Existence of a solution for a general order boundary value problem using the Leray-Schauder fixed point theorem
Abstract
Introduction/purpose: This paper illustrates the existence of a generic Green’s function for a boundary value problem of arbitrary order that appears in many phenomena of heat convection, e.g. in the atmosphere, in the oceans, and on the Sun’s surface.
Methods: A fixed point theorem in the Leray-Schauder form has been used to establish the existence of a fixed point in the problem.
Results: The existence of a solution has been shown for an arbitrary order of the problem. Some practical examples are proposed.
Conclusions: The boundary problem has a solution for an arbitrary order n.
References
Ahmad B. & Ntouyas, S.K. 2012. A study of higher-order nonlinear ordinary differential equations with four-point nonlocal integral boundary conditions. Journal of Applied Mathematics and Computing, 39, pp.97-108. Available at: https://doi.org/10.1007/s12190-011-0513-0.
Bekri Z. & Benaicha, S. 2018. Nontrivial solution of a nonlinear sixth-order boundary value problem. Waves, Wavelets and Fractals, 4(1), pp.10-18. Available at: https://doi.org/10.1515/wwfaa-2018-0002.
Chandrasekhar, S. 1961. Hydrodynamic and Hydromagnetic Stability. New York, NY: Dover. Online ISBN: 780486319209.
Deimling, K. 1985. Nonlinear Functional Analysis. Berlin, Heidelberg: Springer. Available at: https://doi.org/10.1007/978-3-662-00547-7. Online ISBN: 978-3-662-00547-7.
Fabiano, N., Nikolić, N., Shanmugam, T., Radenović, S. & Čitaković, N. 2020. Tenth order boundary value problem solution existence by fixed point theorem. Journal of Inequalities and Applications, art.number:166. Available at: https://doi.org/10.1186/s13660-020-02429-2.
Isac, G. 2006. Leray–Schauder Type Alternatives, Complementarity Problems and Variational Inequalities. Boston, MA: Springer. Available at: https://doi.org/10.1007/0-387-32900-5. Online ISBN: 978-0-387-32900-0.
Ma, R. 2000. Existence and uniqueness theorems for some fourth-order nonlinear boundary value problems. International Journal of Applied Mathematics and Computer Science, 23, art.ID:739631. Available at: https://doi.org/10.1155/S0161171200003057.
Shanmugam, T., Muthiah, M. & Radenović, S. 2019. Existence of Positive Solution for the Eighth-Order Boundary Value Problem Using Classical Version of Leray–Schauder Alternative Fixed Point Theorem. Axioms, 8(4), art.number:129. Available at: https://doi.org/10.3390/axioms8040129.
Zvyagin, V.G. & Baranovskii, E.S. 2010. Topological degree of condensing multi-valued perturbations of the (S)+-class maps and its applications. Journal of Mathematical Sciences, 170, pp.405-422. Available at: https://doi.org/10.1007/s10958-010-0094-8.
Proposed Creative Commons Copyright Notices
Proposed Policy for Military Technical Courier (Journals That Offer Open Access)
Authors who publish with this journal agree to the following terms:
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).