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## Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria

 1 Department of Mathematics and General Sciences, Prince Sultan University-Riyadh-KSA, Saudi Arabia 2 Department of Medical Research, China Medical University, Taichung, Taiwan 3 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan 4 Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan, Iran

* Corresponding author: mesamei@gmail.com

Received  October 2019 Revised  February 2020 Published  November 2020

Fund Project: The second author is supported by Bu-Ali Sina University

Crisis intervention in natural disasters is significant to look at from many different slants. In the current study, we investigate the existence of solutions for
 $q$
-integro-differential equation
 $D_q^{\alpha} u(t) + w\left(t , u(t), u'(t), D_q^{\beta} u(t), \int_0^t f(r) u(r) \, {\mathrm d}r, \varphi(u(t)) \right) = 0,$
with three criteria and under some boundary conditions which therein we use the concept of Caputo fractional
 $q$
-derivative and fractional Riemann-Liouville type
 $q$
-integral. New existence results are obtained by applying
 $\alpha$
-admissible map. Lastly, we present two examples illustrating the primary effects.
Citation: Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020440
##### References:
 [1] T. Abdeljawad and J. Alzabut, The $q$-fractional analogue for Gronwall-type inequality, Journal of Function Spaces and Applications, (2013), Art. ID 543839, 7 pp. doi: 10.1155/2013/543839.  Google Scholar [2] T. Abdeljawad, J. Alzabut and D. Baleanu, A generalized $q$-fractional Gronwall inequality and its applications to nonlinear delay $q$-fractional difference systems, Journal of Inequalities and Applications, (2016), Paper No. 240, 13 pp. doi: 10.1186/s13660-016-1181-2.  Google Scholar [3] C. Adams, The general theory of a class of linear partial $q$-difference equations, Transactions of the American Mathematical Society, 26 (1924), 283-312.  doi: 10.2307/1989141.  Google Scholar [4] R. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Proceedings of the Cambridge Philosophical Society, 66 (1969), 365-370.  doi: 10.1017/S0305004100045060.  Google Scholar [5] R. Agarwal, D. O'Regan and S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 371 (2010), 57-68.  doi: 10.1016/j.jmaa.2010.04.034.  Google Scholar [6] S. Alizadeh, D. Baleanu and S. Rezapour, Analyzing transient response of the parallel RCL circuit by using the Caputo–Fabrizio fractional derivative, Advances in Difference Equations, 2020 (2020), Paper No. 55, 19 pp. doi: 10.1186/s13662-020-2527-0.  Google Scholar [7] R. Almeida, B. Bastos and M. Monteiro, Modeling some real phenomena by fractional differential equations, Mathematical Methods in the Applied Sciences, 39 (2016), 4846-4855.  doi: 10.1002/mma.3818.  Google Scholar [8] J. Alzabut and T. Abdeljawad, Perron's theorem for $q$-delay difference equations, Applied Mathematics and Information Sciences, 5 (2011), 74-84.   Google Scholar [9] M. Annaby and Z. Mansour, $q$-Fractional Calculus and Equations, Springer Heidelberg, 2012. doi: 10.1007/978-3-642-30898-7.  Google Scholar [10] Z. Bai and T. Qiu, Existence of positive solution for singular fractional differential equation, Applied Mathematics and Computation, 215 (2009), 2761-2767.  doi: 10.1016/j.amc.2009.09.017.  Google Scholar [11] D. Baleanu, H. Mohammadi and S. Rezapour, Analysis of the model of HIV-1 infection of $CD4^{+}$ T-cell with a new approach of fractional derivative, Advances in Difference Equations, 2020 (2020), Paper No. 71, 17 pp. doi: 10.1186/s13662-020-02544-w.  Google Scholar [12] D. Baleanu, A. Mousalou and S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations, Boundary Value Problems, 2017 (2017), Paper No. 145, 9 pp. doi: 10.1186/s13661-017-0867-9.  Google Scholar [13] M. Berezowski, Crisis phenomenon in a chemical reactor with recycle, Chemical Engineering Science, 101 (2013), 451-453.  doi: 10.1016/j.ces.2013.07.014.  Google Scholar [14] A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, Journal of Mathematical Analysis and Applications, 389 (2012), 403-411.  