Nondecreasing Continuous Function Research Articles

The Effect of Diffusions and Sources on Semilinear Elliptic Problems

This paper deals with properties of non-negative solutions of the boundary value problem in the presence of diffusion a and source f in a bounded domain Ω ⊂ ℝn, n ≥ 1, where a and f are non-decreasing continuous functions on [0, L0) and f is positive. Part of the results are new even if we restrict ourselves to the Gelfand type case L0 = ∞, a(t) = t and f is a convex function. We study the behavior of related extremal parameters and solutions with respect to L0 and also to a and f in the C0 topology. The work is carried out in a unified framework for 0 < L0 ≤ ∞ under some interactive conditions between a and f

Global bifurcation of positive solutions for a class of superlinear first-order differential systems

We are concerned with the first-order differential system of the form $$\left\{ \begin{array}{ll} u'(t)+a(t)u(t)=\lambda b(t) f(v(t-\tau(t))), &t\in\mathbb{R},\\ v'(t)+a(t)v(t)=\lambda b(t)g(u(t-\tau(t))), &t\in\mathbb{R},\\ \end{array} \right. $$ where $\lambda\in\mathbb{R}$~is a parameter. $a,b\in C(\mathbb{R},[0,+\infty))$ are two $\omega$-periodic functions such that $\int_0^\omega a(t)\text{d}t>0$,~$\int_0^\omega b(t)\text{d}t>0$,~$\tau\in C(\mathbb{R},\mathbb{R})$ is an $\omega$-periodic function. The nonlinearities~$f,g\in C(\mathbb{R},(0,+\infty))$~are two nondecreasing continuous functions and satisfy superlinear conditions at infinity.~By using the bifurcation theory,~we will show the existence of an unbounded component of positive solutions, which is bounded in positive $\lambda$-direction.

Encounter-based model of a run-and-tumble particle

In this paper we extend the encounter-based model of diffusion-mediated surface absorption to the case of an unbiased run-and-tumble particle (RTP) confined to a finite interval [0, L] and switching between two constant velocity states ±v at a rate α. The encounter-based formalism is motivated by the observation that various surface-based reactions are better modeled in terms of a reactivity that is a function of the amount of time that a particle spends in a neighborhood of an absorbing surface, which is specified by a functional known as the boundary local time. The effects of surface reactions are taken into account by identifying the first passage time (FPT) for absorption with the event that the local time crosses some random threshold . In the case of a Brownian particle, the local time ℓ(t) is a continuous non-decreasing function of the time t. Taking to be an exponential distribution, , is equivalent to imposing a Robin boundary condition with a constant rate of absorption κ 0. One major difference in the encounter-based model of an RTP is that the boundary local time ℓ(t) is a now a discrete random variable that counts the number of collisions of the RTP with the boundary. Given this modification, we show that in the case of a geometric distribution Ψ(ℓ) = z ℓ , z = 1/(1 + κ 0/v), we recover the RTP analog of the Robin boundary condition. This allows us to solve the boundary value problem (BVP) for the joint probability density for particle position and the local time, and thus incorporate more general models of absorption based on non-geometric distributions Ψ(ℓ). We illustrate the theory by calculating the mean FPT (MFPT) for absorption at x = L given a totally reflecting boundary at x = 0. We also determine the splitting probability for absorption at x = L when the boundary at x = 0 is totally absorbing.

Positive supersolutions of fourth-order nonlinear elliptic equations: explicit estimates and Liouville theorems

