Second-order Three-point Boundary Value Problem Research Articles

Solutions to nonlinear second-order three-point boundary value problems of dynamic equations on time scales

In this paper, we consider existence criteria of three positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales. To show our main results, we apply the well-known Leggett-Williams fixed point theorem. Moreover, we present some results for the existence of single and multiple positive solutions for boundary value problems on time scales, by applying fixed point theorems in cones. The conditions we used in the paper are different from those in [Dogan A. On the existence of positive solutions for the one-dimensional $ p $-Laplacian boundary value problems on time scales. Dynam Syst Appl 2015; 24: 295-304].

On positive solutions for some second-order three-point boundary value problems with convection term

In this paper, a fixed point theorem in a cone and some inequalities of the associated Green’s function are applied to obtain the existence of positive solutions of second-order three-point boundary value problem with dependence on the first-order derivative \t\t\tx″(t)+f(t,x(t),x′(t))=0,0<t<1,x(0)=0,x(1)=μx(η),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& x''(t) + f\\bigl(t, x(t), x'(t)\\bigr) =0, \\quad 0< t< 1, \\\\& x(0) =0, \\qquad x(1) =\\mu x(\\eta ), \\end{aligned}$$ \\end{document} where f: [0, 1] times [0, infty ) times R rightarrow [0, infty ) is a continuous function, mu >0, eta in (0, 1), mu eta <1. The interesting point is that the nonlinear term is dependent on the convection term.

The lower and upper solution method for three-point boundary value problems with integral boundary conditions on a half-line

This paper deal with the following second-order three-point boundary value problem with integral boundary condition on a half-line u''(x) + q(x) f(x, u(x), u'(x)) = 0, x ? (0,+?), u(0) = ? ??0 u(s)ds, u'(+1)=C, where ? &gt; 0, 0 &lt; ?? &lt; 1 and f : [0,+?)x R2 ? R satisfies a Nagumo?s condition which plays an important role in the nonlinear term depend on the first-order derivative explicitly. By using Schauder?s fixed point theorem, the upper and lower solution method and topological degree theory, first we give sufficient conditions for the existence of at least one solution and next at least three solutions of the above problem. Moreover, an example is included to demonstrate the efficiency of the main results.

Existence theorems for nonlinear second-order three-point boundary value problems involving the distributional Henstock-Kurzweil integral

Existence theorems for nonlinear second-order three-point boundary value problems involving the distributional Henstock-Kurzweil integral

Existence and approximation of solution for a nonlinear second-order three-point boundary value problem

A nonlinear second-order ordinary differential equation with four cases of three-point boundary value conditions is studied by investigating the existence and approximation of solutions. First, the integration method is proposed to transform the considered boundary value problems into Hammerstein integral equations. Second, the existence of solutions for the obtained Hammerstein integral equations is analyzed by using the Schauder fixed point theorem. The contraction mapping theorem in Banach spaces is further used to address the uniqueness of solutions. Third, the approximate solution of Hammerstein integral equations is constructed by using a new numerical method, and its convergence and error estimate are analyzed. Finally, some numerical examples are addressed to verify the given theorems and methods.

Fractional differential equations and inclusions with semiperiodic and three-point boundary conditions

In this article, we investigate the existence of solutions for boundary value problems of fractional differential equations and inclusions with semiperiodic and three-point boundary conditions. The existence results for equations are obtained by applying Banach’s contraction mapping principle, Schaefer-type fixed point theorem, Leray-Schauder degree theory, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative, whereas the existence of solutions for convex and nonconvex set-valued maps (inclusion case) is shown via nonlinear alternative of Leray-Schauder type for multivalued maps and Wegrzyk’s fixed point theorem for generalized contractions, respectively. We emphasize that a variety of fixed point theorems are used to obtain different existence criteria for the problems at hand. Several examples are discussed for illustration of the obtained results. Moreover, an interesting observation related to symmetric second-order three-point boundary value problems is presented.

