• Previous Article
    Sparse regularized learning in the reproducing kernel banach spaces with the $ \ell^1 $ norm
  • MFC Home
  • This Issue
  • Next Article
    Averaging versus voting: A comparative study of strategies for distributed classification
August  2020, 3(3): 195-203. doi: 10.3934/mfc.2020019

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

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.

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.

[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. AhmadJ. J. NietoA. 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. JleliM. 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. KassimK. 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. RajkovićS. 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.

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.

[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. AhmadJ. J. NietoA. 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. JleliM. 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. KassimK. 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. RajkovićS. 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.

[1]

Pavel I. Etingof. Galois groups and connection matrices of q-difference equations. Electronic Research Announcements, 1995, 1: 1-9.

[2]

Xuecheng Wang. Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u}$) type. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5037-5048. doi: 10.3934/dcds.2017217

[3]

Amira Khelifa, Yacine Halim. Global behavior of P-dimensional difference equations system. Electronic Research Archive, 2021, 29 (5) : 3121-3139. doi: 10.3934/era.2021029

[4]

Tran Hong Thai, Nguyen Anh Dai, Pham Tuan Anh. Global dynamics of some system of second-order difference equations. Electronic Research Archive, 2021, 29 (6) : 4159-4175. doi: 10.3934/era.2021077

[5]

Olga Salieva. On nonexistence of solutions to some nonlinear parabolic inequalities. Communications on Pure and Applied Analysis, 2017, 16 (3) : 843-853. doi: 10.3934/cpaa.2017040

[6]

Xiaohong Li, Fengquan Li. Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities. Communications on Pure and Applied Analysis, 2012, 11 (3) : 935-943. doi: 10.3934/cpaa.2012.11.935

[7]

Jin Feng He, Wei Xu, Zhi Guo Feng, Xinsong Yang. On the global optimal solution for linear quadratic problems of switched system. Journal of Industrial and Management Optimization, 2019, 15 (2) : 817-832. doi: 10.3934/jimo.2018072

[8]

Georgia Karali, Takashi Suzuki, Yoshio Yamada. Global-in-time behavior of the solution to a Gierer-Meinhardt system. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2885-2900. doi: 10.3934/dcds.2013.33.2885

[9]

Lingbing He. On the global smooth solution to 2-D fluid/particle system. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 237-263. doi: 10.3934/dcds.2010.27.237

[10]

Dario D. Monticelli, Fabio Punzo. Nonexistence results for elliptic differential inequalities with a potential in bounded domains. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 675-695. doi: 10.3934/dcds.2018029

[11]

Feida Jiang, Xi Chen, Juhua Shi. Nonexistence of entire positive solutions for conformal Hessian quotient inequalities. Electronic Research Archive, 2021, 29 (6) : 4075-4086. doi: 10.3934/era.2021072

[12]

Anna Cima, Armengol Gasull, Francesc Mañosas. Global linearization of periodic difference equations. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1575-1595. doi: 10.3934/dcds.2012.32.1575

[13]

Jorge A. Esquivel-Avila. Nonexistence of global solutions for a class of viscoelastic wave equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4213-4230. doi: 10.3934/dcdss.2021134

[14]

De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431

[15]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064

[16]

Fei Chen, Boling Guo, Xiaoping Zhai. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinetic and Related Models, 2019, 12 (1) : 37-58. doi: 10.3934/krm.2019002

[17]

Masaki Kurokiba, Toshitaka Nagai, T. Ogawa. The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system. Communications on Pure and Applied Analysis, 2006, 5 (1) : 97-106. doi: 10.3934/cpaa.2006.5.97

[18]

Chunxiao Guo, Fan Cui, Yongqian Han. Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1687-1699. doi: 10.3934/dcdss.2016070

[19]

Xiaoqiang Dai, Wenke Li. Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem. Electronic Research Archive, 2021, 29 (6) : 4087-4098. doi: 10.3934/era.2021073

[20]

Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070

 Impact Factor: 

Metrics

  • PDF downloads (207)
  • HTML views (432)
  • Cited by (0)

Other articles
by authors

[Back to Top]