Abstract
In this article, the existence of solution for the first-order nonlinear coupled system of ordinary differential equations with nonlinear coupled boundary condition (CBC for short) is studied using a coupled lower and upper solution approach. Our method for a nonlinear coupled system with nonlinear CBC is new and it unifies the treatment of many different first-order problems. Examples are included to ensure the validity of the results.
Highlights
1 Introduction In this article, we consider the following nonlinear coupled system of ordinary differential equations (ODEs for short): u (t) = f t, v(t) , t ∈ [ , ], ( )
In this article, we consider the following nonlinear coupled system of ordinary differential equations (ODEs for short):u (t) = f t, v(t), t ∈ [, ], ( )v (t) = g t, u(t), t ∈ [, ], subject to the nonlinear CBC h u( ), v( ), u( ), v( ) = (, ), ( )where the nonlinear functions f, g : [, ] × R → R and h : R → R are continuous
Β ( ) ≥ –α ( ), β ( ) ≥ –α ( ), so to generalize the classical results ( ) and ( ) the concept of coupled lower and upper solution is defined, which allows us to obtain a solution in the sector [α, β ] × [α, β ] or [β, α ] × [β, α ] and the inequalities ( )-( ) imply ( ) and ( )
Summary
1 Introduction In this article, we consider the following nonlinear coupled system of ordinary differential equations (ODEs for short): u (t) = f t, v(t) , t ∈ [ , ], ( ) While dealing with nonlinear ordinary differential systems (ODSs for short) mostly authors only focus attention on the differential systems with uncoupled boundary conditions [ – ].
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