Existence of solutions for a system of singular sum fractional q-differential equations via quantum calculus

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 4 Some numerical results of \(\eta_{1}\) and \(\varGamma_{q}(\alpha_{1} - 1)\) from inequality (16) in Example 1 for \(q \in \{ \frac{1}{8}, \frac{1}{2}, \frac{8}{9} \}\). One can check that \(\frac{ \eta_{1}}{\varGamma_{q}(\alpha_{1} - 1)}\) by approximation is smaller than 1

n	\(\vphantom{\sum_{A_{A}}}q =\frac{1}{8}\)			\(q =\frac {1}{2}\)			\(q =\frac{8}{9}\)
n	\(\eta_{1}\)	\(\vphantom{\sum^{A^{A}}}\varGamma_{q}(\alpha_{1} - 1)\)	\(\frac{\eta_{1}}{\varGamma_{q}(\alpha _{1} - 1)}\)	\(\eta_{1}\)	\(\varGamma_{q}(\alpha_{1} - 1)\)	\(\frac{\eta_{1}}{\varGamma_{q}(\alpha_{1} - 1)}\)	\(\eta_{1}\)	\(\varGamma_{q}(\alpha_{1} - 1)\)	\(\frac{\eta_{1}}{\varGamma_{q}(\alpha_{1} - 1)}\)
1	0.6075	1.0323	0.5885	0.5909	1.0035	0.5888	0.4428	0.5904	0.7500
2	0.6079	1.0336	0.5882	0.6062	1.0455	0.5799	0.4683	0.6701	0.6988
3	0.6080	1.0338	0.5881	0.6138	1.0659	0.5759	0.4894	0.7338	0.6670
4	0.6080	1.0338	0.5881	0.6175	1.0759	0.5739	0.5073	0.7861	0.6453
5	0.6080	1.0338	0.5881	0.6194	1.0809	0.5730	0.5225	0.8299	0.6296
6	0.6080	1.0338	0.5881	0.6203	1.0834	0.5726	0.5356	0.8670	0.6178
7	0.6080	1.0338	0.5881	0.6208	1.0847	0.5723	0.5470	0.8986	0.6087
8	0.6080	1.0338	0.5881	0.6210	1.0853	0.5722	0.5569	0.9259	0.6014
9	0.6080	1.0338	0.5881	0.6211	1.0856	0.5722	0.5655	0.9494	0.5956
10	0.6080	1.0338	0.5881	0.6212	1.0858	0.5721	0.5730	0.9699	0.5908
11	0.6080	1.0338	0.5881	0.6212	1.0858	0.5721	0.5797	0.9877	0.5869
12	0.6080	1.0338	0.5881	0.6212	1.0859	0.5721	0.5855	1.0033	0.5836
⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
48	0.6080	1.0338	0.5881	0.6213	1.0859	0.5721	0.6296	1.1187	0.5628
49	0.6080	1.0338	0.5881	0.6213	1.0859	0.5721	0.6296	1.1188	0.5628
50	0.6080	1.0338	0.5881	0.6213	1.0859	0.5721	0.6297	1.1190	0.5627
51	0.6080	1.0338	0.5881	0.6213	1.0859	0.5721	0.6298	1.1191	0.5627
52	0.6080	1.0338	0.5881	0.6213	1.0859	0.5721	0.6298	1.1193	0.5627
53	0.6080	1.0338	0.5881	0.6213	1.0859	0.5721	0.6299	1.1194	0.5627
54	0.6080	1.0338	0.5881	0.6213	1.0859	0.5721	0.6299	1.1195	0.5627