A Generalized Upper and Lower Solution Method for Singular Discrete Boundary Value Problems

This paper presents new existence results for singular discrete boundary value problems. In particular our nonlinearity may be singular in its dependent variable and is allowed to change sign.

SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS

We mainly study the existence of positive solutions for the following third order singular multi-point boundary value problem{x^（3）（t） ＋ f（t, x（t）, x′（t）） = 0, 0 〈 t 〈 1,x（0）-∑i=1^m1 αi x（ξi） = 0, x′（0）-∑i=1^m2 βi x′（ηi） = 0, x′（1）=0,where 0 ≤ ai≤∑i=1^m1 αi 〈 1, i = 1, 2, ···, m1, 0 〈 ξ1〈 ξ2〈 ··· 〈 ξm1〈 1, 0 ≤βj≤∑i^m2=1βi〈1,J=1,2, ···, m2, 0 〈 η1〈 η2〈 ··· 〈 ηm2〈 1. And we obtain some necessa βi 〈=11, j = 1,ry and sufficient conditions for the existence of C^1[0, 1] and C^2[0, 1] positive solutions by constructing lower and upper solutions and by using the comparison theorem. Our nonlinearity f（t, x, y）may be singular at x, y, t = 0 and/or t = 1.

The Existence and Uniqueness of Positive Solutions for a Singular Nonlinear Three-Point Boundary Value Problems

Using the method of lower and upper solutions, we study the following singular nonlinear three-point boundary value problems: , where K ∈ C[0,1] ,0 α η < 1 and λ is a positive parameter and present the existence, uniqueness, and the dependency on parameters of the positive solutions under various assumptions. Our result improves those in the previous literatures.

Existence of Solutions for S-L Singular Boundary Value Problems with p-Laplacian Operators

The existence of solutions for singular nonlinear two point boundary value problems subject to Sturm Liouville boundary conditions with p Laplacian operators is studied by the method of upper and lower solutions. The proof is based on an application of Schauder’s fixed point theorem to a modified problem whose solutions are that of the original one. At the same time, Arzela Ascoli theorem is used to prove that the defined operator N is a compact map.

Mixed Monotone Iterative Technique and Boundary Value Problem for a Class of Impulsive Integro-Differential Equations

In this paper,the existence and uniqueness of iterative solutions to the boundary value problems for a class of first order impulsive integro-differential equations were studied. Under a new concept of upper and lower solutions, a new monotone iterative technique on the boundary value problem of integro-differential equations was proposed. The existence and uniqueness of iterative solutions and the error estimation in certain interval were obtained.An example was also given to illustrate the results.

On Monotone Method for Second Order Periodic Boundary Value Problems and Periodic Solutions of Functional Difference Equations

In this paper, we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions to second order periodic boundary value problems and periodic solutions of functional difference equations. We obtain some new results under the lower solution α and upper solutionβ with α≤β

THE EXISTENCE OF POSITIVE SOLUTION FOR THE NONLINEARSINGULAR BOUNDARY VALUE PROBLEMS

Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1), x’ (1)) = 0, g (z (0), x’(0)) = 0, and x (1) = 0,which extends the result of J. V. Baxley.

Upper and Lower Solutions for Impulsive Differential Equations with Application to ODE

In this paper, we establish the existence of upper and lower solutions for a periodic boundary value problems (PBVP for short) of impulsive differential equations. which guarantees the existence of at least one solution for the problem. As an application, these results are applied to PBVP of ODE and some examples are given to illustrate our results.

EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower solutions. Our results improve someknown works.

THIRD-ORDER NONLINEAR SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM

Third order singulary perturbed boundary value problem by means of differential inequality theories is studied. Based on the given results of second order nonlinear boundary value problem, the upper and lower solutions method of third order nonlinear boundary value problems by making use of Volterra type integral operator was established. Specific upper and lower solutions were constructed, and existence and asymptotic estimates of solutions under suitable conditions were obtained. The result shows that it seems to be new to apply these techniques to solving these kinds of third order singularly perturbed boundary value problem. An example is given to demonstrate the applications.

