Existence results for a coupled system of higher-order nonlinear differential equations with integral-multipoint boundary conditions
DOI:
https://doi.org/10.24193/subbmath.2025.2.05Keywords:
Ordinary differential equations, system, integral-multipoint boundary conditions, nonlocal, fixed pointAbstract
In this paper, we establish the existence and uniqueness criteria for solutions of an integral-multipoint coupled boundary value problem involving a system of nonlinear higher-order ordinary differential equations. We apply the Leray-Schauder’s alternative to prove an existence result for the given problem, while the uniqueness of its solutions is accomplished with the aid of Banach’s fixed point theorem. Examples are constructed for illustrating the obtained results.
Mathematics Subject Classification (2010): 34A34, 34B10, 34B15.
Received 13 October 2024; Accepted 10 January 2025.
References
1. Adomian, G., Adomian, G.E., Cellular systems and aging models, Comput. Math. Appl., 11(1985), 283-291.
2. Ahmad, B., Almalki, A., Ntouyas, S.K., Alsaedi, A., Existence results for a self-adjoint coupled system of nonlinear second-order ordinary differential inclusions with nonlocal integral boundary conditions, J. Inequal. Appl., (2022), Paper No. 111, 41 pp.
3. Ahmad, B., Alsaedi, A., Al-Malki, N., On higher-order nonlinear boundary value problems with nonlocal multipoint integral boundary conditions, Lith. Math. J., 56(2016), 143-163.
4. Akyildiz, F.T., Bellout, H., Vajravelu, K., Van Gorder, R.A., Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces, Nonlinear Anal. Real World Appl., 12(2011), 2919-2930.
5. Alsaedi, A., Almalki, A., Ntouyas, S.K., Ahmad, B., Existence theory for a self-adjoint coupled system of nonlinear ordinary differential equations with nonlocal integral multi-point boundary conditions, Appl. Math. E-Notes, 22(2022), 496-508.
6. Alsaedi, A., Alsulami, M., Srivastava, H.M., Ahmad, B., Ntouyas, S.K., Existence theory for nonlinear third-order ordinary differential equations with nonlocal multi-point and multi-strip boundary conditions, Symmetry, 11(2019), 281.
7. Boukrouche, M., Tarzia, D.A., A family of singular ordinary differential equations of the third order with an integral boundary condition, Bound. Value Probl., (2018), 2018:32.
8. Bressan, A., Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem, Oxford University Press, Oxford, UK, 2000.
9. Cannon, J.R., The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 21(1963), 155-160.
10. Chegis, R.Y., Numerical solution of a heat conduction problem with an integral condition, Litovsk. Mat. Sb., 24 (1984), 209-215.
11. Clark, S., Henderson, J., Uniqueness implies existence and uniqueness criterion for non-local boundary value problems for third-order differential equations, Proc. Amer. Math. Soc., 134(2006), 3363-3372.
12. Deimling, K., Nonlinear Functional Analysis, Springer Verlag, Berlin, Heidelberg, 1985.
13. Elango, S., Govindarao, L., Vadivel, R., A comparative study on numerical methods for Fredholm integro-differential equations of convection-diffusion problem with integral boundary conditions, Appl. Numer. Math., 207 (2025), 323-338.
14. Eloe, P.W., Ahmad, B., Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions, Appl. Math. Lett., 18(2005), 521-527.
15. Feng, M., Zhang, X., Ge, W., Existence theorems for a second order nonlinear differential equation with nonlocal boundary conditions and their applications, J. Appl. Math. Comput., 33 (2010), 137-153.
16. Graef, J.R., Webb, J.R.L., Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal., 71(2009), 1542-1551.
17. Granas, A., Dugundji, J., Fixed Point Theory, Springer-Verlag, New York, NY, 2005.
18. Hayat, T., Aziz, A., Muhammad, T., Ahmad, B., On magnetohydrodynamic flow of second grade nanofluid over a nonlinear stretching sheet, J. Magn. Magn. Mater., 408(2016), 99-106.
19. Hayat, T., Kiyani, M.Z., Ahmad, I., Ahmad, B., On analysis of magneto Maxwell nano- material by surface with variable thickness, Internat. J. Mech. Sci., 131 (2017), 1016-1025.
20. Ionkin, N.I., The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition (Russian), Diff. Uravn., 13(1977), 294-304.
21. Karaca, I.Y., Fen, F.T., Positive solutions of nth-order boundary value problems with integral boundary conditions, Math. Model. Anal., 20(2015), 188-204.
22. Ma, R., A survey on nonlocal boundary value problems, Appl. Math. E-Notes, 7(2007), 257-279.
23. Nicoud, F., Schfonfeld, T., Integral boundary conditions for unsteady biomedical CFD applications, Int. J. Numer. Meth. Fluids, 40(2002), 457-465.
24. Sun, Y., Liu, L., Zhang, J., Agarwal, R.P., Positive solutions of singular three-point boundary value problems for second-order differential equations, J. Comput. Appl. Math., 230(2009), 738-750.
25. Taylor, C., Hughes, T., Zarins, C., Finite element modeling of blood flow in arteries, Comput. Methods Appl. Mech. Engrg., 158(1998), 155-196.
26. Webb, J.R.L., Infante, G., Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Differential Equations Appl., 15(2008), 45-67.
27. Womersley, J.R., Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known, J. Psychol., 127(1955), 553-563.
28. Zheng, L., Zhang, X., Modeling and analysis of modern fluid problems. Mathematics in Science and Engineering, Elsevier/Academic Press, London, 2017.
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