Positive Solutions of Nonlocal Boundary Value Problem for High-Order Nonlinear Fractional q -Difference Equations

Abstract

We study the nonlinear $q$-difference equations of fractional order $(D_q^{\alpha }u)(t)+f(t,u(t))=\mathrm{0}$, , $(D_q^{i}u)(\mathrm{0})=\mathrm{0}$, $(D_q^{\beta }u)(\mathrm{1})=a(D_q^{\beta }u)(\eta )$, $\mathrm{0}\le i\le n-\mathrm{2}$, where $D_q^{\alpha }$ is the fractional $q$-derivative of the Riemann-Liouville type of order $\alpha$, , $\alpha >\mathrm{2}$, $\mathrm{1}\le \beta \le n-\mathrm{2}$, and $\mathrm{0}\le a\le \mathrm{1}$. We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems. Finally, we give examples to illustrate the results.