereg(t) is defined for everyt ∈ ℝ.

On the minimality of the p-harmonic map $$x/\|x\|$$ for weighted energy

In this paper, we study the minimality of the map \(\frac{x}{\|x\|}\) for the weighted energy functional \(E_{f,p}= \int_{\mathbf{B}^n}f(r)\|\nabla u\|^p dx\), where \(f : [0,1] \rightarrow \mathbb{R}^{+}\) is a continuous function. We prove that for any integer \(p \in \{2, \ldots, n-1\}\) and any non-negative, non-decreasing continuous function f, the map \(\frac{x}{\|x\|}\) minimizes Ef,p among the maps in \(W^{1,p}(\mathbf{B}^n, \mathbb{S}^{n-1})\) which coincide with \(\frac{x}{\|x\|}\) on \(\partial \mathbf{B}^n\). The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincare Anal. Non-lineaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥ 7 for any real \(p \in [2,n)\) with \(p \in (n-2\sqrt{n-1},n)\), the map \(\frac{x}{\|x\|}\) minimizes Ef,p among the maps in \(W^{1,p}(\mathbf{B}^n, \mathbb{S}^{n-1})\) which coincide with \(\frac{x}{\|x\|}\) on \(\partial \mathbf{B}^n\).

Application of iterative processes of R-order at least three to operators with unbounded second derivative

In this paper, we apply a family of Newton-like methods, which contains the best known iterative processes, to operator equations where the usual convergence conditions are relaxed. We weaken these conditions by assuming ∥ F″( x 0)∥ ⩽ α and ∥ F″( x) − F″( y)∥ ⩽ ω(∥ x − y∥), with ω a non-decreasing continuous real function. Our results include the ones obtained when the convergence of the family is studied under Lipschitz continuous or Hölder continuous conditions for the second derivative of the operator involved. To finish, we apply the study to boundary value problems.

Existence of explosive positive solutions of quasilinear elliptic equations

In this paper, our main purpose is to consider the quasilinear equation div ( | ∇ u | p - 2 ∇ u ) = m ( x ) f ( u ) on a domain Ω ⊆ R N , N ⩾ 3, where f is a nonnegative, nondecreasing continuous function which vanishes at the origin, and m is a nonnegative continuous function with the property that any zero of m is contained in a bounded domain in Ω such that m is positive on its boundary. For Ω bounded, we show that a nonnegative solution u satisfying u( x) → ∞ as x → ∂ Ω exists. For Ω un-boundary (including Ω = R N ), we show that a similar result holds where u( x) → ∞ as ∣ x∣ → ∞ within Ω and u( x) → ∞ as x → ∂ Ω.

New results on global exponential stability of recurrent neural networks with time-varying delays

This Letter provides new sufficient conditions for the existence, uniqueness and global exponential stability of the equilibrium point of recurrent neural networks with time-varying delays by employing Lyapunov functions and using the Halanay inequality. The time-varying delays are not necessarily differentiable. Both Lipschitz continuous activation functions and monotone nondecreasing activation functions are considered. The derived stability criteria are expressed in terms of linear matrix inequalities (LMIs), which can be checked easily by resorting to recently developed algorithms solving LMIs. Furthermore, the proposed stability results are less conservative than some previous ones in the literature, which is demonstrated via some numerical examples.

Some remarks on elliptic problems with critical growth in the gradient

In this work we analyze existence, nonexistence, multiplicity and regularity of solution to problem (1) - Δ u = β ( u ) | ∇ u | 2 + λ f ( x ) in Ω , u = 0 on ∂ Ω where β is a continuous nondecreasing positive function and f belongs to some suitable Lebesgue spaces.

