## Abstract and Applied Analysis

### On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay

Nai-Sher Yeh

#### Abstract

For each ${x}_{\mathrm{0}}\in [\mathrm{0,2}\pi )$ and $k\in \mathbf{N}$, we obtain some existence theorems of periodic solutions to the two-point boundary value problem ${u}^{\mathrm{\prime }\mathrm{\prime }}(x)+{k}^{\mathrm{2}}u(x-{x}_{\mathrm{0}})+g(x,u(x-{x}_{\mathrm{0}}))=h(x)$ in $(\mathrm{0},\mathrm{2}\pi )$ with $u(\mathrm{0})-u(\mathrm{2}\pi )={u}^{\mathrm{\prime }}(\mathrm{0})-{u}^{\mathrm{\prime }}(\mathrm{2}\pi )=\mathrm{0}$ when $g:(\mathrm{0,2}\pi )\times\mathbf{R}\to \mathbf{R}$ is a Caratheodory function which grows linearly in $u$ as $|u|\to \mathrm{\infty }$, and $h\in {L}^{\mathrm{1}}(\mathrm{0,2}\pi )$ may satisfy a generalized Landesman-Lazer condition $(\mathrm{1}+\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\beta )){\int }_{\mathrm{0}}^{\mathrm{2}\pi }h(x)v(x)dx<{\int }_{v(x)>\mathrm{0}}{g}_{\beta }^{+}(x){|v(x)|}^{\mathrm{1}-\beta }dx+{\int }_{v(x)<\mathrm{0}}{g}_{\beta }^{-}(x){|v(x)|}^{\mathrm{1}-\beta }dx$ for all $v\in N(L)\\{\mathrm{0}\}$. Here $N(L)$ denotes the subspace of ${L}^{\mathrm{1}}(\mathrm{0,2}\pi )$ spanned by $\mathrm{sin}kx$ and $\mathrm{cos}kx$, $-\mathrm{1}<\beta \le \mathrm{0}$, ${g}_{\beta }^{+}(x)={\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}}_{u\to \mathrm{\infty }}(g(x,u)u/{|u|}^{\mathrm{1}-\beta })$, and ${g}_{\beta }^{-}(x)={\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}}_{u\to -\mathrm{\infty }}(g(x,u)u/{|u|}^{\mathrm{1}-\beta })$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2018 (2018), Article ID 5321314, 6 pages.

Dates
Accepted: 28 January 2018
First available in Project Euclid: 8 May 2018

https://projecteuclid.org/euclid.aaa/1525744868

Digital Object Identifier
doi:10.1155/2018/5321314

Mathematical Reviews number (MathSciNet)
MR3786304

Zentralblatt MATH identifier
06929588

#### Citation

Yeh, Nai-Sher. On Solvability Theorems of Second-Order Ordinary Differential Equations with Delay. Abstr. Appl. Anal. 2018 (2018), Article ID 5321314, 6 pages. doi:10.1155/2018/5321314. https://projecteuclid.org/euclid.aaa/1525744868

#### References

• P. Drábek, “Landesman-lazer condition for nonlinear problem with jumping nonlinearity,” Journal of Differential Equations, vol. 85, no. 1, pp. 186–199, 1990.
• P. Drábek and S. Invernizzi, “On the periodic BVP for the forced duffing equation with jumping nonlinearity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 10, no. 7, pp. 643–650, 1986.
• J.-P. Gossez and P. Omari, “Nonresonance with respect to the Fucik spectrum for periodic solutions of second order ordinary differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 14, no. 12, pp. 1079–1104, 1990.
• C. P. Gupta, J. J. Nieto, and L. Sanchez, “Periodic solutions of some lienard and duffing equations,” Journal of Mathematical Analysis and Applications, vol. 140, no. 1, pp. 67–82, 1989.
• C. P. Gupta and J. Mawhin, “Asymptotic conditions at the two first eigenvalues for the periodic solutions of lineard differential equations and an inequality of E. Schmidt,” Zeitschrift Für Analysis Und Ihre Anwendungen, vol. 3, no. 1, pp. 33–42, 1984.
• C. P. Gupta, “Solvability of a forced autonomous duffing's equation with boundary conditions in the presence of damping,” Applications of Mathematics, vol. 38, no. 3, pp. 195–203, 1993.
• R. Iannacci and M. N. Nkashama, “Nonlinear two-point boundary value problems at resonance without landesman-lazer condition,” Proceedings of the American Mathematical Society, vol. 106, no. 4, pp. 943–952, 1989.
• C.-C. Kuo, “Periodic solutions of a two-point boundary value problem at resonance,” Osaka Journal of Mathematics, vol. 37, no. 2, pp. 345–353, 2000.
• C.-C. Kuo, “On the solvability of semilinear differential equations at resonance,” Proceedings of the Edinburgh Mathematical Society, vol. 43, no. 1, pp. 103–112, 2000.
• M. Martelli and J. D. Schuur, “Periodic solutions of linear type second order ordinary differential equations,” Tohoku Mathematical Journal, vol. 32, pp. 337–351, 1983.
• J. Mawhin and J. Ward, “Periodic solutions of some forced lineard differential equations at resonance,” Archiv der Mathematik, vol. 41, no. 4, pp. 337–351, 1983.
• J. Mawhin and J. R. Ward, “Nonuniform nonresonance conditions at the two first eigenvalues for periodic solutions of forced lineard and duffing equations,” Rocky Mountain Journal of Mathematics, vol. 12, no. 4, pp. 643–654, 1982.
• J. R. Ward Jr., “Periodic solutions for systems of second order differential equations,” Journal of Mathematical Analysis and Applications, vol. 81, no. 1, pp. 92–98, 1981.
• R. Iannacci and M. N. Nkashama, “Nonresonance conditions for periodic solutions of lineard and duffing equations with delay,” Annales de la Société scientifique de Bruxelles Série I, vol. 99, no. 1, pp. 29–43, 1985.
• R. Iannacci and M. N. Nkashama, “On periodic solutions of forced second order differential equations with a deviating argument,” in Lecture Notes in Mathematics, vol. 1151, pp. 224–232, Springer, Berlin, Germany, 1985.
• E. De Pascale and R. Iannacci, “Periodic solutions of generalized lineard equations with delay,” in Lecture Notes in Mathematics, vol. 1017, pp. 148–156, Springer, Berlin, Germany, 1983.
• R. Kannan and R. Ortega, “Periodic solutions of pendulum-type equations,” Journal of Differential Equations, vol. 59, no. 1, pp. 123–144, 1985.
• K. Deimling, Nonlinear Functional Analysis, Springer, New York, NY, USA, 1985.
• J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, vol. 40 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1979.
• R. Iannacci and M. N. Nkashama, “Nonlinear boundary value problems at resonance,” Nonlinear Analysis. Theory, Methods & Applications, vol. 11, no. 4, pp. 455–473, 1987. \endinput