Positive solutions of second-order three-point boundary value problems with sign-changing coefficients

Positive solutions of second-order three-point boundary value problems with sign-changing coefficients

Approximate solution of three-point boundary value problems for second-order ordinary differential equations with variable coefficients

Some three-point boundary value problems for a second-order ordinary differential equation with variable coefficients are investigated in the present paper. By using the integration method, the second-order three-point boundary value problems are transformed into a Fredholm integral equation of the second kind. The solutions and Green’s functions for some special cases of the second-order three-point boundary value problems can be determined easily. The existence and uniqueness of the solutions of the given Fredholm integral equations are considered by using the fixed point theorem in Banach spaces. A new numerical method is further proposed to solve the second-kind Fredholm integral equation and an approximate solution is made. The convergence and error estimate of the obtained approximate solution are further analyzed. Numerical results are carried out to verify the feasibility and novelty of the proposed solution procedures.

Existence of solution for a three-point boundary value problem for a second-order impulsive differential equation

In this paper, we study the existence of nontrivial solutions for the second-order three-point boundary value problem with impulses. Under certain growth conditions on the nonlinearity function and by using Leray-Schauder nonlinear alternative, sufficient conditions for the existence of a nontrivial solution are obtained. Furthermore, we illustrate our results with some examples.

Existence of positive pseudo-symmetric solution for second-order three-point boundary value problems

The existence results of at least one positive pseudo-symmetric solution for a nonlinear second-order three-point boundary value problem are established by means of the fixed point theorem due to Avery, Anderson and Krueger.

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&gt;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.

A Variational Principle for Three-Point Boundary Value Problems with Impulse

We construct a variational functional of a class of three-point boundary value problems with impulse. Using the critical points theory, we study the existence of solutions to second-order three-point boundary value problems with impulse.

Positive Solutions to System of Nonlinear Second-Order Three-Point Boundary Value Problem

In this paper,we investigate the following system of nonlinear second-order three-point boundary value problems — u~" = f(t,v),t ∈(0,1),— v″ = g(t,u),t ∈(0,1),u(0) = αu(η),u(1) =βu(η),u(0) = av(η),v(1) = βv(η),where f,g ∈C([0,1]x R~+,R~+),g{t,0)≡ 0,η∈(0,1) and0 β≤α 1.First,Green's function for associated liner boundary value problem is constructed;next,several useful properties of the Green's function are obtained;in the last place,some existence and multiplicity criteria of positive solutions are established by using the fixed point theorems of cone expansion and compression.

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.

New Fixed Point Theorems of Mixed Monotone Operators and Applications to Singular Boundary Value Problems on Time Scales

Some new existence and uniqueness theorems of fixed points of mixed monotone operators are obtained, and then they are applied to a nonlinear singular second-order three-point boundary value problem on time scales. We prove the existence and uniqueness of a positive solution for the above problem which cannot be solved by using previously available methods.

Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order

This article introduces a new concept of upper and lower solutions and studies the existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order. The sufficient conditions for the existence and uniqueness of solutions are obtained by using the monotone iterative method, meantime, the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution are given. Some results of previous literature are extended and improved. A numerical example is also included to illustrate the effectiveness of the proposed results.

Sign-Changing Solutions for Discrete Second-Order Three-Point Boundary Value Problems

We consider the second-order three-point discrete boundary value problem. By using the topological degree theory and the fixed point index theory, we provide sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions. As an application, an example is given to demonstrate our main results.

Multiple solutions for second-order three-point boundary value problems with [formula omitted]-Laplacian operator

In this paper, we study the second-order three-point boundary value problem with a p -Laplacian operator { ( ϕ p ( x ′ ( t ) ) ) ′ + g ( x ( t ) ) = f ( t , x ( t ) , x ′ ( t ) ) , x ( 0 ) = 0 , x ( ξ ) = β x ( 1 ) , where ϕ p ( s ) = | s | p − 2 s , p > 1 , ξ ∈ ( 0 , 1 ) , β ∈ ( 0 , 1 ) ⋃ ( 1 , ∞ ) . We obtain sufficient conditions for the existence of multiple solutions by applying generalized polar coordinates and the Leray–Schauder degree theorem.

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.

Nontrivial symmetric solution of a nonlinear second-order three-point boundary value problem

Nontrivial symmetric solution of a nonlinear second-order three-point boundary value problem