Existence of solutions for a third-order boundary value problem with integral boundary conditions

Abstract

In this study, the existence of bounded or unbounded solutions for the following third order three-point integral conditional boundary value problem on a half line

${\vartheta }^{\prime \prime \prime }\left(t\right)+r\left(t\right)g(t,\vartheta (t),{\vartheta }^{\prime }(t),{\vartheta }^{\prime \prime }(t\left)\right)=0,t\in (0,\infty ),$
$\vartheta \left(0\right)={\int }_0^{\eta }\vartheta \left(s\right)ds,{\vartheta }^{\prime }\left(0\right)=A,{\vartheta }^{\prime \prime }\left(\infty \right)={lim}_{t\to \infty }\vartheta \left(t\right)=B$

where $\eta \in (0,\infty )$ fixed and $g:(0,\infty )\times R^3\to R$ satisfies Nagumo’s condition is proved. The expected result is obtained by applying Schäuder’s fixed point theorem and the lower and upper solutions method. The existence of at least one solution of the problem has been shown by determining suitable and sufficient conditions in our problem. Since it is difficult to work on the semi-infinite interval, this study with integral conditions on the semi-infinite interval will contribute to the literature for studies on this subject. Also, the solutions to this boundary value problem can be unbounded.

Keywords

• Lower and upper solutions
• Nagumo’s condition
• Schäuder’s fixed point
• infinite interval

İntegral sınır koşullu üçüncü mertebeden sınır değer probleminin çözümlerinin varlığı

Abstract

Bu çalışmada, yarı sonsuz aralık üzerinde tanımlı üçüncü mertebeden üç noktalı integral koşullu, $\eta \in (0,\infty )$ sabit ve $g:(0,\infty )\times R^3⟶R$ Nagumo koşullarını sağlayan bir fonksiyon olmak üzere 
${\vartheta }^{\prime \prime \prime }\left(t\right)+r\left(t\right)g(t,\vartheta (t),{\vartheta }^{\prime }(t),{\vartheta }^{\prime \prime }(t\left)\right)=0,t\in (0,\infty ),$
$\vartheta \left(0\right)={\int }_0^{\eta }\vartheta \left(s\right)ds,{\vartheta }^{\prime }\left(0\right)=A,{\vartheta }^{\prime \prime }\left(\infty \right)={lim}_{t\to \infty }\vartheta \left(t\right)=B$
sınır değer probleminin sınırlı veya sınırlı olmayan çözümlerinin varlığı ispatlanmıştır. Schäuder sabit nokta teoremi ve alt ve üst çözümler yöntemi uygulanarak istenilen sonuca ulaşılmıştır. Problemimizde uygun ve yeterli koşullar belirlenerek problemin en az bir çözümünün varlığı gösterilmiştir. Yarı sonsuz aralık üzerinde çalışılması zor olduğundan yarı sonsuz aralık üzerinde integral koşullu bu çalışma bu konuda yapılacak çalışmalar için litaratüre katkı sağlamış olacaktır. Ayrıca, bu sınır değer probleminin çözümleri sınırsız olabilir.

Keywords

• Alt ve üst çözümler
• Naguma koşulu
• Schäuder sabit nokta teoremi
• yarı sonsuz aralık

References

1. Anar, İ.E., Kısmi Diferansiyel Denklemler, Palme, Ankara, (2005).
2. Agarwal, R.P., Boundary value problems for higher order differential equations, World Scientific, Singapore, (1986).
3. Nagumo, M., Über die differentialgleichung, Proc. Phys. Math. Soc., Japan 19:861– 866, (1937).
4. Dragoni, G.S., IL Problema dei valori ai limiti studiato in grande per gli integrali di una equazione differenziale del secondo ordine, Giornale di Mat., Battaglini, 69:77-112, (1931).
5. Jackson, L. and Schrader, K., Third order differential equations, Journal of Differential Equations 9:46-54, (1971).
6. Boucherif, A., Third order boundary value problems with integral boundary conditions, Nonlinear Anal., 70:364–371, (2009).
7. Agarwal, R.P. and Çetin, E., Unbounded solutions of third order three point boundary value problems on a half-line, Advances in Nonlinear Analysis, 5 (2):105-119, (2015).
8. Bai, C. and Li, C., Unbounded upper and lower solution method for third-order boundary value problems on the half-line, Electron J. Differ. Equ., 119:1-2, (2009).

9. Eloe, P.W., Kaufmann, E.R. and Tisdell, C.C., Multiple solutions of a boundary value problem on an unbounded domain. Dyn. Syst. Appl., 15:53-63, (2006).
10. Enguiça, R., Gavioli, A. and Sanchez, L., Solutions of second order and fourth-order ODEs on the half-line, Nonlinear Analysis, 73:2968-2979, (2010).
11. Lian, H., Zha, J. and Agarwal, R.P., Upper and lower solution method for nth-order BVPs on an infinite interval, Boundary Value Problems, 2014:100 17, (2014).
12. Lian, H., Wang, P. and Ge, W., Unbounded upper and lower solutions method for Sturm-Liouville boundary value problem on infinite intervals, Nonlinear Analysis, 70:2627-2633, (2009).
13. Yan, B. and Liu, Y., Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line, Appl. Math. Comput., 147:629-644, (2004).
14. Zhao, Y., Chen, H. and Xu, C., Existence of multiple solutions for three-point boundary-value problems on infinite intervals in Banach spaces, Electron. J. Differ. Equ., 44:1-11, (2012).
15. Ege, Ş.M. ve Çetin, E., Yarı Sonsuz Aralık Üzerinde Dördüncü Mertebeden Üç Noktalı Sınır Değer Problemlerinin Çözümlerinin Varlığı, Ege üniversitesi Bilimsel Araştırma Projesi, Fen Fakültesi Matematik Bölümü, Bornova, İzmir, (2018).
16. Picard, E., Sur I’application des méthodes d’approximations successives á I’étude de certaines équations différentielles ordinaires, J. De Math., 9:217-271, (1893).
17. Akcan, U. ve Çetin, E., The lower and upper solution method for three-point boundary value problems with integral boundary conditions on a half-line, Filomat 32(1):341-353, (2018).
18. Agarwal, R.P. and O’Regan, D., Infinite interval problems for differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, (2001).
19. Smart, D.R., Fixed point theorems. Cambridge University Press, Cambridge, (1974).
20. Fei Yang, F., Lin, Y., Zhang, J. and Lou, Q., Positive Solutions for Third-Order Boundary Value Problems with the Integral Boundary Conditions and Dependence on the First-Order Derivatives, Journal of Applied Mathematics, Article ID 8411318, 6 pages, (2022)
21. Cabada, A. and N.D. Dimitrov, N.D., Third-order differential equations with three-point boundary conditions, Open Mathematics, 19: 11–31, (2021).