doi: 10.1016/j.jmaa.2011.11.065.  Google Scholar [15] R. Carmichael, The general theory of linear $q$-difference equations, American Journal of Mathematics, 34 (1912), 147-168.  doi: 10.2307/2369887.  Google Scholar [16] R. Ferreira, Nontrivials solutions for fractional $q$-difference boundary value problems, Electronic Journal of Qualitative Theory of Differential Equations, 70 (2010), 1-101.   Google Scholar [17] R. Finkelstein and E. Marcus, Transformation theory of the $q$-oscillator, Journal of Mathematical Physics, 36 (1995), 2652-2672.  doi: 10.1063/1.531057.  Google Scholar [18] A. Goswami, J. Singh, D. Kumar and Su shila, An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Physica A: Statistical Mechanics and its Applications, 524 (2019), 563-575.  doi: 10.1016/j.physa.2019.04.058.  Google Scholar [19] V. Hedayati and M. Samei, Positive solutions of fractional differential equation with two pieces in chain interval and simultaneous dirichlet boundary conditions, Boundary Value Problems, 2019 (2019), Paper No. 141, 23 pp. doi: 10.1186/s13661-019-1251-8.  Google Scholar [20] F. Jackson, $q$-difference equations, American Journal of Mathematics, 32 (1910), 305-314.  doi: 10.2307/2370183.  Google Scholar [21] V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.  Google Scholar [22] V. Kalvandi and M. E. Samei, New stability results for a sum-type fractional $q$-integro-differential equation, Journal of Advanced Mathematical Studies, 12 (2019), 201-209.   Google Scholar [23] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, B. V., Amsterdam, 2006.  Google Scholar [24] M. Krasnosel'skij, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. Google Scholar [25] S. Liang and M. E. Samei, New approach to solutions of a class of singular fractional $q$-differential problem via quantum calculus, Advances in Difference Equations, 2020 (2020), Paper No. 14, 22 pp. doi: 10.1186/s13662-019-2489-2.  Google Scholar [26] R. Li, Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition, Advances in Difference Equations, 214 (2014), 292, 12 pp. doi: 10.1186/1687-1847-2014-292.  Google Scholar [27] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999.   Google Scholar [28] S. K. Ntouyas and M. E. Samei, Existence and uniqueness of solutions for multi-term fractional $q$-integro-differential equations via quantum calculus, Advances in Difference Equations, 2019 (2019), Paper No. 475, 20 pp. doi: 10.1186/s13662-019-2414-8.  Google Scholar [29] P. Rajković, S. Marinković and M. Stanković, Fractional integrals and derivatives in $q$-calculus, Applicable Analysis and Discrete Mathematics, 1 (2007), 311-323.   Google Scholar [30] M. E. Samei, Existence of solutions for a system of singular sum fractional $q$-differential equations via quantum calculus, Advances in Difference Equations, 2020 (2020), Paper No. 23, 23 pp. doi: 10.1186/s13662-019-2480-y.  Google Scholar [31] M. Samei and G. Khalilzadeh Ranjbar, Some theorems of existence of solutions for fractional hybrid $q$-difference inclusion, Journal of Advanced Mathematical Studies, 12 (2019), 63-76.   Google Scholar [32] M. E. Samei, G. Khalilzadeh Ranjbar and V. Hedayati, Existence of solutions for a class of caputo fractional $q$-difference inclusion on multifunctions by computational results, Kragujevac Journal of Mathematics, 45 (2021), 543-570.   Google Scholar [33] M. Samei, V. Hedayati and S. Rezapour, Existence results for a fraction hybrid differential inclusion with Caputo–Hadamard type fractional derivative, Advances in Difference Equations, 2019 (2019), Paper No. 163, 15 pp. doi: 10.1186/s13662-019-2090-8.  Google Scholar [34] M. E. Samei, V. Hedayati and G. K. Ranjbar, The existence of solution for $k$-dimensional system of Langevin Hadamard-type fractional differential inclusions with $2k$ different fractional orders, Mediterranean Journal of Mathematics, 17 (2020), Paper No. 37, 23 pp. doi: 10.1007/s00009-019-1471-2.  Google Scholar [35] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for $\alpha$-$\psi$-contractive type mappings, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 2154-2165.  