Nonnegative Function Research Articles

Positive Solutions for Third-Order Boundary Value Problems with the Integral Boundary Conditions and Dependence on the First-Order Derivatives

In this paper, by the use of a new fixed point theorem and the Green function of BVPs, the existence of at least one positive solution for the third-order boundary value problem with the integral boundary conditions is considered,where there is a nonnegative continuous function. Finally, an example which to illustrate the main conclusions of this paper is given.

Weighted Inequalities for a Superposition of the Copson Operator and the Hardy Operator

We study a three-weight inequality for the superposition of the Hardy operator and the Copson operator, namely \begin{equation*} \bigg(\int_a^b \bigg(\int_t^b \bigg(\int_a^s f(\tau)^p v(\tau) \,d\tau \bigg)^\frac{q}{p} u(s) \,ds \bigg)^{\frac{r}{q}} w(t) \,dt \bigg)^{\frac{1}{r}} \leq C \int_a^b f(t)\,dt, \end{equation*} in which $(a,b)$ is any nontrivial interval, $q,r$ are positive real parameters and $p\in(0,1]$. A simple change of variables can be used to obtain any weighted $L^p$-norm with $p\ge1$ on the right-hand side. Another simple change of variables can be used to equivalently turn this inequality into the one in which the Hardy and Copson operators swap their positions. We focus on characterizing those triples of weight functions $(u,v,w)$ for which this inequality holds for all nonnegative measurable functions $f$ with a constant independent of $f$. We use a new type of approach based on an innovative method of discretization which enables us to avoid duality techniques and therefore to remove various restrictions that appear in earlier work. This paper is dedicated to Professor Stefan Samko on the occasion of his 80th birthday.

Comparison principles for solutions to the fractional differential inequalities with the general fractional derivatives and their applications

In this paper, we discuss some comparison principles for the solutions to the fractional differential inequalities with the general fractional derivatives in the Caputo and Riemann-Liouville senses. These general fractional derivatives are defined as compositions of the first order derivative and a convolution integral with a non-negative and non-increasing kernel. First we prove some estimates for these derivatives acting on the non-negative functions. These estimates are employed for derivation of the comparison principles in several different forms. Finally, we consider an application of the comparison principles for analysis of solutions to the initial-value problems for the fractional differential equations with the general fractional derivatives.

Functional John ellipsoids

We introduce a new way of representing logarithmically concave functions on Rd. It allows us to extend the notion of the largest volume ellipsoid contained in a convex body to the setting of logarithmically concave functions as follows. For every s>0, we define a class of non-negative functions on Rd derived from ellipsoids in Rd+1. For any log-concave function f on Rd, and any fixed s>0, we consider functions belonging to this class, and find the one with the largest integral under the condition that it is pointwise less than or equal to f, and we call it the John s-function of f. After establishing existence and uniqueness, we give a characterization of this function similar to the one given by John in his fundamental theorem. We find that John s-functions converge to characteristic functions of ellipsoids as s tends to zero and to Gaussian densities as s tends to infinity.As an application, we prove a quantitative Helly type result: the integral of the pointwise minimum of any family of log-concave functions is at least a constant cd multiple of the integral of the pointwise minimum of a properly chosen subfamily of size 3d+2, where cd depends only on d.

Characterisation of the class of bell-shaped functions

A non-negative function f is said to be bell-shaped if f tends to zero at \(\pm \infty \) and the n-th derivative of f changes its sign n times for every \(n = 0, 1, 2, \ldots \) We provide a complete characterisation of the class of bell-shaped functions, resembling Bernstein’s identification of completely monotone functions with Laplace transforms of non-negative measures. More precisely, we prove that every bell-shaped function is a convolution of a Pólya frequency function and an absolutely monotone-then-completely monotone function. An equivalent condition in terms of the holomorphic extension of the Fourier transform is also given. As a corollary, we find various properties of bell-shaped functions. In particular, we prove that bell-shaped probability distributions are infinitely divisible, and that a random walk \(X_n\) (or a Lévy process \(X_t\)) have bell-shaped distributions if and only if the distribution of \(X_1\) is an extended generalised gamma convolution. We also link the rate of growth of the zeroes of n-th derivative of a bell-shaped function f with numbers p such that \(f + p f'\) is bell-shaped. Our result provides a practical method for checking whether a given distribution is bell-shaped, and it is new even for one-sided distributions.

