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

Abstract

Abstract In this paper, we study, on the interval I = [ a , b ] {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 ) u_{i}^{(m_{i})}(t)=p_{i}(t)u_{i+1}(\tau_{i}(t))+q_{i}(t)\quad(i=1,\ldots,n) and u i ( m i ) ( t ) = f i ( t , u i + 1 ( τ i ( t ) ) ) + q 0 ⁢ i ( t ) ( i = 1 , … , n ) , u_{i}^{(m_{i})}(t)=f_{i}\bigl{(}t,u_{i+1}(\tau_{i}(t))\bigr{)}+q_{0i}(t)\quad(% i=1,\ldots,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 ) , u_{i}^{(j)}(b)-u_{i}^{(j)}(a)=c_{ij}\quad(i=1,\ldots,n,\,j=0,\ldots,m_{i}-1), where u n + 1 = u 1 {u_{n+1}=u_{1}} , n ≥ 2 {n\geq 2} , m i ≥ 1 {m_{i}\geq 1} , p i ∈ L ∞ ⁢ ( I ; R ) {p_{i}\in L_{\infty}(I;R)} , q 0 ⁢ i ∈ L ⁢ ( I ; R ) {q_{0i}\in L(I;R)} , f i : I × R → R {f_{i}:I\times R\to R} are Carathéodory class functions, and τ i : I → I {\tau_{i}:I\to 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.