doi: 10.1016/j.na.2011.10.014.  Google Scholar [36] S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar [37] M. Shabibi, M. Postolache and S. Rezapour, Investigation of a multi-singular point-wise defined fractional integro-differential equation, Journal of Mathematical Analysis, 7 (2016), 61-77.   Google Scholar [38] M. Shabibi, S. Rezapour and S. Vaezpour, A singular fractional integro-differential equation, University Politehnica of Bucharest Scientific Bulletin, Series A, 79 (2017), 109-118.   Google Scholar [39] S. Staněk, The existence of positive solutions of singular fractional boundary value problems, Computers & Mathematics with Applications, 62 (2011), 1379-1388.  doi: 10.1016/j.camwa.2011.04.048.  Google Scholar [40] M. S. Stanković, P. M. Rajković and S. D. Marinković, On $q$-fractional derivatives of Riemann–Liouville and caputo type, C. R. Acad. Bulgare Sci., 63 (2010), 197–-204.  Google Scholar [41] N. Tatar, An impulsive nonlinear singular version of the Gronwall-Bihari inequality, Journal of Inequalities and Applications, 2006 (2006), Art. ID 84561, 12 pp. doi: 10.1155/JIA/2006/84561.  Google Scholar [42] A. Zada, J. Alzabut, H. Waheed and I. L. Popa, Ulam–Hyers stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions, Advances in Difference Equations, 2020 (2020), Paper No. 64, 50 pp. doi: 10.1186/s13662-020-2534-1.  Google Scholar [43] H. Zhou, J. Alzabut and L. Yang, On fractional Langevin differential equations with anti-periodic boundary conditions, The European Physical Journal Special Topics, 226 (2017), 3577-3590.  doi: 10.1140/epjst/e2018-00082-0.  Google Scholar

show all references

##### References:
 [1] T. Abdeljawad and J. Alzabut, The $q$-fractional analogue for Gronwall-type inequality, Journal of Function Spaces and Applications, (2013), Art. ID 543839, 7 pp. doi: 10.1155/2013/543839.  Google Scholar [2] T. Abdeljawad, J. Alzabut and D. Baleanu, A generalized $q$-fractional Gronwall inequality and its applications to nonlinear delay $q$-fractional difference systems, Journal of Inequalities and Applications, (2016), Paper No. 240, 13 pp. doi: 10.1186/s13660-016-1181-2.  Google Scholar [3] C. Adams, The general theory of a class of linear partial $q$-difference equations, Transactions of the American Mathematical Society, 26 (1924), 283-312.  doi: 10.2307/1989141.  Google Scholar [4] R. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Proceedings of the Cambridge Philosophical Society, 66 (1969), 365-370.  doi: 10.1017/S0305004100045060.  Google Scholar [5] R. Agarwal, D. O'Regan and S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, Journal of Mathematical Analysis and Applications, 371 (2010), 57-68.  doi: 10.1016/j.jmaa.2010.04.034.  Google Scholar [6] S. Alizadeh, D. Baleanu and S. Rezapour, Analyzing transient response of the parallel RCL circuit by using the Caputo–Fabrizio fractional derivative, Advances in Difference Equations, 2020 (2020), Paper No. 55, 19 pp. doi: 10.1186/s13662-020-2527-0.  Google Scholar [7] R. Almeida, B. Bastos and M. Monteiro, Modeling some real phenomena by fractional differential equations, Mathematical Methods in the Applied Sciences, 39 (2016), 4846-4855.  doi: 10.1002/mma.3818.  Google Scholar [8] J. Alzabut and T. Abdeljawad, Perron's theorem for $q$-delay difference equations, Applied Mathematics and Information Sciences, 5 (2011), 74-84.   Google Scholar [9] M. Annaby and Z. Mansour, $q$-Fractional Calculus and Equations, Springer Heidelberg, 2012. doi: 10.1007/978-3-642-30898-7.  Google Scholar [10] Z. Bai and T. Qiu, Existence of positive solution for singular fractional differential equation, Applied Mathematics and Computation, 215 (2009), 2761-2767.  doi: 10.1016/j.amc.2009.09.017.  Google Scholar [11] D. Baleanu, H. Mohammadi and S. Rezapour, Analysis of the model of HIV-1 infection of $CD4^{+}$ T-cell with a new approach of fractional derivative, Advances in Difference Equations, 2020 (2020), Paper No. 71, 17 pp. doi: 10.1186/s13662-020-02544-w.  Google Scholar [12] D. Baleanu, A. Mousalou and S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations, Boundary Value Problems, 2017 (2017), Paper No. 145, 9 pp. doi: 10.1186/s13661-017-0867-9.  