Abstract
In this study, we analyze the oscillatory behavior of solutions to a specific class of fractional integro-differential equations. First, we derive sufficient conditions that ensure nonoscillatory solutions exhibit a well-defined asymptotic behavior. Building on this result, we establish a series of oscillation theorems that provide deeper insight into the qualitative nature of solutions. To validate our theoretical findings, we present a concrete example that demonstrates the applicability of our main results. These contributions aim to advance the theoretical framework of fractional equations, offering new perspectives on their dynamic behavior and potential applications in mathematical modeling
Keywords
Fractional integro-differential equation Riemann-Liouville Derivative Riemann-Liouville Integral Oscillation
References
- [1] Singh J., Kumar D., Baleanu D., New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Math. Model. Nat. Phenom., 14(3) (2019).
- [2] Kilbas A.A., Hadamard-type fractional calculus, J. Korean Math. Soc., 38(6) (2001) 1191-1204.
- [3] Katugampola U.N., New approach to generalized fractional integral, Appl. Math. Comput., 218(3) (2011) 860-865.
- [4] Katugampola U.N., A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6(4) (2014) 1-15.
- [5] Jarad F., Abdeljawad T., Baleanu D., On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10(5) (2017) 2607-2619.
- [6] Jarad F., Abdeljawad T., Baleanu D., Caputo-type modification of the Hadamard fractional derivatives, Advances in Difference Equations, 2012 (2012) 1-8.
- [7] Jarad F., Uğurlu E., Abdeljawad T., Baleanu D., On a new class of fractional operators, Adv. Differ. Equ., 247 (2017).
- [8] Grace S.R., Agarwal R.P., Wong P.J.Y., Zafer A., On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal., 15 (2012) 222-231.
- [9] Zhu P., Xiang Q., Oscillation criteria for a class of fractional delay differential equations, Adv. Differ. Equ., 2018:403 (2018). [10] Bolat Y., On the oscillation of fractional order delay differential equations with constant coefficients, Commun. Nonlinear Sci. Numer. Simul., 19 (2014) 3988-3993. [11] Chen D.X., Oscillation criteria of fractional differential equations, Adv. Differ. Equ., 2012:33 (2012).
- [12] Abdalla B., Abdeljawad T., Oscillation criteria for kernel function dependent fractional dynamic equations, Discrete & Continuous Dynamical Systems - S, 14(10) (2021).
- [13] Zhou Y., Ahmad B., Chen F., Alsaedi A., Oscillation for fractional partial differential equations, Bull. Malays. Math. Soc., 42 (2019) 449-465.
- [14] Aphithana A., Ntouyas S.K., Tariboon J., Forced oscillation of fractional differential equations via conformable derivatives with damping term, Bound. Value Probl., 2019 (2019) 1-16.
- [15] Chen D., Qu P., Lan Y., Forced oscillation of certain fractional differential equations, Adv. Differ. Equ., 2013 (2013) 125-134.
- [16] Alzabut J., Abdeljawad T., Sufficient conditions for the oscillation of nonlinear fractional difference equations, J. Fract. Calc. Appl., 5(1) (2014) 177-187.
- [17] Abdalla B., On the oscillation of q-fractional difference equations, Adv. Differ. Equ., 2017:254 (2017) 1-11.
- [18] Abdalla B., Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives, Adv. Differ. Equ., 2018:107 (2018) 1-15.
- [19] Abdalla B., Abdeljawad T., On the oscillation of Hadamard fractional differential equations, Adv. Differ. Equ., 2018:409 (2018).
- [20] Aslıyüce S., Güvenilir A.F., Zafer A., Oscillation criteria for a certain class of fractional order integro-differential equations, Hacet. J. Math. Stat., 46(2) (2017) 199-207.
- [21] Mert R., Bayeğ S., Abdeljawad T., Abdalla B., On the oscillation of kernel function dependent fractional integro-differential equations, Rocky Mt. J. Math., 52(4) (2022) 1451-1460.
- [22] Restrepo J.E., Suragan D., Oscillatory solutions of fractional integro-differential equations, Math. Methods Appl. Sci., 43(15) (2020) 9080-9089.
- [23] Restrepo J.E., Suragan D., Oscillatory solutions of fractional integro-differential equations II, Math. Methods Appl. Sci., 44(8) (2021) 7262-7274.
- [24] Grace S.R., Zafer A., Oscillatory behavior of integro-dynamic and integral equations on time scales, Appl. Math. Lett., 28 (2014) 47-52.
- [25] Kilbas A.A., Srivastava M.H., Trujillo J.J., Theory and Application of Fractional Differential Equations, North Holland Mathematics Studies 204, (2006).
- [26] Jarad F., Abdeljawad T., Generalized fractional derivatives and Laplace transform, Discrete and Continuous Dynamical Systems-S, 13 (2020) 709-722.
- [27] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, (1993).
- [28] Hardy G.H., Littlewood J.E., Polya G., Inequalities, Cambridge University Press, (1988).
