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Positive solutions of nonlocal integral BVPS for the nonlinear coupled system involving high-order fractional differential

  • Kaihong Zhao EMAIL logo and Ping Gong
Published/Copyright: April 28, 2017
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 67 Issue 2

Abstract

In the paper, we investigate a class of four-point integral boundary value problems for the nonlinear coupled system involving higher-order Caputo fractional derivatives and Riemann-Stieltjes integral boundary conditions. By employing Guo-Krasnoselskii fixed point theorem, some sufficient conditions are obtained to guarantee the existence of at least one or two positive solutions for this system. Meanwhile, the eigenvalue intervals of existence for positive solutions are also given. As applications, some examples are provided to illustrate the validity of our main results.

Keywords: coupled fractional differential system; positive solutions; Riemann-Stieltjes integral BVPs; fixed point theorem
MSC 2010: Primary 34B09; 34B10; Secondary 34B15; 34B18

This work was supported by the National Natural Sciences Foundation of People’s Republic of China under Grant No. 11161025, Yunnan Province natural scientific research fund project No. 2011FZ058.

(Communicated by Michal Fečkan)

Acknowledgement

The authors would like to thank the anonymous referees for their useful and valuable suggestions. This work was supported by the National Natural Sciences Foundation of People’s Republic of China under Grant No. 11161025, Yunnan Province natural scientific research fund project No. 2011FZ058.

References

[1] Ahmad, B.—Ntouyas, S.—Alsaedi, A.: New existence results for nonlinear fractional differential equations with three-point integral boundary conditions, Adv. Differ. Equ. 2011 (2011), Article ID 107384.10.1155/2011/107384Search in Google Scholar

[2] Agarwal, R. P.—Liu, Y.—O’Regan, D.—Tian, C.: Positive solutions of two-point boundary value problems for fractional singular differential equations, Differ. Equ. 48 (2012), 619–629.10.1134/S0012266112050011Search in Google Scholar

[3] Cui, Y.—Zou, Y.: Existence results and the monotone iterative technique for nonlinear fractional differential systems with coupled four-point boundary value problems, Abstr. Appl. Anal. 2014 (2014), Article ID 242591.10.1155/2014/242591Search in Google Scholar

[4] Feng, M.—Ji, D.—Ge, W.: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces, J. Comput. Appl. Math. 222 (2008), 351–363.10.1016/j.cam.2007.11.003Search in Google Scholar

[5] Guo, D.—Lakshmikantham, V.—Liu, X.: Nonlinear Integral Equations in Abstract Spaces. In: Mathematics and Its Applications, Vol. 373, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.10.1007/978-1-4613-1281-9Search in Google Scholar

[6] Henderson, J.—Luca, R.: Existence and multiplicity of positive solutions for a system of fractional boundary value problems, Bound. Value Probl. 2014 (2014), Article ID 60.10.1186/1687-2770-2014-60Search in Google Scholar

[7] Chen, F.—Zhou, Y.: Attractivity of fractional functional differential equations, Comput. Math. Appl. 62 (2011), 1359–1369.10.1016/j.camwa.2011.03.062Search in Google Scholar

[8] Chang, Y.—Nieto, J.: Some new existence results for fractional differential inclusions with boundary conditions, Math. Comput. Modelling 49 (2009), 605–609.10.1016/j.mcm.2008.03.014Search in Google Scholar

[9] Jia, M.—Liu, X.: Three nonnegative solutions for fractional differential equations with integral boundary conditions, Comput. Math. Appl. 62 (2011), 1405–1412.10.1016/j.camwa.2011.03.026Search in Google Scholar

[10] Kilbas, A. A.—Trujillo, J. J.: Differential equations of fractional order: methods, results and problems-I, Appl. Anal. 78 (2001), 153–192.10.1080/0003681021000022032Search in Google Scholar

[11] Kilbas, A. A.—Trujillo, J. J.: Differential equations of fractional order: methods, results and problems-II, Appl. Anal. 81 (2002), 435–493.10.1080/0003681021000022032Search in Google Scholar