Google Scholar [13] M. Berezowski, Crisis phenomenon in a chemical reactor with recycle, Chemical Engineering Science, 101 (2013), 451-453.  doi: 10.1016/j.ces.2013.07.014.  Google Scholar [14] A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, Journal of Mathematical Analysis and Applications, 389 (2012), 403-411.  doi: 10.1016/j.jmaa.2011.11.065.  Google Scholar [15] R. Carmichael, The general theory of linear $q$-difference equations, American Journal of Mathematics, 34 (1912), 147-168.  doi: 10.2307/2369887.  Google Scholar [16] R. Ferreira, Nontrivials solutions for fractional $q$-difference boundary value problems, Electronic Journal of Qualitative Theory of Differential Equations, 70 (2010), 1-101.   Google Scholar [17] R. Finkelstein and E. Marcus, Transformation theory of the $q$-oscillator, Journal of Mathematical Physics, 36 (1995), 2652-2672.  doi: 10.1063/1.531057.  Google Scholar [18] A. Goswami, J. Singh, D. Kumar and Su shila, An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma, Physica A: Statistical Mechanics and its Applications, 524 (2019), 563-575.  doi: 10.1016/j.physa.2019.04.058.  Google Scholar [19] V. Hedayati and M. Samei, Positive solutions of fractional differential equation with two pieces in chain interval and simultaneous dirichlet boundary conditions, Boundary Value Problems, 2019 (2019), Paper No. 141, 23 pp. doi: 10.1186/s13661-019-1251-8.  Google Scholar [20] F. Jackson, $q$-difference equations, American Journal of Mathematics, 32 (1910), 305-314.  doi: 10.2307/2370183.  Google Scholar [21] V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.  Google Scholar [22] V. Kalvandi and M. E. Samei, New stability results for a sum-type fractional $q$-integro-differential equation, Journal of Advanced Mathematical Studies, 12 (2019), 201-209.   Google Scholar [23] A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, B. V., Amsterdam, 2006.  Google Scholar [24] M. Krasnosel'skij, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. Google Scholar [25] S. Liang and M. E. Samei, New approach to solutions of a class of singular fractional $q$-differential problem via quantum calculus, Advances in Difference Equations, 2020 (2020), Paper No. 14, 22 pp. doi: 10.1186/s13662-019-2489-2.  Google Scholar [26] R. Li, Existence of solutions for nonlinear singular fractional differential equations with fractional derivative condition, Advances in Difference Equations, 214 (2014), 292, 12 pp. doi: 10.1186/1687-1847-2014-292.  Google Scholar [27] I. Podlubny, Fractional Differential Equations, Academic Press, Inc., San Diego, CA, 1999.   Google Scholar [28] S. K. Ntouyas and M. E. Samei, Existence and uniqueness of solutions for multi-term fractional $q$-integro-differential equations via quantum calculus, Advances in Difference Equations, 2019 (2019), Paper No. 475, 20 pp. doi: 10.1186/s13662-019-2414-8.  Google Scholar [29] P. Rajković, S. Marinković and M. Stanković, Fractional integrals and derivatives in $q$-calculus, Applicable Analysis and Discrete Mathematics, 1 (2007), 311-323.   Google Scholar [30] M. E. Samei, Existence of solutions for a system of singular sum fractional $q$-differential equations via quantum calculus, Advances in Difference Equations, 2020 (2020), Paper No. 23, 23 pp. doi: 10.1186/s13662-019-2480-y.  Google Scholar [31] M. Samei and G. Khalilzadeh Ranjbar, Some theorems of existence of solutions for fractional hybrid $q$-difference inclusion, Journal of Advanced Mathematical Studies, 12 (2019), 63-76.   Google Scholar [32] M. E. Samei, G. Khalilzadeh Ranjbar and V. Hedayati, Existence of solutions for a class of caputo fractional $q$-difference inclusion on multifunctions by computational results, Kragujevac Journal of Mathematics, 45 (2021), 543-570.   Google Scholar [33] M. Samei, V. Hedayati and S. Rezapour, Existence results for a fraction hybrid differential inclusion with Caputo–Hadamard type fractional derivative, Advances in Difference Equations, 2019 (2019), Paper No. 163, 15 pp. doi: 10.1186/s13662-019-2090-8.  