ON POSITIVE SOLUTIONS OF EIGENVALUE PROBLEMS FOR A CLASS OF <i>P</i>-LAPLACIAN FRACTIONAL DIFFERENTIAL EQUATIONS

Xiaofeng Su; Mei Jia; Xianlong Fu

doi:10.11948/2018.152

2018 Volume 8 Issue 1

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Xiaofeng Su, Mei Jia, Xianlong Fu. ON POSITIVE SOLUTIONS OF EIGENVALUE PROBLEMS FOR A CLASS OF P-LAPLACIAN FRACTIONAL DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 152-171. doi: 10.11948/2018.152

Citation:

Xiaofeng Su, Mei Jia, Xianlong Fu. ON POSITIVE SOLUTIONS OF EIGENVALUE PROBLEMS FOR A CLASS OF P-LAPLACIAN FRACTIONAL DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 152-171. doi: 10.11948/2018.152

ON POSITIVE SOLUTIONS OF EIGENVALUE PROBLEMS FOR A CLASS OF P-LAPLACIAN FRACTIONAL DIFFERENTIAL EQUATIONS

1 Department of Mathematics, East China Normal University, Shanghai 200241, China;
2 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Fund Project:

Abstract

Abstract

In this paper, we are concerned with the eigenvalue problem of a class of p-Laplacian fractional differential equations involving integral boundary conditions. New criteria are established for the existence of positive solutions of the problem under some superlinear and suberlinear conditions. The results of the existence of at least one, two and the nonexistence of positive solutions are also obtained by using the fixed point theory. Finally, several examples are provided to illustrate the obtained results.
- Fractional differential equation /
- p-Laplacian operator /
- integral boundary condition /
- fixed point /
- eigenvalue
MSC: 34B15;26A33;47H10;47H11

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References

[1]	Z. Bai, Eigenvalue intervals for a class of fractional boundary value problem, Comp. Math. Appl., 2012, 64, 3253-3257. Google Scholar
[2]	G. Chai, Positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator, Bound. Val. Probl., 2012(2012), 1-20. Google Scholar
[3]	D. Guo and V. Lakshmikanthan, Nonlinear problems in abstract cones, Academic Press, New York, 1988. Google Scholar
[4]	Z. Hao and L. Debnath, On eigenvalue intervals and eigenfunctions of fourthorder singular boundary value problems, Appl. Math. Lett., 2005, 18, 543-553. Google Scholar
[5]	Z. Han, H. Lu, S. Sun and D. Yan, Positive solutions to boundary value problems of p-Laplacian fractional differential equations with a parameter in the boundary conditions, Electr. J. Diff. Equ., 2012(2012), 1-14. Google Scholar
[6]	Z. Han, H. Lu, and C. Zhang, Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian operator, Appl. Math. Comp., 2015, 257, 526-536. Google Scholar
[7]	M. Jia and X. Liu, Three nonnegative solutions for fractional differential equations with integral boundary conditions, Comp. Math. Appl., 2011, 62, 1405-1412. Google Scholar
[8]	M. Jia and X. Liu, The existence of positive solutions for fractional differential equations with integral and disturbance parameter in boundary conditions, Abstr. Appl. Anal., 2014(2014), Article ID 131548, 1-14. Google Scholar
[9]	M. Jia and X. Liu, Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions, Appl. Math. Comp., 2014, 232, 313-323. Google Scholar
[10]	J. Jin, X. Liu, and M. Jia, Existence of positive solutions for singular fractional differential equations with integral boundary conditions, Electr. J. Diff. Equ., 2012(2012), 1457-1470. Google Scholar
[11]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science, Amsterdam, 2006. Google Scholar
[12]	N. Kosmatov, A boundary value problem of fractional order at resonance, Electr. J. Diff. Equ., 2010, 135, 1-10. Google Scholar
[13]	X. Liu, J. Jin and M. Jia, Multipie positive solutions of integral boundary value problems for fractional differential equations, J. Appl. Math. Inf., 2010, 30, 305-320. Google Scholar
[14]	K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional, Differential Equations, Wiley, 1993. Google Scholar
[15]	I. Podlubny, Fraction differential equations, Acad press, New york, 1999. Google Scholar
[16]	J. Wang, H. Xiang and Z. Liu, Positive solutions for three-point boundary value problems of nonlinear fractional differential equations with p-Laplacian operator, Far East J. Appl. Math., 2009, 37, 33-47. Google Scholar
[17]	J. Wang, H. Xiang and Z. Liu, Upper and lower solutions method for a class of singular fractional boundary value problems with p-Laplacian operator, Abstr. Appl. Anal., 2010, 12, 2010 Article ID 971824. Google Scholar
[18]	J. Wang, Eigenvalue intervals for multi-point boundary value problems of fractional differential equation, Appl. Math. Comp., 2013, 219, 4570-4575. Google Scholar
[19]	H. Wang, On the number of positive solutions of nonlinear systems, J. Math. Anal., 2003, 281, 287-306. Google Scholar
[20]	C. Yang and J. Yan, Positive solutions for third-order Sturm-Liouville boundary value problems with p-Laplacian, Comput. Math. Appl., 2010, 59, 2059-2066. Google Scholar
[21]	S. Zhang and X. Su, The existence of a solution for a fractional equation with nonlinear boundary conditions considdered using upper and lower solutions in reverse order, Comp. Math. Appl., 2011, 62, 1269-1274. Google Scholar
[22]	X. Zhang, L. Liu, B. Wiwatanapataphee and Y. Wu, The eigenvalue for a class of singular p-Laplacian fractional differential equation involving the RiemannStieltjes integral boundary condition, Appl. Math. Comp., 2014, 235, 412-422. Google Scholar
[23]	X. Zhang, L. Liu and Y. Wu, The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives, Appl. Math. Comp., 2012, 218, 8526-8536. Google Scholar