Uniqueness and properties of positive solutions for infinitepoint fractional differential equation with pLaplacian and a parameter

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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 infinitepoint fractional differential equation with pLaplacian 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
 infinitepoint fractional differential equation
 mixed monotone operator
MSC
References

[1]
D. Baleanu, S. D. Purohit, J. C. Prajapati, Integral inequalities involving generalized ErdélyiKober 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 higherorder fractional differential equations with infinitepoint boundary conditions , Bound. Value Probl., 2016 (2016), 22 pages.

[6]
L. Hu , Existence of solutions to a coupled system of fractional differential equations with infinitepoint 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 differentialdifference 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 infinitepoint 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 mpoint 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 existenceuniqueness 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 infinitepoint boundary value conditions, Appl. Math. Lett., 39 (2015), 22–27.

[17]
Q.Y. Zhong, X.Q Zhang, Positive solution for higherorder singular infinitepoint fractional differential equation with pLaplacian, Adv. Difference Equ., 2016 (2016), 11 pages.