Google Scholar [34] M. E. Samei, V. Hedayati and G. K. Ranjbar, The existence of solution for $k$-dimensional system of Langevin Hadamard-type fractional differential inclusions with $2k$ different fractional orders, Mediterranean Journal of Mathematics, 17 (2020), Paper No. 37, 23 pp. doi: 10.1007/s00009-019-1471-2.  Google Scholar [35] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for $\alpha$-$\psi$-contractive type mappings, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 2154-2165.  doi: 10.1016/j.na.2011.10.014.  Google Scholar [36] S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar [37] M. Shabibi, M. Postolache and S. Rezapour, Investigation of a multi-singular point-wise defined fractional integro-differential equation, Journal of Mathematical Analysis, 7 (2016), 61-77.   Google Scholar [38] M. Shabibi, S. Rezapour and S. Vaezpour, A singular fractional integro-differential equation, University Politehnica of Bucharest Scientific Bulletin, Series A, 79 (2017), 109-118.   Google Scholar [39] S. Staněk, The existence of positive solutions of singular fractional boundary value problems, Computers & Mathematics with Applications, 62 (2011), 1379-1388.  doi: 10.1016/j.camwa.2011.04.048.  Google Scholar [40] M. S. Stanković, P. M. Rajković and S. D. Marinković, On $q$-fractional derivatives of Riemann–Liouville and caputo type, C. R. Acad. Bulgare Sci., 63 (2010), 197–-204.  Google Scholar [41] N. Tatar, An impulsive nonlinear singular version of the Gronwall-Bihari inequality, Journal of Inequalities and Applications, 2006 (2006), Art. ID 84561, 12 pp. doi: 10.1155/JIA/2006/84561.  Google Scholar [42] A. Zada, J. Alzabut, H. Waheed and I. L. Popa, Ulam–Hyers stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions, Advances in Difference Equations, 2020 (2020), Paper No. 64, 50 pp. doi: 10.1186/s13662-020-2534-1.  Google Scholar [43] H. Zhou, J. Alzabut and L. Yang, On fractional Langevin differential equations with anti-periodic boundary conditions, The European Physical Journal Special Topics, 226 (2017), 3577-3590.  doi: 10.1140/epjst/e2018-00082-0.  Google Scholar
Numerical results of $A_1(q)$ where $q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9}$ in Example 1
Numerical results of $A_2(q)$ where $q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9}$ in Example 2
Numerical results of $\Gamma_q(\alpha -1)$ where $q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9}$ in Example 2
Some numerical results for calculation of $\Gamma_q(x)$ with $q = \frac{1}{8}$ which is constant, for $x = 9.5, 65,110,780$ in Algorithm 2
 $n$ $x=9.5$ $x=65$ $x=110$ $x=780$ $1$ $2.679786$ $4432.545834$ $1804225.634753$ $1.29090809480473E+45$ $2$ $2.674552$ $4423.888518$ $1800701.756560$ $1.28838678993206E+45$ $3$ $2.673899$ $4422.808467$ $1800262.132108$ $1.28807224237593E+45$ $4$ $2.673818$ $4422.673494$ $1800207.192468$ $1.28803293353064E+45$ $5$ $2.673808$ $4422.656623$ $1800200.325222$ $1.28802802007493E+45$ $6$ 2.673806 $4422.654514$ $1800199.466820$ $1.28802740589531E+45$ $7$ $2.673806$ $4422.654250$ $1800199.359519$ $1.28802732912289E+45$ $8$ $2.673806$ $4422.654217$ $1800199.346107$ $1.28802731952634E+45$ $9$ $2.673806$ $4422.654213$ $1800199.344430$ $1.28802731832677E+45$ $10$ $2.673806$ $4422.654213$ $1800199.344221$ $1.28802731817683E+45$ $11$ $2.673806$ 4422.654212 $1800199.344195$ $1.28802731815808E+45$ $12$ $2.673806$ $4422.654212$ 1800199.344191 $1.28802731815574E+45$ $13$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815545E+45$ $14$ $2.673806$ $4422.654212$ $1800199.344191$ 1.28802731815541E+45 $15$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $16$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $17$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $18$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $19$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$