Joint flight scheduling and task allocation for secure data collection in UAV-aided IoTs

With the flexible mobility and deployment, Unmanned Aerial Vehicles (UAVs) have created a new dimension to provide data collection services for a large number of distributed wireless sensory devices. Unfortunately, the broadcast nature of wireless channels make it hard to guarantee the security of data transmission in Internet of Things (IoTs). In this paper, focusing on the scenario where potential eavesdroppers intend to intercept the data uploading of legitimate nodes, we take the advantages of the UAV technology and the physical-layer security approach to study the Secure Data Collection Problem (SDCP). We first explore the flight scheduling and task allocation, that is, determining which areas to be visited and who to be served by UAVs, to establish favorable communication channels. Then we set up the utility function with secrecy rates and non-negative budget constraints. Due to the coupling between the UAV scheduling and task allocation optimization variables in the SDCP, we construct a surrogate function to achieve decoupling and reformulate SDCP into the SDCP-m problem, which is analyzed and proved to be a nonnegative monotone submodular function subject to a general constraint. Finally, an efficient Cost–Benefit Greedy (CBG) algorithm, which is theoretically analyzed to seek the quantitative value of approximation ratio, is proposed. The simulation results show that the CBG algorithm not only outperforms benchmark algorithms in terms of the average secrecy rate and the iteration number, but also generates a nearly optimal trajectory.

On positivity of orthogonal series and its applications in probability

We give necessary and sufficient conditions for an orthogonal series to converge in the mean-squares to a nonnegative function. We present many examples and applications, in analysis and probability. In particular, we give necessary and sufficient conditions for a Lancaster-type of expansion sum _{nge 0}c_{n}alpha _{n}(x)beta _{n}(y) with two sets of orthogonal polynomials left{ alpha _{n}right} and left{ beta _{n}right} to converge in means-squares to a nonnegative bivariate function. In particular, we study the properties of the set C(alpha ,beta ) of the sequences left{ c_{n}right} , for which the above-mentioned series converge to a nonnegative function and give conditions for the membership to it. Further, we show that the class of bivariate distributions for which a Lancaster type expansion can be found, is the same as the class of distributions having all conditional moments in the form of polynomials in the conditioning random variable.

Polynomial-time Algorithm for Maximum Weight Independent Set on P 6 -free Graphs

In the classic Maximum Weight Independent Set problem, we are given a graph G with a nonnegative weight function on its vertices, and the goal is to find an independent set in G of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any P 6 -free graph, that is, a graph that has no path on 6 vertices as an induced subgraph. This improves the polynomial-time algorithm on P 5 -free graphs of Lokshtanov et al. [ 15 ] and the quasipolynomial-time algorithm on P 6 -free graphs of Lokshtanov et al. [ 14 ]. The main technical contribution leading to our main result is enumeration of a polynomial-size family ℱ of vertex subsets with the following property: For every maximal independent set I in the graph, ℱ contains all maximal cliques of some minimal chordal completion of G that does not add any edge incident to a vertex of I .

Nontrivial solutions for a class of Hamiltonian strongly degenerate elliptic system

In this paper, we study the existence of nontrivial to the following Hamiltonian strongly degenerate elliptic system in whole space where are nonnegative functions with and V might unbounded or vanish at infinity. Here is the strongly degenerate operator and the nonlinearity terms are continuous functions and satisfy some assumptions that will be specified later. Due to the unbounded or vanishing of potentials and degeneracy of the operator, some new compact embedding theorems are used in the proof. Our results extend and generalize some existing results.

Numerical aspects of a Godunov-type stabilization scheme for the Boltzmann transport equation

We discuss the numerical aspects of the Boltzmann transport equation (BE) for electrons in semiconductor devices, which is stabilized by Godunov’s scheme. The k-space is discretized with a grid based on the total energy to suppress spurious diffusion in the stationary case. Band structures of arbitrary shape can be handled. In the stationary case, the discrete BE yields always nonnegative distribution functions and the corresponding system matrix has only eigenvalues with positive real parts (diagonally dominant matrix) resulting in an excellent numerical stability. In the transient case, this property yields an upper limit for the time step ensuring the stability of the CPU-efficient forward Euler scheme and a positive distribution function. Similar to the Monte-Carlo (MC) method, the discrete BE can be solved in time together with the Poisson equation (PE), where the time steps for the PE are split into shorter time steps for the BE, which can be performed at minor additional computational cost. Thus, similar to the MC method, the transient approach is matrix-free and the solution of memory and CPU intensive large systems of linear equations is avoided. The numerical properties of the approach are demonstrated for a silicon nanowire NMOSFET, for which the stationary I–V characteristics, small-signal admittance parameters and the switching behavior are simulated with and without strong scattering. The spurious damping introduced by Godunov’s (upwind) scheme is found to be negligible in the technically relevant frequency range. The inherent asymmetry of the upwind scheme results in an error for very strong scattering that can be alleviated by a finer grid in transport direction.

