Abstract
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.
Highlights
Nonlocal boundary value problems for linear and nonlinear ordinary differential equations are arising in the theory of mathematical physics and some engineering applications [ – ]
This paper generally focuses on the nonlinear second-order ordinary differential equation with four cases of threepoint boundary value conditions in ( )-( )
5 Conclusions Four cases of nonlinear second-order three-point boundary value problems have been investigated and they are transformed into the Hammerstein integral equations by using the integration method
Summary
Nonlocal boundary value problems for linear and nonlinear ordinary differential equations are arising in the theory of mathematical physics and some engineering applications [ – ]. Theorem When (b – a) + k(ξ – a) = , the three-point boundary value problem φ (x) + ψ(x, φ(x)) = g(x), x ∈ [a, b], φ(a) = α, φ(b) + kφ(ξ ) = β, ξ ∈ (a, b), can be transformed into the Hammerstein integral equation as follows:. The proof can be completed similar to that of Theorem It is seen from Theorems - that the nonlinear second-order three-point boundary value problems have been transformed into Hammerstein integral equations. According to Theorem , the nonlinear ordinary differential equation with three-point boundary value conditions in ( ) can be transformed into the following Hammerstein integral equation: φ(x) + K (x, t)e–tφ (t) dt = f (x), where. Making use of Theorem , the boundary value problem ( ) can be transformed into the following Hammerstein integral equation: φ(x) + K (x, t) sin t · eφ(t) dt = f (x), t(x– ). The observation is in accordance with the theoretical analysis in Theorem and that in [ ]
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