Nonnegative Continuous Function Research Articles

Multi‐bump solutions for the nonlinear magnetic Schrödinger equation with logarithmic nonlinearity

AbstractIn this paper, we study the following nonlinear magnetic Schrödinger equation with logarithmic nonlinearity where the magnetic potential , is a parameter and the nonnegative continuous function has the deepening potential well. Using the variational methods, we obtain that the equation has at least multi‐bump solutions when is large enough.

Застосування методу двобічних наближень до знаходження додатних аксіально-симетричних розв’язків крайових задач з монотонними нелінійностями

This paper considers the application of the method of two-sided approximations for finding positive solutions to boundary value problems for nonlinear elliptic differential equations with axial symmetry. The case of Dirichlet boundary conditions (first kind) is considered, with a power-type monotone nonlinearity, where the exponent ranges from 0 to 1. By transitioning to polar coordinates in the boundary value problem for the elliptic equation, due to the axial symmetry of the solution, the problem is reduced to a boundary value problem for an ordinary differential equation on an interval with respect to a function that depends only on the polar radius, thus eliminating the dependence on the angle. The pole of the polar coordinate system becomes a singular point, where a boundedness condition must be imposed. For the boundary value problem, a Green's function is constructed to further reduce the problem to a Hammerstein integral equation. The integral equation is treated as a nonlinear operator equation in a Banach space of continuous functions on the interval, partially ordered by a cone of non-negative continuous functions on this interval. The operator is examined for properties such as monotonicity (isotonicity), positivity, boundedness, and concavity. Next, the initial approximation is found as the endpoints of the invariant conical segment for the isotone operator in such a way as to ensure the highest convergence rate of the iterative process. The following iterative sequences of two-sided approximations are constructed: the first sequence, which is non-decreasing with respect to the cone, and the second sequence, which is non-increasing with respect to the cone. At each iteration, the arithmetic mean of the upper and lower approximations is taken as the next approximation. The iterative process is terminated when the error estimate of the solution satisfies the specified accuracy. The theoretical results obtained in this work were validated through a computational experiment. The dependence of the solution on the parameters in the right-hand side was analyzed and illustrated with corresponding graphs

Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces

In this paper we investigate the existence of positive solution for a class of quasilinear problem on an Orlicz-Sobolev space that can be nonreflexive $$ - \Delta_{\Phi} u +V(x)\phi(|u|)u= K(x)f(u)\quad\mbox{in } \mathbb{R}^{N}, $$ where $ N \geq 2 $, $ V, K $ are nonnegative continuous functions and $f$ is a continuous function with a quasicritical growth. Here we extend the Hardy-type inequalities presented in \cite{AlvesandMarco} to nonreflexive Orlicz spaces. Through inequalities together with a variational method for non-differentiable functionals we will obtain a ground state solution. We analyze also the problem with $V=0$.

Efficient α-Dense Curve Strategies for Multiple Integrals over Hyper-rectangle Regions

Abstract In this paper, we propose an approximation technique to compute multiple integrals of a non-negative real continuous function over a hyper-rectangle Ω of ℝ n . The main idea is to use a reducing transformation procedure obtained by using α-dense curves. First, the region Ω f whose measure represents the value of the integral, is densified by a specific curve ℓ α(t) of finite length. Therefore, the multiple integral can be approached by a simple integral corresponding to ℓ α (t). Some numerical examples are given.

A necessary and sufficient condition for the global existence of solutions to nonlinear reaction‐diffusion equations on the half‐spaces in ℝN

In this paper, we study the existence and nonexistence of the global solutions to nonlinear reaction‐diffusion equations where is the half‐space , is a nonnegative continuous function, and is a locally Lipschitz function with some additional properties. The purpose of this paper is to give a necessary and sufficient condition for the existence of global solutions as follows: There is no global solution for any nonnegative and nontrivial initial data if and only if for every . In fact, we introduce a very special curve in to obtain the lower bound of decay of the heat semigroup, which is essential to prove the main result.

IDEALS OF FUNCTION SPACE IN THE LIGHT OF AN EXPONENTIAL ALGEBRA

Exponential algebra is a new algebraic structure consisting of a semigroup structure, a scalar multiplication, an internal multiplication and a partial order [introduced in [4]]. This structure is based on the structure `exponential vector space' which is thoroughly developed by Priti Sharma et. al. in [11] [This structure was actually proposed by S. Ganguly et. al. in [1] with the name `quasi-vector space'] Exponential algebra can be considered as an algebraic ordered extension of the concept of algebra. In the present paper we have shown that the function space $ C^+(\mathbf X) $ of all non-negative continuous functions on a topological space $\mathbf X$ is a topological exponential algebra under the compact open topology. Also we have discussed the ideals and maximal ideals of $ C^+(\mathbf X) $. We find an ideal of $ C^+(\mathbf X)$ which is not a maximal ideal in general; actually maximality of that ideal depends on the topology of $\mathbf{X}$. The concept of ideals of exponential algebra was introduced by us in [4].

