Menu: Home :: go to Journal :: switch to Russian :: switch to English
You are here: all Journals and Issues→ Journal→ Issue→ Article

Antiperiodic boundary value problem for an implicit ordinary differential equation

Annotation

The paper is devoted to the investigation of the antiperiodic boundary value problem for an implicit nonlinear ordinary differential equation f(t; x; x ̇ )=0,x(0)+x(τ)=0. We assume that the mapping f:R×R^n×R^n→R^k defining the equation under consideration is smooth and satisfies the condition of uniform nondegeneracy of the first derivative "inf" {"cov" f_v^' (t,x,v):(t,x,v)∈R×R^n×R^n }>0. Here "cov" A is the Banach constant of the linear operator A. The assumption of uniform nondegeneracy holds, in particular, for the mapping f defining an explicit ordinary differential equation. For implicit equations, sufficient conditions for the existence of a solution to an antiperiodic boundary value problem are obtained, and estimates for solutions are found. Corollaries for normal ordinary differential equations are formulated. To prove the main result, the original implicit equation is reduced to an explicit differential equation by applying a nonlocal implicit function theorem. Then we prove an auxiliary assertion on the solvability of the equation x+ψ(x)=0, which is an analog of Brouwer’s fixed point theorem. It is shown that the mapping ψ, that assigns the value of the solution of the Cauchy problem at the point τ to an arbitrary initial point x_0, is well defined and satisfies the assumptions of the auxiliary statement. This reasoning completes the proof of the existence of a solution to the boundary value problem.

Keywords

antiperiodic boundary value problem, implicit ordinary differential equation, implicit function theorem

Full-text in one file

Download

DOI

10.20310/2686-9667-2022-27-139-205-213

UDC

517.927.4

Pages

205-213

References

[1] E.R. Avakov, A.V. Arutyunov, E.S. Zhukovskii, “Covering mappings and their applications to differential equations unsolved for the derivative”, Differential Equations, 45:5 (2009), 627–649. [2] A.V. Arutyunov, S.E. Zhukovskiy, “Application of methods of ordinary differential equations to global inverse function theorems”, Differential Equations, 55:4 (2019), 437–448. [3] L. Nirenberg, Topics in Nonlinear Functional Analysis, American Mathematical Society, New York–London, 2001. [4] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York–London, 1972.

Received

2022-06-28

Section of issue

Scientific articles

Для корректной работы сайта используйте один из современных браузеров. Например, Firefox 55, Chrome 60 или более новые.