Second-order M-point Boundary Value Problem Research Articles

Existence of solution for a resonant p-Laplacian second-order m-point boundary value problem on the half-line with two dimensional kernel

The existence of a solution for a second-order p-Laplacian boundary value problem at resonance with two dimensional kernel will be considered in this paper. A semi-projector, the Ge and Ren extension of Mawhin’s coincidence degree theory, and algebraic processes will be used to establish existence results, while an example will be given to validate our result.

Existence of symmetric positive solutions for a multipoint boundary value problem with sign-changing nonlinearity on time scales

In this paper, we make use of the four functionals fixed point theorem to verify the existence of at least one symmetric positive solution of a second-order m-point boundary value problem on time scales such that the considered equation admits a nonlinear term f whose sign is allowed to change. The discussed problem involves both an increasing homeomorphism and homomorphism, which generalizes the p-Laplacian operator. An example which supports our theoretical results is also indicated.MSC:34B10, 39A10.

Existence of Positive Solutions of Nonlinear Second-Order M-Point Boundary Value Problem

Under suitable conditions on, the nonlinear second-order m-point boundary value problem has at least one positive solution. In this paper, the authors examine the positive solutions of nonlinear second-order m-point boundary value problem.

Global Structure of Nodal Solutions for Second-Order m-Point Boundary Value Problems with Superlinear Nonlinearities

We consider the nonlinear eigenvalue problems , , , , where , and for with and satisfies for , and , where . We investigate the global structure of nodal solutions by using the Rabinowitz's global bifurcation theorem.

Two and three positive solutions of m-point boundary value problems for functional dynamic equations on time scales

In this paper, by using fixed point theorems in a cone, we study the existence of at least two and three positive solutions of a nonlinear second-order m-point boundary value problem for functional dynamic equations on time scales. As an application, we also give an example to demonstrate our results.

Exact solutions of a class of second-order nonlocal boundary value problems and applications

In this paper, solutions of a class of second-order differential equations with some multi-point boundary conditions are studied. We give exact expressions of the solutions for the linear m-point boundary problems by the Green’s functions. As applications, we study uniqueness and iteration of the positive solutions for a nonlinear singular second-order m-point boundary value problem.

Existence of positive solutions for a system of second-order m-point BVPs with variable parameters

In this paper, we study the existence of positive solutions for a class of second-order nonlinear m-point boundary value problems of differential system. By using the fixed point theory of cone expansion and compression with norm type, we show the sufficient conditions for the existence results.

Existence of three positive solutions for some second-order m-point boundary value problems

By using fixed-point theorems, some new results for multiplicity of positive solutions for some second order m-point boundary value problems are obtained.The associated Green’s function of these problems are also given.

Solvability of Second-Order m-Point Boundary Value Problems with Impulses

By Leray-Schauder continuation theorem and the nonlinear alternative of Leray-Schauder type, the existence of a solution for an-point boundary value problem with impulses is proved.

Nodal solutions for a second-order m-point boundary value problem

We study the existence of nodal solutions of the m-point boundary value problem $$\begin{gathered} u'' + f(u) = 0,0 < t < 1, \hfill u'(0) = 0,u(1) = \sum\limits_{i = 1}^{m - 2} {\alpha _i u(\eta i)} \hfill \end{gathered} $$ where ηi ∈ ℚ (i = 1, 2, ..., m − 2) with 0 0 and \(\sum\nolimits_{i = 1}^{m - 2} {\alpha _i } \) < 1. We give conditions on the ratio f(s)/s at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.

Multiple positive solutions for second-order m-point boundary value problems

In this paper, the second-order m-point boundary value problem u ″ ( t ) + f ( t , u ) = 0 , 0 ⩽ t ⩽ 1 , u ( 0 ) = 0 , u ′ ( 1 ) − ∑ i = 1 m − 2 k i u ′ ( ξ i ) = 0 is studied, where k i > 0 ( i = 1 , 2 , … , m − 2 ) , 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 . We give at first the associated Green's function and its properties for the m-point boundary value problem, and growth conditions are imposed on f which yield the existence of at least three positive solutions by using the five functionals fixed point theorem and the properties.