Abstract
Bir sınıf kesirli integral-türev denkleminin çözümlerinin salınımlı davranışını inceledik. İlk olarak, her titreşimsiz çözümün belirli bir asimptotik davranış sergilemesi için yeterli koşulları sunduk. Daha sonra, bu asimptotik sonucu kullanarak bazı salınım teoremleri ve ispatlarını verdik. Son olarak, ana sonucumuzun geçerliliğini göstermek için bir örnek verdik.
Keywords
Kesirli integral-türev denklemi Riemann-Liouville Türevi Riemann-Liouville İntegrali Salınım
References
- [1] Singh J., Kumar D., Baleanu D., New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Math. Model. Nat. Phenom., 14(3) (2019).
- [2] Kilbas A.A., Hadamard-type fractional calculus, J. Korean Math. Soc., 38(6) (2001) 1191-1204.
- [3] Katugampola U.N., New approach to generalized fractional integral, Appl. Math. Comput., 218(3) (2011) 860-865.
- [4] Katugampola U.N., A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl., 6(4) (2014) 1-15.
- [5] Jarad F., Abdeljawad T., Baleanu D., On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10(5) (2017) 2607-2619.
- [6] Jarad F., Abdeljawad T., Baleanu D., Caputo-type modification of the Hadamard fractional derivatives, Advances in Difference Equations, 2012 (2012) 1-8.
- [7] Jarad F., Uğurlu E., Abdeljawad T., Baleanu D., On a new class of fractional operators, Adv. Differ. Equ., 247 (2017).
- [8] Grace S.R., Agarwal R.P., Wong P.J.Y., Zafer A., On the oscillation of fractional differential equations, Fract. Calc. Appl. Anal., 15 (2012) 222-231.
- [9] Zhu P., Xiang Q., Oscillation criteria for a class of fractional delay differential equations, Adv. Differ. Equ., 2018:403 (2018). [10] Bolat Y., On the oscillation of fractional order delay differential equations with constant coefficients, Commun. Nonlinear Sci. Numer. Simul., 19 (2014) 3988-3993. [11] Chen D.X., Oscillation criteria of fractional differential equations, Adv. Differ. Equ., 2012:33 (2012).
- [12] Abdalla B., Abdeljawad T., Oscillation criteria for kernel function dependent fractional dynamic equations, Discrete & Continuous Dynamical Systems - S, 14(10) (2021).
- [13] Zhou Y., Ahmad B., Chen F., Alsaedi A., Oscillation for fractional partial differential equations, Bull. Malays. Math. Soc., 42 (2019) 449-465.
- [14] Aphithana A., Ntouyas S.K., Tariboon J., Forced oscillation of fractional differential equations via conformable derivatives with damping term, Bound. Value Probl., 2019 (2019) 1-16.
- [15] Chen D., Qu P., Lan Y., Forced oscillation of certain fractional differential equations, Adv. Differ. Equ., 2013 (2013) 125-134.
- [16] Alzabut J., Abdeljawad T., Sufficient conditions for the oscillation of nonlinear fractional difference equations, J. Fract. Calc. Appl., 5(1) (2014) 177-187.
- [17] Abdalla B., On the oscillation of q-fractional difference equations, Adv. Differ. Equ., 2017:254 (2017) 1-11.
- [18] Abdalla B., Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives, Adv. Differ. Equ., 2018:107 (2018) 1-15.
- [19] Abdalla B., Abdeljawad T., On the oscillation of Hadamard fractional differential equations, Adv. Differ. Equ., 2018:409 (2018).
- [20] Aslıyüce S., Güvenilir A.F., Zafer A., Oscillation criteria for a certain class of fractional order integro-differential equations, Hacet. J. Math. Stat., 46(2) (2017) 199-207.
- [21] Mert R., Bayeğ S., Abdeljawad T., Abdalla B., On the oscillation of kernel function dependent fractional integro-differential equations, Rocky Mt. J. Math., 52(4) (2022) 1451-1460.
- [22] Restrepo J.E., Suragan D., Oscillatory solutions of fractional integro-differential equations, Math. Methods Appl. Sci., 43(15) (2020) 9080-9089.
- [23] Restrepo J.E., Suragan D., Oscillatory solutions of fractional integro-differential equations II, Math. Methods Appl. Sci., 44(8) (2021) 7262-7274.
- [24] Grace S.R., Zafer A., Oscillatory behavior of integro-dynamic and integral equations on time scales, Appl. Math. Lett., 28 (2014) 47-52.
- [25] Kilbas A.A., Srivastava M.H., Trujillo J.J., Theory and Application of Fractional Differential Equations, North Holland Mathematics Studies 204, (2006).
- [26] Jarad F., Abdeljawad T., Generalized fractional derivatives and Laplace transform, Discrete and Continuous Dynamical Systems-S, 13 (2020) 709-722.
- [27] Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, (1993).
- [28] Hardy G.H., Littlewood J.E., Polya G., Inequalities, Cambridge University Press, (1988).
Details
| Primary Language | English |
|---|---|
| Subjects | Ordinary Differential Equations, Difference Equations and Dynamical Systems, Applied Mathematics (Other) |
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | March 25, 2025 |
| Submission Date | December 24, 2024 |
| Acceptance Date | March 1, 2025 |
| Published in Issue | Year 2025Volume: 46 Issue: 1 |
