Abstract
An analog of the classical Vallée-Poussin theorem about differential inequality in the theory of ordinary differential equations is developed for fractional functional differential equations. The main results are obtained in a form of a theorem about several equivalent assertions. Among them solvability of two-point boundary value problems with fractional functional differential equation, negativity of Green’s function, and its derivatives and existence of a function v(t) satisfying a corresponding differential inequality. Thus the Vallée-Poussin theorem presents one of the possible “entrances” to assertions on nonoscillating properties and assertions about the negativity of Green’s functions and their derivatives for various two-point problems. Choosing the function v(t) in the condition, we obtain explicit tests of sign-constancy of Green’s functions and their derivatives. It can be stressed that a choice of a corresponding function in the Vallée-Poussin theorem leads to explicit criteria in the form of algebraic inequalities, which, as we demonstrate with examples, cannot be improved. Replacing strict inequalities with non-strict ones will already lead to incorrect statements. In some cases, the well-known results obtained in the form of the Lyapunov inequalities for fractional differential equations can be improved based on ours. Another development, we propose, is a concept to consider fractional functional differential equations. The basis of this concept is to reduce boundary value problems for fractional functional differential equations to operator equations in the space of essentially bounded functions. Fractional functional differential equations can appear in various applications and in the process of construction of equations for a corresponding component of a solution-vector of a system of fractional equations.
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References
Agarwal, R.P., Bohner, M., Özbekler, A.: Lyapunov Inequalities and Applications. Springer, Berlin (2021)
Agarwal, R.P., Lakshmikantham, V., Nieto, J.J.: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. Theory Methods Appl. 72(6), 2859–2862 (2010)
Azbelev, N.V., Domoshnitsky, A.: On de la Vallée Poussin’s differential inequality. Differ. Uravn. 22(12), 2041–5 (1986)
Azbelev, N.V., Domoshnitsky, A.: A question concerning linear-differential inequalities. 1. Differ. Equ. 27(3), 257–263 (1991)
Azbelev, N.V., Domoshnitsky, A.: A question concerning linear-differential inequalities. 2. Differ. Equ. 27(6), 641–647 (1991)
Azbelev, N.V., Maksimov, V.P., Rakhmatullina, L.F.: Introduction to the Theory of Functional Differential Equations. Hindawi Publishing, London (2007)
Benmezai, A., Saadi, A.: Existence of positive solutions for a nonlinear fractional differential equations with integral boundary conditions. J. Fract. Calc. Appl. 7(2), 145–152 (2016)
Berezansky, L., Domoshnitsky, A., Koplatadze, R.: Oscillation, Nonoscillation, Stability and Asymptotic Properties for Second and Higher Order Functional Differential Equations. CRC Press, Boca Raton (2020)
Bhalekar, S., Daftardar-Gejji, V., Baleanu, D., Magin, R.: Fractional Bloch equation with delay. Comput. Math. Appl. 61(5), 1355–1365 (2011)
Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389(1), 403–411 (2012)
de La Vallée Poussin, C.-J.: Sur le’quation differentielle line’aire du second ordre. De’termination d’une inte’grale par deux valeurs assigne’es. Extension aux e’quations d’orde \(n\). J. Math. Pures Appl. 8, 125–144 (1929). (in French)
Feliu-Batlle, V., Rivas-Perez, R., Castillo-Garcia, F.J.: Fractional order controller robust to time delay variations for water distribution in an irrigation main canal pool. Comput. Electron. Agric. 69(2), 185–197 (2009)
Ferreira, R.: A Lyapunov-type inequality for a fractional boundary value problem. Fract. Calc. Appl. Anal. 16(4), 978–984 (2013). https://doi.org/10.2478/s13540-013-0060-5
Ferreira, R.A.: Existence and uniqueness of solutions for two-point fractional boundary value problems. Electron. J. Differ. Equ. 2016(202), 1–5 (2016)
Ferreira, R.A.: Fractional de la Vallée Poussin inequalities. arXiv preprint (2018). arXiv:1805.09765
Henderson, J., Luca, R.: Boundary Value Problems for Systems of Differential, Difference and Fractional Equations: Positive Solutions. Academic, New York (2015)
Henderson, J., Luca, R.: Nonexistence of positive solutions for a system of coupled fractional boundary value problems. Bound. Value Probl. 2015(1), 1–12 (2015)
Henderson, J., Luca, R.: Positive solutions for a system of semipositone coupled fractional boundary value problems. Bound. Value Probl. 2016(1), 1–23 (2016)
Jankowski, T.: Positive solutions to fractional differential equations involving Stieltjes integral conditions. Appl. Math. Comput. 241, 200–213 (2014)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Krasnosel’skii, M.A., Vainikko, G.M., Zabreyko, R.P., Ruticki, Y.B., Stet’senko, V.V.: Approximate Solution of Operator Equations. Springer, Dordrecht (2012)
Latha, V.P., Rihan, F.A., Rakkiyappan, R., Velmurugan, G.: A fractional-order delay differential model for Ebola infection and CD8+ T-cells response: stability analysis and Hopf bifurcation. Int. J. Biomath. 10(08), 1750111 (2017)
Mawhin, J.: The legacy of de La Vallée Poussin’s work on boundary value problems for ordinary differential equations: a survey and a bibliography. Acad. R. Belg. Ch.-J. Val. Poussin Collect. Works II, 357–401 (2001)
Padhi, S., Graef, J.R., Pati, S.: Multiple positive solutions for a boundary value problem with nonlinear nonlocal Riemann–Stieltjes integral boundary conditions. Fract. Calc. Appl. Anal. 21(3), 716–745 (2018). https://doi.org/10.1515/fca-2018-0038
Podlubny, I.: Fractional Differential Equations. Academic, San Diego (1999)
Qiao, Y., Zhou, Z.: Existence of positive solutions of singular fractional differential equations with infinite-point boundary conditions. Adv. Differ. Equ. 2017(1), 1–9 (2017)
Sun, W., Wang, Y.: Multiple positive solutions of nonlinear fractional differential equations with integral boundary value conditions. Fract. Calc. Appl. Anal. 17(3), 605–616 (2014). https://doi.org/10.2478/s13540-014-0188-y
Wang, G., Zhang, L., Agarwal, R.: Nonlocal integral boundary value problems with causal operators and fractional derivatives. Funct. Differ. Equ. 27(1–2), 39–50 (2020). https://doi.org/10.26351/FDE/27/1-2/5
Wang, Y.: Positive solutions for fractional differential equation involving the Riemann–Stieltjes integral conditions with two parameters. J. Nonlinear Sci. Appl. 9(11), 5733–5740 (2016)
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This paper is a part of the third author’s Ph.D. Thesis, which is being carried out in the Department of Mathematics at Ariel University.
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Domoshnitsky, A., Padhi, S. & Srivastava, S.N. Vallée-Poussin theorem for fractional functional differential equations. Fract Calc Appl Anal 25, 1630–1650 (2022). https://doi.org/10.1007/s13540-022-00061-z
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DOI: https://doi.org/10.1007/s13540-022-00061-z
Keywords
- Fractional differential equations
- Riemann–Liouville derivative
- Boundary value problems
- Positive solutions
- Comparison of solution
- Lyapunov inequality
- Sign constancy of Green’s function
- Differential inequality