Nondecreasing Continuous Function Research Articles

The entire large solutions for a quasilinear Schrödinger elliptic equation by the dual approach

In this paper, we consider a Modified Nonlinear Schrödinger equation Δu−Δ(|u|2γ)|u|2γ−2u=q(x)F(u) on Rn(n≥3), where γ>12, q is a nonnegative continuous function on Rn, F is a continuous positive and non-decreasing function on [0,∞). We establish sufficient conditions for the existence and nonexistence of positive entire large solutions for the given equation by using the dual approach.

On the stationary tail index of iterated random Lipschitz functions

Let Ψ,Ψ1,Ψ2,… be a sequence of i.i.d. random Lipschitz maps from a complete separable metric space (X,d) with unbounded metric d to itself and let Xn=Ψn∘⋯∘Ψ1(X0) for n=1,2,… be the associated Markov chain of forward iterations with initial value X0 which is independent of the Ψn. Provided that (Xn)n≥0 has a stationary law π and picking an arbitrary reference point x0∈X, we will study the tail behavior of d(x0,X0) under Pπ, viz. the behavior of Pπ(d(x0,X0)>t) as t→∞, in cases when there exist (relatively simple) nondecreasing continuous random functions F,G:R≥→R≥ such that F(d(x0,x))≤d(x0,Ψ(x))≤G(d(x0,x)) for all x∈X and n≥1. In a nutshell, our main result states that, if the iterations of i.i.d. copies of F and G constitute contractive iterated function systems with unique stationary laws πF and πG having power tails of order ϑF and ϑG at infinity, respectively, then lower and upper tail index of ν=Pπ(d(x0,X0)∈⋅) (to be defined in Section 2) are falling in [ϑG,ϑF]. If ϑF=ϑG, which is the most interesting case, this leads to the exact tail index of ν. We illustrate our method, which may be viewed as a supplement of Goldie’s implicit renewal theory, by a number of popular examples including the AR(1)-model with ARCH errors and random logistic transforms.

Noncommutative ordered spaces: examples and counterexamples

In order to introduce the notion of causality in noncommutative geometry, it is necessary to extend Gelfand theory to the context of ordered spaces. In a previous work we have already given an algebraic characterization of the set of non-decreasing continuous functions on a certain class of topological ordered spaces. Such a set is called an isocone, and there exist at least two versions of them (strong and weak) which coincide in the commutative case. In this paper, we introduce yet another breed of isocones, ultraweak isocones, which has a simpler definition with a clear physical meaning. We show that ultraweak and weak isocones are in fact the same, and completely classify those that live in a finite-dimensional -algebra, hence corresponding to finite noncommutative ordered spaces. We also give some examples in infinite dimension.

Boundary singularities of solutions of semilinear elliptic equations with critical Hardy potentials

We study the boundary behavior of positive functions u satisfying (E) −Δu−κd2(x)u+g(u)=0 in a bounded domain Ω of RN, where 0<κ≤14, g is a continuous nondecreasing function and d(.) is the distance function to ∂Ω. We first construct the Martin kernel associated to the linear operator Lκ=−Δ−κd2(x) and give a general condition for solving equation (E) with any Radon measure μ for boundary data. When g(u)=|u|q−1u we show the existence of a critical exponent qc=qc(N,κ)>1 with the following properties: when 0<q<qc any measure is eligible for solving (E) with μ for boundary data; if q≥qc, a necessary and sufficient condition is expressed in terms of the absolute continuity of μ with respect to some Besov capacity. The same capacity characterizes the removable compact boundary sets. At end any positive solution (F) −Δu−κd2(x)u+|u|q−1u=0 with q>1 admits a boundary trace which is a positive outer regular Borel measure. When 1<q<qc we prove that to any positive outer regular Borel measure we can associate a positive solutions of (F) with this boundary trace.

