Sign-changing solutions and multiplicity results for elliptic problems via lower and upper solutions

In the first part of this work, we recall variational methods related to invariant sets in ${C^1_0}$ . In the second part of the work, we consider an elliptic Dirichlet problem in a situation where the origin is a solution around which the nonlinearity has a slope between two consecutive eigenvalues of order larger than 2 and near + infinity the slope of the nonlinearity is smaller than the first eigenvalue. Then we discuss the conditions needed near - infinity in order to ensure the existence of a positive solution and two sign-changing solutions.     

The method of upper and lower solutions for diffraction problems of quasilinear elliptic reaction-diffusion systems

The aim of this paper is to show the existence of solutions of the n-dimensional diffraction problem for weakly coupled quasilinear elliptic reaction-diffusion system. The coefficients of the equations under consideration are allowed to be discontinuous. We extend the method of upper and lower solutions for reaction-diffusion equations with continuous coefficients to the elliptic diffraction problem. An application of these results is given to the steady-state problem of Lotka-Volterra cooperation model with two cooperating species.     

Existence of non-spurious solutions to discrete Dirichlet problems with lower and upper solutions

This paper investigates the solvability of discrete Dirichlet boundary value problems by the lower and upper solution method. Here, the second-order difference equation with a nonlinear right hand side f$f$ is studied and f(t,u,v)$f(t,u,v)$ can have a superlinear growth both in u$u$ and in v$v$. Moreover, the growth conditions on f$f$ are one-sided. We compute a priori bounds on solutions to the discrete problem and then obtain the existence of at least one solution. It is shown that solutions of the discrete problem will converge to solutions of ordinary differential equations.     

Lower and upper solutions for elliptic problems in nonsmooth domains

In this paper we prove some existence results of semilinear Dirichlet problems in nonsmooth domains in presence of lower and upper solutions well-ordered or not. We first prove existence results in an abstract setting using degree theory. We secondly apply them for domains with conical points.     

Singular problems motivated from classical upper and lower solutions

General existence criteria are presented for nonlinear singular boundary value problems. Our nonlinearity may be singular in both the dependent and independent variable. This revised version was published online in June 2006 with corrections to the Cover Date.     

Maximum principles and the method of upper and lower solutions for time-periodic problems of the telegraph equations

This paper deals with the existence of bounded time-periodic solutions for the nonlinear telegraph equation

     

Analytical sequences of upper and lower solutions for a class of elliptic equations

Comparison arguments are applied to derive decreasing sequences of upper solutions and increasing sequences of lower solutions for a class of nonlinear elliptic equations. The monotonicity of the two sequences is proven. These polynomial sequences are obtained by applying new algorithms and solving linear differential equations. The obtained upper and lower solutions are analytic and have closed forms. Different examples are presented to explore the effectiveness of the new algorithms. The presented ideas and algorithms can be extended to deal with different classes of equations.     

Nonlinear parabolic boundary value problems with upper and lower solutions

The existence of at least one periodic solution of a very general second order nonlinear parabolic boundary value problem is proved under the assumption that a lower solution ϕ and an upper solution ψ with ϕ≦ψ are known.     

Some new nonlinear inequalities and applications to boundary value problems

In this paper, we establish some new nonlinear integral inequalities of the Gronwall–Bellman–Ou-Iang-type in two variables. These on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations. We illustrate this by applying our new results to certain boundary value problem.     

The method of lower and upper solutions for third order singular four point boundary value problems

We mainly study the existence of positive solutions for the following third order singular four point boundary value problem $$\begin{cases}x^{(3)}(t)+f(t,x,x',-x'')=0,\quad 0<t<1,\\x(0)-\alpha x(\xi)=0,\quad x'(1)-\beta x'(\eta)=0,\quad x''(0)=0.\end{cases}$$ where 0≤α<1, 0≤β<1, 0<ξ<1,0<η<1. And we obtain some necessary and sufficient conditions for the existence of C ²[0,1] positive solutions by means of the lower and upper solution method. Our nonlinearity f(t,x,y,z) may be singular at x,y,z,t=0 and/or t=1.     

Boundary value problems involving upper and lower solutions in reverse order

In this paper, we discuss the existence of extreme solutions of the boundary value problem for a class of first-order functional equations with a nonlinear boundary condition. In the presence of a lower solution α$\alpha$ and an upper solution β$\beta$ with β≤α$\beta \le \alpha$, we establish existence results of extreme solutions by using the method of upper and lower solutions and a monotone iterative technique.     

Some new Jacobi elliptic function solutions for the short-pulse equation via a direct symbolic computation method

Applying a direct symbolic computation method combined with variable transformations, some new Jacobi elliptic function solutions are obtained to the short-pulse equation in nonlinear optics. When Jacobi elliptic function modulus m??1 or?0, the travelling wave solutions degenerate to two types of solutions, namely, the loop-like soliton solution and the trigonometric function solution.     

Some sufficient conditions for lower semicontinuity in SBD and applications to minimum problems

We provide some lower semicontinuity results in the space of special functions of bounded deformation for energies of the type and give some examples and applications to minimum problems, arising in Fracture Mechanics for linearly elastic materials. Copyright © 2011 John Wiley & Sons, Ltd.     

The method of lower and upper solutions for fourth order singular m-point boundary value problems

This paper investigates the existence of positive solutions for fourth order singular m-point boundary value problems. Firstly, we establish a comparison theorem, then we define a partial ordering in C²[0,1]∩C⁴(0,1) and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C²[0,1] as well as C³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x, y, t=0 and/or t=1.