On the Solvability of Iterative Systems of Fractional-Order Differential Equations with Parameterized Integral Boundary Conditions

Year 2024, Volume: 7 Issue: 1, 46 - 58, 18.03.2024

Muralee Bala Krushna Boddu , Mahammad Khuddush

https://doi.org/10.32323/ujma.1387528

Abstract

The aim of this paper is to determine the eigenvalue intervals of $\mu_{\mathtt{k}},~1\le \mathtt{k}\le \mathtt{n}$ for which an iterative system of a class of fractional-order differential equations with parameterized integral boundary conditions (BCs) has at least one positive solution by means of standard fixed point theorem of cone type. To the best of our knowledge, this will be the first time that we attempt to reach such findings for the topic at hand in the literature. The obtained results in the paper are illustrated with an example of their feasibility.

Keywords

Boundary value problem, Fixed-point theorems, Fractional derivative, Kernel, Positive solution

References

• [1] H. G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213–231.
• [2] A. A. Kilbas, H. M. Srivasthava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, 2006.
• [3] I. Podulbny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [4] E. F. D. Goufo, A biomathematical view on the fractional dynamics of cellulose degradation, Fract. Calc. Appl. Anal., 18(3) (2015), 554–564.
• [5] H. H. Sherief, M. A. el-Hagary, Fractional order theory of thermo-viscoelasticity and application, Mech. Time-Depend Mater., 24 (2020), 179–195.
• [6] G. Alotta, E. Bologna, G. Failla, M. Zingales, A fractional approach to Non-Newtonian blood rheology in capillary vessels, J. Peridyn Nonlocal Model, 1 (2019), 88–96.
• [7] H. Sun, W. Chen, C. Li, Y. Q. Chen, Fractional differential models for anomalous diffusion, Physica A Stat. Mech. Appl., 389 (2010), 2719–2724.
• [8] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70.
• [9] M. A. Hammad, R. Khalil, Abel’s formula and Wronskian for conformable fractional differential equations, Int. J. Diff. Equ. Appl., 13 (2014), 177–183.
• [10] U. Katugampola, A new fractional derivative with classical properties, J. American Math. Soc., arXiv:1410.6535v2.
• [11] D. R. Anderson, R. I. Avery, Fractional order boundary value problem with Sturm–Liouville boundary conditions, Electron. J. Differential Equations, 2015 (2015), 1–10.
• [12] K. R. Prasad, B. M. B. Krushna, Multiple positive solutions for a coupled system of Riemann–Liouville fractional order two-point boundary value problems, Nonlinear Stud., 20 (2013), 501–511.
• [13] K. R. Prasad, B. M. B. Krushna, Eigenvalues for iterative systems of Sturm–Liouville fractional order two-point boundary value problems, Fract. Calc. Appl. Anal., 17 (2014), 638–653.
• [14] K. R. Prasad, B. M. B. Krushna, Solvability of p-Laplacian fractional higher order two-point boundary value problems, Commun. Appl. Anal., 19 (2015), 659–678.
• [15] K. R. Prasad, B. M. B. Krushna, V. V. R. R. B. Raju, Y. Narasimhulu, Existence of positive solutions for systems of fractional order boundary value problems with Riemann–Liouville derivative, Nonlinear Stud., 24 (2017), 619–629.
• [16] K. R. Prasad, B. M. B. Krushna, L.T. Wesen, Existence results for positive solutions to iterative systems of four-point fractional-order boundary value problems in a Banach space, Asian-Eur. J. Math., 13 (2020), 1–17.
• [17] B. Zhou, L. Zhang, Uniqueness and iterative schemes of positive solutions for conformable fractional differential equations via sum-type operator method, J. Math. Res. Appl., 42(4) (2022), 349–362.
• [18] M. Bouaouid, Mild solutions of a class of conformable fractional differential equations with nonlocal conditions, Acta Math. Appl. Sin., Engl. Ser., 39(2) (2023), 249–261.
• [19] G. Xiao, J. Wang, D. O’Regan, Existence and stability of solutions to neutral conformable stochastic functional differential equations, Qual. Theory Dyn. Syst., 21(1) (2022), 1–22.
• [20] A. Jaiswal, D. Bahuguna, Semilinear conformable fractional differential equations in Banach spaces, Differ. Equ. Dyn. Syst., 27 (2019), 313–325.
• [21] A. Gokdogan, E. Unal, E. Celik, Existence and uniqueness theorems for sequential linear conformable fractional differential equations, Miskolc Math. Notes, 17 (2016), 267–279.
• [22] M. Khuddush, K. R. Prasad, Infinitely many positive solutions for an iterative system of conformable fractional order dynamic boundary value problems on time scales, Turk. J. Math., 46(2) (2022), 338-359. https://doi.org/10.3906/mat-2103-117,
• [23] W. Zhong, L. Wang, Positive solutions of conformable fractional differential equations with integral boundary conditions, Bound. Value Probl., 2018 (2018), 1–12.
• [24] B. Bendouma, A. Cabada, A. Hammoudi, Existence results for systems of conformable fractional differential equations, Arch. Math., 55(2) (2019), 69–82.
• [25] F. Haddouchi, Existence of positive solutions for a class of conformable fractional differential equations with parameterized integral boundary conditions, Kyungpook Math. J., 61 (2021), 139–153.
• [26] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, 1988.
• [27] M. A. Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
• [28] J. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math., 21 (1963), 155–160.
• [29] R. Chegis, Numerical solution of a heat conduction problem with an integral boundary condition, Litovsk. Mat. Sb., 24 (1984), 209–215.
• [30] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66.
• [31] S. Bhatter, S. D. Purohit, K. S. Nisar, S. R. Munjam, Some fractional calculus findings associated with the product of incomplete À-function and Srivastava polynomials, Int. J. Math. Comput. Eng., 2(1) (2024), 97–116.
• [32] E. Bas, M. Karaoglan, Representation of the solution of the M-Sturm–Liouville problem with natural transform, Int. J. Math. Comput. Eng., 1(2) (2023), 243–252.
• [33] S. T. Abdulazeez, M. Modanli, Analytic solution of fractional order Pseudo-Hyperbolic Telegraph equation using modified double Laplace transform method, Int. J. Math. Comput. Eng., 1(1) (2023), 105–114.
• [34] R. Singh, J. Mishra, V. K. Gupta, Dynamical analysis of a Tumor Growth model under the effect of fractal fractional Caputo-Fabrizio derivative, Int. J. Math. Comput. Eng., 1(1) (2023), 115–126.
• [35] E. Ata, I. O. Kiymaz, New generalized Mellin transform and applications to partial and fractional differential equations, Int. J. Math. Comput. Eng., 1(1) (2023), 45–66.