The existence of nonnegative solutions for a nonlinear fractional q-differential problem via a different numerical approach.

This paper deals with the existence of nonnegative solutions for a class of boundary value problems of fractional q-differential equation D q σ c [ k ] ( t ) = w ( t , k ( t ) , c D q ζ [ k ] ( t ) ) with three-point conditions for t ∈ ( 0 , 1 ) on a time scale T t 0 = { t : t = t 0 q n } ∪ { 0 } , where n ∈ N , t 0 ∈ R , and 0 < q < 1 , based on the Leray-Schauder nonlinear alternative and Guo-Krasnoselskii theorem. Moreover, we discuss the existence of nonnegative solutions. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.