Confining a non-negative solution between a lower and upper solution for a sixth-degree boundary value problem

  • Zouaoui Bekri University of Oran 1, Laboratory of Fundamental and Applied Mathematics, Oran; Nour Bachir University Center, Institute of Sciences, Department of Sciences and Technology, El-Bayadh; People’s Democratic Republic of Algeria https://orcid.org/0000-0002-2430-6499
  • Nicola Fabiano University of Belgrade, “Vinča” Institute of Nuclear Sciences - National Institute of the Republic of Serbia, Belgrade, Republic of Serbia https://orcid.org/0000-0003-1645-2071
  • Mohammad Esmael Samei Bu-Ali Sina University, Faculty of Basic Science, Department of Mathematic, Hamedan, Islamic Republic of Iran https://orcid.org/0000-0002-5450-3127
  • Stojan Radenović University of Belgrade, Faculty of Mechanical Engineering, Belgrade, Republic of Serbia https://orcid.org/0000-0001-8254-6688
Keywords: Leray-Schauder nonlinear alternative, Green’s function, fixed point theorem, lower and upper solutions, boundary value problem

Abstract


Introduction/purpose: The aim of the paper is to prove the existence of solutions for a special case of the sixth-order boundary value problem.

Methods: The Leray-Schauder fixed point theorem is used in order to determine lower and upper bound solutions.

Results: Lower and upper bound solutions have been found.

Conclusions: The sixth-order boundary value problem admits solutions.

References

Agarwal, R.P., Kovacs, B. & O’Regan, D. 2013. Positive solutions for a sixth-order boundary value problem with four parameters. Boundary Value Problems, 2013, art.number:184. Available at: https://doi.org/10.1186/1687-2770-2013-184.

Amiri, P. & Samei, M.E. 2022. Existence of Urysohn and Atangana-Baleanu fractional integral inclusion systems solutions via common fixed point of multi-valued operators. Chaos, Solitons & Fractals, 165(2, art.number:112822). Available at: https://doi.org/10.1016/j.chaos.2022.112822.

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.

Bekri, Z. & Benaicha, S. 2020. Positive solutions for boundary value problem of sixth-order elastic beam equation. Open Journal of Mathematical Sciences, 4, pp. 9–17. Available at: https://doi.org/10.30538/oms2020.0088.

Boutiara, A., Benbachir, M., Alzabut, J. & Samei, M.E. 2021. Monotone iterative and upper-lower solutions techniques for solving nonlinear ψ−Caputo fractional boundary value problem. Fractal and Fractional, 5(9, art.number:194). Available at: https://doi.org/10.3390/fractalfract5040194.

Boutiara, A., Matar, M.M., Alzabut, J., Samei, M.E. & Khan, H. 2023. On ABC coupled Langevin fractional differential equations constrained by Perov’s fixed point in generalized Banach spaces. AIMS Mathematics, 8(5), pp. 12109–12132. Available at: https://doi.org/10.3934/math.2023610.

Chabanea, F., Benbachir, M., Hachama, M. & Samei, M.E. 2022. Existence of positive solutions for p-Laplacian boundary value problems of fractional differential equationss. Boundary Value Problems, 2022, art.number:65. Available at: https://doi.org/10.1186/s13661-022-01645-7.

Deimling, K. 1985. Nonlinear Functional Analysis. Berlin: Springer. Available at: https://doi.org/10.1007/978-3-662-00547-7.

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, 2020, art.number:166. Available at: https://doi.org/10.1186/s13660-020-02429-2.

Fabiano, N. & Parvaneh, V. 2021. Existence of a solution for a general or- der boundary value problem using the Leray-Schauder fixed point theorem. Vojnotehnički glasnik/Military Technical Courier, 69(2), pp. 323–337. Available at: https://doi.org/10.5937/vojtehg69-29703.