Multiple positive solutions of singular second-order m-point boundary value problems

The existence of multiple positive solutions about the singular second-order m-point boundary value problem φ ″ ( x ) + h ( x ) f ( φ ( x ) ) = 0 , 0 < x < 1 , subject to some m-point boundary value conditions is considered. h ( x ) is allowed to be singular at x = 0 and x = 1 .

Multiplicity results for second-order m-point boundary value problem

By using fixed-point theorems, some new results for multiplicity of positive solutions for a class of second-order m-point boundary value problem are obtained. The associated Green's function of the problem is also given.

Singular second-order multipoint dynamic boundary value problems with mixed derivatives

We study a certain singular second-order m-point boundary value problem on a time scale and establish the existence of a solution. The proof of our main result is based upon the Leray-Schauder continuation theorem.

Nodal solutions for second-order m-point boundary value problems with nonlinearities across several eigenvalues

We study the existence of nodal solutions of the m-point boundary value problem u ″ + f ( u ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m - 2 α i u ( η i ) , where η i ∈ Q ( i = 1 , 2 , … , m - 2 ) with 0 < η 1 < η 2 < ⋯ < η m - 2 < 1 , and α i ∈ R ( i = 1 , 2 , … , m - 2 ) with α i > 0 and 0 < ∑ i = 1 m - 2 α i ⩽ 1 . We give conditions on the ratio f ( s ) / s at infinity and zero that guarantee the existence of nodal solutions. The proofs of our main results are based upon bifurcation techniques.

Positive solutions of nonlinear second-order m-point boundary value problem

In this paper, by using Krasnoselskii's fixed point theorem of cone expansion–compression type and under suitable conditions, we present the existence of single and multiple positive solutions to the nonlinear second-order m-point boundary value problem u ″ ( t ) + λ a ( t ) f ( u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m - 2 a i u ( ξ i ) , where λ is a positive parameter, a i ⩾ 0 for i = 1 , 2 , … , m - 3 and a m - 2 > 0 , ξ i satisfy 0 < ξ 1 < ξ 2 < ⋯ < ξ m - 2 < 1 and ∑ i = 1 m - 2 a i ξ i < 1 . We derive an explicit interval of λ such that for any λ in this interval, the existence of at least one positive solution to the boundary value problem is guaranteed, and the existence of at least two solutions for λ in an appropriate interval is also discussed.

Positive solutions of m-point boundary value problems

The second-order m-point boundary value problem ϕ″(x)+h(x)f(ϕ(x))=0, 0<x<1, ϕ(0)=0, ϕ(1)= ∑ i=1 m−2 a iϕ(ξ i), is considered under some conditions concerning the first eigenvalue of the relevant linear operator, where ξ i ∈(0,1) with 0< ξ 1< ξ 2<⋯< ξ m−2 <1, a i ∈[0,∞) with ∑ i=1 m−2 a i <1. h( x) is allowed to be singular at x=0 and x=1. The existence of positive solutions and multiple positive solutions is obtained by means of fixed point index theory. Similarly conclusions hold for some other m-point boundary value conditions.

Three positive solutions for second-order m-point boundary value problems

For the second-order m-point boundary value problem u ″(t)+f(t,u)=0, 0⩽t⩽1, u(0)=0,u(1)−∑ i=1 m−2k iu(ξ i)=0, where k i>0 (i=1,2,…,m−2) , 0< ξ 1< ξ 2<⋯< ξ m−2 <1, growth conditions are imposed on f which yield the existence of at least three positive solutions by using the Leggett–Williams fixed point theorem. The associated Green's function for the m-point boundary value problem is also given.

Positive solutions for second-order m-point boundary value problems

By using the Schauder fixed point theorem and analysis method, we establish the existence of solutions for the m-point boundary value problem u″(t)+a(t)f(u)=0, u(0)=0, u(1)− ∑ i=1 m−2 k iu(ξ i)=b, where b,k i>0 (i=1,2,…,m−2),0<ξ 1<ξ 2<⋯<ξ m−2<1 , a( t) is allowed to be singular at t=0,1. Under some conditions, we show that there exists a positive number b ∗ such that the problem has at least one positive solution for 0<b<b ∗ and no solution for b>b ∗ .