# Existence of positive solutions for fractional differential systems with multi point boundary conditions

Existence of positive solutions for fractional differential systems with multi point boundary... In this paper, we study the existence of positive solutions to the boundary value problem for the fractional differential system $$\left\{\begin{array}{lll} D_{0^+}^\beta \phi_p(D_{0^+}^\alpha u) (t) = f_1 (t, u (t), v (t)),\quad t \in (0, 1),\\ D_{0^+}^\beta \phi_p(D_{0^+}^\alpha v) (t) = f_2 (t, u (t), v(t)), \quad t \in (0, 1),\\ D_{0^+}^\alpha u(0)= D_{0^+}^\alpha u(1)=0,\; u (0) = 0, \quad u (1)-\Sigma_{i=1}^{m-2} a_{1i}\;u(\xi_{1i})=\lambda_1,\\ D_{0^+}^\alpha v(0)= D_{0^+}^\alpha v(1)=0,\; v (0) = 0, \quad v (1)-\Sigma_{i=1}^{m-2} a_{2i}\; v(\xi_{2i})=\lambda_2, \end{array}\right.$$ where $${1<\alpha,\beta\leq 2, 2 <\alpha + \beta\leq 4, D_{0^+}^\alpha}$$ is the Riemann–Liouville fractional derivative of order α. By using the Leggett–Williams fixed point theorem in a cone, the existence of three positive solutions for nonlinear singular boundary value problems is obtained. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ANNALI DELL'UNIVERSITA' DI FERRARA Springer Journals

# Existence of positive solutions for fractional differential systems with multi point boundary conditions

, Volume 59 (2) – Jun 26, 2012
18 pages

/lp/springer-journals/existence-of-positive-solutions-for-fractional-differential-systems-LhiThXNF1e
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general; Analysis; Geometry; History of Mathematical Sciences; Numerical Analysis; Algebraic Geometry
ISSN
0430-3202
eISSN
1827-1510
DOI
10.1007/s11565-012-0160-x
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we study the existence of positive solutions to the boundary value problem for the fractional differential system $$\left\{\begin{array}{lll} D_{0^+}^\beta \phi_p(D_{0^+}^\alpha u) (t) = f_1 (t, u (t), v (t)),\quad t \in (0, 1),\\ D_{0^+}^\beta \phi_p(D_{0^+}^\alpha v) (t) = f_2 (t, u (t), v(t)), \quad t \in (0, 1),\\ D_{0^+}^\alpha u(0)= D_{0^+}^\alpha u(1)=0,\; u (0) = 0, \quad u (1)-\Sigma_{i=1}^{m-2} a_{1i}\;u(\xi_{1i})=\lambda_1,\\ D_{0^+}^\alpha v(0)= D_{0^+}^\alpha v(1)=0,\; v (0) = 0, \quad v (1)-\Sigma_{i=1}^{m-2} a_{2i}\; v(\xi_{2i})=\lambda_2, \end{array}\right.$$ where $${1<\alpha,\beta\leq 2, 2 <\alpha + \beta\leq 4, D_{0^+}^\alpha}$$ is the Riemann–Liouville fractional derivative of order α. By using the Leggett–Williams fixed point theorem in a cone, the existence of three positive solutions for nonlinear singular boundary value problems is obtained.

### Journal

ANNALI DELL'UNIVERSITA' DI FERRARASpringer Journals

Published: Jun 26, 2012

### References

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