Oscillation Criteria for Certain Fourth-Order Nonlinear Delay Differential Equations

In this article, we establish some new criteria for the oscillation of fourth-order nonlinear delay differential equations of the form

$$(r_2(t)(r_1(t)(y''(t))^\alpha)')' + p(t)(y''(t))^\alpha + q(t)f(y(g(t))) = 0$$

provided that the second-order equation

$$(r_2(t)z'(t))') + \frac{p(t)}{r_1(t)}z(t) = 0$$

is nonoscillatory or oscillatory.

     

Multiple symmetric positive solutions of fourth-order two point boundary value problem

In this paper, by using the Krasnoselskii fixed point theorem, we study the existence of 2m or 2m + 1 symmetric positive solutions of fourth-order two point boundary value problem $y^{(4)} (t) - f(t,y(t),y''(t)) = 0,y(0) = y(1) = y'(0) = y''(1) = 0$ .     

Oscillation theorems for Sturm-Liouville problems with distribution potentials

The Sturm-Liouville problem

$\begin{array}{*{20}c} { - y'' + q(x)y = \lambda y,} \\ {y(0) = y(1) = 0} \\ \end{array} $

is considered with a singular potential q(x) representing the derivative of a real function from the space L ₂[0, 1] in the distributional sense. Two approaches are developed for the study of oscillation properties of eigenfunctions of this problem. The first approach is based on generalization of methods of the Sturm theory. The second one is based on development of variational principles.

     

Bifurcations of a parametrized family of boundary value problems for second order differential inclusions

We study the existence of non-trivial solutions of the following family of differential inclusions of second order (S) $$\left\{ \begin{gathered} y''(t) \in F(p, t, y(t), y'(t)) t \in [0,a] , \hfill \\ (y(0), y'(0), y(a), y'(a)) \in b(p) , \hfill \\ \end{gathered} \right.$$ where \(F:P \times [0,a] \times \mathbb{R}^n \times \mathbb{R}^n \to 2^{\mathbb{R}^n } \) is a Carathéodory multifunction with non-empty compact convex values and b: P→G_2n(?⁴ⁿ) is a continuous map from a CW-complex P to the Grassmann manifold G_2n(?⁴ⁿ). We show that if (X,A) is a finite CW-pair in P, A contractible in X, b: (X, A)→(G_2n(?⁴ⁿ), pt) is such that and F satisfies the Nagumo growth conditions at some point p₀ ε X, then the system (S) has a bifurcation from infinity in X; i.e. there exists a sequence of non-trivial solutions of S whose norms in the space C¹ tend to infinity.     

Inverse Sturm-Liouville problems with homogeneous delay

We study the inverse Sturm-Liouville problem with a delay $$- y''(x) + q(x)y(\alpha x), q \in L_2 [0,\pi ], \alpha \in (0,1],$$ and the boundary conditions y(0) = y(π) = 0.     

Triple positive solutions for a class of third-order p-Laplacian singular boundary value problems

In this work, we study the existence of triple positive solutions for one-dimensional p-Laplacian singular boundary value problems $$\begin{array}{l}(\phi_p(y''(t)))'+f(t)g(t,\,y(t),\,y'(t),\,y''(t))=0,\quad 0<t<1,\\[3pt]ay(0)-by'(0)=0,\qquad cy(1)+dy'(1)=0,\qquad y''(0)=0,\end{array}$$ where φ _p (s)=|s| ^p?2 s,?p>1, g:[0,?1]×[0,?+∞)×R ²?[0,?+∞) and f:(0,?1)?[0,?+∞) are continuous. The nonlinear term f may be singular at t=0 and/or t=1. Firstly, Green’s function for the associated linear boundary value problem is constructed. Then, by making use of a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of triple positive solutions to the above boundary value problem. The interesting point is that the nonlinear term g involved with the first-order and second-order derivatives explicitly.     