Reduced measures on the boundary

We study the existence of solutions of the nonlinear problem (0.1) - Δ u + g ( u ) = 0 in Ω , u = μ on ∂ Ω , where μ is a bounded measure and g : R → R is a nondecreasing continuous function with g ( t ) = 0 , ∀ t ⩽ 0 . Problem (0.1) admits a solution for every μ ∈ L 1 ( ∂ Ω ) , but this need not be the case when μ is a general bounded measure. We introduce a concept of reduced measure μ * (in the spirit of Brezis et al. (Ann. Math. Stud., to appear)); this is the “closest” measure to μ for which (0.1) admits a solution.

On the global output convergence of a class of recurrent neural networks with time-varying inputs

This paper studies the global output convergence of a class of recurrent neural networks with globally Lipschitz continuous and monotone nondecreasing activation functions and locally Lipschitz continuous time-varying inputs. We establish two sufficient conditions for global output convergence of this class of neural networks. Symmetry in the connection weight matrix is not required in the present results which extend the existing ones.

Symmetry properties of solutions of semilinear elliptic equations in the plane

If g is a nondecreasing nonnegative continuous function we prove that any solution of −Δu+g(u)=0 in a half plane which blows-up locally on the boundary, in a fairly general way, depends only on the normal variable. We extend this result to problems in the complement of a disk. Our main application concerns the exponential nonlinearity g(u)=eau, or power–like growths of g at infinity. Our method is based upon a combination of the Kelvin transform and moving plane method.

A new concept of reduced measure for nonlinear elliptic equations

We study the existence of solutions of the nonlinear problem (i) − Δ u + g ( u ) = μ in Ω , u = 0 on ∂ Ω , where μ is a Radon measure and g : R → R is a nondecreasing continuous function with g ( t ) = 0 , ∀ t ⩽ 0 . Given g, Eq. (i) need not have a solution for every measure μ, and we say that μ is a good measure if (i) admits a solution. We show that for every μ there exists a largest good measure μ * ⩽ μ . This reduced measure μ * has a number of remarkable properties. To cite this article: H. Brezis et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Global Output Convergence of a Class of Continuous-Time Recurrent Neural Networks With Time-Varying Thresholds

This paper discusses the global output convergence of a class of continuous-time recurrent neural networks (RNNs) with globally Lipschitz continuous and monotone nondecreasing activation functions and locally Lipschitz continuous time-varying thresholds. We establish one sufficient condition to guarantee the global output convergence of this class of neural networks. The present result does not require symmetry in the connection weight matrix. The convergence result is useful in the design of recurrent neural networks with time-varying thresholds.

Kato's inequality when Δ u is a measure

We extend the classical version of Kato's inequality in order to allow functions u∈ L 1 loc such that Δ u is a Radon measure. This inequality has been recently applied by Brezis, Marcus, and Ponce to study the existence of solutions of the nonlinear equation −Δ u+ g( u)= μ, where μ is a measure and g : R→ R is a nondecreasing continuous function. To cite this article: H. Brezis, A.C. Ponce, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Existence of entire large positive solutions of a semilinear elliptic system

We show that large positive radial solutions exist for the semilinear elliptic system Δ u= p(| x|) f( v), Δ v= q(| x|) g( u) on R N ( N⩾3) for nonnegative, nondecreasing continuous functions f and g. Provided the nonnegative functions p and q are continuous, C-positive and satisfy the decay conditions ∫ ∞ 0tp(t) dt<∞ and ∫ ∞ 0tq(t) dt<∞ , and f, g satisfy ∫ ∞ 1[∫ s 0f(t) dt] −(1/2) ds<∞ , ∫ ∞ 1[∫ s 0g(t) dt] −(1/2) ds<∞ .

