Singular Three-point Boundary Value Problem Research Articles

Positive solution for a fractional singular boundary value problem with p-Laplacian operator

In this paper, we consider a fractional singular three-point boundary value problem with p-Laplacian operator. The nonlinearity f(t,u) may be singular at t = 0,1 and u = 0. Some properties of the associated Green function are obtained. By using the upper and lower solutions method and a fixed point theorem, the existence result of positive solution is established.

Existence of solutions of boundary value problems for singular fractional q-difference equations

In this paper, we study the existence and uniqueness of solutions for a class of singular three-point boundary value problems of fractional q-difference equations invovling fractional q-derivative of Riemann–Liouville type. Based on the generalization of Banach contraction principle, we obtain a sufficient condition for existence and uniqueness of solutions of the problem. By applying the Krasnoselskii’s fixed point theorem, we establish a sufficient condition for the existence of at least one solution of the problem. As applications, two examples are presented to illustrate our main results.

Multiplicity of Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem with a Parameter

This paper is concerned with the following second-order three-point boundary value problemu″t+β2ut+λqtft,ut=0,t∈0 , 1,u0=0,u(1)=δu(η), whereβ∈(0,π/2),δ>0,η∈(0,1), andλis a positive parameter. First, Green’s function for the associated linear boundary value problem is constructed, and then some useful properties of Green’s function are obtained. Finally, existence, multiplicity, and nonexistence results for positive solutions are derived in terms of different values ofλby means of the fixed point index theory.

Multiple Solutions for Nonlinear Doubly Singular Three-Point Boundary Value Problems with Derivative Dependence

We study the existence of multiple nonnegative solutions for the doubly singular three-point boundary value problem with derivative dependent data function-(p(t)y′(t))′=q(t)f(t,y(t),p(t)y′(t)),0<t<1,y(0)=0,y(1)=α1y(η). Here,p∈C[0,1]∩C1(0,1]withp(t)>0on(0,1]andq(t)is allowed to be discontinuous att=0. The fixed point theory in a cone is applied to achieve new and more general results for existence of multiple nonnegative solutions of the problem. The results are illustrated through examples.

Eigenvalue for a singular second order three-point boundary value problem

In this paper, the existence of positive solutions for a singular second-order three-point boundary value problem is investigated. By using Krasnoselskii’s fixed point theorem, several sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exist positive solutions are obtained. Finally, two examples are given to illustrate the importance of results obtained.

Nonexistence and existence of positive solutions for second order singular three-point boundary value problems with derivative dependent and sign-changing nonlinearities

This paper discusses the nonexistence and the existence of positive solutions for second order singular three-point boundary value problems when the nonlinear term f(t,x,y) is sign-changing and may be singular at t=0, x=0, y=0.

Positive solutions to singular third-order three-point boundary value problems on time scales

In this paper, we are concerned with singular third-order three-point boundary value problems on time scales. Some theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is also given to illustrate our results.

A necessary and sufficient condition for the existence of positive solutions to a kind of singular three-point boundary value problem

In this paper, we are concerned with the following three point boundary value problem { u ″ ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , δ u ′ ( 0 ) − u ( ξ ) = 0 , u ( 1 ) = 0 , where δ ≥ ξ , 0 < ξ < 1 , and the nonlinear term f may be singular at t = 1 or/and t = 0 . A necessary and sufficient condition for the existence of C [ 0 , 1 ] as well as C 1 [ 0 , 1 ] positive solution is given by constructing lower and upper solutions, and with the maximal theorem. Also, the uniqueness, iterative methods, convergence rate and computational methods of the positive solutions are considered.

Multiplicity of positive solutions for singular three-point boundary value problems at resonance

In this paper, we study the existence and multiplicity of positive solutions for the three-point boundary value problems with singular nonlinear perturbations at resonance. The existence results are applicable to both the case of a strong singularity and the case of a weak singularity. The proof is based on a nonlinear alternative Leray–Schauder principle and a fixed point theorem in cones for completely continuous operators.

Positive solutions of singular three-point boundary value problems for second-order differential equations

In this paper, we study the existence of positive solutions of a singular three-point boundary value problem for the following second-order differential equation { y ″ + μ a ( t ) f ( t , y ( t ) ) = 0 , t ∈ ( 0 , 1 ) , y ( 0 ) − β y ′ ( 0 ) = 0 , y ( 1 ) = α y ( η ) , where μ > 0 is a parameter, β > 0 , 0 < η < 1 , 0 < α η < 1 , ( 1 − α η ) + β ( 1 − α ) > 0 . By constructing an available integral operator and combining fixed point index theory with properties of Green’s function under some conditions concerning the first eigenvalues corresponding to the relevant linear operator, the sufficient conditions of the existence of positive solutions for the boundary value problems are established. The interesting point of the results is that the term a ( t ) may be singular at t = 0 and/or t = 1 , moreover the nonlinear f ( t , x ) is also allowed to have singularity at x = 0 .

Positive solutions of second order semipositone singular three-point boundary value problems

In this paper we prove the existence of positive solutions for a class of second order semipositone singular three-point boundary value problems. The results are obtained by the use of a GuoKrasnoselskii’s fixed point theorem in cones.

