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.     

Existence and Relaxation Theorems for Nonlinear Multivalued Boundary Value Problems

In this paper we consider a general nonlinear boundary value problem for second-order differential inclusions. We prove two existence theorems, one for the ``convex' problem and the other for the ``nonconvex' problem. Then we show that the solution set of the latter is dense in the C ¹ (T,R ^N ) -norm to the solution set of the former (relaxation theorem). Subsequently for a Dirichlet boundary value problem we prove the existence of extremal solutions and we show that they are dense in the solutions of the convexified problem for the C ¹ (T,R ^N ) -norm . Our tools come from multivalued analysis and the theory of monotone operators and our proofs are based on the Leray—Schauder principle. Accepted 18 September 1997     

On linear and nonlinear fourth-order eigenvalue problems with indefinite weight

We determine the principal eigenvalues of the linear indefinite weight problem Moreover, we investigate the existence of positive solutions for the corresponding nonlinear indefinite weight problem, where g:[0,1]→R is a continuous function which attains both positive and negative values, f∈C(R,R), and r is a parameter.     

Multipoint boundary value problems for nonlinear ordinary differential equations

In this paper we provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations. We consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2. When this solution space is trivial, we establish existence results via the Schauder Fixed Point Theorem. In the resonance case, we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem. We accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem.     

Nonlinear second-order equations with functional implicit impulses and nonlinear functional boundary conditions

In this paper we deal with nonlinear second-order boundary value problems with impulses. The impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear. In both cases functional dependence on the solution is allowed. The existence results follow from the existence of a pair of well-ordered lower and upper solutions.     

First- and second-order discontinuous functional differential equations with impulses at fixed moments

We give sufficient conditions for the existence of extremal solutions to discontinuous and functional differential equations with impulses. Our main results are new even for ordinary differential equations without impulses.     

On Sturm–Liouville boundary value problems for second-order nonlinear functional finite difference equations

Motivated by the interesting paper [I. Karaca, Discrete third-order three-point boundary value problem, J. Comput. Appl. Math. 205 (2007) 458–468], this paper is concerned with a class of boundary value problems for second-order functional difference equations. Sufficient conditions for the existence of at least one solution of a Sturm–Liouville boundary value problem for second-order nonlinear functional difference equations are established. We allow f to be at most linear, superlinear or sublinear in obtained results.     

Multiplicity of sign-changing solutions for some four-point boundary value problem

In this paper we obtain using Leray–Schauder degree theory some multiplicity results for sign-changing solutions of a four-point boundary value problem. We assume the existence of a pair of strict lower and upper solutions and some additional conditions on the nonlinear term f$f$.     

The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method

In this paper, we study the existence and multiplicity of classical solutions for a second-order impulsive differential equation with periodic boundary conditions. By using a variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution, two solutions and infinitely many solutions when the parameter pair (c,λ) lies in different intervals, respectively. Some examples are given in this paper to illustrate the main results.     

Nagumo and Landesman-Lazer type conditions for nonlinear second order systems

We study the existence of solutions for a nonlinear second order system of ordinary differential equations under various boundary conditions. Assuming suitable Nagumo type conditions we prove the existence of at least one solution applying the method of upper and lower solutions. Moreover, using topological degree methods we prove the existence of solutions under Landesman-Lazer type conditions.     

Positive solutions for second-order four-point boundary value problems with alternating coefficient

In this paper, we investigate the existence of positive solutions for a class of nonlinear second-order four-point boundary value problems with alternating coefficient. Our approach relies on the Krasnosel’skii fixed point theorem. The result of this paper is new and extent the previously known result.     

The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order

In this paper, we show that the monotone technique produces two monotone sequences that converge uniformly to extremal solutions of second order functional differential equations and ?$?$-Laplacian equations with Neumann boundary value conditions. Moreover, we obtain existence results assuming upper and lower solutions in the reverse order.     

Triple positive solutions for boundary value problems on infinite intervals

This paper deals with the existence of triple positive solutions for Sturm–Liouville boundary value problems of second-order nonlinear differential equation on a half line. By using a fixed point theorem in a cone due to Avery–Peterson, we show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term.     

The existence of asymptotically stable solutions for a nonlinear functional integral equation with values in a general Banach space

Motivated by the recent known results about the solvability and existence of asymptotically stable solutions for nonlinear functional integral equations in spaces of functions defined on unbounded intervals with values in the n-dimensional real space, we establish asymptotically stable solutions for a nonlinear functional integral equation in the space of all continuous functions on R₊ with values in a general Banach space, via a fixed point theorem of Krasnosel’skii type. In order to illustrate the result obtained here, an example is given.     

A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations

This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities.     

Boundary value problems for the one-dimensional Willmore equation

The one-dimensional Willmore equation is studied under Navier as well as under Dirichlet boundary conditions. We are interested in smooth graph solutions, since for suitable boundary data, we expect the stable solutions to be among these. In the first part, classical symmetric solutions for symmetric boundary data are studied and closed expressions are deduced. In the Navier case, one has existence of precisely two solutions for boundary data below a suitable threshold, precisely one solution on the threshold and no solution beyond the threshold. This effect reflects that we have a bending point in the corresponding bifurcation diagram and is not due to that we restrict ourselves to graphs. Under Dirichlet boundary conditions we always have existence of precisely one symmetric solution. In the second part, we consider boundary value problems with nonsymmetric data. Solutions are constructed by rotating and rescaling suitable parts of the graph of an explicit symmetric solution. One basic observation for the symmetric case can already be found in Euler’s work. It is one goal of the present paper to make Euler’s observation more accessible and to develop it under the point of view of boundary value problems. Moreover, general existence results are proved.     

Positive solutions of four-point boundary value problems for higher-order p-Laplacian operator

In this paper, we study the existence of positive solutions for the nonlinear four-point singular boundary value problem for higher-order with p-Laplacian operator. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear singular boundary value problem with p-Laplacian operator are obtained.     

The existence of positive solutions for second-order multi-point BVPs with the first derivative

By using the fixed-point index theory, several theorems of existence of positive solutions for second-order m$m$-point boundary value problem with the first derivative are obtained, where the nonlinear term may be noncontinuous. An example illustrating the main results is given.     

Remark on the global existence and large time asymptotics of solutions for the quadratic NLS

We study the global in time existence of small solutions to the nonlinear Schrödinger equation with quadratic interactions (0.1) We prove that if the initial data u₀ satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore, we prove the existence of the usual scattering states and find the large time asymptotics of the solutions.