Singular perturbation of a second-order three-point boundary value problem for nonlinear systems

This paper deals with the existence of solutions to a singularly perturbed second-order three-point boundary value problem for nonlinear differential systems. The authors construct an appropriate generalized lower- and upper-solution pair, a concept defined in this paper, and employ the Nagumo conditions and algebraic boundary layer functions to ensure the existence of solutions of the problem. The uniformly valid asymptotic estimate of the solutions is given as well. The differential systems have nonlinear dependence on all order derivatives of the unknown.

EXISTENCE AND UNIQUENESS FOR SECOND-ORDER VECTOR BOUNDARY VALUE PROBLEM OF NONLINEAR SYSTEMS

This paper is concerned with the following second-order vector boundary value problem ：x^R=f（t,Sx,x,x＇）,0〈t〈1,x（0）=A,g（x（1）,x＇（1））=B,where x,f,g,A and B are n-vectors. Under appropriate assumptions,existence and uniqueness of solutions are obtained by using upper and lower solutions method.

Existence of Solutions for Boundary Value Problems of Conformable Fractional Differential Equations

In this paper, we study a class of boundary value problems for conformable fractional differential equations under a new definition. Firstly, by using the monotone iterative technique and the method of coupled upper and lower solution, the sufficient condition for the existence of the boundary value problem is obtained, and the range of the solution is determined. Then the existence and uniqueness of the solution are proved by the proof by contradiction. Finally, a concrete example is given to illustrate the wide applicability of our main results.

A Class of Singularly Perturbed Boundary Value Problems Arising from Catalytic Reactions

A class of singularly perturbed boundary value problems arising from the catalytic reactions in chemical engineering is observed. That kind of p roblems exhibits the behavior of nonexponentially decayed boundary layer, and he nce the study of asymptotic behavior of their solutions seems more diffcult. The uniformly valid asymptotic expansions of solutions as well as their derivatives are given via the upper and lower solutions method, and those estimates seem qu ite accurate.

ON THE EXISTENCE OF SOLUTION OF A NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEM ARISING FROM A LIQUID METAL FLOW

In this paper, we discuss the existence of solution of a nonlinear two-point boundary value problem with a positive parameter Q arising in the study of surfacetension-induced flows of a liquid metal or semiconductor. By applying the Schauder's fixed-point theorem, we prove that the problem admits a solution for 0 ≤ Q ≤ 14.306.It improves the result of 0 ≤ Q ＜ 1 in [2] and 0 ≤ Q ≤ 13.213 in [3].

Periodic boundary value problem for the first order functional differential equations with impulses

This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is presented and the previous results are extended.

Positive Solutions to Singular Boundary Value Problems with Sign Changing Nonlinearities on the Half-Line via Upper and Lower Solutions

This paper presents a lower and upper solution technique for singular second order boundary value problems on the half line.

Periodic Boundary Value Problem for Functional Differential Equations with Impulses

Aim To investigate the periodic boundary value problem for functional differential equations with impulses. Methods The method of upper and lower solutions and the monotone iterative technique were used to establish our results. Results and Conclusion The results of the existence of maximal and minimal solutions of the periodic boundary value problem for functional differential equations with impulses are obtained.

Second Order Periodic Boundary Value Problems Involving the Distributional Henstock-Kurzweil Integral

We apply the distributional derivative to study the existence of solutions of the second order periodic boundary value problems involving the distributional Henstock-Kurzweil integral. The distributional Henstock-Kurzweil integral is a general intergral, which contains the Lebesgue and Henstock-Kurzweil integrals. And the distributional derivative includes ordinary derivatives and approximate derivatives. By using the method of upper and lower solutions and a fixed point theorem, we achieve some results which are the generalizations of some previous results in the literatures.

EXISTENCE OF BOUNDARY VALUE PROBLEMS FOR TFD EQUATION IN QUANTUM MECHANICS

TFD ( Thomas-Fermi-Dirac) equation in quantum mechanics is established. The existence theorems of the solutions are obtained by singular boundary value problem theory of ordinary differential equation and upper and lower solution method.