HL-nets: Physics-informed neural networks for hydrodynamic lubrication with cavitation

Recently, physics-informed neural networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs). In this study, we establish a deep learning computational framework, HL-nets, for computing the flow field of hydrodynamic lubrication involving cavitation effects. The Swift-Stieber (SS) and the Jakobsson-Floberg-Olsson (JFO) cavitation conditions are implemented in the PINNs to solve the Reynolds equation. For the non-negativity constraint of the SS cavitation condition, a penalizing scheme with a residual of the non-negativity and an imposing scheme with a continuous differentiable non-negative function are proposed. For the complementarity constraint of the JFO cavitation condition, the pressure and cavitation fraction are taken as the neural network outputs, and the residual of the Fischer-Burmeister (FB) equation constrains their complementary relationships. Multi-task learning (MTL) methods are applied to balance the loss terms of functions and constraints described above. To estimate the accuracy of HL-nets, we present a numerical solution of the Reynolds equation for oil-lubricated bearings involving cavitation. The results indicate that the proposed HL-nets can highly accurately simulate hydrodynamic lubrication involving cavitation phenomena. The imposing scheme can effectively improve the accuracy of the training results of PINNs, and it is expected to have great potential to be applied to different fields where the non-negativity constraint is needed.

Localized growth speed of the digits in Engel expansions

We introduce a localized version of a Galambos's question about the growth speed of the digits in Engel expansion, namely the set{x∈(0,1]:limn→∞⁡1nlog⁡dn(x)=α(x)}, where α:[0,1]→[0,∞] is a nonnegative continuous function and dn(x) denotes the nth digits in the Engel expansion of x. The Hausdorff dimension is shown to be irrelevant of the function α(x). As applications, this answers Galambos's question and strengthens some results by Liu & Wu.

Nonlinear disjointness/supplement preservers of nonnegative continuous functions

Let F(X),F(Y) be sufficiently large sets of nonnegative continuous real-valued functions defined on completely regular spaces X,Y, respectively. Let Φ:F(X)→F(Y) be a surjective map satisfying thatf∨g>0⟺Φ(f)∨Φ(g)>0,∀f,g∈F(X). In many cases, we show that there is a homeomorphism τ:Y→X such thatΦ(f)(y)≠0⟺f(τ(y))≠0,∀f∈F(X),∀y∈Y.Assume X,Y are locally compact Hausdorff (resp. separable and metrizable) and Φ:C0(X)+→C0(Y)+ (resp. Φ:Cb(X)+→Cb(Y)+) is a surjective map. We show that Φ preserves the norms of infima, i.e.,‖Φ(f)∧Φ(g)‖=‖f∧g‖,∀f,g∈C0(X)+ (resp.Cb(X)+), if and only if there is a homeomorphism τ:Y→X such thatΦ(f)(y)=f(τ(y)),∀f∈C0(X)+ (resp.Cb(X)+), ∀y∈Y.

SOME RESULTS IN CONNECTION WITH SUM AND PRODUCT THEOREMS RELATED TO GENERALIZED RELATIVE ORDER (α, β) AND GENERALIZED RELATIVE TYPE (α, β) OF ENTIRE FUNCTIONS IN THE UNIT DISC

Orders and types of entire functions have been actively investigated by many authors. In this paper, we investigate some basic properties in connection with sum and product of generalized relative order (α, β), generalized relative type (α, β) and generalized relative weak type (α, β) of entire functions in the unit disc D with respect to another entire function where α, β are continuous non-negative functions on (−∞,+∞).

A stochastic delayed SIS epidemic model with Holling type II incidence rate

In this article, a stochastic SIS epidemic model with constant time delay and Holling type II incidence rate is investigated. We firstly show the existence, uniqueness, and moment boundedness of the global positive solution. Then we extend the initial value space to a complete nonnegative continuous function space and obtain the existence of invariant measures for this system. Furthermore, the analysis of the asymptotic behavior around the disease-free equilibrium is given. To demonstrate, some numerical examples are provided to illustrate our results.

On a diffusion on finite adeles and the Feynman-Kac integral

Let K be an algebraic number field. With K, we associate the ring of finite adeles AK. Following a recent result of Weisbart on diffusions on finite rational adeles AQ, we define the Vladimirov operator ΔAK on AK and define the Brownian motion on the group AK. We also consider the Schrödinger operator −HAK=−ΔAK+V with a potential operator V given by a non-negative continuous function v on AK. We prove a version of the Feynman–Kac formula for the Schrödinger semigroup generated by −HAK.

Systems of Riemann–Liouville Fractional Differential Equations with ρ-Laplacian Operators and Nonlocal Coupled Boundary Conditions

In this paper, we study the existence of positive solutions for a system of fractional differential equations with ρ-Laplacian operators, Riemann–Liouville derivatives of diverse orders and general nonlinearities which depend on several fractional integrals of differing orders, supplemented with nonlocal coupled boundary conditions containing Riemann–Stieltjes integrals and varied fractional derivatives. The nonlinearities from the system are continuous nonnegative functions and they can be singular in the time variable. We write equivalently this problem as a system of integral equations, and then we associate an operator for which we are looking for its fixed points. The main results are based on the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type.

Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

In this paper we are interested in positive classical solutions of(1){−Δu=a(x)up−1 in Ω,u>0 in Ω,u=0 on ∂Ω, where Ω is a bounded annular domain (not necessarily an annulus) in RN(N≥3) and a(x) is a nonnegative continuous function. We show the existence of a classical positive solution for a range of supercritical values of p when the problem enjoys certain mild symmetry and monotonicity conditions. As a consequence of our results, we shall show that (1) has ⌊N2⌋ (the floor of N2) positive nonradial solutions when a(x)=1 and Ω is an annulus with certain assumptions on the radii. We also obtain the existence of positive solutions in the case of toroidal domains. Our approach is based on a new variational principle that allows one to deal with supercritical problems variationally by limiting the corresponding functional on a proper convex subset instead of the whole space at the expense of a mild invariance property.

Permitted Sets and Convex Coding in Nonthreshold Linear Networks.

Hebbian theory proposes that ensembles of neurons form a basis for neural processing. It is possible to gain insight into the activity patterns of these neural ensembles through a binary analysis, regarding neurons as either active or inactive. The framework of permitted and forbidden sets, introduced by Hahnloser, Seung, and Slotine (2003), is a mathematical model of such a binary analysis: groups of coactive neurons can be permitted or forbidden depending on the network's structure. In order to widen the applicability of the framework of permitted sets, we extend the permitted set analysis from the original threshold-linear regime. Specifically, we generalize permitted sets to firing rate models in which Φ is a nonnegative continuous piecewise C1 activation function. In our framework, the focus is shifted from a neuron's firing rate to its responsiveness to inputs; if a neuron's firing rate is sufficiently sensitive to changes in its input, we say that the neuron is responsive. The algorithm for categorizing a neuron as responsive depends on thresholds that a user can select arbitrarily and that are independent of the dynamics. Given a synaptic weight matrix W, we say that a set of neurons is permitted if it is possible to find a stimulus where those neurons, and no others, remain responsive. The main coding property we establish about PΦ(W), the collection of all permitted sets of the network, is that PΦ(W) is a convex code when W is almost rank one. This means that PΦ(W) in the low-rank regime can be realized as a neural code resulting from the pattern of overlaps of receptive fields that are convex.

Generalized relative Nevanlinna order (α,β) and generalized relative Nevanlinna type (α,β) based some growth properties of composite analytic functions in the unit disc

Our aim in this paper is to introduce some idea about generalized relative Nevanlinna order (α,β) and generalized relative Nevanlinna type (α,β) of an analytic function with respect to another analytic function in the unit disc where α and β are continuous non-negative functions on (-∞,+∞). So we discuss about some growth properties relating to the composition of two analytic functions in the unit disc on the basis of generalized relative Nevanlinna order (α,β) and generalized relative Nevanlinna type (α,β) as compared to the growth of their corresponding left and right factors.

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.

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.

Multi-bump solutions for the nonlinear magnetic Choquard equation with deepening potential well

In this paper, using variational methods, we study multiplicity of multi-bump solutions for the following nonlinear magnetic Choquard equation{−(∇+iA(x))2u+(λV(x)+1)u=(1|x|μ⁎|u|p)|u|p−2ux∈RN,u∈H1(RN,C), where N≥2, λ>0 is a real parameter, 0<μ<2, i is the imaginary unit, p∈(2,2⁎(2(N−μ)2N)), where 2⁎=2NN−2 if N≥3, 2⁎=+∞, if N=2. The magnetic potential A∈Lloc2(RN,RN) and V:RN→R is a nonnegative continuous function. We show that if the zero set of V has several isolated connected components Ω1,⋯,Ωk such that the interior of Ωj is non-empty and ∂Ωj is smooth, then for λ>0 large enough, the above equation has at least 2k−1 multi-bump solutions.

Blow-up in a quasilinear parabolic–elliptic Keller–Segel system with logistic source

This paper deals with the quasilinear parabolic–elliptic Keller–Segel system with logistic source, ut=Δ(u+1)m−χ∇⋅(u(u+1)α−1∇v)+λ(|x|)u−μ(|x|)uκ,x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0,where Ω≔BR(0)⊂Rn(n≥3) is a ball with some R>0; m>0, χ>0, α>0 and κ≥1; λ and μ are continuous nonnegative functions. About this problem, Winkler (2018) found the condition for κ such that solutions blow up in finite time when m=α=1. In the case that m=1 and α∈(0,1) as well as λ and μ are constants, some conditions for α and κ such that blow-up occurs were obtained in a previous paper (Tanaka and Yokota, 2020). Moreover, in the case that m≥1 and α=1 Black et al. (2021) showed that there exist initial data such that the corresponding solution blows up in finite time under some conditions for m and κ. The purpose of the present paper is to give conditions for m≥1, α>0 and κ≥1 such that solutions blow up in finite time.