 $n$ $x=9.5$ $x=65$ $x=110$ $x=780$ $1$ $2.679786$ $4432.545834$ $1804225.634753$ $1.29090809480473E+45$ $2$ $2.674552$ $4423.888518$ $1800701.756560$ $1.28838678993206E+45$ $3$ $2.673899$ $4422.808467$ $1800262.132108$ $1.28807224237593E+45$ $4$ $2.673818$ $4422.673494$ $1800207.192468$ $1.28803293353064E+45$ $5$ $2.673808$ $4422.656623$ $1800200.325222$ $1.28802802007493E+45$ $6$ 2.673806 $4422.654514$ $1800199.466820$ $1.28802740589531E+45$ $7$ $2.673806$ $4422.654250$ $1800199.359519$ $1.28802732912289E+45$ $8$ $2.673806$ $4422.654217$ $1800199.346107$ $1.28802731952634E+45$ $9$ $2.673806$ $4422.654213$ $1800199.344430$ $1.28802731832677E+45$ $10$ $2.673806$ $4422.654213$ $1800199.344221$ $1.28802731817683E+45$ $11$ $2.673806$ 4422.654212 $1800199.344195$ $1.28802731815808E+45$ $12$ $2.673806$ $4422.654212$ 1800199.344191 $1.28802731815574E+45$ $13$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815545E+45$ $14$ $2.673806$ $4422.654212$ $1800199.344191$ 1.28802731815541E+45 $15$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $16$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $17$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $18$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$ $19$ $2.673806$ $4422.654212$ $1800199.344191$ $1.28802731815541E+45$
Some numerical results for calculation of $\Gamma_q(x)$ with $q = \frac{1}{8}, \frac{1}{2}, \frac{4}{5}, \frac{8}{9}$ for $x = 9.5$ of Algorithm 2
 $n$ $q=\frac{1}{8}$ $q=\frac{1}{2}$ $q=\frac{4}{5}$ $q=\frac{8}{9}$ $1$ $2.679786$ $136.046206$ $79062.138227$ $6301918.338883$ $2$ $2.674552$ $119.081545$ $41793.335091$ $2528395.395827$ $3$ $2.673899$ $111.658224$ $26290.733638$ $1232715.590371$ $4$ $2.673818$ $108.178242$ $18589.881264$ $689176.848061$ $5$ $2.673808$ $106.492553$ $14278.326587$ $426538.394173$ $6$ 2.673806 $105.662861$ $11650.586796$ $285518.687713$ $7$ $2.673806$ $105.251251$ $9946.3508930$ $203363.796571$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $26$ $2.673806$ $104.841780$ $5522.283831$ $25842.863721$ $27$ $2.673806$ $104.841780$ $5513.202433$ $25230.371788$ $28$ $2.673806$ 104.841779 $5505.949683$ $24699.649904$ $29$ $2.673806$ $104.841779$ $5500.155385$ $24238.446645$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $106$ $2.673806$ $104.841779$ $5477.048235$ $20879.606269$ $107$ $2.673806$ $104.841779$ 5477.048234 $20879.566792$ $108$ $2.673806$ $104.841779$ $5477.048234$ $20879.531702$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $118$ $2.673806$ $104.841779$ $5477.048234$ $20879.337427$ $119$ $2.673806$ $104.841779$ $5477.048234$ $20879.327822$ $120$ $2.673806$ $104.841779$ $5477.048234$ 20879.319284
 $n$ $q=\frac{1}{8}$ $q=\frac{1}{2}$ $q=\frac{4}{5}$ $q=\frac{8}{9}$ $1$ $2.679786$ $136.046206$ $79062.138227$ $6301918.338883$ $2$ $2.674552$ $119.081545$ $41793.335091$ $2528395.395827$ $3$ $2.673899$ $111.658224$ $26290.733638$ $1232715.590371$ $4$ $2.673818$ $108.178242$ $18589.881264$ $689176.848061$ $5$ $2.673808$ $106.492553$ $14278.326587$ $426538.394173$ $6$ 2.673806 $105.662861$ $11650.586796$ $285518.687713$ $7$ $2.673806$ $105.251251$ $9946.3508930$ $203363.796571$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $26$ $2.673806$ $104.841780$ $5522.283831$ $25842.863721$ $27$ $2.673806$ $104.841780$ $5513.202433$ $25230.371788$ $28$ $2.673806$ 104.841779 $5505.949683$ $24699.649904$ $29$ $2.673806$ $104.841779$ $5500.155385$ $24238.446645$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $106$ $2.673806$ $104.841779$ $5477.048235$ $20879.606269$ $107$ $2.673806$ $104.841779$ 5477.048234 $20879.566792$ $108$ $2.673806$ $104.841779$ $5477.048234$ $20879.531702$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $118$ $2.673806$ $104.841779$ $5477.048234$ $20879.337427$ $119$ $2.673806$ $104.841779$ $5477.048234$ $20879.327822$ $120$ $2.673806$ $104.841779$ $5477.048234$ 20879.319284
Some numerical results for calculation of $\Gamma_q(x)$ with $q = \frac{1}{8}, \frac{1}{2}, \frac{4}{5}, \frac{8}{9}$ for $x = 110$ of Algorithm 2