Density topologies on the plane between ordinary and strong. II

Abstract Let C0 denote a set of all non-decreasing continuous functions f : (0, 1] → (0, 1] such that limx→0+f(x) = 0 and f(x) ≤ x for every x ∊ (0, 1], and let A be a measurable subset of the plane. The notions of a density point of A with respect to f and the mapping defined on the family of all measurable subsets of the plane were introduced in Wagner-Bojakowska, E. Wilcziński, W.: Density topologies on the plane between ordinary and strong, Tatra Mt. Math. Publ. 44 (2009), 139 151. This mapping is a lower density, so it allowed us to introduce the topology Tf , analogously to the density topology. In this note, properties of the topology Tf and functions approximately continuous with respect to f are considered. We prove that (ℝ2, Tf) is a completely regular topological space and we study conditions under which topologies generated by two functions f and g are equal.

Measure boundary value problems for semilinear elliptic equations with critical Hardy potentials

Let Ω⊂RN be a bounded C2 domain and Lκ=−Δ−κd2 where d=dist(.,∂Ω) and 0<κ≤14. Let α±=1±1−4κ, λκ the first eigenvalue of Lκ with corresponding positive eigenfunction ϕκ. If g is a continuous nondecreasing function satisfying ∫1∞(g(s)+|g(−s)|)s−22N−2+α+2N−4+α+ds<∞, then for any Radon measures ν∈Mϕκ(Ω) and μ∈M(∂Ω) there exists a unique weak solution to problem Pν,μ: Lκu+g(u)=ν in Ω, u=μ on ∂Ω. If g(r)=|r|q−1u (q>1), we prove that, in the supercritical range of q, a necessary and sufficient condition for solving P0,μ with μ>0 is that μ is absolutely continuous with respect to the capacity associated with the space B2−2+α+2q′,q′(RN−1). We also characterize the boundary removable sets in terms of this capacity. In the subcritical range of q we classify the isolated singularities of positive solutions.

Convergence for a class of multi-point modified Chebyshev-Halley methods under the relaxed conditions

In this paper, the semilocal convergence for a class of multi-point modified Chebyshev-Halley methods in Banach spaces is studied. Different from the results in reference [11], these methods are more general and the convergence conditions are also relaxed. We derive a system of recurrence relations for these methods and based on this, we prove a convergence theorem to show the existence-uniqueness of the solution. A priori error bounds is also given. The R-order of these methods is proved to be 5+q with ??conditioned third-order Frechet derivative, where ?(μ) is a non-decreasing continuous real function for μ > 0 and satisfies ?(0) ? 0, ?(tμ) ≤ t q ?(μ) for μ > 0,t ? [0,1] and q ? [0,1]. Finally, we give some numerical results to show our approach.

Existence of Strictly Positive Solutions for Sublinear Elliptic Problems in Bounded Domains

Abstract Let Ω be a smooth bounded domain in RN and let m be a possibly discontinuous and unbounded function that changes sign in Ω. Let f : [0,∞) → [0,∞) be a nondecreasing continuous function such that k1ξp ≤ f (ξ) ≤ k2ξp for all ξ ≥ 0 and some k1, k2 &gt; 0 and p ∈ (0, 1). We study existence and nonexistence of strictly positive solutions for nonlinear elliptic problems of the form −Δu = m(x) f (u) in Ω, u = 0 on ∂Ω.

Optimal burn-in and warranty for a product with post-warranty failure penalty

This paper develops an optimization model to investigate the lengths of the optimal burn-in and warranty period, so that the mean of total product servicing cost is minimised. It is assumed that the cost of a minimal repair to a component at age t is a continuous non-decreasing function of t. Moreover, we model the customer dissatisfaction with product failures after the warranty within the product useful lifetime by introducing a post-warranty failure penalty cost to the manufacturer. The properties of the optimal burn-in time and optimal warranty policy are also analysed. Finally, under different product lifetime distributions, numerical examples and sensitivity analysis with respects to the values of the model parameters are provided.

An approximation theorem for non-decreasing functions on compact posets

In this short note, we prove a theorem of the Stone–Weierstrass sort for subsets of the cone of non-decreasing continuous functions on compact partially ordered sets.