Existence of a solution of a difference scheme for one variational problem

The problem of minimizing the functional (A) $${}_a\smallint ^b \varphi (x,y,y',y'')dx$$ under the conditions (B) $$y(a) = a_0 ,y'(a) = a_1 ,y(b) = b_0 ,y'(b) = b_1$$ is replaced by the problem of finding the vector (y₁,y₂,...,y_n?1) on which the sum (C) $$\sum\limits_{\kappa = 0}^n {C_\kappa \varphi (x_\kappa ,y_\kappa ,\left. {\frac{{y_{\kappa + 1} - y_\kappa }}{h},\frac{{y_{\kappa + 1} - 2y_\kappa + y_{\kappa + 1} )}}{{h^2 }}} \right)}$$ takes a minimal value. Under certain conditions on ? andC _k it is proved that a solution exists for the difference scheme constructed. The method of differentiation with respect to a parameter is used for the proof.     

Riesz bases formed by root functions of a functional-differential equation with a reflection operator

We find conditions under which the system of root functions of the operator

$$L_y = l[y] = ay'(x) + y'(1 - x) + p_1 (x)y(x) + p_2 (x)y(1 - x),x \in [0,1],U_1 (y) = \int\limits_0^1 {y(t)d\sigma (t) = 0,} $$

is a Riesz basis in L ₂[0, 1].

     

Variable-mesh difference scheme for singularly-perturbed boundary-value problems using splines

A second-order variable-mesh difference scheme via cubic splines for singularly-perturbed boundary-value problems of the form $$y'' = \rho (x)y' + q(x)y + r(x), y(a) = a_0 , y(b) = a_1 ,$$ is presented. The convergence analysis is given and the method is shown to have quadratic convergence. Several test examples are solved to demonstrate the efficiency of the method.     

Asymptotics of the eigenvalues of the Sturm-Liouville problem with discrete self-similar weight

We study the asymptotics of the spectrum of the boundary-value problem $$ - y'' - \lambda \rho y = 0,y(0) = y(1) = 0 $$ for the case in which the weight ρ ∈ W? ₂ ^?1 [0, 1] is the generalized (in the sense of distributions) derivative of a self-similar function P ∈ L ₂[0, 1] of zero spectral order.     

带阻尼项的二阶奇异耦合系统的正周期解

We establish the existence of positive periodic solutions of the second-order singular coupled systems{x′′+ p_1(t)x′+ q_1(t)x = f_1(t, y(t)) + c_1(t),y′′+ p_2(t)y′+ q_2(t)y = f_2(t, x(t)) + c_2(t),where pi, qi, ci ∈ C(R/T Z; R), i = 1, 2; f_1, f_2 ∈ C(R/T Z ×(0, ∞), R) and may be singular near the zero. The proof relies on Schauder's fixed point theorem and anti-maximum principle.Our main results generalize and improve those available in the literature.     

一类两点边值问题的多重非负解

该文考虑两点边值问题[1/q(t)][q(t)y′(t)]′+p(t)f(y(t))= 0,λ_1 y(α)-λ_2y′(α)=0 and y(β)=B非负解的存在性, 其中p(t)可能在t=α或t=β附近具有奇异性, f(0)≥0, lim_(y→+∞)f(y)/y=+∞, 并且存在y>0, 使得f(y)<0.      

一维奇异p-Laplace方程的上下解方法[英文]

本文讨论了一维奇异 p Laplace方程( φp( y′) )′+ q(t) f(t,y) =0 ,0 相似文献   

Spectral singularities of Klein-Gordon s-wave equations with an integral boundary condition

We investigate the spectral singularities and the eigenvalues of the boundary value problem $$\begin{gathered} y'' + \left[ {\lambda - Q\left( x \right)} \right]^2 y = 0,x \in R_ + = [0,\infty ), \hfill \\ \quad \int\limits_0^\infty {K\left( x \right)y\left( x \right)dx + \alpha y'\left( 0 \right) - \beta y\left( 0 \right) = 0,} \hfill \\ \end{gathered}$$ where Q and K are complex valued functions, K∈L ²(R ₊), α,β∈C with |α|+|β|≠0 and λ is a spectral parameter.     

High-Accuracy P-Stable Methods with Minimal Phase-Lag for $y "= f(t,y) ^*$

In this paper, we develop a one-parameter family of P-stable sixth-order and eighth-order two-step methods with minimal phase-lag errors for numerical integration of second order periodic initial value problems: $$ y''=f(t,y), \quad y(t_0)=y_0, \quad y'(t_0)=y'_0. $$ We determine the parameters so that the phase-lag (frequency distortion) of these methods are minimal. The resulting methods are P-stable methods with minimal phase-lag errors. The superiority of our present P-stable methods over the P-stable methods in [1-4] is given by comparative studying of the phase-lag errors and illustrated with numerical examples.     

