Twin solutions to semipositone boundary value problems for fractional differential equations with coupled integral boundary conditions


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.


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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


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