In this paper, we consider positive supersolutions of the semilinear fourth-order problem { ( − Δ ) 2 u = ρ ( x ) f ( u ) in Ω , − Δ u > 0 in Ω , where Ω is a domain in R N (bounded or not), f : D f = [ 0 , a f ) → [ 0 , ∞ ) ( 0 < a f ⩽ + ∞ ) is a non-decreasing continuous function with f ( u ) > 0 for u > 0 and ρ : Ω → R is a positive function. Using a maximum principle-based argument, we give explicit estimates on positive supersolutions that can easily be applied to obtain Liouville-type results for positive supersolutions either in exterior domains, or in unbounded domains Ω with the property that sup x ∈ Ω ⁡ dist ( x , ∂ Ω ) = ∞ . In particular, we consider the above problem with f ( u ) = u p ( p > 0 ) and with different weights ρ ( x ) = | x | a , e a x 1 or x 1 m ( m is an even integer). Also, when f is convex and ρ : Ω → ( 0 , ∞ ) is smooth with Δ ( ρ ) > 0 , then under an extra condition between f and ρ we show that every positive supersolution u of this problem with u = 0 on ∂Ω (Ω bounded) satisfies the inequality − Δ u ≥ 2 ρ ( x ) F ( u ) for all x ∈ Ω , where F ( t ) : = ∫ 0 t ( f ( s ) − f ( 0 ) ) d s .

Boundary singularities of semilinear elliptic equations with Leray-Hardy potential

We study existence and uniqueness of solutions of ([Formula: see text]) [Formula: see text] in [Formula: see text], [Formula: see text] on [Formula: see text], where [Formula: see text] is a bounded smooth domain such that [Formula: see text], [Formula: see text] is a constant, [Formula: see text] a continuous nondecreasing function satisfying some integral growth condition and [Formula: see text] and [Formula: see text] two Radon measures, respectively, in [Formula: see text] and on [Formula: see text]. We show that the situation differs considerably according the measure is concentrated at [Formula: see text] or not. When [Formula: see text] is a power we introduce a capacity framework which provides necessary and sufficient conditions for the solvability of problem ([Formula: see text]).

On a mean method of summability

Let $p(x)$ be a nondecreasing real-valued continuous function&#x0D; on $R_+:=[0,\infty)$ such that $p(0)=0$ and $p(x) \to \infty$ as $x \to \infty$.&#x0D; Given a real or complex-valued integrable function $f$ in Lebesgue's sense on every bounded interval $(0,x)$&#x0D; for $x&gt;0$, in symbol $f \in L^1_{loc} (R_+)$, we set&#x0D; $$&#x0D; s(x)=\int _{0}^{x}f(u)du&#x0D; $$&#x0D; and&#x0D; $$&#x0D; \sigma _{p}(s(x))=\frac{1}{p(x)}\int_{0}^{x}s(u)dp(u),\,\,\,\,x&gt;0&#x0D; $$&#x0D; provided that $p(x)&gt;0$.&#x0D; &#x0D; A function $s(x)$&#x0D; is said to be summable to $l$ by the weighted mean method determined&#x0D; by the function $p(x)$, in short, $(\overline{N},p)$ summable to $l$,&#x0D; if&#x0D; $$&#x0D; \lim_{x \to \infty}\sigma _{p}(s(x))=l.&#x0D; $$&#x0D; &#x0D; If the limit $\lim _{x \to \infty} s(x)=l$&#x0D; exists, then $\lim _{x \to \infty} \sigma _{p}(s(x))=l$ also exists. However, the converse is not true in general.&#x0D; In this paper, we give an alternative proof a Tauberian theorem stating that convergence follows from summability by weighted mean method on $R_+:=[0,\infty)$ and a Tauberian condition of slowly decreasing type with respect to the weight function due to Karamata. These Tauberian conditions are one-sided or two-sided if $f(x)$ is a real or complex-valued function, respectively. Alternative proofs of some well-known Tauberian theorems given for several important summability methods can be obtained by choosing some particular weight functions.

Training Neural Networks by Lifted Proximal Operator Machines.

