A fourth-order iterative boundary value problem with conjugate boundary conditions
Main Article Content
Abstract
We establish conditions on the function $f$ for the existence and uniqueness of solutions for the fourth-order iterative differential equation
\begin{displaymath}
x^{(4)}(t) = f(t, x(t), x^{[2]}(t), ..., x^{[m]}(t)), \quad a < t < b
\end{displaymath}
$m \ge 2$, with solutions subject to one of the boundary conditions
\begin{eqnarray*}
x(a) = c, \ x'(a) = 0, \ x''(a) = 0, \ x(b) = d,\\
x(a) = c, \ x'(a) = 0, \ x(b) = d, \ x'(b) = 0,\\
x(a) = c, \ x(b) = d, \ x'(b) = 0, \ x''(b) = 0.
\end{eqnarray*}
We assume that $a, b, c, d$ are constants such that $a < c < d < b$. The main tool employed is Schauder's Fixed-Point Theorem.
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References
[1] F. F. Bonsall, Lectures on some Fixed Point Theorems of Function Analysis, Tata Institute of Fundamental Research, Bombay, 1962. Google Scholar
[2] Y. Chen, J. E. Zhang, Stability analysis of a class of neural networks with state-dependent state delay, Discrete Dyn. Nat. Soc. 2020 (2020), 1–9. https://doi.org/10.1155/2020/4820351 Google Scholar
[3] S. Cheraiet, A. Bouakkaz, R. Khemis, Bounded positive solutions of an iterative three-point boundary-value problem with integral boundary conditions, J. Appl. Math. Comput. 65 (2021), 597–610. https://doi.org/10.1007/s12190-020-01406-8 Google Scholar
[4] D. Driver, A two-body problem of classical electrodynamics: The one-dimensional case, Ann. Phys. 21 (1963), 122–142. https://doi.org/10.1016/0003-4916(63)90227-6 Google Scholar
[5] E. Eder, The functional-differential equation x′(t) = x(x(t)), J. Differential Equations 54 (1984), 390–400. https://doi.org/10.1016/0022-0396(84)90150-5 Google Scholar
[6] S. Erman, Stability analysis of some dynamic economic systems modeled by state-dependent delay differential equations, Springer, Cham (2018), 227–240. https://doi.org/10.1007/978-3-319-78494-6_11 Google Scholar
[7] M. Fečkan, On a certain type of functional-differential equations, Math. Slovaca 43 (1993), 39–43. http://eudml.org/doc/31986 Google Scholar
[8] S. Greenhalgh, T. Day, Time-varying and state-dependent recovery rates in epidemiological models, Infect. Dis. Model. 2 (2017), 419–430. https://doi.org/10.1016/j.idm.2017.09.002 Google Scholar
[9] R. Johnson, Functional equations, approximations, and dynamic response of systems with variable time-delay, IEEE Trans. Automatic Control 17 (1972), 398–401. https://doi.org/10.1109/TAC.1972.1099999 Google Scholar
[10] E. R. Kaufmann, A fourth order boundary value problem with Lidstone boundary conditions, Differ. Equ. Appl. 14 (2022), 305–312. https://doi.org/10.7153/dea-2022-14-21 Google Scholar
[11] E. R. Kaufmann, N. Kosmatov, Solutions of second order iterative boundary value problems with nonhomogeneous boundary conditions, preprint. Google Scholar
[12] E. R. Kaufmann, Z. Whaley, Existence and uniqueness of solutions of a fourth-order boundary value problem with non-homogeneous boundary conditions, Electron. J. Qual. Theory Differ. Equ. 59 (2023), 1–11. https://doi.org/10.14232/ejqtde.2023.1.59 Google Scholar
[13] A. Keane, B. Krauskopf, H. A. Dijkstra, The effect of state dependence in a delay differential equation model for the El Niño Southern Oscillation, Phil. Trans. R. Soc. A 377 (2019). https://doi.org/10.1098/rsta.2018.0121 Google Scholar
[14] F. M. G. Magpantay, N. Kosovalić, An age-structured population model with state-dependent delay: Dynamics, IFAC-PapersOnLine 48 (2015), 99–104. https://doi.org/10.1016/j.ifacol.2015.09.360 Google Scholar
[15] V. R. Petuhov, On a boundary value problem (in Russian with English summary), Trudy Sem. Teor. Differencial. Uravnenii s Otklon. Argumentom Univ. Druzhby Narodov Patrisa Lumumby 3 (1965), 252–255. Google Scholar
[16] J. Si, X. Wang, Smooth solutions of a nonhomogeneous iterative functional-differential equation with variable coefficients, J. Math. Anal. Appl. 226 (1998), 377–392. https://doi.org/10.1006/jmaa.1998.6086 Google Scholar
[17] K. Wang, On the equation x′(t) = f(x(x(t))), Funkcial. Ekvac. 33 (1990), 405–425. Google Scholar
[18] H. Y. Zhao, J. Liu, Periodic solutions of an iterative functional differential equation with variable coefficients, Math. Methods Appl. Sci. 40 (2017), 286–292. https://doi.org/10.1002/mma.3991 Google Scholar