## Abstract and Applied Analysis

### Positive Solutions to Fractional Boundary Value Problems with Nonlinear Boundary Conditions

#### Abstract

We consider a system of boundary value problems for fractional differential equation given by ${D}_{{0}^{+}}^{\beta }{\varphi }_{p}({D}_{{0}^{+}}^{\alpha }u)(t)={\lambda }_{1}{a}_{1}(t){f}_{1}(u(t),v(t))$, $t\in (0,1)$, ${D}_{{0}^{+}}^{\beta }{\varphi }_{p}({D}_{{0}^{+}}^{\alpha }v)(t)={\lambda }_{2}{a}_{2}(t){f}_{2}(u(t),v(t))$, $t\in (0,1)$, where $1<\alpha$, $\beta \le 2$, $2<\alpha +\beta \le 4$, ${\lambda }_{1}$, ${\lambda }_{2}$ are eigenvalues, subject either to the boundary conditions ${D}_{{0}^{+}}^{\alpha }u(0)={D}_{{0}^{+}}^{\alpha }u(1)=0$, $u(0)=0$, ${D}_{{0}^{+}}^{{\beta }_{1}}u(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{1\mathrm{i }}{D}_{{0}^{+}}^{{\beta }_{1}}u({\xi }_{1i})=0$, ${D}_{{0}^{+}}^{\alpha }v(0)={D}_{{0}^{+}}^{\alpha }v(1)=0$, $v(0)=0$, ${D}_{{0}^{+}}^{{\beta }_{1}}v(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{2\mathrm{i }}{D}_{{0}^{+}}^{{\beta }_{1}}v({\xi }_{2i})=0$ or ${D}_{{0}^{+}}^{\alpha }u(0)={D}_{{0}^{+}}^{\alpha }u(1)=0$, $u(0)=0$, ${D}_{{0}^{+}}^{{\beta }_{1}}u(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{1\mathrm{i }}{D}_{{0}^{+}}^{{\beta }_{1}}u({\xi }_{1i})={\psi }_{1}(u)$, ${D}_{{0}^{+}}^{\alpha }v(0)={D}_{{0}^{+}}^{\alpha }v(1)=0$, $v(0)=0$, ${D}_{{0}^{+}}^{{\beta }_{1}}v(1)-{\mathrm{\Sigma }}_{i=1}^{m-2}{a}_{2\mathrm{i }}{D}_{{0}^{+}}^{{\beta }_{1}}v({\xi }_{2i})={\psi }_{2}(v)$, where $0<{\beta }_{1}<1$, $\alpha -{\beta }_{1}-1\ge 0$ and ${\psi }_{1}$, ${\psi }_{2}:C([0,1])\to [0$, $\mathrm{\infty })$ are continuous functions. The Krasnoselskiis fixed point theorem is applied to prove the existence of at least one positive solution for both fractional boundary value problems. As an application, an example is given to demonstrate some of main results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 579740, 20 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393450403

Digital Object Identifier
doi:10.1155/2013/579740

Mathematical Reviews number (MathSciNet)
MR3070200

Zentralblatt MATH identifier
1296.34027

#### Citation

Nyamoradi, Nemat; Baleanu, Dumitru; Bashiri, Tahereh. Positive Solutions to Fractional Boundary Value Problems with Nonlinear Boundary Conditions. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 579740, 20 pages. doi:10.1155/2013/579740. https://projecteuclid.org/euclid.aaa/1393450403