Systems of discrete fractional boundary value problems with nonlinearities satisfying no growth conditions

Abstract

We consider the coupled system of discrete fractional boundary value problems

$-{\displaystyle {\Delta }_{\nu -2}^{\nu }}x\left(t\right)={\lambda }_{1}f(t+\nu -1,y(t+\mu -1\left)\right),t\in [0,b+1]{}_{\mathbb{N}{}_0}-{\displaystyle {\Delta }_{\mu -2}^{\mu }}y\left(t\right)={\lambda }_{2}g(t+\mu -1,x(t+\nu -1\left)\right),t\in [0,b+1]{}_{\mathbb{N}{}_0}x(\nu -2)=H_1({\displaystyle \sum _{i=1}^n} a_{i}y({\xi }_i\left)\right),x(\nu +b+1)=0y(\mu -2)=H_2({\displaystyle \sum _{j=1}^m} b_{j}x({\zeta }_j\left)\right),y(\mu +b+1)=0,$

where $1<\nu \le 2$ and $1<\mu \le 2$. In particular, we consider the case of nonlocal and possibly nonlinear boundary conditions, and demonstrate that this problem has at least one positive solution by imposing growth conditions only on the nonlocal terms $H_1$ and $H_2$. Of special note is that no growth conditions of any kind are imposed on the nonlinearities $f$ and $g$.

Keywords::

• discrete fractional calculus
• positive solution
• nonlocal boundary condition
• nonlinear boundary condition
• coupled boundary value problem

MSC (2010) Classification::

• Primary: 39A05
• 39A12
• 39A99
• Secondary: 26A33
• 47H07

Disclosure statement

No potential conflict of interest was reported by the author.