Existence of the positive solutions for a tripled system of fractional differential equations via integral boundary conditions

Abstract

The purpose of this paper, is studying the existence and
nonexistence of positive solutions to a class of a following tripled
system of fractional differential equations.
\begin{eqnarray*} \left\{ \begin{array}{ll}
D^{\alpha}u(\zeta)+a(\zeta)f(\zeta,v(\zeta),\omega(\zeta))=0, \quad
\quad u(0)=0,\quad u(1)=\int_0^1\phi(\zeta)u(\zeta)d\zeta, \\ \\
D^{\beta}v(\zeta)+b(\zeta)g(\zeta,u(\zeta),\omega(\zeta))=0, \quad
\quad v(0)=0,\quad v(1)=\int_0^1\psi(\zeta)v(\zeta)d\zeta,\\ \\
D^{\gamma}\omega(\zeta)+c(\zeta)h(\zeta,u(\zeta),v(\zeta))=0,\quad
\quad \omega(0)=0,\quad
\omega(1)=\int_0^1\eta(\zeta)\omega(\zeta)d\zeta,\\ \end{array}
\right.\end{eqnarray*} \\ where $0\leq \zeta \leq 1$, $1<\alpha,
\beta, \gamma \leq 2$, $a,b,c\in C((0,1),[0,\infty))$, $ \phi, \psi,
\eta \in L^1[0,1]$ are nonnegative and $f,g,h\in
C([0,1]\times[0,\infty)\times[0,\infty),[0,\infty))$ and $D$ is the
standard Riemann-Liouville fractional derivative.\\
Also, we provide some examples to demonstrate the validity of our
results.

Keywords

• Tripled System
• fractional differential equation
• integral boundary conditions
• existence and nonexistence of positive solutions

References

1. [1] M. S. ABDO, Further results on the existence of solutions for generalized fractional quadratic functional integral equations, Journal of Mathematical Analysis and Modeling, (2020)1(1) : 33-46, doi:10.48185/jmam.v1i1.2.
2. [2] B. Ahmad, J. Nieto, Existence results for a coupled system of nonlinear fractional di?erential equations with three-point boundary conditions, Comput. Math. Appl. 58 (2009) 1838-1843.
3. [3] H. Afshari, M. Atapour, E. Karapinar, A discussion on a generalized Geraghty multi-valued mappings and applications. Adv. Differ. Equ. 2020, 356 (2020).
4. [4] H., Afshari, D., Baleanu, Applications of some fixed point theorems for fractional differential equations with Mittag-Leffler kernel, Advances in Difference Equations, 140 (2020), Doi:10.1186/s13662-020-02592-2.
5. [5] H., Afshari, S., Kalantari, D., Baleanu, Solution of fractional differential equations via α−φ-Geraghty type mappings. Adv. Di?er. Equ. 2018, 347(2018), https://doi.org/10.1186/s13662-018-1807-4.
6. [6] H. Afshari, Solution of fractional differential equations in quasi-b-metric and b-metric-like spaces, Adv. Differ. Equ. 2018, 285(2018), https://doi.org/10.1186/s13662-019-2227-9.
7. [7] H. Afshari, M. Sajjadmanesh, D. Baleanu, Existence and uniqueness of positive solutions for a new class of coupled system via fractional derivatives. Advances in Difference Equations. 2020 Dec;2020(1):1-8, https://doi.org/10.1186/s13662-020-02568-2.
8. [8] H. Afshari, F. Jarad, and T., Abdeljawad, On a new fixed point theorem with an application on a coupled system of fractional di?erential equations. Advances in Difference Equations 2020.1 (2020): 1-13, https://doi.org/10.1186/s13662-020-02926-0.