 $n$ $q=\frac{1}{8}$ $q=\frac{1}{2}$ $q=\frac{4}{5}$ $q=\frac{8}{9}$ $1$ $1804225.634753$ $2.43388915243820E+32$ $1.10933564801075E+75$ $2.3996994906237E+102$ $2$ $1800701.756560$ $2.12965300838343E+32$ $5.41355796236824E+74$ $7.1431517307455E+101$ $3$ $1800262.132108$ $1.99654969535946E+32$ $3.19616462101800E+74$ $2.6837217226512E+101$ $4$ $1800207.192468$ $1.93415751737948E+32$ $2.14884539802207E+74$ $1.1944485864825E+101$ $5$ $1800200.325222$ $1.90393630617042E+32$ $1.58553847001434E+74$ $6.0526350536381E+100$ $6$ $1800199.466820$ $1.88906180377847E+32$ $1.25302695267477E+74$ $3.3987862057282E+100$ $7$ $1800199.359519$ $1.88168265610746E+32$ $1.04280391429109E+74$ $2.0741306563269E+100$ $8$ $1800199.346107$ $1.87800749466975E+32$ $9.02841142168746E+73$ $1.3555712905453E+100$ $9$ $1800199.344430$ $1.87617350297573E+32$ $8.05899312693661E+73$ $9.38129101307050E+99$ $10$ $1800199.344221$ $1.87525740263248E+32$ $7.36673088857628E+73$ $6.81335603265770E+99$ $11$ $1800199.344195$ $1.87479957611817E+32$ $6.86049299667128E+73$ $5.15556440821410E+99$ $12$ 1800199.344191 $1.87457071874804E+32$ $6.48333340557523E+73$ $4.04051908444650E+99$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $48$ $1800199.344191$ 1.87434189862553E+32 $5.18960499065178E+73$ $6.66324790738213E+98$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $90$ $1800199.344191$ $1.87434189862553E+32$ 5.18923469131315E+73 $6.50025876524830E+98$ $91$ $1800199.344191$ $1.87434189862553E+32$ $5.18923468501255E+73$ $6.50013085733126E+98$ $92$ $1800199.344191$ $1.87434189862553E+32$ $5.18923467997207E+73$ $6.50001716364224E+98$ $93$ $1800199.344191$ $1.87434189862553E+32$ $5.18923467593968E+73$ $6.49991610435300E+98$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $118$ $1800199.344191$ $1.87434189862553E+32$ $5.18923465987107E+73$ $6.49915022957670E+98$ $119$ $1800199.344191$ $1.87434189862553E+32$ $5.18923465985889E+73$ $6.49914550293450E+98$ $120$ $1800199.344191$ $1.87434189862553E+32$ $5.18923465984914E+73$ 6.49914130147782E+98
 $n$ $q=\frac{1}{8}$ $q=\frac{1}{2}$ $q=\frac{4}{5}$ $q=\frac{8}{9}$ $1$ $1804225.634753$ $2.43388915243820E+32$ $1.10933564801075E+75$ $2.3996994906237E+102$ $2$ $1800701.756560$ $2.12965300838343E+32$ $5.41355796236824E+74$ $7.1431517307455E+101$ $3$ $1800262.132108$ $1.99654969535946E+32$ $3.19616462101800E+74$ $2.6837217226512E+101$ $4$ $1800207.192468$ $1.93415751737948E+32$ $2.14884539802207E+74$ $1.1944485864825E+101$ $5$ $1800200.325222$ $1.90393630617042E+32$ $1.58553847001434E+74$ $6.0526350536381E+100$ $6$ $1800199.466820$ $1.88906180377847E+32$ $1.25302695267477E+74$ $3.3987862057282E+100$ $7$ $1800199.359519$ $1.88168265610746E+32$ $1.04280391429109E+74$ $2.0741306563269E+100$ $8$ $1800199.346107$ $1.87800749466975E+32$ $9.02841142168746E+73$ $1.3555712905453E+100$ $9$ $1800199.344430$ $1.87617350297573E+32$ $8.05899312693661E+73$ $9.38129101307050E+99$ $10$ $1800199.344221$ $1.87525740263248E+32$ $7.36673088857628E+73$ $6.81335603265770E+99$ $11$ $1800199.344195$ $1.87479957611817E+32$ $6.86049299667128E+73$ $5.15556440821410E+99$ $12$ 1800199.344191 $1.87457071874804E+32$ $6.48333340557523E+73$ $4.04051908444650E+99$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $48$ $1800199.344191$ 1.87434189862553E+32 $5.18960499065178E+73$ $6.66324790738213E+98$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $90$ $1800199.344191$ $1.87434189862553E+32$ 5.18923469131315E+73 $6.50025876524830E+98$ $91$ $1800199.344191$ $1.87434189862553E+32$ $5.18923468501255E+73$ $6.50013085733126E+98$ $92$ $1800199.344191$ $1.87434189862553E+32$ $5.18923467997207E+73$ $6.50001716364224E+98$ $93$ $1800199.344191$ $1.87434189862553E+32$ $5.18923467593968E+73$ $6.49991610435300E+98$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $118$ $1800199.344191$ $1.87434189862553E+32$ $5.18923465987107E+73$ $6.49915022957670E+98$ $119$ $1800199.344191$ $1.87434189862553E+32$ $5.18923465985889E+73$ $6.49914550293450E+98$ $120$ $1800199.344191$ $1.87434189862553E+32$ $5.18923465984914E+73$ 6.49914130147782E+98
Some numerical results for calculation of $\ell$, $\ell'$ and $A_1(q)$ in Example 1 for $q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9}$