Bifurcation and asymptotics for the Lane–Emden–Fowler equation

We are concerned with the Lane–Emden–Fowler equation −Δ u= λf( u)+ a( x) g( u) in Ω, subject to the Dirichlet boundary condition u=0 on ∂Ω, where Ω⊂ R N is a smooth bounded domain, λ is a positive parameter, a : Ω→[0,∞) is a Hölder function, and f is a positive nondecreasing continuous function such that f( s)/ s is nonincreasing in (0,∞). The singular character of the problem is given by the nonlinearity g which is assumed to be unbounded around the origin. In this Note we discuss the existence and the uniqueness of a positive solution of this problem and we also describe the precise decay rate of this solution near the boundary. The proofs rely essentially on the maximum principle and on elliptic estimates. To cite this article: M. Ghergu, V.D. Rădulescu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Nonradial Large Solutions of Sublinear Elliptic Equations

We show that the equation Δu = p(x)f(u) has a positive solution on R N , N ≥ 3, satisfying <artwork name="GAPA31011ei1"> <artwork name="GAPA31011ei2"> if and only if <artwork name="GAPA31011ei3"> when ψ(r) = min{p(x): |x| = r}. The nondecreasing continuous function f satisfies f(0) = 0, f (s) > 0 for s > 0, and sup s ≥ 1 f(s)/s<∞, and the nonnegative continuous function p is required to be asymptotically radial. This extends previous results which required the function p to be constant or radial.

Absolute exponential stability of a class of continuous-time recurrent neural networks

This paper presents a new result on absolute exponential stability (AEST) of a class of continuous-time recurrent neural networks with locally Lipschitz continuous and monotone nondecreasing activation functions. The additively diagonally stable connection weight matrices are proven to be able to guarantee AEST of the neural networks. The AEST result extends and improves the existing absolute stability and AEST ones in the literature.

Global stability of a class of continuous-time recurrent neural networks

This paper investigates global asymptotic stability (GAS) and global exponential stability (GES) of a class of continuous-time recurrent neural networks. First, we introduce a necessary and sufficient condition for the existence and uniqueness of equilibrium of the neural networks with Lipschitz continuous activation functions. Next, we present two sufficient conditions to ascertain the GAS of the neural networks with globally Lipschitz continuous and monotone nondecreasing activation functions. We then give two GES conditions for the neural networks whose activation functions may not be monotone nondecreasing. We also provide a Lyapunov diagonal stability condition, without the nonsingularity requirement for the connection weight matrices, to ascertain the GES of the neural networks with globally Lipschitz continuous and monotone nondecreasing activation functions. This Lyapunov diagonal stability condition generalizes and unifies many of the existing GAS and GES results. Moreover, two higher exponential convergence rates are estimated.

Global stability of a class of discrete-time recurrent neural networks

This paper presents several analytical results on global asymptotic stability (GAS) and global exponential stability (GES) for the equilibrium states of a general class of discrete-time recurrent neural networks (DTRNNS) with asymmetric connection weight matrices and globally Lipschitz continuous and monotone nondecreasing activation functions. A necessary and sufficient condition is formulated to guarantee the existence and uniqueness of equilibria of such DTRNNS. The obtained results are less restrictive, different from, and improve upon the existing ones on GAS and GES of neural networks with special classes of activation functions.

Global asymptotic stability and global exponential stability of continuous-time recurrent neural networks

This paper presents new results on global asymptotic stability (GAS) and global exponential stability (GES) of a general class of continuous-time recurrent neural networks with Lipschitz continuous and monotone nondecreasing activation functions. We first give three sufficient conditions for the GAS of neural networks. These testable sufficient conditions differ from and improve upon existing ones. We then extend an existing GAS result to GES one and also extend the existing GES results to more general cases with less restrictive connection weight matrices and/or partially Lipschitz activation functions.

Strong Comparison Principle for Radial Solutions of Quasi-Linear Equations

Let Ω be either a ball or an annulus centered about the origin in RN and Δp the usual p-Laplace operator in RN. Let f1,f2∈L1loc(Ω) be two radial functions on Ω with f1≤f2,f1≢f2. Let b:R→R be a non-decreasing continuous function. Let u1,u2∈C1,β(Ω),β∈(0,1] be any two radial weak solutions of −Δpui=b(ui)+fi in Ω. We then show that u1≤u2 in Ω implies u1<u2 in Ω and also that appropriate versions of Hopf boundary point principle hold.