On bounded solutions of a second-order iterative boundary value problem
On bounded solutions of a second-order iterative boundary value problem
In this article, we investigate a second-order iterative differential equation with boundary conditions. The use of the principle of contraction mappings and the Schauder’s fixed point theorem allows us to prove some existence and uniqueness results. Finally, an example is given to check the validity of our findings, which are new, and complete some published manuscripts to some degree.
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- [1] Bai C, Xu X. Positive solutions for a functional delay second-order three-point boundary value problem. Electronic Journal of Differential Equations 2006; 41: 1-10.
- [2] Benchohra M, Nieto J, Ouahab A. Second-order boundary value problem with integral boundary conditions. Boundary Value Problems 2011; 6: 2011.
- [3] Berinde V. Existence and approximation of solutions of some first order iterative differential equations. Miskolc Mathematical Notes 2010; 11: 13-26.
- [4] Bouakkaz A, Ardjouni A, Djoudi, A. Existence of positive periodic solutions for a second-order nonlinear neutral differential equation by the Krasnoselskii’s fixed point theorem. Nonlinear Dynamics and Systems Theory 2017; 17: 230-238.
- [5] Bouakkaz A, Ardjouni A, Djoudi A. Periodic solutions for a second order nonlinear functional differential equation with iterrative terms by Schauder’s fixed point theorem. Acta Mathematica Universitatis Comenianae 2018; 87 (2): 223-235.
- [6] Bouakkaz A, Ardjouni A, Khemis R, Djoudi A. Periodic solutions of a class of third-order functional differential equations with iterative source terms. Boletín de la Sociedad Matemática Mexicana 2020; 26: 443-458.
- [7] Bouakkaz A, Khemis R. Positive periodic solutions for a class of second-order differential equations with statedependent delays. Turkish Journal of Mathematics 2020; 44 (4): 1412-1426.
- [8] Bouakkaz A, Khemis R. Positive periodic solutions for revisited Nicholson’s blowflies equation with iterative harvesting term. Journal of Mathematical Analysis and Applications 2021; 494 (2): 124663.
- [9] Boucherif A. Second-order boundary value problems with integral boundary conditions. Nonlinear Analysis: Theory, Methods and Applications 2009; 70: 364-371.
- [10] Cheraiet S, Bouakkaz A, Khemis R. Bounded positive solutions of an iterative three-point boundary-value problem with integral boundary conditions. Journal of Applied Mathematics and Computing 2021; 65: 597-610.
- [11] Cooke KL. Functional differential systems: Some models and perturbation problems. International Symposium on Differential Equations and Dynamical Systems. Puerto Rico, 1965. New York, NY, USA: Academic Press, 1967.
- [12] Fečkan M. On a certain type of functional-differential equations. Mathematica Slovaca 1993; 43: 39-43.
- [13] Fite W.B. Properties of the solutions of certain functional differential equations. Transactions of the American Mathematical Society 1921; 22 (3): 311-319.
- [14] Galvis J, Rojas EM, Sinitsyn AV. Existence of positive solutions of a nonlinear second-order boundary-value problem with integral boundary conditions. Electronic Journal of Differential Equations 2015; 236: 1-7.
- [15] Infante G. Positive solutions of nonlocal boundary value problems with singularities. Discrete and Continuous Dynamical Systems 2009; 377-384.
- [16] Jiang DQ. Multiple positive solutions for boundary-value problems of second-order delay differential equations. Applied Mathematics Letters 2002; 15: 575-583.
- [17] Kaufmann ER. Existence and uniqueness of solutions for a second-order iterative boundary-value problem functional differential equation. Electronic Journal of Differential Equations 2018; 150: 1-6.
- [18] Khemis R, Ardjouni A, Djoudi A. Existence of periodic solutions for a second-order nonlinear neutral differential equation by the Krasnoselskii’s fixed point technique. Le Matematiche 2017; 72: 145-156.
- [19] Khemis R, Ardjouni A, Bouakkaz A, Djoudi A. Periodic solutions of a class of third-order differential equations with two delays depending on time and state. Commentationes Mathematicae Universitatis Carolinae 2019; 60: 379-399.
- [20] Stephan BH. On the existence of periodic solutions of z′(t) = −az (t − r + µk (t, z (t))) + F (t). Journal of Differential Equations 1969; 6: 408-419.
- [21] Wang W. Positive pseudo almost periodic solutions for a class of differential iterative equations with biological background. Applied Mathematics Letters 2015; 46: 106-110.
- [22] Yang D, Zhang W. Solutions of equivariance for iterative differential equations. Applied Mathematics Letters 2004; 17: 759-765.
- [23] Yankson E. Positive periodic solutions for second-order neutral differential equations with functional delay. Electronic Journal of Differential Equations 2012; 14: 1-6.
- [24] Zhao HY, Liu J. Periodic solutions of an iterative functional differential equation with variable coefficients. Mathematical Methods in the Applied Sciences 2017; 40 (1): 286-292.