# Some two-point boundary value problems for systems of higher order functional differential equations

## DOI:

https://doi.org/10.7146/math.scand.a-126021## Abstract

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

\begin{equation*} u_i^{(m_i)}(t)=p_i(t)u_{i+1}(\tau _{i}(t))+ q_i(t), (i=\overline {1, n}), \text {for $t\in I:=[a, b]$},

\end{equation*}

and

\begin{equation*}u_i^{(m_i)} (t)=F_{i}(u)(t)+q_{0i}(t), (i = \overline {1, n}), \text {for $ t\in I$},

\end{equation*}

under the conjugate

$u_i^{(j_1-1)}(a)=a_{i j_1}$, $u_i^{(j_2-1)}(b)=b_{i j_2}$, $j_1=\overline {1, k_i}$, $j_2=\overline {1, m_i-k_i}$, $i=\overline {1, n}$,

and the right-focal

$u_i^{(j_1-1)}(a)=a_{i j_1}$, $u_i^{(j_2-1)}(b)=b_{i j_2}$, $j_1=\overline {1, k_i}$, $j_2=\overline {k_i+1,m_i}$, $i=\overline {1, n}$,

boundary conditions, where $u_{n+1}=u_1, $ $n\geq 2, $ $m_i\geq 2, $ $p_i \in L_{\infty }(I; R), $ $q_i, q_{0i}\in L(I; R), $ $\tau _i\colon I\to I$ 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.

## References

Azbelev, N. V., Maksimov, V. P. and Rakhmatullina, L. F., Introduction to the theory of functional differential equations: methods and applications, Hindawi Publishing Corporation, Cairo, 2007. https://doi.org/10.1155/9789775945495

Agarwal, R. P., Focal boundary value problems for differential and difference equations, Mathematics and its Applications 436, Kluwer Academic Publishers, Dordrecht, 1998. https://doi.org/10.1007/978-94-017-1568-3

Bai, C., Fang, J., On positive solutions of boundary value problems for second-order functional differential equations on infinite intervals, J. Math. Anal. Appl. 282 (2003), no. 2, 711–731. https://doi.org/10.1016/S0022-247X(03)00246-4

Bravyi, E., A note on the Fredholm property of boundary value problems for linear functional differential equations, Mem. Differential Equations Math. Phys. 20 (2000), 133–135.

Domoshnitsky, A., Sign properties of Green's matrices of periodic and some other problems for system of functional-differential equations, Funct. Differential Equations Israel Sem. 2 (1994), 39–57.

Domoshnitsky, A., Hakl, R. and Sremr, J., Component-wise positivity of solutions to periodic boundary value problem for linear functional differential systems, J. Inequal. Appl. 2012 2012:112. https://doi.org/10.1186/1029-242X-2012-112

Domoshnitsky, A., Hakl, R. and Puža, B., Multi-point boundary value problems for linear functional-differential equations, Georgian Math. J. 24 (2017), no. 2, 193–206. https://doi.org/10.1515/gmj-2016-0076

Erbe, L., Kong, Q., Boundary value problems for singular second-order functional differential equations, J. Comput. Appl. Math. 53 (1994), no. 3, 377–388. https://doi.org/10.1016/0377-0427(94)90065-5

Hardy, G., Littlewood, J. and Polya, G., Inequalities, 2d ed. Cambridge, at the University Press 1952.

Jiang, D., Wang, J., On boundary value problems for singular second-order functional differential equations, J. Comput. Appl. Math. 116 (2000), no. 2, 231–241. https://doi.org/10.1016/S0377-0427(99)00314-3

Ronto, A., Ronto, M., Existence results for three-point boundary value problems for systems of linear functional differential equations, Carpathian J. Math. 28 (2012), no. 1, 163–182.

Kiguradze, I., Kusano, T., On periodic solutions of higher-order nonautonomous ordinary differential equations, Differential Equations 1 (1999), no.1, 71–77.

Kiguradze, I., Kusano, T., On periodic solutions of even-order ordinary differential equations, Ann. Mat. Pura Appl. (4) 180 (2001), no. 3, 285–301. https://doi.org/10.1007/s102310100012

Kiguradze, I., Boundary value problems for systems of ordinary differential equations, J. Soviet Math. 43 (1988), 2259–2339.

Kiguradze, I., Puža, B., On boundary value problems for systems of linear functional differential equations, Czechoslovak Math. J. 47 (122) (1997), no. 2, 341–373. https://doi.org/10.1023/A:1022829931363

Kiguradze, I., Puža, B., and Stavroulakis, I., On singular boundary value problems for functional differential equations of higher order, Georgian Math. J. 8 (2001), no. 4, 791–814.

Lomtatidze, A., Mukhigulashvili, S.,Some two-poit boundary value problems for second order functional differential equations, Folia Facultatis Scientiarium Naturalium Universitatis Masarykianae Brunensis. Mathematica, 8, Masaryk University, Brno 2000.

Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, Topological methods for ordinary differential equations (Montecatini Terme, 1991), 74–142, Lecture Notes in Math. 1537, Springer, Berlin, 1993. https://doi.org/10.1007/BFb0085076

Ma, R., Positive solutions for boundary value problems of functional differential equations, Appl. Math. Comput. 193 (2007), no. 1, 66–72. https://doi.org/10.1016/j.amc.2007.03.039

Mukhigulashvili, S.,Two-point boundary value problems for second order functional differential equations, Mem. Differential Equations Math. Phys. 20 (2000), 1–112.

Nieto, J., Rodriguez-Lopez, R., Boundary value problems for a class of impulsive functional equations, Comput. Math. Appl. 55 (2008), no. 12, 2715–2731. https://doi.org/10.1016/j.camwa.2007.10.019

Waltman, P., and Wong, J. S. W., Two Point Boundary Value Problems for Nonlinear Functional Differential Equations, Trans. Am. Math. Soc. 164 (1972), 39–54. https://doi.org/10.2307/1995958

## Downloads

## Published

## How to Cite

*MATHEMATICA SCANDINAVICA*,

*127*(2), 382–404. https://doi.org/10.7146/math.scand.a-126021