Degenerate elliptic boundary value problems with asymptotically linear nonlinearity

The purpose of this paper is to study a class of semilinear degenerate elliptic boundary value problems with asymptotically linear nonlinearity which include as particular cases the Dirichlet and Robin problems. Our approach is based on the global inversion theorems between Banach spaces, and is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. By making use of the variational method, we prove existence and uniqueness theorems for our problem. The results here extend three earlier theorems due to Ambrosetti and Prodi to the degenerate case.     

Existence of sign-changing solutions for some asymptotically linear three-point boundary value problems

In this paper, by using the topological degree theory and the fixed point index theory, the existence of three kinds of solutions (i.e., sign-changing solutions, positive solutions and negative solutions) for asymptotically linear three-point boundary value problems is obtained.     

Multiple solutions for an asymptotically linear Duffing equation with Newmann boundary value conditions

We investigate multiple solutions of an asymptotically linear Duffing equation satisfying Newmann boundary value conditions with resonance using index theory and Morse theory and obtain some new results.     

On linear transformations preserving at least one eigenvalue

Let be an algebraically closed field and be a linear transformation. In this paper we show that if preserves at least one eigenvalue of each matrix, then preserves all eigenvalues of each matrix. Moreover, for any infinite field (not necessarily algebraically closed) we prove that if is a linear transformation and for any with at least an eigenvalue in , and have at least one common eigenvalue in , then preserves the characteristic polynomial.

     

Multiple solutions for discrete boundary value problems

A recent multiplicity theorem for the critical points of a functional defined on a finite-dimensional Hilbert space, established by Ricceri, is extended. An application to Dirichlet boundary value problems for difference equations involving the discrete p-Laplacian operator is presented.     

Multiple solutions for indefinite functionals and applications to asymptotically linear problems

In this paper, by means of Morse theory of isolated critical points (orbits) we study further the critical points theory of asymptotically quadratic functionals and give some results concerning the existence of multiple critical points (orbits) which generalize a series of previous results due to Amann, Conley, Zehnder and K.C. Chang. As applications, the existence of multiple periodic solutions for asymptotically linear Hamiltonian systems is investigated. And our results generalize some recent ones due to Coti-Zelati, J.Q.Liu, S.Li, etc.This research was supported in part by the National Postdoctoral Science Fund.     

Multiple positive solutions for second-order m-point boundary value problems

In this paper, the second-order m-point boundary value problem

     

Multiple solutions for sub-linear impulsive three-point boundary value problems

In this article we establish multiplicity results for impulsive three-point boundary value problems if we assume a sub-linear condition at infinity. The proof involves Leray–Schauder degree and the lower and upper solution method.     

Multiple positive solutions for discrete nonlocal boundary value problems

In this paper, we investigate a second-order nonlinear difference equation with sign-changing nonlinearity subject to two different sets of nonlocal boundary conditions. The explicit expressions of the associated Green's functions are presented. By using a recently developed fixed point theorem, we establish sufficient conditions for the existence of multiple positive solutions of the boundary value problem.     

Multiple positive solutions for Kirchhoff type problems with singularity and asymptotically linear nonlinearities

In this paper, we devote ourselves to investigating a nonlocal problem involving singularity and asymptotically linear nonlinearities. By using the variational and perturbation methods, we obtain the existence of two positive solutions which improve the existing result in the literature.     

Positive solutions for one-dimensional -Laplacian boundary value problems with sign changing nonlinearity

This paper deals with the existence of positive solutions for the one-dimensional p-Laplacian

subject to the boundary value conditions:

where _p(s)=|s|^p−2s,p>1. We show that it has at least one or two positive solutions under some assumptions by applying the fixed point theorem. The interesting points are that the nonlinear term f is involved with the first-order derivative explicitly and f may change sign.     

Multiple nontrivial solutions for nonlinear eigenvalue problems

In this paper we study a nonlinear eigenvalue problem driven by the -Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposite constant sign solutions. The approach relies on a combination of variational and minimization methods coupled with the construction of upper-lower solutions. The framework of the paper incorporates problems with concave-convex nonlinearities.

     

Multiple solutions of some fourth-order boundary value problems

For some fourth-order boundary value problems, several new existence theorems on multiple positive, negative and sign-changing solutions are obtained. The critical point theory and the supersolution and subsolution method are employed to discuss this problem.     

Multiple solutions of nonlinear functional boundary value problems

A class of nonlinear functional boundary conditions for the system of functional differential equations x"(t)=(F(x,y))(t)x'(t)=(F(x,y))(t), y"(t)=(H(x,y))(t)y'(t)=(H(x,y))(t) is introduced. Here F, H:C¹([a,b]) ×C¹([a,b]) ? L₁([a,b])F,\,H:C^1([a,b]) \times C^1([a,b]) \rightarrow L_1([a,b]) are nonlinear continuous operators. Sufficient conditions for the existence of at least four solutions are given. Results are proved by the Bihari lemma, the Leray-Schauder degree theory and the Borsuk theorem.     

Multiple solutions of two-point nonlinear boundary value problems

A quasi-linear boundary value problem has a solution with properties induced by oscillatory properties of the linear part of an equation. This result is proved for two dimensional systems. Consequences for Φ-Laplacian equations and problems with resonant linear parts are discussed.     

Multiple solutions of singular boundary value problems for differential systems

By constructing a special cone and using fixed point index theory in cone, this paper investigates the existence of multiple solutions of singular boundary value problems for differential systems.     

Multiple solutions of impulsive boundary value problems on the half-line

Existence results of multiple solutions are obtained under suitable conditions for impulsive integrodifferential boundary value problems on the half-line which may be singular at the boundary.