Positive solutions for Caputo fractional differential equations involving integral boundary conditions

Volume 8, Issue 2, pp 99--109 http://dx.doi.org/10.22436/jnsa.008.02.03

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Authors

Yong Wang - School of Science, Jiangnan University, Wuxi 214122, China. Yang Yang - School of Science, Jiangnan University, Wuxi 214122, China.

Abstract

In this work we study integral boundary value problem involving Caputo differentiation \[ \begin{cases} ^c D^q_t u(t)= f(t,u(t)),\,\, 0<t<1,\\ \alpha u(0)-\beta u(1)=\int^1_0 h(t)u(t)dt, \gamma u'(0)-\delta u'(1)\int^1_0 g(t)u(t)dt, \end{cases} \] where \(\alpha,\beta,\gamma,\delta\) are constants with \(\alpha>\beta>0,\gamma>\delta>0, f\in C([0,1]\times \mathbb{R}^+,\mathbb{R}), g,h\in C([0,1],\mathbb{R}^+)\) and \( ^c D^q_t\) is the standard Caputo fractional derivative of fractional order \(q(1 < q < 2)\). By using some fixed point theorems we prove the existence of positive solutions.

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Keywords

• Caputo fractional boundary value problem
• fixed point theorem
• positive solution.

MSC

•  34B15
•  34B25

References

• [1] B. Ahmad, S. Ntouyas, A. Alsaedi , Existence of solutions for fractional q-integro-difference inclusions with fractional q-integral boundary conditions, Advances in Difference Equations, 2014 (2014), 18 pages.


• [2] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., 389 (2012), 403-411.


• [3] J. Henderson, H. Thompson, Multiple symmetric positive solutions for second order boundary-value problems, Proc. Amer. Math. Soc., 128 (2000), 2373-2379.


• [4] M. Jia, X. Liu, Three nonnegative solutions for fractional differential equations with integral boundary conditions , Comput. Math. Appl., 62 (2011), 1405-1412.


• [5] M. Jia, X. Liu, Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions, Appl. Math. Comput., 232 (2014), 313-323.


• [6] M. Krasnoselskii,. Zabreiko , Geometrical methods of nonlinear analysis, Springer-verlag, New York (1984)


• [7] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier Science, Amsterdam, The Netherlands (2006)


• [8] R. Leggett, R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana. Univ. Math. J., 28 (1979), 673-688.


• [9] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering , vol. 198, Academic Press, San Diego, Calif, USA (1999)


• [10] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon (1993)


• [11] J. Tariboon, S. Ntouyas, W. Sudsutad , Positive solutions for fractional differential equations with three-point multi-term fractional integral boundary conditions, Advances in Difference Equations, 2014 (2014), 17 pages.


• [12] S. Vong, Positive solutions of singular fractional differential equations with integral boundary conditions, Math. Comput. Modelling, 57 (2013), 1053-1059.


• [13] Z. Wei, C. Pang, Y. Ding , Positive solutions of singular Caputo fractional differential equations with integral boundary conditions , Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3148-3160.


• [14] S. Westerlund , Dead Matter Has Memory, Physica Scripta., 43 (1991), 174-179.


• [15] X. Xu, D. Jiang, C. Yuan, Multiple positive solutions to singular positone and semipositone Dirichlet-type boundary value problems of nonlinear fractional differential equations, Nonlinear Anal., 74 (2011), 5685-5696.


• [16] W. Yang , Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions , J. Appl. Math. Comput., 44 (2014), 39-59.


• [17] W. Yang , Positive solutions for nonlinear semipositone fractional q-difference system with coupled integral boundary conditions, Appl. Math. Comput., 244 (2014), 702-725.


• [18] X. Zhang, Positive solution for a class of singular semipositone fractional differential equations with integral boundary conditions , Boundary Value Problems, 2012 (2012), 17 pages.