Upper and Lower Semicontinuity of Solution Sets for Parametric Generalized Vector Quasi-equilibrium Problems

In this paper, two kinds of parametric generalized vector quasi-equilibrium problems are introduced and the relations between them are studied. The upper and lower semicontinuity of their solution sets to parameters are investigated.

THE APPLICATION OF WEINSTEIN-CHIEN′S METHOD——THE UPPER AND LOWER LIMITS OF FUNDAMENTAL FREQUENCY OF RECTANGULAR PLATES WITH EDGES ARE THE MIXTURE OF SIMPLY SUPPORTED PORTIONS AND CLAMPED PORTIONS

In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundamental frequency of such plates is evaluated. A kind of polynomial satisfying the displacement boundary conditions is designed, os that it is enabled to evaluate the upper limit of fundamental frequency by Ritz' method. The practical calculation examples solved by these methods have given satisfactory results. At the end of this paper, it is pointed out that the socalled exact solution of such plates usually evaluated by the force superposition method is essentially a kind of lower limit of solution, if the truncated error of series which occurs in actual calculation is considered.

Eigenfunction expansion method of upper triangular operator matrixand application to two-dimensional elasticity problems based onstress formulation

This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.

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

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.

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.

Upper bound analytical solution for surrounding rock pressure of shallow unsymmetrical loading tunnels

By combining the results of laboratory model tests with relevant flow rules, the failure mode of shallow unsymmetrical loading tunnels and the corresponding velocity field were established. According to the principle of virtual power, the upper bound solution for surrounding rock pressure of shallow unsymmetrical loading tunnel was derived and verified by an example. The results indicate that the calculated results of the derived upper bound method for surrounding rock pressure of shallow unsymmetrical loading tunnels are relatively close to those of the existing "code method" and test results, which means that the proposed method is feasible. The current code method underestimates the unsymmetrical loading feature of surrounding rock pressure of shallow unsymmetrical loading tunnels, so it is unsafe; when the burial depth is less or greater than two times of the tunnel span and the unsymmetrical loading angle is less than 45°, the upper bound method or the average value of the results calculated by the upper bound method and code method respectively, is comparatively reasonable. When the burial depth is greater than two times of the tunnel span and the unsymmetrical loading angle is greater than 45°, the code method is more suitable.

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.

NEW EXISTENCE RESULTS OF POSITIVE SOLUTION FOR A CLASS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

In this article, the existence and uniqueness of positive solution for a class of nonlinear fractional differential equations is proved by constructing the upper and lower control functions of the nonlinear term without any monotone requirement. Our main method to the problem is the method of upper and lower solutions and Schauder fixed point theorem. Finally, we give an example to illuminate our results.

ON MONOTONE ITERATIVE METHOD FOR INITIAL VALUE PROBLEMS OF NONLINEAR SECOND-ORDER INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

Using the monotone iterative method and Monch Fixed point theorem, the existence of solutions and coupled minimal and maximal quasisolutions of initial value problems for mixed monotone second-order integro-differential equations in Banach spaces are studied. Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding steady state solutions. Then, the above results are extended to a semilinear parabolic equation with nonlinear boundary condition by analyzing the corresponding eigenvalue problem and using the method of upper and lower solutions.

Charge density at the nucleus and radial behavior of ground state for lithium-like ions with Z=21 to 30

By using the full-core plus correlation （FCPC） type wave functions, the accurate charge densities p（0） at the nucleus and the radial expectation values of the ground states for the lithium-like systems with Z =- 21 to 30 are obtained. The determinantal conditions and the electron-nucleus cusp condition are used to calculate the inequalities of the upper and the lower bounds to p（0） with two or more expectation values. These inequalities, derived by Angulo and Dehesa [Phys. Rev. A 44 1516 （1991）], are verified to be also valid for these ions with higher nuclear charge. The present results show that the wave functions used in this paper are satisfactory in the whole configuration space for these ions with higher nuclear charge.

Eigenfunction expansion method and its application to two-dimensional elasticity problems based on stress formulation

This paper proposes an eigenfunction expansion method to solve twodimensional （2D） elasticity problems based on stress formulation. By introducing appropriate state functions, the fundamental system of partial differential equations of the above 2D problems is rewritten as an upper triangular differential system. For the associated operator matrix, the existence and the completeness of two normed orthogonal eigenfunction systems in some space are obtained, which belong to the two block operators arising in the operator matrix. Moreover, the general solution to the above 2D problem is given by the eigenfunction expansion method.

Double Symplectic Eigenfunction Expansion Method of Free Vibration of Rectangular Thin Plates

The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrixdifferential system.For the associated operator matrix,we find that the two diagonal block operators are Hamiltonian.Moreover,the existence and completeness of normed symplectic orthogonal eigenfunction systems of these two blockoperators are demonstrated.Based on the completeness,the general solution of the free vibration of rectangular thinplates is given by double symplectic eigenfunction expansion method.

Dynamics Behaviors of a Reaction-Diffusion Predator-Prey System with Beddington-DeAngelis Functional Response and Delay

This paper is concerned with the existence of traveling wave solutions in a reaction-diffusion predator-prey system with Beddington-DeAngelis functional response and a discrete time delay. By introducing a partial quasi-monotonicity condition and constructing a pair of upper-lower solutions, we establish the existence of traveling wave solutions. Moreover, a numerical simulation is carried out to illustrate the theoretical results.

Existence of positive solutions for fourth order singular differential equations with Sturm-Liouville boundary conditions

By using the upper and lower solutions method and fixed point theory,we investigate a class of fourth-order singular differential equations with the Sturm-Liouville Boundary conditions.Some sufficient conditions are obtained for the existence of C2[0,1] positive solutions and C3[0,1] positive solutions.

UPPER AND LOWER SOLUTIONS METHODS FOREXISTENCE OF PERIODIC SOLUTIONS OFDUFFING EQUATION

This paper gives upper and lower solutions methods for the existence of solutions of periodic BVP for second order Duffing equation x" + kx’+ g(t, x) = 0, x(0) = x(T), x’(0) = x’(T),where g is a Caratheodory function. As an application, some existence theorems are given for the equation x" + kx’ + g(t, x) = s, x(0) = x(T), x’(0) = x’(T),with respect to a real parameter s.

A class of singular perturbation solutions to semilinear equations of fourth order

A class of singularly perturbed boundary value problems for semilinear equations of fourth order with two parameters are considered. Under suitable conditions, using the method of lower and upper solutions, the existence and the asymptotic behavior of the solution to the boundary value problem are studied, In the present paper, the solution to the original singularly perturbed problem with two parameters has only one boundary layer.

ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED PROBLEM FOR A NONLINEAR SINGULAR EQUATION

The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.

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.