We present the lifted proximal operator machine (LPOM) to train fully-connected feed-forward neural networks. LPOM represents the activation function as an equivalent proximal operator and adds the proximal operators to the objective function of a network as penalties. LPOM is block multi-convex in all layer-wise weights and activations. This allows us to develop a new block coordinate descent (BCD) method with convergence guarantee to solve it. Due to the novel formulation and solving method, LPOM only uses the activation function itself and does not require any gradient steps. Thus it avoids the gradient vanishing or exploding issues, which are often blamed in gradient-based methods. Also, it can handle various non-decreasing Lipschitz continuous activation functions. Additionally, LPOM is almost as memory-efficient as stochastic gradient descent and its parameter tuning is relatively easy. We further implement and analyze the parallel solution of LPOM. We first propose a general asynchronous-parallel BCD method with convergence guarantee. Then we use it to solve LPOM, resulting in asynchronous-parallel LPOM. For faster speed, we develop the synchronous-parallel LPOM. We validate the advantages of LPOM on various network architectures and datasets. We also apply synchronous-parallel LPOM to autoencoder training and demonstrate its fast convergence and superior performance.

Nonlinear boundary value problems relative to harmonic functions

We study the problem of finding a function u verifying −Δu=0 in Ω under the boundary condition ∂u∂n+g(u)=μ on ∂Ω where Ω⊂RN is a smooth domain, n is the normal unit outward vector to Ω, μ is a measure on ∂Ω and g a continuous nondecreasing function. We give sufficient condition on g for this problem to be solvable for any measure. When g(r)=|r|p−1r, p>1, we give conditions in order an isolated singularity on ∂Ω to be removable. We also give capacitary conditions on a measure μ in order the problem with g(r)=|r|p−1r to be solvable for some μ. We also study the isolated singularities of functions satisfying −Δu=0inΩ and ∂u∂n+g(u)=0 on ∂Ω∖{0}.

On the local existence for a weakly parabolic system in Lebesgue spaces

We consider the parabolic system ut−aΔu=f(v),vt−bΔv=g(u) in Ω×(0,T), where a,b>0, f,g:[0,∞)→[0,∞) are non-decreasing continuous functions and either Ω is a bounded domain with smooth boundary ∂Ω or the whole space RN. We characterize the functions f and g so that the system has a local solution for every initial data (u0,v0)∈Lr(Ω)×Ls(Ω), u0,v0≥0, r,s∈[1,∞).

Lifted Proximal Operator Machines

We propose a new optimization method for training feedforward neural networks. By rewriting the activation function as an equivalent proximal operator, we approximate a feedforward neural network by adding the proximal operators to the objective function as penalties, hence we call the lifted proximal operator machine (LPOM). LPOM is block multiconvex in all layer-wise weights and activations. This allows us to use block coordinate descent to update the layer-wise weights and activations. Most notably, we only use the mapping of the activation function itself, rather than its derivative, thus avoiding the gradient vanishing or blow-up issues in gradient based training methods. So our method is applicable to various non-decreasing Lipschitz continuous activation functions, which can be saturating and non-differentiable. LPOM does not require more auxiliary variables than the layer-wise activations, thus using roughly the same amount of memory as stochastic gradient descent (SGD) does. Its parameter tuning is also much simpler. We further prove the convergence of updating the layer-wise weights and activations and point out that the optimization could be made parallel by asynchronous update. Experiments on MNIST and CIFAR-10 datasets testify to the advantages of LPOM.

Some Tauberian theorems for weighted means of double integrals

Let p(x) and q(y) be nondecreasing continuous functions on [0,∞) such that p(0)=q(0)=0 and p(x),q(y)→∞ as x,y→∞. For a locally integrable function f(x,y) on R₊²=[0,∞)×[0,∞), we denote its double integral by F(x,y)=∫₀^{x}∫₀^{y}f(t,s)dtds and its weighted mean of type (α,β) by t_{α,β}(x,y)=∫₀^{x}∫₀^{y}(1-((p(t))/(p(x))))^{α}(1-((q(s))/(q(y))))^{β}f(t,s)dtds where α>-1 and β>-1. We say that ∫₀^{∞}∫₀^{∞}f(t,s)dtds is integrable to L by the weighted mean method of type (α,β) determined by the functions p(x) and q(x) if lim_{x,y→∞}t_{α,β}(x,y)=L exists. We prove that if lim_{x,y→∞}t_{α,β}(x,y)=L exists and t_{α,β}(x,y) is bounded on R₊² for some α>-1 and β>-1, then lim_{x,y→∞}t_{α+h,β+k}(x,y)=L exists for all h>0 and k>0. Finally, we prove that if ∫₀^{∞}∫₀^{∞}f(t,s)dtds is integrable to L by the weighted mean method of type (1,1) determined by the functions p(x) and q(x) and conditions [displaystyle] \displaystyle ((p(x))/(p′(x)))∫₀^{y}f(x,s)ds=O(1) and [displaystyle] \displaystyle ((q(y))/(q′(y)))∫₀^{x}f(t,y)dt=O(1) hold, then lim_{x,y→∞}F(x,y)=L exists.

