The nonexistence of global solution for system of q-difference inequalities

Yaoyao Luo; Run Xu

doi:10.3934/mfc.2020019

Article Contents

2020, Volume 3, Issue 3: 195-203. Doi: 10.3934/mfc.2020019

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The nonexistence of global solution for system of q-difference inequalities

Yaoyao Luo and
Run Xu^,

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China

^* Corresponding author: xurun2005@163.com
^* Corresponding author: xurun2005@163.com

Received: September 2019

Revised: May 2020

Published: August 2020

The first author is supported by National Science Foundation of China (11671227, 11971015) and the Natural Science Foundation of Shandong Province (ZR2019MA034)

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Abstract

In this paper, we obtain sufficient conditions for the nonexistence of global solutions for the system of $ q $-difference inequalities. Our approach is based on the weak formulation of the problem, a particular choice of the test function, and some $ q $-integral inequalities.

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Mathematics Subject Classification: Primary: 39A12; Secondary: 26A33.

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References

[1]	R. P. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Proc. Cambridge Philos. Soc., 66 (1969), 365-370. doi: 10.1017/S0305004100045060.
[2]	P. N. Agrawal and H. S. Kasana, On simultaneous approximation by Szász-Mirakian operators, Bull. Inst. Math. Acad. Sinica, 22 (1994), 181-188.
[3]	B. Ahmad and S. Ntouyas, Boundary value problems for $q$-difference inclusion, Abstr. Appl. Anal., 2011 (2011), Article ID 292860, 15 pages.
[4]	B. Ahmad, A. Alsaedi and S. K. Ntouyas, A study of second-order $q$-difference equations with boundary conditions, Adv. Difference Equ., 2012 (2012), 1-10. doi: 10.1186/1687-1847-2012-35.
[5]	B. Ahmad, J. J. Nieto, A. Alsaedi and H. Al-Hutami, Existence of solutions for nonlinear fractional $q$-difference integral equations with two fractional orders and nonlocal four-point boundary conditions, J. Franklin Inst., 351 (2014), 2890-2909. doi: 10.1016/j.jfranklin.2014.01.020.
[6]	W. A. Al-Salam, Some fractional $q$-integrals and $q$-derivatives, Proc. Edinburgh Math. Soc., 2 (1966/67), 135-140. doi: 10.1017/S0013091500011469.
[7]	H. Aydi, M. Jleli and B. Samet, On the absence of global solutions for some $q$-difference inequalities, Adv. Difference Equ., 2019 (2019), 9 pages. doi: 10.1186/s13662-019-1985-8.
[8]	A. De Sole and V. G. Kac, On integral representations of $q$-gamma and $q$-beta functions, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 16 (2005), 11-29.
[9]	T. Ernst, A method for $q$-calculus, J. Nonlinear Math. Phys., 10 (2003), 487-525.
[10]	F. H. Jackson, On $q$-functions and a certain difference operator, Trans. R. Soc. Edinburgh, 46 (1909), 253-281. doi: 10.1017/S0080456800002751.
[11]	R. A. C. Ferreira, Positive solutions for a class of boundary value problems with fractional $q$-differences, Comput. Math. Appl., 61 (2011), 367-373. doi: 10.1016/j.camwa.2010.11.012.
[12]	M. N. Islam and J. T. Neugebauer, Existence of periodic solutions for a quantum Volterra equation, Adv. Dyn. Syst. Appl., 11 (2016), 67-80.
[13]	F. H. Jackson, On $q$-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-200.
[14]	L. Jia, J. Cheng and Z. Feng, A $q$-analogue of Kummer's equation, Electron. J. Differential Equations, (2017), Paper No. 31, 1-20.
[15]	M. Jleli, M. Kirane and B. Samet, On the absence of global solutions for quantum versions of Schrödinger equations and systems, Comput. Math. Appl., 77 (2019), 740-751. doi: 10.1016/j.camwa.2018.10.010.
[16]	V. Kac and P. Cheung, Quantum Calculus, Universitext. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.
[17]	M. D. Kassim, K. M. Furati and N.-E. Tatar, Non-existence for fractionally damped fractional differential problems, Acta Math. Sci. Ser. B, 37 (2017), 119-130. doi: 10.1016/S0252-9602(16)30120-5.
[18]	N. Khodabakhshi and S. M. Vaezpour, Existence and uniqueness of positive solution for a class of boundary value problems with fractional $q$-differences, J. Nonlinear Convex Anal., 16 (2015), 375-384.
[19]	M. Kirane and N.-E. Tatar, Nonexistence of solutions to a hyperbolic equation with a time fractional damping, Z. Anal. Anwend., 25 (2006), 131-142. doi: 10.4171/ZAA/1281.
[20]	M. Kirane and N.-E. Tatar, Absence of local and global solutions to an elliptic system with time-fractional dynamical boundary conditions, Sib. Math. J., 48 (2007), 477-488. doi: 10.1007/s11202-007-0050-0.
[21]	È. Mitidieri and S. I. Pohozaev, A priori estimates and nonexistence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
[22]	M. D. Qassim, K. M. Furati and N.-E. Tatar, On a differential equation involving Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal., 2012 (2012), Article ID 391062, 17 pages. doi: 10.1155/2012/391062.
[23]	P. M. Rajković, S. D. Marinković and M. S. Stanković, Fractional integrals and derivatives in $q$-calculus, Appl. Anal. Discrete Math., 1 (2007), 311-323.
[24]	W. Yang, Positive solutions for boundary value problems involving nonlinear fractional $q$-difference equations, Differ. Equ. Appl., 5 (2013), 205-219. doi: 10.7153/dea-05-13.