Existence and uniqueness of solutions for a nonlinear differential equation with one-parameter nonlocal condition: A numerical study via MATLAB
DOI:
https://doi.org/10.65419/albahit.v5i2.142Keywords:
nonlinear differential equation, nonlocal condition, existence and uniqueness of solution, MATLABAbstract
The primary focus of this study is to explore a nonlinear differential equation with one-parameter nonlocal condition. Our analysis establishes rigorous proofs for both the existence and uniqueness of the solutions. Furthermore, we investigate the solution's stability by examining its continuous dependence on the initial data and the parameter. A numerical example is presented with MATLAB tools to validate the theoretical findings and to demonstrate the relationship between the unique solution and governing parameter and initial data.
References
M. Benchohra , S. Hamani and S. Ntouyas , Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Anal. 71 (2009) 2391-2396. https://doi.org/10.1016/j.na.2009.01.073
E. O. Bin-taher, Maximal and minimal positive solutions for a boundary value problem with a nonlocal conditions, Journal of Fractional Calculus and Applications, 4(6) (2013) 1- 9.
E. O. Bin-taher, The Existence and Uniqueness of Positive Solutions of an Ordinary Differential Equation with a Nonlocal Conditions, J. Phys.: Conf. Ser. 1900(2021) 012010. https://doi.org/10.1088/1742-6596/1900/1/012010
A. Boucherif and R. Precup, On the nonlocal initial value problem for first order differential equations, Fixed Point Theory, 4(2)(2003)205-212.
A.M. A. El-Sayed, R.O.Abd El-Rahmanand and M. El-Gendy. M , Existence of solution of a coupled system of differential equations with nonlocal conditions, Malaya J. Mat., 2(4)(2014) 345-351.
A. M. A. El-Sayed, E. M Hamdallah and Kh. W. El-Kadeky, Solutions of a class of internal nonlocal Cauchy problems for the differential equation acute{x}=f(t,xleft(tright), acute{x}left(tright)), Fixed Point Theory, 15(2)(2014) 441-448.
A.M. A El-Sayed, M. SH. Mohamed and E. M. AL-Barg, On a nonlinear differential equation with two point nonlocal conditions with parameters, J. Math Anal, 4(2)(2020) 64-73. https://doi.org/10.30538/psrp-oma2020.0063
K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.
E. M. A. Hamd-Allah, On the existence of solutions of two differential equations with a nonlocal condition, Journal of the Egyptian Mathematics Society, 24(3) (2016) 367-372. https://doi.org/10.1016/j.joems.2015.10.002
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Prentice-Hall, Englewood Cliffs, NJ. USA, 1970.
B.Liu, A note on a nonlocal boundary value problems, Applied Mathematics and Computation, 154 (2004) 871-880. https://doi.org/10.1016/S0096-3003(03)00756-2
R. Ma, Existence and uniqueness of solutions of first-order three-point boundary value problem, Applied Mathematics Letters, 15(2002) 211-216.
https://doi.org/10.1016/j.aml.2003.08.014
R. Ma, A survey on nonlocal boundary value problems Applied Mathematics E-Notes 7(2007) 257-279. http://eudml.org/doc/55525
A. S. Shaikh and M. A. Sajjan, Existence and uniqueness of solution of fractional differential equation using non-local operator, Indian Journal of Science and Technology, 17(37) (2024) 3881-3888. https://doi.org/10.17485/IJST/v17i37.2251
B. Shoimov and A. Mukumov, Existence and uniqueness of the solution of the nonlocal problem, AIP Conf. Proc. 3377(1) (2025). https://doi.org/10.1063/5.0299951
E.A.A. Ziada, Solution of a nonlocal Cauchy problem of a differential equation by ADomian decomposition method, Journal of Fractional Calculus and Applications, 3(8) (2012) 1-10.
Fathiyah Abraheem Abdullah Ali, Ambark Ashat, & Sawsan Mustafa Ali Saeed. (2026). Oscillation Criteria for Fractional-Order Third-Order Neutral Differential Equations with Damping. Journal of Libyan Academy Bani Walid, 2(1), 103–113. https://doi.org/10.61952/jlabw.v2i1.421
Ambark Ashat. (2025). Using the fuzzy center method to solve linear fractional fuzzy differential equations of order n with initial conditions. Journal of Libyan Academy Bani Walid, 1(4), 253–270. https://doi.org/10.61952/jlabw.v1i4.347
Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Heterogeneous Deviating Arguments and Distributed Kernels. (2025). Albahit Journal of Applied Sciences, 4(2), 87-99. https://doi.org/10.65419/albahit.v4i2.85
