Positive solutions for nonlinear nonhomogeneous Dirichlet problems with concave-convex nonlinearities

We consider a nonlinear parametric Dirichlet equation driven by a nonhomogeneous differential operator involving a reaction exhibiting the competing effects of concave and convex terms. Using variational methods combined with truncation and comparison techniques we prove a bifurcation near zero theorem describing the dependence of the positive solutions on the parameter \(\lambda >0\).     

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.     

On the multiplicity of radial solutions to superlinear Dirichlet problems in bounded domains

In this paper we are concerned with the existence and multiplicity of nodal solutions to the Dirichlet problem associated to the elliptic equation Δu+q(|x|)g(u)=0 in a ball or in an annulus in .The nonlinearity g has a superlinear and subcritical growth at infinity, while the weight function q is nonnegative in [0,1] and strictly positive in some interval [r₁,r₂]⊂[0,1].By means of a shooting approach, together with a phase-plane analysis, we are able to prove the existence of infinitely many solutions with prescribed nodal properties.     

Existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order

This paper studies the existence and uniqueness of solutions of second-order three-point boundary value problems with lower and upper solutions in the reversed order, obtains the sufficient conditions for the existence and uniqueness of solutions by use of the monotone iterative method, and gives the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution.     

Positive solutions for semi-positone three-point boundary value problems

Using a fixed point theorem of generalized cone expansion and compression we present in this paper criteria which guarantee the existence of at least two positive solutions for semi-positone three-point boundary value problems with parameter λ>0$\lambda >0$ belonging to a certain interval.     

Existence of multiple positive solutions of singular nonlinear boundary value problems

For a given positive integer N, we provide conditions on the nonlinear function f which guarantee that the boundary value problem

     

Positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions

In this paper, we study the existence, multiplicity and nonexistence of positive solutions for 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions. The results are based upon the fixed-point theorem of cone expansion and compression type due to Krasnosel’skill. Moreover, it generalizes and includes some known results.     

Nodal solutions of multi-point boundary value problems

We study the nonlinear boundary value problem consisting of the equation y^″+w(t)f(y)=0 on [a,b] and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes in the existence question for different types of nodal solutions as the problem changes.     

Some fixed point theorems and existence of positive solutions of two-point boundary-value problems

Three theorems are obtained for the existence of at least one or three fixed points for a completely continuous mapping, which extend the Krasnoselskii’s compression–expansion theorem in cones. Based on them two theorems for the existence of positive solutions of two-point boundary-value problems are proved under a quite relaxed condition compared with the existing literature.