Boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations

Under certain conditions, solutions of the boundary value problem, y^″=f(x,y,y^′), y(x₁)=y₁, and , are differentiated with respect to boundary conditions, where a<x₁<η₁<?<ηm<x₂<b, r₁,…,rm∈R, and y₁,y₂∈R.     

Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations

For the nth order differential equation, y(n)=f(x,y,y^′,…,y(n−1)), we consider uniqueness implies existence results for solutions satisfying certain nonlocal (k+2)-point boundary conditions, 1?k?n−1. Uniqueness of solutions when k=n−1 is intimately related to uniqueness of solutions when 1?k?n−2. These relationships are investigated as well.     

On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order

In this paper, we prove the existence and uniqueness of solutions for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q ∈ (1, 2] by applying some standard fixed point theorems. An illustrative example is also presented.     

EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

Abstract. In this Paper, the existence and uniqueness of solutions for boundary valueproblem     

Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions

We use a fixed point theorem due to Avery and Peterson to establish the existence of at least three non-negative solutions of some nonlocal boundary value problems to third order differential equations with advanced arguments. An example is given to illustrate the main results.     

THE SINGULARLY PERTURBED NONLOCAL BOUNDARY VALUE PROBLEMS FOR HIGHER ORDER NONLINEAR ELLIPTIC EQUATIONS

In this paper,a class of singular perturbation of noidocal boundary value problems forelliptic partial differentia[ equations of higher order is considered by using the differential in-equalities. The uniformly valid asymptotic expansion of solution is obtained.     

On solvability of boundary value problems for higher order nonlinear hyperbolic equations

In the rectangle Ω=[0,a]×[0,b] for the nonlinear hyperbolic equation

the boundary value problems of the type

are considered, where and are linear bounded functionals.Sufficient conditions of solvability and unique solvability of the general problem and its particular cases (Nicoletti type, Dirichlet, Lidstone and Periodic problems) are established.     

Positive solutions and eigenvalue intervals of nonlocal boundary value problems with delays

The paper is concerned with the delay differential equation u^″+λb(t)f(u(t−τ))=0 satisfying u(t)=0 for −τ?t?0 and , where denotes the Riemann-Stieltjes integral. By applying the fixed point theorem in cones, we show the relationship between the asymptotic behaviors of the quotient (at zero and infinity) and the open intervals (eigenvalue intervals) of the parameter λ such that the problem has zero, one and two positive solution(s). If g(t)=t, by using a property of the Riemann-Stieltjes integral, the above nonlocal boundary value problem educes a three-point boundary value problem with delay, for which some similar results are established.     

Well-posedness of general nonlocal nonhomogeneous boundary value problems for pseudodifferential equations with partial derivatives

In the article we study the questions of well-posedness of general nonlocal boundary value problems for pseudodifferential equations in the Besov-type limit spaces.     

An existence and uniqueness theorem for a second order nonlinear system

In this paper we give an existence and uniqueness theorem for a nonlinear second order homogeneous Dirichlet system.     

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.     

On initial-boundary value problems in bounded and unbounded domains for a class of nonlinear hyperbolic equations of the third order

For the nonlinear hyperbolic equation

u(2,1)=f(x,t,u,u(1,0),u(2,0),u(0,1),u(1,1))     

Noncontinuous data boundary value problems for Schrödinger equation in Lipschitz domains

The noncontinuous data boundary value problems for Schrödinger equations in Lipschitz domains and its progress are pointed out in this paper. Particularly, the L ^p boundary value problems with p > 1, and H ^p boundary value problems with p < 1 have been studied. Some open problems about the Besov-Sobolev and Orlicz boundary value problems are given.     

Uniqueness implies existence and uniqueness criterion for nonlocal boundary value problems for third order differential equations

For the third order differential equation, we consider uniqueness implies existence results for solutions satisfying the nonlocal -point boundary conditions, Uniqueness of solutions of such boundary value problems is intimately related to solutions of the third order equation satisfying certain nonlocal -point boundary conditions. These relationships are investigated as well.

     

Integral boundary value problems for first order impulsive integro-differential equations of mixed type

This paper considers existence of solutions for a class of first order impulsive differential equation with integral boundary value conditions. We present a new comparison theorem and show that the monotone iterative technique coupled with lower and upper solutions is still valid. The results we obtained generalize and improve many recent results.