Existence and uniqueness of solutions for the first-order non-linear differential equations with three-point boundary conditions

Abstract

In this paper the existence and uniqueness of the solutions to boundary value problems for the first order non-linear system of the ordinary differential equations with three-point boundary conditions are investigated. For the first time the Green function is constructed and the considered problem is reduced to the equivalent integral equations that allow us to prove the existence and uniqueness theorems in differ from existing works, applying the Banach contraction mapping principle and Schaefer's fixed point theorem. An example is given to illustrate the obtained results.

Existence of solutions of a second order equation defined on unbounded intervals with integral conditions on the boundary

“…Many authors have deduced existence, uniqueness and multiplicity of solutions for different kind of differential equations defined on bounded intervals and coupled to suitable integral boundary conditions, see [10,11,13,[19][20][21]26] and references therein. The used tools are related to continuation methods.…”

Non-Linear First-Order Differential Boundary Problems with Multipoint and Integral Conditions

“…on the interval [0, T].Some more problems for the two-point and the three-point boundary value conditions are studied in[5,[30][31][32][33][34].…”