Optimal conditions of solvability of periodic problem for systems of differential equations with argument deviation

Sulkhan Mukhigulashvili

doi:10.1515/gmj-2022-2182

Article

Optimal conditions of solvability of periodic problem for systems of differential equations with argument deviation

Sulkhan Mukhigulashvili

Published/Copyright: September 30, 2022

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From the journal Georgian Mathematical Journal Volume 30 Issue 1

Abstract

In this paper, we study, on the interval I = [ a , b ] , the problem of the solvability and unique solvability of systems of higher-order differential equations with argument deviation

u i ( m i ) ( t ) = p i ( t ) u i + 1 ( τ i ( t ) ) + q i ( t ) ( i = 1 , … , n )

and

u i ( m i ) ( t ) = f i ( t , u i + 1 ( τ i ( t ) ) ) + q 0 ⁢ i ( t ) ( i = 1 , … , n ) ,

under the periodic boundary conditions

u i ( j ) ( b ) - u i ( j ) ( a ) = c i ⁢ j ( i = 1 , … , n , j = 0 , … , m i - 1 ) ,

where u n + 1 = u 1 , n ≥ 2 , m i ≥ 1 , p i ∈ L ∞ ⁢ ( I ; R ) , q 0 ⁢ i ∈ L ⁢ ( I ; R ) , f i : I × R → R are Carathéodory class functions, and τ i : I → I are measurable functions. The optimal conditions are obtained, which guarantee the unique solvability of the linear problem and take into account the effect of argument deviation. Based on these results, the optimal conditions of the solvability and unique solvability are proved for the nonlinear problem.

Keywords: Higher-order system; periodic problem; argument deviations; unique solvability

MSC 2010: 34K13; 34B05

Funding statement: The research was supported by Grant FP-S-20-6376 of the Internal Grant Agency at Brno University of Technology.

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Received: 2021-11-30

Accepted: 2022-06-07

Published Online: 2022-09-30

Published in Print: 2023-02-01

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Articles in the same Issue

https://doi.org/10.1515/gmj-2022-2182

Keywords for this article

Higher-order system; periodic problem; argument deviations; unique solvability