Robust backstepping non‐smooth practical tracking for nonlinear systems with mismatched uncertainties

AbstractThis article investigates the robust finite‐time output tracking control problem for a class of uncertain nonlinear systems subject to mismatched uncertainties and external disturbances. The nonlinear systems with nonstrict‐feedback form are considered, which possess unknown control coefficients and uncertainties bounded by nonnegative convex functions. Based on signal compensation theory and backstepping methodology, a new robust finite‐time controller is proposed, which incorporates a non‐smooth time‐invariant controller to achieve finite‐time convergence property and a robust compensator introduced to restrain the influences of uncertainties. With the help of finite‐time Lyapunov theorem, the robust finite‐time tracking performance of the closed‐loop system is proved. Finally, two simulation examples are given to validate the effectiveness of the proposed control approach.

On the stability and behavior of solutions in mixed differential equations with delays and advances

In this study, some new results are obtained on asymptotic behavior and stability analysis of the mixed type differential equation where , and are real nonnegative continuous functions on , and and are real valued functions of bounded variation on . These results were obtained by using an appropriate real root of the characteristic equation. Moreover, by the use of two appropriate distinct real roots of the corresponding characteristic equation, a new result on the behavior of solutions is established. Five examples are also given to illustrate our results. We also presented the application of the obtained results in the special case of constant coefficients and have given three different cases in one example. The results obtained in this article show that real roots play an important role.

On the robustness of the approximate price of anarchy in generalized congestion games

One of the main results shown through Roughgarden's notions of smooth games and Robust Price of Anarchy is that, for any sum-bounded utilitarian social function, the worst-case Price of Anarchy of Coarse Correlated Equilibria coincides with that of Pure Nash Equilibria in the class of weighted congestion games with non-negative and non-decreasing latency functions and that such a value can always be derived through the, so called, smoothness argument. We significantly extend this result by proving that, for a variety of (even non-sum-bounded) utilitarian and egalitarian social functions, and for a broad generalization of the class of weighted congestion games with non-negative (and possibly decreasing) latency functions, the worst-case Price of Anarchy of ϵ-approximate Coarse Correlated Equilibria still coincides with that of ϵ-approximate Pure Nash Equilibria, for any ϵ≥0. As a byproduct of our proof, it also follows that such a value can always be determined by making use of the primal-dual method we introduced in a previous work.

A Simple Method to Find All Solutions to the Functional Equation of the Smoothing Transform

Given a nonincreasing null sequence \(T = (T_j)_{j \geqslant 1}\) of nonnegative random variables satisfying some classical integrability assumptions and \({\mathbb {E}}(\sum _{j}T_{j}^{\alpha })=1\) for some \(\alpha >0\), we characterize the solutions of the well-known functional equation $$\begin{aligned} f(t)\,=\,\textstyle {\mathbb {E}}\left( \prod _{j\geqslant 1}f(tT_{j})\right) ,\quad t\geqslant 0, \end{aligned}$$related to the so-called smoothing transform and its min-type variant. In order to do so within the class of nonnegative and nonincreasing functions, we provide a new three-step method whose merits are that (i) it simplifies earlier approaches in some relevant aspects; (ii) it works under weaker, close to optimal conditions in the so-called boundary case when \({\mathbb {E}}\big (\sum _{j\geqslant 1}T_{j}^{\alpha }\log T_{j}\big )=0\); (iii) it can be expected to work as well in more general setups like random environment. At the end of this article, we also give a one-to-one correspondence between those solutions that are Laplace transforms and thus correspond to the fixed points of the smoothing transform and certain fractal random measures. The latter are defined on the boundary of a weighted tree related to an associated branching random walk.

The Persistence Exponents of Gaussian Random Fields Connected by the Lamperti Transform

The (fractional) Brownian sheet is the simplest example of a Gaussian random field $$X$$ whose covariance is the tensor product of a finite number (d) of nonnegative correlation functions of self- similar Gaussian processes. We consider the homogeneous Gaussian field $$Y$$ obtained by applying to X the exponential change of time (more precisely, the Lamperti transform). Under some assumptions, we prove the existence of the persistence exponents for both fields, $$X$$ and $$Y$$ , and find the relation between them. The exponent for any random function $$Z$$ is $$\psi (T)$$ if lim $$\ln P(Z({\mathbf{t}}) \le ,{\mathbf{t}} \in TG)/\psi (T) = - 1$$ , $$T > > {1}$$ , where $$G$$ is a d-dimensional region containing 0, and $$T$$ is a similarity coefficient. The considered problem was raised by Li and Shao [Ann. Probab. 32:1, 2004] and originally concerned the Brownian sheet.