 $n$ $q = \frac{1}{7}$ $q = \frac{1}{2}$ $q = \frac{8}{9}$ $\ell, \ell'$ $A_1(q)$ $\ell, \ell'$ $A_1(q)$ $\ell, \ell'$ $A_1(q)$ $1$ $1.0124$ $0$ $0.9946$ $0$ $0.8133$ $0$ $2$ $1.0132$ $0.1243$ $1.0132$ $0.2154$ $0.8462$ $0.0164$ $3$ $1.0133$ $0.1406$ $1.0224$ $0.3091$ $0.873$ $0.0376$ $4$ $1.0133$ $0.1429$ $1.0269$ $0.3539$ $0.8954$ $0.0621$ $5$ 1.0133 0.1433 $1.0291$ $0.3759$ $0.9143$ $0.0888$ $6$ $1.0133$ $0.1433$ $1.0302$ $0.3868$ $0.9305$ $0.1165$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $13$ $1.0133$ $0.1433$ $1.0313$ $0.3975$ $0.9972$ $0.2933$ $14$ $1.0133$ $0.1433$ 1.0314 0.3976 $1.0026$ $0.313$ $15$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0073$ $0.3312$ $16$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0115$ $0.3478$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $80$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0442$ $0.4985$ $81$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0442$ $0.4985$ $82$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ 1.0442 0.4986 $83$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0442$ $0.4986$
 $n$ $q = \frac{1}{7}$ $q = \frac{1}{2}$ $q = \frac{8}{9}$ $\ell, \ell'$ $A_1(q)$ $\ell, \ell'$ $A_1(q)$ $\ell, \ell'$ $A_1(q)$ $1$ $1.0124$ $0$ $0.9946$ $0$ $0.8133$ $0$ $2$ $1.0132$ $0.1243$ $1.0132$ $0.2154$ $0.8462$ $0.0164$ $3$ $1.0133$ $0.1406$ $1.0224$ $0.3091$ $0.873$ $0.0376$ $4$ $1.0133$ $0.1429$ $1.0269$ $0.3539$ $0.8954$ $0.0621$ $5$ 1.0133 0.1433 $1.0291$ $0.3759$ $0.9143$ $0.0888$ $6$ $1.0133$ $0.1433$ $1.0302$ $0.3868$ $0.9305$ $0.1165$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $13$ $1.0133$ $0.1433$ $1.0313$ $0.3975$ $0.9972$ $0.2933$ $14$ $1.0133$ $0.1433$ 1.0314 0.3976 $1.0026$ $0.313$ $15$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0073$ $0.3312$ $16$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0115$ $0.3478$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $80$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0442$ $0.4985$ $81$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0442$ $0.4985$ $82$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ 1.0442 0.4986 $83$ $1.0133$ $0.1433$ $1.0314$ $0.3976$ $1.0442$ $0.4986$
Some numerical results for calculation of $A_3(q)$ in Example 2 for $q = \frac{1}{7}, \frac{1}{2}, \frac{8}{9}$. One can verify that $A_3(q) < \Gamma_q(\alpha-1)$ for all $n$ when $q$ changes
 $n$ $q = \frac{1}{7}$ $q = \frac{1}{2}$ $q = \frac{8}{9}$ $A_3(q)$ $\Gamma_q(\alpha-1)$ $A_3(q)$ $\Gamma_q(\alpha-1)$ $A_3(q)$ $\Gamma_q(\alpha-1)$ $1$ $1.3246$ $1.6802$ $1.3085$ $8.774$ $1.393$ $1799.5494$ $2$ $1.2731$ $1.6753$ $1.1294$ $7.7199$ $1.0942$ $913.1535$ $3$ 1.2666 1.6746 $1.0755$ $7.2574$ $0.9855$ $542.374$ $4$ $1.2656$ $1.6745$ $1.0539$ $7.0403$ $0.9363$ $358.4859$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $19$ $1.2655$ $1.6745$ $1.0350$ $6.8321$ $0.8635$ $46.4785$ $20$ $1.2655$ $1.6745$ 1.0350 6.832 $0.8632$ $44.7825$ $21$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8629$ $43.3383$ $22$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8627$ $42.1023$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $113$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8612$ $33.6244$ $114$ $1.2655$ $1.6745$ $1.035$ $6.832$ 0.8612 33.6243 $115$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8612$ $33.6243$ $116$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8612$ $33.6243$
 $n$ $q = \frac{1}{7}$ $q = \frac{1}{2}$ $q = \frac{8}{9}$ $A_3(q)$ $\Gamma_q(\alpha-1)$ $A_3(q)$ $\Gamma_q(\alpha-1)$ $A_3(q)$ $\Gamma_q(\alpha-1)$ $1$ $1.3246$ $1.6802$ $1.3085$ $8.774$ $1.393$ $1799.5494$ $2$ $1.2731$ $1.6753$ $1.1294$ $7.7199$ $1.0942$ $913.1535$ $3$ 1.2666 1.6746 $1.0755$ $7.2574$ $0.9855$ $542.374$ $4$ $1.2656$ $1.6745$ $1.0539$ $7.0403$ $0.9363$ $358.4859$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $19$ $1.2655$ $1.6745$ $1.0350$ $6.8321$ $0.8635$ $46.4785$ $20$ $1.2655$ $1.6745$ 1.0350 6.832 $0.8632$ $44.7825$ $21$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8629$ $43.3383$ $22$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8627$ $42.1023$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $113$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8612$ $33.6244$ $114$ $1.2655$ $1.6745$ $1.035$ $6.832$ 0.8612 33.6243 $115$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8612$ $33.6243$ $116$ $1.2655$ $1.6745$ $1.035$ $6.832$ $0.8612$ $33.6243$
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