Existence and stability analysis for a class of fractional pantograph q-difference equations with nonlocal boundary conditions

Boundary Value Problems

Table 2 Numerical results of \(\Gamma _{q}(\theta +1)\) and \(\mathrm{k}^{\ast}\) of \(\mathbb{F}\mathrm{D}q-\mathbb{DP}\) (21) with \(q\in \{ \frac{3}{10}, \frac{1}{2}, \frac{9}{10} \}\) in Example 5.1

n	\({q = \frac{3}{10}}\)		\({q = \frac{1}{2}}\)		\({q = \frac{9}{10}}\)
n	\(\Gamma _{q}(\theta +1)\)	\(\mathrm{k}^{\ast}\)	\(\Gamma _{q}(\theta +1)\)	\(\mathrm{k}^{\ast}\)	\(\Gamma _{q}(\theta +1)\)	\(\mathrm{k}^{\ast}\)
1	2.2631	0.4544	4.8692	0.2054	515.9756	0.0017
2	2.2049	0.4664	4.3568	0.2299	317.9299	0.0028
3	2.1879	0.4701	4.1301	0.2427	220.5109	0.0041
4	2.1829	0.4712	4.0233	0.2492	165.3122	0.0055
5	2.1813	0.4715	3.9714	0.2525	130.9255	0.0070
6	2.1809	0.4716	3.9458	0.2541	107.9976	0.0086
7	2.1808	0.4716	3.9331	0.2550	91.9110	0.0101
⋮	⋮	⋮	⋮	⋮	⋮	⋮
8	2.1807	0.4716	3.9268	0.2554	80.1699	0.0116
9	2.1807	0.4716	3.9237	0.2556	71.3277	0.0131
10	2.1807	0.4716	3.9221	0.2557	64.4973	0.0145
11	2.1807	0.4716	3.9213	0.2558	59.1099	0.0159
12	2.1807	0.4716	3.9209	0.2558	54.7861	0.0172
⋮	⋮	⋮	⋮	⋮	⋮	⋮
45	2.1807	0.4716	3.9205	0.2558	30.6949	0.0312
46	2.1807	0.4716	3.9205	0.2558	30.6444	0.0313
47	2.1807	0.4716	3.9205	0.2558	30.5990	0.0313
48	2.1807	0.4716	3.9205	0.2558	30.5583	0.0314
49	2.1807	0.4716	3.9205	0.2558	30.5217	0.0314
50	2.1807	0.4716	3.9205	0.2558	30.4888	0.0314
51	2.1807	0.4716	3.9205	0.2558	30.4592	0.0315
52	2.1807	0.4716	3.9205	0.2558	30.4326	0.0315
53	2.1807	0.4716	3.9205	0.2558	30.4087	0.0315