Regularization effect of the mixed-type damping in a higher-dimensional logarithmic Keller-Segel system related to crime modeling.

We study a logarithmic Keller-Segel system proposed by Rodríguez for crime modeling as follows: $ \begin{equation*} \left\{ \begin{split} &u_t = \Delta u-\chi\nabla\cdot\left(u\nabla\ln v\right)- \kappa uv+ h_1,\\ &v_t = \Delta v- v+ u+h_2, \end{split} \right. \end{equation*} $ in a bounded and smooth spatial domain $ \Omega\subset \mathbb R^n $ with $ n\geq3 $, with the parameters $ \chi > 0 $ and $ \kappa > 0 $, and with the nonnegative functions $ h_1 $ and $ h_2 $. For the case that $ \kappa = 0 $, $ h_1\equiv0 $ and $ h_2\equiv0 $, recent results showed that the corresponding initial-boundary value problem admits a global generalized solution provided that $ \chi < \chi_0 $ with some $ \chi_0 > 0 $. In the present work, our first result shows that for the case of $ \kappa > 0 $ such problem possesses global generalized solutions provided that $ \chi < \chi_1 $ with some $ \chi_1 > \chi_0 $, which seems to confirm that the mixed-type damping $ -\kappa uv $ has a regularization effect on solutions. Besides the existence result for generalized solutions, a statement on the large-time behavior of such solutions is derived as well.

Semilinear heat equation with singular terms

The main goal of this paper is to analyze the existence and nonexistence as well as the regularity of positive solutions for the following initial parabolic problem { ∂ t u − Δ u = μ u | x | 2 + f u σ in Ω T := Ω × ( 0 , T ) , u = 0 on ∂ Ω × ( 0 , T ) , u ( x , 0 ) = u 0 ( x ) in Ω , where Ω ⊂ R N , N ≥ 3 , is a bounded open, σ ≥ 0 and μ &gt; 0 are real constants and f ∈ L m ( Ω T ) , m ≥ 1 , and u 0 are nonnegative functions. The study we lead shows that the existence of solutions depends on σ and the summability of the datum f as well as on the interplay between μ and the best constant in the Hardy inequality. Regularity results of solutions, when they exist, are also provided. Furthermore, we prove uniqueness of finite energy solutions.

Decay rates for the damped wave equation with finite regularity damping

Decay rates for the energy of solutions of the damped wave equation on the torus are studied. In particular, damping invariant in one direction and equal to a sum of squares of nonnegative functions with a particular number of derivatives of regularity is considered. For such damping energy decays at rate $1/t^{2/3}$. If additional regularity is assumed the decay rate improves. When such a damping is smooth the energy decays at $1/t^{4/5-\delta}$. The proof uses a positive commutator argument and relies on a pseudodifferential calculus for low regularity symbols.

Global solutions of a doubly tactic resource consumption model with logistic source

We study a doubly tactic resource consumption model (ut = Δu − ∇ · (u∇w), vt = Δv − ∇ · (v∇u) + v(1 − vβ−1), wt = Δw − (u + v)w − w + r) in a smooth bounded domain Ω∈R2 with homogeneous Neumann boundary conditions, where r∈C1(Ω̄×[0,∞))∩L∞(Ω×(0,∞)) is a given non-negative function fulfilling ∫tt+1∫Ω|∇r|2&amp;lt;∞ for all t ≥ 0. It is shown that, first, if β &amp;gt; 2, then the corresponding Neumann initial-boundary problem admits a global bounded classical solution. Second, when β = 2, the Neumann initial-boundary problem admits a global generalized solution.

Monotonicity and extremality analysis of difference operators in Riemann-Liouville family

&lt;abstract&gt;&lt;p&gt;In this paper, we will discuss the monotone decreasing and increasing of a discrete nonpositive and nonnegative function defined on $ \mathbb{N}_{r_{0}+1} $, respectively, which come from analysing the discrete Riemann-Liouville differences together with two necessary conditions (see Lemmas 2.1 and 2.3). Then, the relative minimum and relative maximum will be obtained in view of these results combined with another condition (see Theorems 2.1 and 2.2). We will modify and reform the main two lemmas by replacing the main condition with a new simpler and stronger condition. For these new lemmas, we will establish similar results related to the relative minimum and relative maximum again. Finally, some examples, figures and tables are reported to demonstrate the applicability of the main lemmas. Furthermore, we will clarify that the first condition in the main first two lemmas is solely not sufficient for the function to be monotone decreasing or increasing.&lt;/p&gt;&lt;/abstract&gt;