Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions
Abstract
:1. Introduction
2. Preliminaries
3. Existence and Uniqueness Results
3.1. Existence Result Via Leray–Schauder Nonlineear Alternative
- There exist functions , with and nondecreasing functions , such that and , for all .
- There exists a constant such that
3.2. Existence Result via Krasnoselskii’s Fixed Point Theorem
- , and for all , , with , where is given by (13), and .
- , , for all , and .
3.3. Existence and Uniqueness Result
4. Analogue Problems
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Coffey, W.T.; Kalmykov, Y.P.; Waldron, J.T. The Langevin Equation, 2nd ed.; World Scientific: Singapore, 2004. [Google Scholar]
- Jakubowska, A.; Walczak, J. Analysis of the transient state in a series circuit of the class RLβCα. Circuits Syst. Signal Process 2016, 35, 1831–1853. [Google Scholar] [CrossRef]
- Webb, J.R.L.; Infante, G. Positive solutions of nonlocal boundary value problems: A unified approach. J. Lond. Math. Soc. 2006, 74, 673–693. [Google Scholar] [CrossRef]
- Ahmad, B.; Alsaedi, A.; Alghamdi, B.S. Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Anal. Real World Appl. 2008, 9, 1727–1740. [Google Scholar] [CrossRef]
- Čiegis, R.; Bugajev, A. Numerical approximation of one model of the bacterial self-organization. Nonlinear Anal. Model. Control 2012, 17, 253–270. [Google Scholar]
- Graef, J.R.; Kong, L. Existence of positive solutions to a higher order singular boundary value problem with fractional q-derivatives. Fract. Calc. Appl. Anal. 2013, 16, 695–708. [Google Scholar] [CrossRef]
- O’Regan, D.; Stanek, S. Fractional boundary value problems with singularities in space variables. Nonlinear Dyn. 2013, 71, 641–652. [Google Scholar] [CrossRef]
- Zhai, C.; Xu, L. Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter. Commun. Nonlinear Sci. Numer. Simul. 2014, 19, 2820–2827. [Google Scholar] [CrossRef]
- Henderson, J.; Kosmatov, N. Eigenvalue comparison for fractional boundary value problems with the Caputo derivative. Fract. Calc. Appl. Anal. 2014, 17, 872–880. [Google Scholar] [CrossRef]
- Li, B.; Sun, S.; Li, Y.; Zhao, P. Multi-point boundary value problems for a class of Riemann-Liouville fractional differential equations. Adv. Differ. Equ. 2014, 2014, 151. [Google Scholar] [CrossRef] [Green Version]
- Henderson, J.; Luca, R. Nonexistence of positive solutions for a system of coupled fractional boundary value problems. Bound. Value Probl. 2015, 2015, 138. [Google Scholar] [CrossRef] [Green Version]
- Ntouyas, S.K.; Etemad, S. On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions. Appl. Math. Comput. 2015, 266, 235–243. [Google Scholar] [CrossRef]
- Qarout, D.; Ahmad, B.; Alsaedi, A. Existence theorems for semilinear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions. Fract. Calc. Appl. Anal. 2016, 19, 463–479. [Google Scholar] [CrossRef]
- Zou, Y.; He, G. On the uniqueness of solutions for a class of fractional differential equations. Appl. Math. Lett. 2017, 74, 68–73. [Google Scholar] [CrossRef]
- Agarwal, R.P.; Ahmad, B.; Garout, D.; Alsaedi, A. Existence results for coupled nonlinear fractional differential equations equipped with nonlocal coupled flux and multi-point boundary conditions. Chaos Solitons Fractals 2017, 102, 149–161. [Google Scholar] [CrossRef]
- Wang, G.; Pei, K.; Agarwal, R.P.; Zhang, L.; Ahmad, B. Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line. J. Comput. Appl. Math. 2018, 343, 230–239. [Google Scholar] [CrossRef]
- Ahmad, B.; Luca, R. Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions. Fract. Calc. Appl. Anal. 2018, 21, 423–441. [Google Scholar] [CrossRef]
- Fernandez, A.; Baleanu, D.; Fokas, A. Solving PDEs of fractional order using the unified transform method. Appl. Math. Comput. 2018, 339, 738–749. [Google Scholar] [CrossRef] [Green Version]
- Mahmudov, N.; Emin, S. Fractional-order boundary value problems with Katugampola fractional integral conditions. Adv. Differ. Equ. 2018, 2018, 81. [Google Scholar] [CrossRef] [Green Version]
- Ahmad, B.; Alsaedi, A.; Salem, S. On a nonlocal integral boundary value problem of nonlinear Langevin equation with different fractional orders. Adv. Differ. Equ. 2019, 2019, 57. [Google Scholar] [CrossRef] [Green Version]
- Wang, Y. Necessary conditions for the existence of positive solutions to fractional boundary value problems at resonance. Appl. Math. Lett. 2019, 97, 34–40. [Google Scholar] [CrossRef]
- Magin, R.L. Fractional Calculus in Bioengineering; Begell House Publishers: Danbury, CT, USA, 2006. [Google Scholar]
- Klafter, J.; Lim, S.C.; Metzler, R. (Eds.) Fractional Dynamics in Physics; World Scientific: Singapore, 2011. [Google Scholar]
- Povstenko, Y.Z. Fractional Thermoelasticity; Springer: New York, NY, USA, 2015. [Google Scholar]
- Petras, I.; Magin, R.L. Simulation of drug uptake in a two compartmental fractional model for a biological system. Commun. Nonlinear Sci. Numer. Simul. 2011, 16, 4588–4595. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Ding, Y.; Wang, Z.; Ye, H. Optimal control of a fractional-order HIV-immune system with memory. IEEE Trans. Control Syst. Technol. 2012, 20, 763–769. [Google Scholar] [CrossRef]
- Javidi, M.; Ahmad, B. Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton– zooplankton system. Ecol. Model. 2015, 318, 8–18. [Google Scholar] [CrossRef]
- Carvalho, A.; Pinto, C.M.A. A delay fractional order model for the co-infection of malaria and HIV/AIDS. Int. J. Dyn. Control 2017, 5, 168–186. [Google Scholar] [CrossRef]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; North-Holland Mathematics Studies; Elsevier Science B.V.: Amsterdam, The Netherlands, 2006; Volume 204. [Google Scholar]
- Zhou, Y. Basic Theory of Fractional Differential Equations; World Scientific Publishing Co. Pte. Ltd.: Hackensack, NJ, USA, 2014. [Google Scholar]
- Ahmad, B.; Alsaedi, A.; Ntouyas, S.K.; Tariboon, J. Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities; Springer: Cham, Switzerland, 2017. [Google Scholar]
- Granas, A.; Dugundji, J. Fixed Point Theory; Springer: New York, NY, USA, 2003. [Google Scholar]
- Krasnoselskii, M.A. Two remarks on the method of successive approximations. Uspekhi Mat. Nauk 1955, 10, 123–127. [Google Scholar]
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Ahmad, B.; Alsaedi, A.; Salem, S.; Ntouyas, S.K. Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions. Fractal Fract. 2019, 3, 34. https://doi.org/10.3390/fractalfract3020034
Ahmad B, Alsaedi A, Salem S, Ntouyas SK. Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions. Fractal and Fractional. 2019; 3(2):34. https://doi.org/10.3390/fractalfract3020034
Chicago/Turabian StyleAhmad, Bashir, Ahmed Alsaedi, Sara Salem, and Sotiris K. Ntouyas. 2019. "Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions" Fractal and Fractional 3, no. 2: 34. https://doi.org/10.3390/fractalfract3020034