Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions
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Authors
Daliang Zhao
- School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, P. R. China.
Yansheng Liu
- School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, P. R. China.
Abstract
This paper investigates the existence of at least two positive solutions for the following high-order fractional semipositone
boundary value problem (SBVP, for short) with coupled integral boundary value conditions:
\[
\begin{cases}
D^\alpha_0+u(t)+\lambda f(t,u(t),v(t))=0,\quad t\in (0,1),\\
D^\alpha_0+v(t)+\lambda g(t,u(t),v(t))=0,\quad t\in (0,1),\\
u^{(j)}(0)= v^{(j)}(0)=0,\quad j=0,1,2,...,n-2,\\
D^{\alpha-1}_{0^+}u(1)=\lambda_1\int^{\eta_1}_0 v(t)dt,\\
D^{\alpha-1}_{0^+}v(1)=\lambda_2\int^{\eta_2}_0 u(t)dt,
\end{cases}
\]
where \(n - 1 < \alpha\leq n, n \geq 3, 0 < \eta_1,\eta_2\leq 1, \lambda,\lambda_1,\lambda_2\) are parameters and satisfy \(\lambda_1\lambda_2(\eta_1\eta_2)^\alpha<\Gamma^2(\alpha+1), D^\alpha_{0^+}\) is the standard
Riemann-Liouville derivative, and f, g are continuous and semipositone. By using the nonlinear alternative of Leray-Schauder
type, Krasnoselskii’s fixed point theorems, and the theory of fixed point index on cone, we establish some existence results of
multiple positive solutions to the considered fractional SBVP. As applications, two examples are presented to illustrate our main
results.
Share and Cite
ISRP Style
Daliang Zhao, Yansheng Liu, Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 7, 3544--3565
AMA Style
Zhao Daliang, Liu Yansheng, Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions. J. Nonlinear Sci. Appl. (2017); 10(7):3544--3565
Chicago/Turabian Style
Zhao, Daliang, Liu, Yansheng. "Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions." Journal of Nonlinear Sciences and Applications, 10, no. 7 (2017): 3544--3565
Keywords
- Fractional differential equations
- semipositone boundary value problem
- coupled integral boundary value conditions
- fixed point index.
MSC
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