Eigenvalue and spectral intervals for a nonlinear algebraic system

This paper studies the nonlinear algebraic system of the form x=λAFx, where λ>0,A is a positive n×n square matrix,x=x1,x2,...,xnT,Fx=fx1,fx2,...,fxnT,the nondecreasing continuous function f is defined on [0,∞) and fu>0 for u>0.The system covers many problems that arise in applications such as difference equations, boundary value problems and dynamical networks. Letf0=limu→0cfuuandf∞=limu→∞fuu.We classify the system according to six pairs of possible values of f0 and f∞ with the assumption that both f0 and f∞ exist (including ∞). For each case, R+=(0,∞) is divided into intervals for the value of λ (λ-intervals) that corresponds to existence, multiplicity and nonexistence of positive solutions of the system respectively. We then obtain intervals that contain only the eigenvalues of the nonlinear operator T=AF. Also, the resolvent and the spectrum of T are determined. Two examples are given to show the applications of the results.

Radial solutions for a quasilinear elliptic system of Schrödinger type

In this paper we analyze the existence of entire radially symmetric solutions for the Schrödinger system Δpiui+hi(r)|∇ui|pi−1=ai(r)fi(u1,…,ud)fori=1,…,d on RN, where pi>1,d∈{1,2,3,…},hi and ai are nonnegative radial continuous functions and fi are nonnegative non-decreasing continuous functions on [0,∞)d in all variables.

Large viscosity solutions for some fully nonlinear equations

We study existence, uniqueness and asymptotic behavior near the boundary of solutions of the problem $$\left\{\begin{array}{ll}-F(D^{2} u) + \beta (u) = f \quad {\rm in} \, \Omega, \\ u = + \infty \quad \quad \quad \quad \quad \quad \,\,\,\, {\rm on}\, \partial \Omega, \end{array} \right.\quad \quad \quad \quad \quad {\rm (P)}$$ where Ω is a bounded smooth domain in \({{\mathbb R}^N, N >1 , F}\) is a fully nonlinear elliptic operator and β is a nondecreasing continuous function. Assuming that β satisfies the Keller–Osserman condition, we obtain existence results which apply to \({f \in L^\infty_{loc}(\Omega)}\) or f having only local integrability properties where viscosity solutions are well defined, i.e. \({f \in L^N_{loc}(\Omega)}\). Besides, we find the asymptotic behavior near the boundary of solutions of (P) for a wide class of functions \({f \in \mathcal{C}(\Omega)}\). Based in this behavior, we also prove uniqueness.

Entropy formulation of degenerate parabolic equation with zero-flux boundary condition

We consider the general degenerate hyperbolic-parabolic equation: $$u_t + {\rm div} f(u) - \Delta \phi(u) = 0\; {\rm in} Q = (0, T) \times \Omega, \quad T > 0, \quad \Omega \subset \mathbb{R}^N;$$ with initial condition and the zero flux boundary condition. Here \({\phi}\) is a continuous non-decreasing function. Following Burger et al. (J Math Anal Appl 326:108–120, 2007), we assume that f is compactly supported (this is the case in several applications), and we define an appropriate notion of entropy solution. Using vanishing viscosity approximation, we prove existence of entropy solution for any space dimension N ≥ 1 under a partial genuine nonlinearity assumption on f. Uniqueness is shown for the case N = 1, using the idea of Andreianov and Bouhsiss (J Evol Equ 4:273–295, 2004), nonlinear semigroup theory and a specific regularity result for one dimension.

Pure jump increasing processes and the change of variables formula

Given an increasing process $(A_t)_{t\geq 0}$, we characterize the right continuous non-decreasing functions $f: \mathbb{R}_+\to \mathbb{R}_+$ that map $A$ to a pure jump process. As an example of application, we show for instance that functions with bounded variations belong to the domain of the extended generator of any subordinator with no drift and infinite Levy measure.

The (C, α) integrability of functions by weighted mean methods

Let p(x) be a nondecreasing continuous function on [0, ?) such that p(0) = 0 and p(t) ? ? as t ? ?. For a continuous function f (x) on [0, ?), we define s(t)= ?0t f(u)du and ?? (t) =?0t? (1- p(u)/p(t))? f(u)du. We say that a continuous function f (x) on [0, ?) is (C, ?) integrable to a by the weighted mean method determined by the function p(x) for some ? &gt; ?1 if the limit limt?? ?? (t) = a exists. We prove that if the limit limt?? ?? (t) = a exists for some ? &gt; ?1, then the limit limt?? ??+h (t) = a exists for all h &gt; 0. Next, we prove that if the limit limt?? ?? (t) = a exists for some ? &gt; 0 and p(t)/p?(t) f(t)= O(1), t ? ?, then the limit limt?? ???1 (t) = a exists.