[12] Kilbas, A.—Srivastava, H.—Trujillo, J.: Theory and Applications of Fractional Differential Equations. In: North-Holland Mathematics Studies, Vol. 204, Elsevier, Amsterdam, 2006.Search in Google Scholar

[13] Lakshmikantham, V.—Leela, S.: Nagumo-type uniqueness result for fractional differential equations, Nonlinear Anal. 71 (2009), 2886–2889.10.1016/j.na.2009.01.169Search in Google Scholar

[14] Podlubny, I.: Fractional Differential Equations, Academic Press, New York, 1993.Search in Google Scholar

[15] Tian, C.—Liu, Y.: Multiple positive solutions for a class of fractional singular boundary value problem, Mem. Differ. Equ. Math. Phys. 56 (2012), 115–131.Search in Google Scholar

[16] Webb, J. R. L.: Positive solutions of some higher order nonlocal boundary value problems, Electron. J. Qual. Theory Differ. Equ. 29 (2009), 1–15.10.14232/ejqtde.2009.4.29Search in Google Scholar

[17] Webb, J. R. L.—Infante, G.: Nonlocal boundary value problems of arbitrary order, J. Lond. Math. Soc. 79(2) (2009), 238–258.10.1112/jlms/jdn066Search in Google Scholar

[18] Zhang, H. F.: Multiple positive solutions of nonlinear BVPs for differential systems involving integral conditions, Bound. Value Probl. 2014 (2014), Article ID 61.10.1186/1687-2770-2014-61Search in Google Scholar

[19] Zhao, K. H.—Gong, P.: Existence of positive solutions for a class of higher-order Caputo fractional differential equation, Qual. Theory Dyn. Syst. (2014). 10.1007/s12346-014-0121-0Search in Google Scholar

[20] Zhao, K. H.—Gong, P.: Positive solutions of Riemann-Stieltjes integral boundary problems for the nonlinear coupling system involving fractional-order differential, Adv. Difference Equ. 2014(254) (2014), 1–18.10.1186/1687-1847-2014-254Search in Google Scholar

[21] Zhao, K. H.—Gong, P.: Positive solutions for impulsive fractional differential equations with generalized periodic boundary value conditions, Adv. Difference Equ. 2014(255) (2014), 1–19.10.1186/1687-1847-2014-326Search in Google Scholar

Received: 2014-9-19
Accepted: 2014-11-28
Published Online: 2017-4-28
Published in Print: 2017-4-25

© 2017 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2016-0281
Keywords for this article
coupled fractional differential system; positive solutions; Riemann-Stieltjes integral BVPs; fixed point theorem

Articles in the same Issue

  2. 10.1515/ms-2015-0200
  4. Zero-divisor graphs of lower dismantlable lattices I
  6. Some results on the intersection graph of submodules of a module
  8. Class number parities of compositum of quadratic function fields
  10. Examples of beurling prime systems
  12. Connection between multiplication theorem for Bernoulli polynomials and first factor hp−
  14. On permutational invariance of the metric discrepancy results
  16. Evaluation of sums containing triple aerated generalized Fibonomial coefficients
  18. Linear algebraic proof of Wigner theorem and its consequences
  20. A note on groups with finite conjugacy classes of subnormal subgroups
  22. Groups with the same complex group algebras as some extensions of psl(2, pn)
  24. Klee-Phelps convex groupoids
  26. On analytic functions with generalized bounded Mocanu variation in conic domain with complex order
  28. Weak interpolation for the lipschitz class
  30. Generalized Padé approximants for plane condenser and distribution of points
  32. Three-variable symmetric and antisymmetric exponential functions and orthogonal polynomials
  34. Positive solutions of nonlocal integral BVPS for the nonlinear coupled system involving high-order fractional differential
  36. Existence of positive solutions for a nonlinear nth-order m-point p-Laplacian impulsive boundary value problem
  38. Dirichlet boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses
  40. On the oscillation of certain third order nonlinear dynamic equations with a nonlinear damping term
  42. Homoclinic solutions for ordinary (q, p)-Laplacian systems with a coercive potential
  44. Semi-equivelar maps on the torus and the Klein bottle with few vertices
  46. A problem considered by Friedlander & Iwaniec and the discrete Hardy-Littlewood method
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