# zbMATH — the first resource for mathematics

On positive solutions of a nonlocal fractional boundary value problem. (English) Zbl 1187.34026
Summary: We investigate the existence and uniqueness of positive solutions for a nonlocal boundary value problem
\begin{aligned} & D^\alpha_{0+}u(t)+f(t,u(t))=0,\quad 0<t<1,\\ & u(0)=0,\quad \beta u(\eta)=u(1),\end{aligned}
where $$1<\alpha\leq 2$$, $$0 <\beta\eta^{\alpha-1}< 1.0 < \eta < 1$$, $$D^\alpha_{0+}$$ is the standard Riemann-Liouville differentiation. The function is continuous on $$[0,1]\times [0,\infty)$$.
Firstly, we give Green’s function and prove its positivity; secondly, the uniqueness of positive solution is obtained by the use of contraction map principle and some Lipschitz-type conditions; thirdly, by means of the fixed point index theory, we obtain some existence results of positive solution. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34A08 Fractional ordinary differential equations and fractional differential inclusions 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
Full Text:
##### References:
 [1] Delbosco, D., Fractional calculus and function spaces, J. fract. calc., 6, 45-53, (1994) · Zbl 0829.46018 [2] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002 [3] Podlubny, I., Fractional differential equations, mathematics in science and engineering, (1999), Academic Press New York, London, Toronto [4] Kilbas, A.A.; Trujillo, J.J., Differential equations of fractional order: methods, results and problems-I, Appl. anal., 78, 153-192, (2001) · Zbl 1031.34002 [5] Kilbas, A.A.; Trujillo, J.J., Differential equations of fractional order: methods, results and problems-II, Appl. anal., 81, 435-493, (2002) · Zbl 1033.34007 [6] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier B.V Netherlands · Zbl 1092.45003 [7] Babakhani, A.; Gejji, V.D., Existence of positive solutions of nonlinear fractional differential equations, J. math. anal. appl., 278, 434-442, (2003) · Zbl 1027.34003 [8] Bai, Z.B.; Lü, H.S., Positive solutions of boundary value problems of nonlinear fractional differential equation, J. math. anal. appl., 311, 495-505, (2005) · Zbl 1079.34048 [9] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. math. anal. appl., 204, 609-625, (1996) · Zbl 0881.34005 [10] Eidelman, S.D.; Kochubei, A.N., Cauchy problem for fractional diffusion equations, J. differential equations, 199, 211-255, (2004) · Zbl 1129.35427 [11] Gejji, V.D.; Babakhani, A., Analysis of a system of fractional differential equations, J. math. anal. appl., 293, 511-522, (2004) · Zbl 1058.34002 [12] Jafari, H.; Gejji, V.D., Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. math. comput., 180, 700-706, (2006) · Zbl 1102.65136 [13] Yu, C.; Gao, G.Z., On the solution of nonlinear fractional order differential equation, Nonlinear anal. TMA, 63, e971-e976, (1998) · Zbl 1224.34005 [14] Zhang, S.Q., Positive solutions for boundary value problems of nonlinear fractional differential equations, Electr. J. differential equations, 2006, 1-12, (2006) [15] Zhang, S.Q., Monotone method for initial value problem for fractional diffusion equation, Sci. China ser. A, 49, 1223-1230, (2006) · Zbl 1109.35064 [16] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Gronignen [17] Guo, D.; Sun, J., Nonlinear integral equations, (1987), Shandong Science and Technology Press Jinan, (in Chinese)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.