Uniqueness and properties of positive solutions for infinite-point fractional differential equation with p-Laplacian and a parameter

Volume 10, Issue 10, pp 5156--5164

Publication Date: 2017-10-12

http://dx.doi.org/10.22436/jnsa.010.10.03

Authors

Li Wang - School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, P. R. China
Chengbo Zhai - School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, P. R. China

Abstract

Using new methods for dealing with an infinite-point fractional differential equation with p-Laplacian and a parameter, we study the existence of unique positive solution for any given positive parameter \(\lambda\), and then give some clear properties of positive solutions which depend on the parameter \(\lambda>0\), that is, the positive solution \(u_\lambda^{*}\) is continuous, strictly increasing in \(\lambda\) and \(\lim_{\lambda\rightarrow +\infty}\|u_\lambda^*\|=+\infty,\lim_{\lambda\rightarrow 0^+}\|u_\lambda^*\|=0.\) Our analysis relies on some new theorems for operator equations \(A(x,x)=x\) and \(A(x,x)=\lambda x\), where \(A\) is a mixed monotone operator.

Keywords

Uniqueness, positive solution, \(p\)-Laplacian, infinite-point fractional differential equation, mixed monotone operator

References

[1] D. Baleanu, S. D. Purohit, J. C. Prajapati, Integral inequalities involving generalized Erd´elyi-Kober fractional integral operators, Open Math., 14 (2016), 89–99.
[2] D. Baleanu, S. D. Purohit, F. Uçar, On Grüss type integral inequality involving the Saigo’s fractional integral operators, J. Comput. Anal. Appl., 19 (2015), 480–489.
[3] H.-L. Gao, X.-L. Han, Existence of positive solutions for fractional differential equation with nonlocal boundary condition, Int. J. Differ. Equ., 2011 (2011), 10 pages.
[4] L.-M. Guo, L.-S. Liu, Y.-H. Wu, Existence of positive solutions for singular fractional differential equations with infinitepoint boundary conditions, Nonlinear Anal. Model. Control, 5 (2016), 635–650.
[5] L.-M. Guo, L.-S. Liu, Y.-H. Wu, Existence of positive solutions for singular higher-order fractional differential equations with infinite-point boundary conditions, Bound. Value Probl., 2016 (2016), 22 pages.
[6] L. Hu, Existence of solutions to a coupled system of fractional differential equations with infinite-point boundary value conditions at resonance, Adv. Difference Equ., 2016 (2016), 13 pages.
[7] D. Kumar, S. D. Purohit, A. Secer, A. Atangana, On generalized fractional kinetic equations involving generalized Bessel function of the first kind, Math. Probl. Eng., 2015 (2015), 7 pages.
[8] D. Kumar, J. Singh, D. Baleanu, Numerical computation of a fractional model of differential-difference equation, J. Comput. Nonlinear Dyn., 11 (2016), 6 pages.
[9] B.-X. Li, S.-R. Sun, Y. Sun, Existence of solutions for fractional Langevin equation with infinite-point boundary conditions, J. Appl. Math. Comput., 53 (2017), 683–692.
[10] X. Y. Lu, X. Q. Zhang, L. Wang, Existence of positive solutions for a class of fractional differential equations with m-point boundary value conditions, (Chinese) J. Systems Sci. Math. Sci., 34 (2014), 218–230.
[11] I. Podlubny, Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, (1999).
[12] S. D. Purohit, Solution of fractional partial differential equations related to quantum mechanics, Adv. Appl. Math. Mech., 5 (2013), 639–651.
[13] S. D. Purohit, S. L. Kalla, On fractional partial differential equations related to quantum mechanics, J. Phys. A, 44 (2011), 8 pages.
[14] H. M. Srivastava, D. Kumar, J. Singh, An efficient analytical technique for fractional model of vibration equation, Appl. Math. Model., 45 (2017), 192–204.
[15] C.-B. Zhai, L.-L. Zhang, New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems, J. Math. Anal. Appl., 382 (2011), 594–614.
[16] X.-Q. Zhang, Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions, Appl. Math. Lett., 39 (2015), 22–27.
[17] Q.-Y. Zhong, X.-Q Zhang, Positive solution for higher-order singular infinite-point fractional differential equation with p-Laplacian, Adv. Difference Equ., 2016 (2016), 11 pages.

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