Some two-point boundary value problems for systems of higher order functional differential equations - Mukhigulashvili, Sulkhan (MATHEMATICA SCANDINAVICA, 2021)
In the paper we study the question of the solvability and unique solvability of systems of the higher order differential equations with the argument deviations equation* u_i^(m_i)(t)=p_i(t)u_i+1(τ _i(t))+ q_i(t), (i=1, n), for tI:=[a, b], equation* and equation*u_i^(m_i) (t)=F_i(u)(t)+q_0i(t), (i = 1, n), for tI, equation* under the conjugate u_i^(j_1-1)(a)=a_i j_1, u_i^(j_2-1)(b)=b_i j_2, j_1=1, k_i, j_2=1, m_i-k_i, i=1, n, and the right-focal u_i^(j_1-1)(a)=a_i j_1, u_i^(j_2-1)(b)=b_i j_2, j_1=1, k_i, j_2=k_i+1,m_i, i=1, n, boundary conditions, where u_n+1=u_1, n≥ 2, m_i≥ 2, p_i L_∞ (I; R), q_i, q_0iL(I; R), τ _iI→ I are the measurable functions, F_i are the local Caratheodory’s class operators, and k_i is the integer part of the number m_i/2. In the paper are obtained the efficient sufficient conditions that guarantee the unique solvability of the linear problems and take into the account explicitly the effect of argument deviations, and on the basis of these results are proved new conditions of the solvability and unique solvability for the nonlinear problems.