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

Yıl 2021, Cilt: 4 Sayı: 3, 186 - 199, 30.09.2021

Hojjat Afshari Hadi Shojaat Mansoureh Siahkali Moradi

https://doi.org/10.53006/rna.938851

Cited By: 14

Öz

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.

Anahtar Kelimeler

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

Kaynakça

• [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] 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] H. Afshari, M. Atapour, E. Karapinar, A discussion on a generalized Geraghty multi-valued mappings and applications. Adv. Differ. Equ. 2020, 356 (2020).
• [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] 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] 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] 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] 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] H. Aydi, E. Karapinar, W. Shatanawi, Tripled fixed point results in generalized metric spaces. J. Appl. Math. 10 (2012). Article ID 314279.
• [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] E. Karapinar, Fixed point theorems in cone Banach spaces, Fixed Point Theory Appl, (2009):9.
• [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] 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] 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] 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] S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl. 59 (2010) 1300-1309.
• [17] Y. Zhao, et al., Positive solutions for boundary value problems of nonlinear fractional differential equations, Appl. Math. Comput. 217 (2011) 6950-6958.
• [18] V. Daftardar-Gejji, Positive solutions of a system of non-autonomous fractional differential equations, J. Math. Anal. Appl. 302 (2005) 56-64.
• [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] C. Goodrich, Existence of a positive solution to a class of fractional di?erential equations, Appl. Math. Lett. 23 (2010) 1050-1055.
• [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] 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.
• [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.
• [25] W. Feng, et al., Existence of solutions for a singular system of nonlinear fractional differential equations, Comput. Math. Appl. 62 (2011) 1370-1378.
• [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.
• [27] X. Su, Boundaryvalue problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22 (2009) 64-69.
• [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.
• [29] Podlubny, I. (1999), Fractional Differential Equations, Academic Press, New york.
• [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.
• [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.
• [32] E. Zeidler, Nonlinear Functional Analysis and Its Applications-I: Fixed-Point Theorems, Springer, New York, NY, USA, 1986.
• [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.