The period function of a delay differential equation and an application

We consider the delay differential equation $$\dot x(t) = - \mu x(t) + f(x(t - \tau ))$$ , where µ, τ are positive parameters and f is a strictly monotone, nonlinear C 1-function satisfying f(0) = 0 and some convexity properties. It is well known that for prescribed oscillation frequencies (characterized by the values of a discrete Lyapunov functional) there exists τ* > 0 such that for every τ > τ* there is a unique periodic solution. The period function is the minimal period of the unique periodic solution as a function of τ > τ*. First we show that it is a monotone nondecreasing Lipschitz continuous function of τ with Lipschitz constant 2. As an application of our theorem we give a new proof of some recent results of Yi, Chen and Wu [14] about uniqueness and existence of periodic solutions of a system of delay differential equations.

Estimates of operator moduli of continuity

In Aleksandrov and Peller (2010) [2] we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in Aleksandrov and Peller (2010) [2] for certain special classes of functions. In particular, we improve estimates of Kato (1973) [18] and show that ‖ | S | − | T | ‖ ⩽ C ‖ S − T ‖ log ( 2 + log ‖ S ‖ + ‖ T ‖ ‖ S − T ‖ ) for all bounded operators S and T on Hilbert space. Here | S | = def ( S ⁎ S ) 1 / 2 . Moreover, we show that this inequality is sharp. We prove in this paper that if f is a nondecreasing continuous function on R that vanishes on ( − ∞ , 0 ] and is concave on [ 0 , ∞ ) , then its operator modulus of continuity Ω f admits the estimate Ω f ( δ ) ⩽ const ∫ e ∞ f ( δ t ) d t t 2 log t , δ > 0 . We also study the problem of sharpness of estimates obtained in Aleksandrov and Peller (2010) [2,3]. We construct a C ∞ function f on R such that ‖ f ‖ L ∞ ⩽ 1 , ‖ f ‖ Lip ⩽ 1 , and Ω f ( δ ) ⩾ const δ log 2 δ , δ ∈ ( 0 , 1 ] . In the last section of the paper we obtain sharp estimates of ‖ f ( A ) − f ( B ) ‖ in the case when the spectrum of A has n points. Moreover, we obtain a more general result in terms of the ε-entropy of the spectrum that also improves the estimate of the operator moduli of continuity of Lipschitz functions on finite intervals, which was obtained in Aleksandrov and Peller (2010) [2].

Multiplicity Results for Classes of Infinite Positone Problems

We study positive solutions to the singular boundary value problem \left\{\begin{alignedat}{2}-\Delta_p u &amp;= \lambda \frac{f(u)}{u^\beta}&amp; \quad &amp;\text{in}~\Omega \\ u &amp;= 0 &amp; \quad &amp;\text{on}~\partial \Omega, \end{alignedat}\right. where \Delta_p u = div (|\nabla u|^{p-2}\nabla u) , p &gt; 1, \lambda &gt; 0, \beta \in (0,1) and \Omega is a bounded domain in \mathbb{R}^{N}, N \geq 1. Here f:~[0, \infty)\rightarrow (0, \infty) is a continuous nondecreasing function such that \lim_{u\rightarrow \infty} \frac{f(u)}{u^{\beta+p-1}}= 0. We establish the existence of multiple positive solutions for certain range of \lambda when f satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is f(u)=e^{\frac{\alpha u}{\alpha+u}} for \alpha \gg 1. We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-supersolutions.

Existence results for nonlinear pseudoparabolic problems

In this paper, we are interested in nonlinear pseudoparabolic problems of type: f ( t , x , ∂ t u ) − Div [ a ( x , u , ∂ t u ) ∇ u ] − Div [ b ( x , u , ∂ t u ) ∇ ∂ u t ] = g . f is a nondecreasing continuous function with respect to its third argument, a is bounded and b is positive and bounded. The result of existence is proved thanks to a time discretization scheme. Then, we derive some applications to the equation of Barenblatt, a degenerate case and differential inclusions. Finally, some numerical illustrations are proposed.