Dynamics of a frictional system, accounting for hereditary-type friction and the mobility of the vibration limiter

Dynamics of a frictional system consisting of a rough body situated on a rough belt moving at a constant velocity is studied. The vibrations are limited by an elastic obstacle. The Coulomb-Hammonton dry friction characteristic is chosen, according to the hypothesis of Ishlinskiy and Kragelskiy, in the form of hereditary-type friction, where the coefficient of friction of relative rest (CFRR) is a monotone non-decreasing continuous function of the time of relative rest at the previous analogous time interval. A mathematical model and the structure of its phase space are presented, as well as the equations of point maps of the Poincare surface and the results of studying the dynamic characteristics of the parameters of the system (velocity of the belt, type of the functional relation of CFRR, rigidity of the elastic obstacle, etc.). A software product has been developed which makes it possible to find complex periodic motion regimes, as well as to calculate bifurcation diagrams used for determining the main variations of the motion regime from periodic to chaotic (the period doubling scenario).

On the Rate of Convergence of P-Iteration, SP-Iteration, and D-Iteration Methods for Continuous Nondecreasing Functions on Closed Intervals

We introduce a new iterative method called D-iteration to approximate a fixed point of continuous nondecreasing functions on arbitrary closed intervals. The purpose is to improve the rate of convergence compared to previous work. Specifically, our main result shows that D-iteration converges faster than P-iteration and SP-iteration to the fixed point. Consequently, we have that D-iteration converges faster than the others under the same computational cost. Moreover, the analogue of their convergence theorem holds for D-iteration.

Approximation of entropy solutions to degenerate nonlinear parabolic equations

We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion function (including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a family of systems of the same number of ODEs in the Banach space $$L^\infty $$ , whose solutions constitute a weak asymptotic solution of the original system of PDEs. We establish well posedness, monotonicity and $$L^1$$ -stability. We prove that the sequence of approximate solutions is strongly $$L^1$$ -precompact and that it converges to an entropy solution of the original equation in the sense of Carrillo. This result contributes to justify the use of this original method for the Cauchy problem to standard multidimensional systems of fluid dynamics for which a uniqueness result is lacking.

Shape preserving properties of univariate Lototsky–Bernstein operators

The main goal of this paper is to study shape preserving properties of univariate Lototsky–Bernstein operators Ln(f) based on Lototsky–Bernstein basis functions. The Lototsky–Bernstein basis functions bn,k(x)(0≤k≤n) of order n are constructed by replacing x in the ith factor of the generating function for the classical Bernstein basis functions of degree n by a continuous nondecreasing function pi(x), where pi(0)=0 and pi(1)=1 for 1≤i≤n. These operators Ln(f) are positive linear operators that preserve constant functions, and a non-constant function γnp(x). If all the pi(x) are strictly increasing and strictly convex, then γnp(x) is strictly increasing and strictly convex as well. Iterates LnM(f) of Ln(f) are also considered. It is shown that LnM(f) converges to f(0)+(f(1)−f(0))γnp(x) as M→∞. Like classical Bernstein operators, these Lototsky–Bernstein operators enjoy many traditional shape preserving properties. For every (1,γnp(x))-convex function f∈C[0,1], we have Ln(f;x)≥f(x); and by invoking the total positivity of the system {bn,k(x)}0≤k≤n, we show that if f is (1,γnp(x))-convex, then Ln(f;x) is also (1,γnp(x))-convex. Finally we show that if all the pi(x) are monomial functions, then for every (1,γn+1p(x))-convex function f, Ln(f;x)≥Ln+1(f;x) if and only if p1(x)=⋯=pn(x)=x.

Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term

AbstractMotivated by the work [9], in this paper we investigate the infinite boundary value problem associated with the semilinear PDE{Lu=f(u)+h(x)}on bounded smooth domains{\Omega\subseteq\mathbb{R}^{n}}, whereLis a non-divergence structure uniformly elliptic operator with singular lower-order terms. In the equation,fis a continuous non-decreasing function that satisfies the Keller–Osserman condition, whilehis a continuous function in Ω that may change sign, and which may be unbounded on Ω. Our purpose is two-fold. First we study some sufficient conditions onfandhthat would ensure existence of boundary blow-up solutions of the above equation, in which we allow the lower-order coefficients to be singular on the boundary. The second objective is to provide sufficient conditions onfandhfor the uniqueness of boundary blow-up solutions. However, to obtain uniqueness, we need the lower-order coefficients ofLto be bounded in Ω, but we still allowhto be unbounded on Ω.

Complete study of the existence and uniqueness of solutions for semilinear elliptic equations involving measures concentrated on boundary

The purpose of this paper is to study the weak solutions of the fractional elliptic problem (Section.Display) where , or , with is the fractional Laplacian defined in the principle value sense, is a bounded open set in with , is a bounded Radon measure supported in and is defined in the distribution sense, i.e. here denotes the unit inward normal vector at . In this paper, we prove that (0.1) with admits a unique weak solution when g is a continuous nondecreasing function satisfying Our interest then is to analyse the properties of weak solution when with , including the asymptotic behaviour near and the limit of weak solutions as . Furthermore, we show the optimality of the critical value in a certain sense, by proving the non-existence of weak solutions when . The final part of this article is devoted to the study of existence for positive weak solutions to (0.1) when and is a bounded nonnegative Radon measure supported in . We employ the Schauder’s fixed point theorem to obtain positive solution under the hypothesis that g is a continuous function satisfying -pagination

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.

Global bifurcation of positive solutions for a class of superlinear elliptic systems

We consider a system of semilinear equations of the form−Δu=λf(v)inΩ;−Δv=λg(u)inΩ;u=0=von∂Ω,} where λ∈R is the bifurcation parameter, Ω⊂RN; N≥2 is a bounded domain with smooth boundary ∂Ω. The nonlinearities f,g:R→(0,+∞) are nondecreasing continuous functions that have superlinear growth at infinity. We use bifurcation theory, combined with an approximation scheme, to establish the existence of an unbounded branch of positive solutions, emanating from the origin, which is bounded in positive λ-direction.If in addition, Ω is convex and f,g∈C1 satisfy certain subcriticality condition, we show that the branch must bifurcate from infinity at λ=0.

Multiple positive radial solutions to some Kirchhoff equations

In this paper, we discuss the existence and multiplicity of positive radial solutions to some Kirchhoff equations and elliptic equations. By using the Leggett–Williams three solutions theorem skillfully, we obtain two positive solutions to Kirchhoff equations under some appropriate conditions. As a direct corollary, we also obtain the same result to elliptic equations. In order to facilitate the proof of main results, we first establish a new simple three solutions theorems for the equation [a+bφ(x)]x=Ax, where A is a completely continuous operator, a>0, b⩾0 and φ(x)⩽k(‖x‖) for all x∈P, where k is a nondecreasing nonnegative continuous function, then apply it to prove the main results. In addition, we discover and prove a new inequality in the article. In the end of the paper, we present an example which makes the elliptic equation has infinite many positive radial solutions and give the approximate images of the nonlinear term.