Green's function and positive solutions of a singular nth-order three-point boundary value problem on time scales

In this paper, we investigate the existence of positive solutions for a class of singular nth-order three-point boundary value problem. The associated Green’s function for the boundary value problem is given at first, and some useful properties of the Green’s function are obtained. The main tool is fixed-point index theory. The results obtained in this paper essentially improve and generalize some well-known results.

Positive Solutions for a Class of Coupled System of Singular Three-Point Boundary Value Problems

Existence of positive solutions for a coupled system of nonlinear three-point boundary value problems of the type , , , , , , , is established. The nonlinearities , are continuous and may be singular at , and/or , while the parameters , satisfy . An example is also included to show the applicability of our result.

Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales

We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result.

Necessary and sufficient conditions for the existence of quasi-symmetric positive solutions of singular boundary value problem

This paper investigates the existence of quasi-symmetric positive solutions for fourth-order singular three-point boundary value problem. We first give definitions of quasi-symmetric lower and upper solutions. By constructing quasi-symmetric lower and upper solutions and using the Schauder fixed point theorem, necessary and sufficient conditions for existence results are obtained.

Positive solutions for three-point semipositone boundary value problems with convex nonlinearity

In this paper, we consider the existence of positive solutions for a class of singular three-point semipositone boundary value problems with convex nonlinearity. By employing a well-known fixed point theorem, some new existence results are given under the case where nonlinearity can be sign-changing and have finitely many singularities.

Multiple solutions of singular three-point boundary value problems on [formula omitted

In this paper, we study the existence of single and multiple solutions of three-point boundary value problems for the following nonlinear singular second-order differential equations { x ″ ( t ) − p x ′ ( t ) − q x ( t ) + h ( t ) f ( t , x ( t ) ) = 0 , t ∈ ( 0 , + ∞ ) , a x ( 0 ) − b x ′ ( 0 ) − k x ( ξ ) = c ≥ 0 , lim t → + ∞ x ( t ) e r t = d ≥ 0 , where p , b ≥ 0 , a > k > 0 , q > 0 , 0 < ξ < + ∞ , r ∈ [ 0 , p + p 2 + 4 q 2 ] , h : ( 0 , + ∞ ) → ( 0 , + ∞ ) is continuous and may be singular at t = 0 , f : [ 0 , + ∞ ) × [ 0 , + ∞ ) → ( − ∞ , + ∞ ) is continuous and may take a negative value. By applying the technique of lower and upper solutions and the theory of topological degree, we obtain the conditions for the existence of at least one solution and at least three solutions respectively.

The existence of countably many positive solutions for some nonlinear three-point boundary problems on the half-line

In this paper, we study the existence of countably many positive solutions for some nonlinear singular three-point boundary problems on the half-line ( ϕ ( u ′ ) ( t ) ) ′ + a ( t ) f ( t , u ( t ) ) = 0 , 0 < t < + ∞ , u ( 0 ) − B 0 ( u ′ ( η ) ) = 0 , u ′ ( ∞ ) = 0 , where ϕ ( s ) : R → R is an increasing homeomorphism and positive homomorphism and ϕ ( 0 ) = 0 and η ∈ ( 0 , + ∞ ) , a : [ 0 , + ∞ ) → [ 0 , + ∞ ) and has countably many singularities on [ 0 , + ∞ ) . By using the fixed-point index theorem and a new fixed-point theorem in cones, the existence of countably many solutions for singular three-point boundary value problem are obtained under conditions weaker than those used by Liu and Zhang [B.F. Liu, J.H. Zhang, The existence of positive solutions for some nonlinear boundary value problems with linear mixed boundary conditions, J. Math. Anal. Appl. 309 (2005) 505–516].

Optimal existence criteria for symmetric positive solutions to a singular three-point boundary value problem

In this paper we are concerned with the existence of symmetric positive solutions of the following singular second order three-point boundary value problem u ″ ( t ) + h ( t ) f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( t ) = u ( 1 − t ) , α u ′ ( 0 ) − β u ′ ( 1 ) = γ u ( 1 / 2 ) , where α , β , γ > 0 , α + β > γ / 2 , h : ( 0 , 1 ) → [ 0 , ∞ ) is symmetric on ( 0 , 1 ) and may be singular at t = 0 and t = 1 . First, the Green’s function for associated linear boundary value problem is constructed, and some useful properties of the Green’s function are obtained. Then by applying the fixed-point index theory, we establish some optimal criteria for the existence of one or two symmetric positive solutions which involve the principal eigenvalue of a related linear operator. Finally we illustrate our results by several examples, none of which can be handled using the existing results.

Positive solutions for a singular second-order three-point boundary value problem

This paper presents the existence of positive solutions for the singular three-point boundary value problem u ″ ( t ) + a ( t ) u ′ ( t ) + b ( t ) u ( t ) + h ( t ) f ( t , u ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = α u ( η ) , where h ( t ) is allowed to be singular at t = 0 , 1 and f may be singular at u = 0. The existence criteria for positive solutions of the above problem is established by applying the fixed point index theorem under some weaker conditions concerning the first eigenvalue corresponding to the relevant linear operator.