## Positive solutions for first-order nonlinear Caputo-Hadamard fractional relaxation differential equations.(English)Zbl 1499.34170

Summary: This article concerns the existence and uniqueness of positive solutions of the first-order nonlinear Caputo-Hadamard fractional relaxation differential equation $\begin{cases} \mathfrak{D}_1^{\alpha}(x(t) -g(t,x(t))) +wx (t) = f(t,x(t)), \quad 1<t \leq e,\\ x(1) = x_0 > g(1,x_0) >0, \end{cases}$ where $$0<\alpha \leq 1$$. In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ the Krasnoselskii fixed point theorem and the method of upper and lower solutions to show the existence of a positive solution of this equation. We also use the Banach fixed point theorem to show the existence of a unique positive solution. Finally, an example is given to illustrate our results.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34A08 Fractional ordinary differential equations 34B08 Parameter dependent boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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### References:

 [1] S. Abbas,Existence of solutions to fractional order ordinary and delay differential equations and applications, Electron. J. Differential Equations2011(9) (2011), 1-11. · Zbl 1211.34096 [2] R. P. Agarwal, Y. Zhou and Y. He,Existence of fractional functional differential equations, Comput. Math. Appl.59(2010), 1095-1100. · Zbl 1189.34152 [3] B. Ahmad and S. K. Ntouyas,Existence and uniqueness of solutions for Caputo-Hadamard sequential fractional order neutral functional differential equations, Electron. J. Differential Equations 2017(36) (2017), 1-11. · Zbl 1360.34159 [4] H. Boulares, A. Ardjouni and Y. Laskri,Positive solutions for nonlinear fractional differential equations, Positivity21(2017), 1201-1212. · Zbl 1377.26006 [5] H. Boulares, A. Ardjouni and Y. Laskri,Stability in delay nonlinear fractional differential equations, Rend. Circ. Mat. Palermo65(2016), 243-253. · Zbl 1373.34114 [6] A. Chidouh, A. Guezane-Lakoud and R. Bebbouchi,Positive solutions of the fractional relaxation equation using lower and upper solutions, Vietnam J. Math.44(4) (2016), 739-748. · Zbl 1358.34009 [7] F. Ge and C. Kou,Stability analysis by Krasnoselskii’s fixed point theorem for nonlinear fractional differential equations, Appl. Math. Comput.257(2015), 308-316. · Zbl 1338.34103 [8] F. Ge and C. Kou,Asymptotic stability of solutions of nonlinear fractional differential equations of order1< α <2, Journal of Shanghai Normal University44(3) (2015), 284-290. [9] A. A. Kilbas, H. H. Srivastava and J. J. Trujillo,Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006. · Zbl 1092.45003 [10] C. Kou, H. Zhou and Y. Yan,Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis, Nonlinear Anal.74(2011), 5975-5986. · Zbl 1235.34022 [11] V. Lakshmikantham and A. S. Vatsala,Basic theory of fractional differential equations, Nonlinear Anal.69(2008) 2677-2682. · Zbl 1161.34001 [12] N. Li and C. Wang,New existence results of positive solution for a class of nonlinear fractional differential equations, Acta Math. Sci.33(2013), 847-854. · Zbl 1299.34015 [13] I. Podlubny,Fractional Differential Equations, Academic Press, San Diego, 1999. · Zbl 0924.34008 [14] W. R. Schneider,Completely monotone generalized Mittag-Leffler functions, Expo. Math.14 (1996), 3-16. · Zbl 0843.60024 [15] D. R. Smart,Fixed Point Theorems, Cambridge Tracts in Mathematics66, Cambridge University Press, London, New York, 1974. · Zbl 0297.47042 [16] S. Zhang,The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl.252(2000) · Zbl 0972.34004
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