Hammad, H.A., Rashwan, R.A., Nafea, A., Samei, M.E. & De la Sen, M. 2022. Stability and Existence of Solutions for a Tripled Problem of Fractional Hybrid Delay Differential Equations. Symmetry, 14(12, art.number:2579). Available at: https://doi.org/10.3390/sym14122579.

Houas, M. & Samei, M.E. 2023. Existence and Stability of Solutions for Linear and Nonlinear Damping of q−Fractional Duffing–Rayleigh Problem. Mediterranean Journal of Mathematics, 20, art.number:148. Available at: https://doi.org/10.1007/s00009-023-02355-9.

Ji, Y., Guo, Y., Yao, Y. & Feng, Y. 2012. Nodal solutions for sixth-order m-point boundary-value problems using bifurcation methods. Electronic Journal of Differential Equations, 2012(217), pp. 1–18 [online]. Available at: https://ejde.math.txstate.edu/ [Accessed: 7 February 2024].

Kovács, B. & Guedda, M. 2014. Positive solution for m-point sixth-order boundary value problem with variable parameter. Scientific Bulletin of the ”Petru Maior” University of Tîrgu Mureș (Acta Marisiensis. Seria Technologica), 11(2), pp. 50–61 [online]. Available at: https://amset.umfst.ro/papers/2014-2/09%20Positive%20s olution%20Kovacs.pdf [Accessed: 7 February 2024].

Li, W. 2012. The existence and multiplicity of positive solutions of nonlinear sixth-order boundary value problem with three variable coefficients. Boundary Value Problems, 2012, art.number:22. Available at: https://doi.org/10.1186/1687-2770-2012-22.

Mirzaei, H. 2016. Existence and nonexistence of positive solution for sixth-order boundary value problems. Bulletin of the Iranian Mathematical Society, 42(6), pp. 1451–1458 [online]. Available at: http://bims.iranjournals.ir/article_902_2e9751112c0f6a8a0ad4c17431e11af8.pdf [Accessed: 7 February 2024].

Möller, M. & Zinsou, B. 2013. Sixth order differential operators with eigenvalue dependent boundary conditions. Applicable Analysis and Discrete Mathematics, 7(2), pp. 378–389. Available at: https://doi.org/10.2298/AADM130608010M.

Samei, M.E., Ahmadi, A., Hajiseyedazizi, S.N., Mishra, S.K. & Ram, B. 2021a. The existence of non-negative solutions for a nonlinear fractional q-differential problem via a different numerical approach. Journal of Inequalities and Applications, 2021, art.number:75. Available at: https://doi.org/10.1186/s13660-021-02612-z.

Samei, M.E., Ahmadi, A., Maria Selvam, A.G., Alzabut, J. & Rezapour, S. 2021b. Well-posed conditions on a class of fractional q-differential equations by using the Schauder fixed point theorem. Advances In Difference Equations, 2021, art.number:482. Available at: https://doi.org/10.1186/s13662-021-03631-2.

Santra, S.S., Mondal, P., Samei, M.E., Alotaibi, H., Altanjii, M. & Botmart, T. 2023. Study on the oscillation of solution to second-order impulsive systems. AIMS Mathematics, 8(9), pp. 22237–22255. Available at: https://doi.org/10.3934/math.20231134.

Tersian, S. & Chaparova, J. 2002. Periodic and homoclinic solutions of some semilinear sixth-order differential equations. Journal of Mathematical Analysis and Applications, 272(1), pp. 223–239. Available at: https://doi.org/10.1016/S0022-247X(02)00153-1.

Yang, B. 2019. Positive solutions to a nonlinear sixth order boundary value problem. Differential Equations & Applications, 11(2), pp. 307–317. Available at: https://doi.org/10.7153/dea-2019-11-13.

Zhang, L. & An, Y. 2010. Existence and Multiplicity of Positive Solutions of a Boundary-Value Problem for Sixth-Order ODE with Three Parameters. Boundary Value Problems, 2010, art.number:878131. Available at: https://doi.org/10.1155/2010/878131.

Published
2024/06/10
Section
Original Scientific Papers