二阶微分系统奇异正定超线性周期边值问题的多重正解

主要研究了二阶微分系统具有奇异正定超线性周期边值问题多重正解的存在性问题,利用Leray-Schauder抉择定理和锥不动点定理给出了奇异正定超线性周期边值问题-(p(t)x′)′+q1(t)x=f1(t,x,y),t∈I=[0,1]-(p(t)y′)′+q2(t)y=f2(t,x,y)x(0)=x(1),x[1](0)=x[1](1)y(0)=y(1),y[1](0)=y[1](1)(1.1)的多重正解的存在性,其中非线性项fi(t,x,y)(i=1,2)在x=∞,y=∞点处超线性,在(x,y)=(0,0)处具有奇性.这里定义x[1](t)=p(t)x′(t),y[1](t)=p(t)y′(t)为准导数,其中系数p(t),qi(t)(i=1,2)是定义在[0,1]上的可测函数,且p(t)>0,qi(t)>0(i=1,2),a.e[0,1],fi(t,x,y)∈C(I×R×R,R+),R+=(0,+∞).     

Dependence of solutions and eigenvalues of third order linear measure differential equations on measures

This paper deals with a complex third order linear measure differential equation id(y')~·+ 2 iq(x)y'dx + y(idq(x) + dp(x)) = λydx on a bounded interval with boundary conditions presenting a mixed aspect of the Dirichlet and the periodic problems. The dependence of eigenvalues on the coefficients p and q is investigated. We prove that the n-th eigenvalue is continuous in p and q when the norm topology of total variation and the weak*topology are considered. Moreover, the Fr′echet differentiability of the n-th eigenvalue in p and q with the norm topology of total variation is also considered. To deduce these conclusions, we investigate the dependence of solutions of the above equation on the coefficients p and q with different topologies and establish the counting lemma of eigenvalues according to the estimates of solutions.     

MULTIPLE POSITIVE SOLUTIONS TO SINGULAR BOUNDARY VALUE PROBLEMS FOR SUPERLINEAR SECOND ORDER ODES

11皿roduction and Main ResultsBoundary vlue probl。s(abbr．as BVP)associated with smgtllar second order ditherelltiale叩atiolls h出柏a IOllg hEtofy nd m聊dlfferem methods nd techniques h出用been sed皿ddev咖pod m OTder to obtmn——Oils qualltatn旧properties of the solutions．nr detmls,see/forInstance,papers [l－6] and the references therein．Ho。。r,only few。rks on singul。boundmpMu。Problems for suPe山near ODEs【of}6]．As》as。thorh。s,the worh on the。stenceof mlmltiple posit讨esoluti…     

The Higher Order Approximation of Solution of Quasilinear Second Order Systems for Singular Perturbation

In this paper, using the theory of invariant region, the author considers the existence and the asymptotic behavior of solution of vector second order quasi-linear boundary value problem: $\epsilon y''=f(x,y,\epsilon)y''+g(x,y,\epsilon)$ $y(0,\epsilon)=A(\epsilon),y(1,\epsilon)=B(\epsilon)$ as the positive perturbation parameter e tends to zero, where y, g, A and B are vector-valued and f is a matrix function. Under the appropriate assumptions the author obtains, involving the boundary layer, uniformly valid asymptotic solution of higher order approximation.     

On solvability of derivative dependent doubly singular boundary value problems

In this paper we establish existence of solutions of singular boundary value problem ?(p(x)y ^′(x))^′=q(x)f(x,y,py′) for 0<x≤b and $\lim_{x\rightarrow0^{+}}p(x)y^{\prime}(x)=0$ , α ₁ y(b)+β ₁ p(b)y ^′(b)=γ ₁ with p(0)=0 and q(x) is allowed to have integrable discontinuity at x=0. So the problem may be doubly singular. Here we consider $\lim_{x\rightarrow0^{+}}\frac{q(x)}{p'(x)}\neq0$ therefore $\lim_{x\rightarrow0^{+}}p(x)y'(x)=0$ does not imply y′(0)=0 unless $\lim_{x\rightarrow0^{+}}f(x,y(x),p(x)y'(x))=0$ .