On positive solution for a nonlocal fractional boundary value problem

Yıl 2019, Cilt: 2 Sayı: 1, 27 - 33, 02.08.2019

Assia Guezane-lakoud , Samia Kouachi

Öz

We discuss the existence and uniqueness of positive solution for a fractional boundary value problem by the help of some fixed point theorems and using the method of upper and lower solutions. An example is also given to illustrat the obtained results.

Anahtar Kelimeler

Fractional boundary value problem, positivity of solution, fixed point theorem

Kaynakça

• 1 : R. P. Agarwal, D. O'Regan and P. J. Y. Wong, Positive Solutions of differential difference and integral equations, Kluwer Academic Publisher, Boston, 1999. 2 : B. Ahmed and J. J. Nieto, Anti-periodic fractional boundary value problems, Computers Mathematics with Applications, vol. 62 no. 3, 1150--1156, 2011. 3 : B. Ahmed, J. J. Nieto and J. Pimentel, Some boundary value problems of fractional differential equations and inclusions, Computers Mathematics with Applications , Vol. 62 no. 3, 1238--1250, 2011. 4 : Z. Bai, On positive solutions of nonlocal fractional boundary value problem, Nonlinear Analysis, vol. 72, no. 2, pp. 916-924, 2010. 5 : B. Bonilla, M. Rivero, L. Rodriguez-Germa, and J. J. Trujillo, Fractional differential equations as alternative models to nonlinear differential equations, Applied Mathematics and Computation, vol. 187, no. 1, pp. 79-88, 2007. 6 : M. El-Shahed, Positive solutions for boundary value problem of nonlinear fractional differential equation, Abstract and Applied Analysis, vol. 2007, Article ID 10368, 8 pages, 2007. 7 : A. Guezane-Lakoud, S. Kouachi and F. Ellaggoune, Positive solutions for a fractional boundary value problem, Commun.Fac.Sci.Univ.Ank.Series A1 Volume 63, Number 2, Pages 177--187 (2014). 8 : A. Guezane-Lakoud, R. Khaldi and A. Kılıçman; Solvability of a boundary value problem at resonance, SpringerPlus, 2016, 5:1504. 9 : R. Khaldi, A. Guezane-Lakoud, Upper and lower solutions method for higher order boundary value problems, Progress in Fractional Differentiation and Applications, Progr. Fract. Differ. Appl. 3, No. 1, 53--57 (2017). 10 : A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. 11 : S. Kouachi, A. Guezane-Lakoud, F. Ellagoune, Existence and localization of positive solutions for a fractional boundary value problem at resonance, Advances in Difference Equations 2015, 2015:316. 12 : V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Analysis, vol. 69, no. 8, pp. 2677-2682, 2008. 13 : K.S. Miller, B. Ross, An introduction to the fractional calculus and differential equations, John Wiley, New York, 1993 14 : M. Matar, On existence of positive solution for initial value problem of nonlinear fractional differential equations of order 1<α≤2, Acta Math. Univ. Comenianae, vol. LXXXIV, 1 (2015), pp. 51-57. 15 : I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. 16 : C. Wang, H. Zhang, SWang, Positive solution of a nonlinear fractional differential equation involving Caputo derivative. Discret. Dyn. Nat. Soc. 2012, 1--16 (2012) (Art ID425408) 17 : S. Zhang, Existence results of positive solutions to boundary value problem for fractional differential equation. Positivity 13(3), 583--599 (2009)