9. [9] H. Aydi, E. Karapinar, W. Shatanawi, Tripled fixed point results in generalized metric spaces. J. Appl. Math. 10 (2012). Article ID 314279.
10. [10] E. Karapinar, Couple fixed point theorems for nonlinear contractions in cone metric spaces Computers and Mathematics With Applications Volume: 59 Issue: 12 Pages: 3656-3668 Published: JUN 2010.
11. [11] E. Karapinar, Fixed point theorems in cone Banach spaces, Fixed Point Theory Appl, (2009):9.
12. [12] E. Karapinar, H.D. Binh, N.H. Luc, and N.H., Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Advances in Difference Equations 2021, no. 1 (2021): 1-24.
13. [13] E. Karapinar, S.I. Moustafa, A. Shehata, R.P. Agarwal, Fractional Hybrid Di?erential Equations and Coupled Fixed-Point Results for α-Admissible F(ψ 1 ,ψ 1 )-Contractions in M-Metric Spaces, Discrete Dynamics in Nature and Society, Volume 2020, Article ID 7126045, 13 pages https://doi.org/10.1155/2020/7126045,2020.
14. [14] C. Li, X. Luo, Y. Zhou, Existence of positive solutions of the boundary value problem for nonlinear fractional di?erential equations, Comput. Math. Appl. 59 (2010) 1363-1375.
15. [15] H.R. Marasi, H. Afshari, M. Daneshbastam, C.B. Zhai, Fixed points of mixed monotone operators for existence and uniqueness of nonlinear fractional differential equations, Journal of Contemporary Mathematical Analysis, vol. 52, p. 8C13, (2017).
16. [16] S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl. 59 (2010) 1300-1309.
17. [17] Y. Zhao, et al., Positive solutions for boundary value problems of nonlinear fractional differential equations, Appl. Math. Comput. 217 (2011) 6950-6958.
18. [18] V. Daftardar-Gejji, Positive solutions of a system of non-autonomous fractional differential equations, J. Math. Anal. Appl. 302 (2005) 56-64.
19. [19] J. Henderson, et al., Positive solutions for systems of generalized three-point nonlinear boundary value problems, Comment. Math. Univ. Carolin. 49 (2008) 79-91.
20. [20] C. Goodrich, Existence of a positive solution to a class of fractional di?erential equations, Appl. Math. Lett. 23 (2010) 1050-1055.
21. [21] H. Salem, On the existence of continuous solutions for a singular system of nonlinear fractional differential equations, Appl. Math. Comput. 198 (2008) 445-452.
22. [22] X. Su, Existence of solution of boundary value problem for coupled system of fractional differential equations, Engrg. Math. 26 (2009) 134-137. [23] C. Bai, J. Fang, The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Appl. Math. Comput. 150 (2004) 611-621.
23. [24] M. Rehman, R. Khan, A note on boundary value problems for a coupled system of fractional di?erential equations, Comput. Math. Appl. 61 (2011) 2630-2637.
24. [25] W. Feng, et al., Existence of solutions for a singular system of nonlinear fractional differential equations, Comput. Math. Appl. 62 (2011) 1370-1378.
25. [26] H. Shojaat, H. Afshari, M.S. Asgari, A new class of mixed monotone operators with concavity and applications to fractional di?erential equations, TWMS J. App. and Eng. Math. V.11, N.1, 2021, pp. 122-133.
26. [27] X. Su, Boundaryvalue problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22 (2009) 64-69.
27. [28] A.A., Kilbas, H.M., Srivastava, j.j., Trujillo, (2006), Theory and applications of fractiona differential equations, North- Holland Mathematics Studies. 204(204) 7-10.
28. [29] Podlubny, I. (1999), Fractional Differential Equations, Academic Press, New york.
29. [30] J. Wang, H. Xiang, Z. Liu, Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional di?erential equations, Internat. J. Differ. Equ. 2010 (2010) 12. Article ID 186928.
30. [31] W. Yang, Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions, Computers and Mathematics with Applications 63 (2012) 288-297.
31. [32] E. Zeidler, Nonlinear Functional Analysis and Its Applications-I: Fixed-Point Theorems, Springer, New York, NY, USA, 1986.
32. [33] D. Guo, V. Lakshmikantham, X. Liu, Nonlinear Integral Equations in Abstract Spaces, in: Mathematics and Its Applications, vol. 373, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.