Solvability of singular second order m-point boundary value problems

Let be a function satisfying Carathéodory's conditions and (1−t)e(t)∈L¹(0,1). Let ξi∈(0,1), ai∈R, i=1,…,m−2, 0<ξ₁<ξ₂<?<ξm−2<1 be given. This paper is concerned with the problem of existence of a C¹[0,1) solution for the m-point boundary value problem

     

Positive solution of singular Dirichlet boundary value problems for second order differential equation system

This paper investigates the existence of positive solutions of singular Dirichlet boundary value problems for second order differential system. A necessary and sufficient condition for the existence of C[0,1]×C[0,1] positive solutions as well as C¹[0,1]×C¹[0,1] positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems. Our nonlinearity fi(t,x₁,x₂) may be singular at x₁=0, x₂=0, t=0 and/or t=1, i=1,2.     

A note on singular nonlinear boundary value problems

In this paper we study the existence and multiplicity of solutions of the following operator equation in Banach space E:

u=λAu,0<λ<+∞,u∈P?{θ},     

Positive solutions for singular boundary value problems of a coupled system of differential equations

By using fixed point theorem of cone expansion and compression, this paper investigates the existence of multiple positive solutions for singular boundary value problems of a coupled system of nonlinear ordinary differential equations.     

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.     

Positive solution to a special singular second-order boundary value problem

Let λ be a nonnegative parameter. The existence of a positive solution is studied for a semipositone second-order boundary value problem

where d>0,α≥0,β≥0,α+β>0, q(t)f(t,u,v)≥0 on a suitable subset of [0,1]×[0,+∞)×(−∞,+∞) and f(t,u,v) is allowed to be singular at t=0,t=1 and u=0. The proofs are based on the Leray–Schauder fixed point theorem and the localization method.     

A CLASS OF SINGULAR PERTURBATIONS FOR SECOND ORDER QUASI－LINEAR BOUNDARY VALUE PROBLEMS ON INFINITE INTERVAL

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Global structure of solutions for a class of two-point boundary value problems involving singular and convex or concave nonlinearities

This paper investigates the following two-point singular boundary value problems (BVP):

     

Positive solutions of a singular nonlinear third order two-point boundary value problem

In this paper some existence results of positive solutions for the following singular nonlinear third order two-point boundary value problem:

     

Positive solutions of singular third-order three-point boundary value problems

The positive solutions of a class of singular third-order three-point boundary value problems are considered by using the Guo-Krasnosel'skii fixed point theorem of cone expansion-compression type. In this class of problems, the nonlinear term is allowed to be singular. Main results show that this class of problems can have n positive solutions provided that the conditions on the nonlinear term on some bounded sets are appropriate.     

Structure of a class of singular boundary value problem with superlinear effect

This paper investigates the structure of solutions of singular boundary value problem with superlinear effect. It is proved that the closure of positive solution set possesses a maximal subcontinuum C (i.e., a maximal closed connected subset of solutions), which comes from (0,θ) and tends to (0,+∞) finally. As a corollary, the existence of multiple positive solutions and the behavior of solutions according to parameter λ are obtained.     

The method of lower and upper solutions for fourth order singular m-point boundary value problems

This paper investigates the existence of positive solutions for fourth order singular m-point boundary value problems. Firstly, we establish a comparison theorem, then we define a partial ordering in C²[0,1]∩C⁴(0,1) and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C²[0,1] as well as C³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x, y, t=0 and/or t=1.     

Positive solutions of superlinear semipositone singular Dirichlet boundary value problems

In this paper, we study a class of superlinear semipositone singular second order Dirichlet boundary value problem. A sufficient condition for the existence of positive solution is obtained under the more simple assumptions.     

Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method

This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

Positive solutions for singular semi-positone boundary value problems

In this paper, some new results about the existence of positive solutions for singular semi-positone boundary value problems are obtained. The results of this paper partially improve the former corresponding work.     

A singular boundary value problem for odd-order differential equations

The odd-order differential equation (−1)ⁿx⁽²ⁿ⁺¹⁾=f(t,x,…,x⁽²ⁿ⁾) together with the Lidstone boundary conditions x^(2j)(0)=x^(2j)(T)=0, 0?j?n−1, and the next condition x⁽²ⁿ⁾(0)=0 is discussed. Here f satisfying the local Carathéodory conditions can have singularities at the value zero of all its phase variables. Existence result for the above problem is proved by the general existence principle for singular boundary value problems.     

Study of singular boundary value problems for second order impulsive differential equations

This paper studies the existence of extremal solutions for a class of singular boundary value problems of second order impulsive differential equations. By using the method of upper and lower solutions and the monotone iterative technique, criteria of the existence of extremal solutions are established.     

Positive solutions for singular m-point boundary value problems with positive parameter

In this paper, some new existence results for positive solutions for a class of singular m-point boundary value problems with parameter be obtained.     

Existence of solutions to the nonlinear,singular second order Bohr boundary value problems

This paper investigates the existence of solutions for nonlinear systems of second order, singular boundary value problems (BVPs) with Bohr boundary conditions. A key application that arises from this theory is the famous Thomas–Fermi equations for the model of the atom when it is in a neutral state. The methodology in this paper uses an alternative and equivalent BVP, which is in the class of resonant singular BVPs, and thus this paper obtains novel results by implementing an innovative differential inequality, Lyapunov functions and topological techniques. This approach furnishes new results in the area of singular BVPs for a priori bounds and existence of solutions, where the BVP has unrestricted growth conditions and subject to the Bohr boundary conditions. In addition, the results can be relaxed and hold for the non-singular case too.