## Positive solutions of nonlocal boundary value problem for high-order nonlinear fractional $$q$$-difference equations.(English)Zbl 1470.39021

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

### MSC:

 39A13 Difference equations, scaling ($$q$$-differences) 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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### References:

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