Nonlinear Second-order Three-point Boundary Value 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].

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.

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.

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.

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

Existence of Positive Solution to Second-Order Three-Point BVPs on Time Scales

We are concerned with the following nonlinear second-order three-point boundary value problem on time scales , , , , where with and . A new representation of Green's function for the corresponding linear boundary value problem is obtained and some existence criteria of at least one positive solution for the above nonlinear boundary value problem are established by using the iterative method.

Multiple positive solutions for nonlinear three-point boundary value problems on time scales

In this paper, we consider the nonlinear second-order three-point boundary value problem u Δ ∇ ( t ) + h ( t ) f ( t , u ( t ) ) = 0 , t ∈ [ t 1 , t 3 ] ⊂ T , u Δ ( t 3 ) = 0 , α u ( t 1 ) − β u Δ ( t 1 ) = u Δ ( t 2 ) , where T is a time scale, 0 ≤ t 1 < t 2 < t 3 , α > 0 and β ≥ 0 are given constants. By using fixed-point theorems in cones, some new results are obtained for the existence of at least one,two and three positive solutions of the above problem.

Positive solutions for singular systems of three-point boundary value problems

This paper investigates the following singular systems 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 ) = v ( 0 ) = 0 , u ( 1 ) = α u ( η ) , v ( 1 ) = α v ( η ) , where η ∈ ( 0 , 1 ) , 0 < α η < 1 , f and g may be singular at t = 0 and/or t = 1 . Under some weaker conditions the existence of positive solutions is obtained by applying the fixed point theorem of cone expansion and compression. Two examples are then presented to demonstrate the application of our main results.

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

In this paper, for a second-order three-point boundary value problem $$\begin{gathered} u'' + f(t,u) = 0, 0 < t < 1, \hfill \\ au(0) - bu'(0) = 0, u(1) - \alpha u(\eta ) = 0, \hfill \\ \end{gathered} $$ where η ∈ (0, 1), a, b, α ∈ R with a2 + b2 > 0, the existence of its nontrivial solution is studied. The conditions on f which guarantee the existence of nontrivial solution are formulated. As an application, some examples to demonstrate the results are given.

Existence of positive solutions for nonlinear three-point problems on time scales

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

Successive iteration and positive solution for nonlinear second-order three-point boundary value problems

In this paper, we study the existence of positive solution for two classes of nonlinear second-order three-point boundary value problems using some monotone iterative schemes. In both cases, such schemes start off with known constant functions and, therefore, are useful to computation purpose.

Solvability for a nonlinear second-order three-point boundary value problem

This paper investigates the existence of nontrivial solution for the three-point boundary value problem u″+f(t,u)=0, 0<t<1, u′(0)=0, u(1)=αu(η), where η∈(0,1), α∈ R , α≠1, f∈C([0,1]× R , R ) . Under certain growth conditions on the nonlinearity f, several sufficient conditions for the existence of nontrivial solution are obtained by using Leray–Schauder nonlinear alternative. As an application, some examples to demonstrate our results are given.

Solutions to Second-order Three-point Problems on Time Scales

In the first part of the paper, we establish the existence of multiple positive solutions to the nonlinear second-order three-point boundary value problem on time scales, u ▵▿ (t)+f(t,u(t))=0, u(0)=0, 𝛂u(𝛈)=u(T) for t∈[0,T]⊂╥, where ╥ is a time scale, 𝛂>0, η∈(0,p(T)⊂╥, and 𝛂η<T. We employ the Leggett-Williams fixed-point theorem in an appropriate cone to guarantee the existence of at least three positive solutions to this nonlinear problem. In the second part, we establish the existence of at least one positive solution to the related problem u ▵▿(t)+a(t)f(u(t))=0, u(0)=0, 𝛂u(η)=u(T), again using a fixed-point theorem for operators.