Positive solution to a special singular second-order boundary value problem

Let λ be a nonnegative parameter. The existence of a positive solution is studied for a semipositone second-order boundary value problem

where d>0,α≥0,β≥0,α+β>0, q(t)f(t,u,v)≥0 on a suitable subset of [0,1]×[0,+∞)×(−∞,+∞) and f(t,u,v) is allowed to be singular at t=0,t=1 and u=0. The proofs are based on the Leray–Schauder fixed point theorem and the localization method.     

Multiple positive solutions to a singular boundary value problem for a superlinear Emden-Fowler equation

A multiplicity result for the singular ordinary differential equation y^″+λx⁻²yσ=0, posed in the interval (0,1), with the boundary conditions y(0)=0 and y(1)=γ, where σ>1, λ>0 and γ?0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ^?∈(0,σ/2] such that a solution to the above problem is possible if and only if λγσ−1?Σ^?. For 0<λγσ−1<Σ^?, there are multiple positive solutions, while if γ=(λ⁻¹Σ^?)^1/(σ−1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x→⁰⁺ is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem −Δu=d⁻²(x)uσ in Ω, where Ω⊂RN, N?2, is a smooth bounded domain and d(x)=dist(x,∂Ω).     

A singular boundary value problem for odd-order differential equations

The odd-order differential equation (−1)ⁿx⁽²ⁿ⁺¹⁾=f(t,x,…,x⁽²ⁿ⁾) together with the Lidstone boundary conditions x^(2j)(0)=x^(2j)(T)=0, 0?j?n−1, and the next condition x⁽²ⁿ⁾(0)=0 is discussed. Here f satisfying the local Carathéodory conditions can have singularities at the value zero of all its phase variables. Existence result for the above problem is proved by the general existence principle for singular boundary value problems.     

Structure of a class of singular boundary value problem with superlinear effect

This paper investigates the structure of solutions of singular boundary value problem with superlinear effect. It is proved that the closure of positive solution set possesses a maximal subcontinuum C (i.e., a maximal closed connected subset of solutions), which comes from (0,θ) and tends to (0,+∞) finally. As a corollary, the existence of multiple positive solutions and the behavior of solutions according to parameter λ are obtained.     

Multiple positive solutions to third-order three-point singular semipositone boundary value problem

By using a specially constructed cone and the fixed point index theory, this paper investigates the existence of multiple positive solutions for the third-order threepoint singular semipositone BVP:

Positive solutions of singular third-order three-point boundary value problem

In this paper we investigate the problem of existence of positive solutions for the nonlinear singular third-order three-point boundary value problem

     

An initial boundary value problem in ideal Magneto-Hydrodynamics

We consider the initial-boundary value problem for the system of equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall boundary condition. Because of the characteristic boundary, where a loss of regularity in the normal direction to the boundary may occur, the natural functional setting is provided by anisotropic Sobolev spaces, which take account of such singular behavior. We show the existence of the regular solution in the anisotropic Sobolev space for . The result improves previous results of the author and of Yanagisawa - Matsumura. RID="h1" ID="h1"Research of the project Cofin. MURST 1998 "Problems and Methods in the Theory of Hyperbolic Equations".     

Boundary behavior of solutions to some singular elliptic boundary value problems

Let Ω be a bounded domain with smooth boundary in . For the more general weight b, some nonlinearities f and singularities g, by two kinds of nonlinear transformations, a new perturbation method, which was introduced by García Melián in [J. García Melián, Boundary behavior of large solutions to elliptic equations with singular weights, Nonlinear Anal. 67 (2007) 818–826], and comparison principles, we show that the boundary behavior of solutions to a boundary blow-up elliptic problem Δw=b(x)f(w),w>0,xΩ,w|_∂Ω=∞ and a singular Dirichlet problem −Δu=b(x)g(u),u>0,xΩ,u|_∂Ω=0 has the same form under the nonlinear transformations, which can be determined in terms of the inverses of the transformations.     

Uniformly convergent difference schemes for a singularly perturbed third order boundary value problem

In this paper we consider a numerical approximation of a third order singularly perturbed boundary value problem by an upwind finite difference scheme on a Shishkin mesh. The behavior of the solution, and the stability of the continuous problem are discussed. The proof of the uniform convergence of the proposed numerical method is based on the strongly uniform stability and a weak consistency property of the discrete problem. Numerical experiments verify our theoretical results.     

Existence-uniqueness results for a class of singular boundary value problems-II

In this paper we establish existence-uniqueness of solution of a class of singular boundary value problem −(p(x)y^′′(x))=q(x)f(x,y) for 0<x?b and y(0)=a, α₁y(b)+β₁y^′(b)=γ₁, where p(x) satisfies (i) p(x)>0 in (0,b), (ii) p(x)∈C¹(0,r), and for some r>b, (iii) is analytic in and q(x) satisfies (i) q(x)>0 in (0,b), (ii) q(x)∈L¹(0,b) and for some r>b, (iii) is analytic in with quite general conditions on f(x,y). Region for multiple solutions have also been determined.     

二阶差分方程的奇性边值问题

主要采用上下解方法,研究了一类带有奇性的二阶差分方程的两点边值问题,给出了一系列正解的存在条件.     

Positive solutions of a singular nonlinear third order two-point boundary value problem

In this paper some existence results of positive solutions for the following singular nonlinear third order two-point boundary value problem:

     

Positive solutions of singular boundary value problem of negative exponent Emden—Fowler equation

This paper investigates the existence of positive solutions of a singular boundary value problem with negative exponent similar to standard Emden-Fowler equation. A necessary and sufficient condition for the existence of C[0, 1] positive solutions as well as C¹[0, 1] positive solutions is given by means of the method of lower and upper solutions with the Schauder fixed point theorem.     

A highly unstable initial value boundary value problem

In this paper we consider a two-dimensional diffusion equation on the closed right halfspace satisfying a boundary condition for which the minimum principle fails. As a consequence, the associated Cauchy initial value problem fails to be well-posed. In particular, solutions need not exist and, when they do exist, they may do so for only a finite length of time. Among other things, we provide a necessary and sufficient condition on the initial data in order that the solution exist for all time and remain non-negative.     

Analytical substrate flux approximation for the Monod boundary value problem

We present an analytical approximation for the diffusive flux of a substrate into a reactive layer, in which the substrate is degraded according to Monod kinetics. This problem is described by a nonlinear two-point boundary value problem. The approximation is derived based on a Homotopy Analysis Method idea and verified computationally, by comparison against a numerical solution of the problem. The analytical approximation is easy to evaluate and depends only on model parameters.     

Positive solutions for singular boundary value problems of a coupled system of differential equations

By using fixed point theorem of cone expansion and compression, this paper investigates the existence of multiple positive solutions for singular boundary value problems of a coupled system of nonlinear ordinary differential equations.     

Positive solutions for nonlinear singular boundary value problems

Some sufficient conditions for the existence of positive solutions to Dirichlet boundary value problems of a class of nonlinear second order differential equations are given.     

A note on singular nonlinear boundary value problems

In this paper we study the existence and multiplicity of solutions of the following operator equation in Banach space E:

u=λAu,0<λ<+∞,u∈P?{θ},