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August  2020, 3(3): 195-203. doi: 10.3934/mfc.2020019

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

Received  September 2019 Revised  May 2020 Published  June 2020

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

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.

Keywords: Global solution, nonexistence, system of q-difference inequalities.
Mathematics Subject Classification: Primary: 39A12; Secondary: 26A33.
Citation: Yaoyao Luo, Run Xu. The nonexistence of global solution for system of q-difference inequalities. Mathematical Foundations of Computing, 2020, 3 (3) : 195-203. doi: 10.3934/mfc.2020019
References:
[1]

R. P. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Proc. Cambridge Philos. Soc., 66 (1969), 365-370.  doi: 10.1017/S0305004100045060.  Google Scholar

[2]

P. N. Agrawal and H. S. Kasana, On simultaneous approximation by Szász-Mirakian operators, Bull. Inst. Math. Acad. Sinica, 22 (1994), 181-188.   Google Scholar

[3]

B. Ahmad and S. Ntouyas, Boundary value problems for $q$-difference inclusion, Abstr. Appl. Anal., 2011 (2011), Article ID 292860, 15 pages. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[9]

T. Ernst, A method for $q$-calculus, J. Nonlinear Math. Phys., 10 (2003), 487-525.   Google Scholar

[10]

F. H. Jackson, On $q$-functions and a certain difference operator, Trans. R. Soc. Edinburgh, 46 (1909), 253-281.  doi: 10.1017/S0080456800002751.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

F. H. Jackson, On $q$-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-200.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

V. Kac and P. Cheung, Quantum Calculus, Universitext. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. P. Agarwal, Certain fractional $q$-integrals and $q$-derivatives, Proc. Cambridge Philos. Soc., 66 (1969), 365-370.  doi: 10.1017/S0305004100045060.  Google Scholar

[2]

P. N. Agrawal and H. S. Kasana, On simultaneous approximation by Szász-Mirakian operators, Bull. Inst. Math. Acad. Sinica, 22 (1994), 181-188.   Google Scholar

[3]

B. Ahmad and S. Ntouyas, Boundary value problems for $q$-difference inclusion, Abstr. Appl. Anal., 2011 (2011), Article ID 292860, 15 pages. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[9]

T. Ernst, A method for $q$-calculus, J. Nonlinear Math. Phys., 10 (2003), 487-525.   Google Scholar

[10]

F. H. Jackson, On $q$-functions and a certain difference operator, Trans. R. Soc. Edinburgh, 46 (1909), 253-281.  doi: 10.1017/S0080456800002751.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

F. H. Jackson, On $q$-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-200.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

V. Kac and P. Cheung, Quantum Calculus, Universitext. Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0071-7.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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