Positive solutions for some Riemann-Liouville fractional boundary value problems
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2005
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Authors
Imed Bachar
- Mathematics Department, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia.
Habib Mâagli
- Department of Mathematics, College of Sciences and Arts, Rabigh Campus, King Abdulaziz University, P. O. Box 344, Rabigh 21911, Saudi Arabia.
- Department of Mathematics, Faculte des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia.
Abstract
We study the existence and global asymptotic behavior of positive continuous solutions to the following
nonlinear fractional boundary value problem
\[
(p_\lambda)
\begin{cases}
D^\alpha u(t)=\lambda f(t,u(t)),\,\,\,\,\, t\in (0,1),\\
\lim_{t\rightarrow 0^+}t^{2-\alpha} u(t)=\mu, \quad u(1)=\nu,
\end{cases}
\]
where \(1 < \alpha\leq 2; D^\alpha\) is the Riemann-Liouville fractional derivative, and \(\lambda,\mu\) and \(\nu\) are nonnegative constants
such that \(\mu + \nu > 0\).
Our purpose is to give two existence results for the above problem, where \(f(t; s)\) is a nonnegative
continuous function on \((0; 1)\times[0;\infty)\); nondecreasing with respect to the second variable and satisfying some
appropriate integrability condition. Some examples are given to illustrate our existence results.
Share and Cite
ISRP Style
Imed Bachar, Habib Mâagli, Positive solutions for some Riemann-Liouville fractional boundary value problems, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 7, 5093--5106
AMA Style
Bachar Imed, Mâagli Habib, Positive solutions for some Riemann-Liouville fractional boundary value problems. J. Nonlinear Sci. Appl. (2016); 9(7):5093--5106
Chicago/Turabian Style
Bachar, Imed, Mâagli, Habib. "Positive solutions for some Riemann-Liouville fractional boundary value problems." Journal of Nonlinear Sciences and Applications, 9, no. 7 (2016): 5093--5106
Keywords
- Fractional differential equation
- positive solutions
- Green's function
- perturbation arguments
- Schäuder fixed point theorem.
MSC
- 34B27
- 34A08
- 34B18
- 34B15
- 47N20
References
-
[1]
R. P. Agarwal, D. O'Regan, S. Staněk, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl., 371 (2010), 57--68
-
[2]
I. Bachar, H. Mâagli, Positive solutions for superlinear fractional boundary value problemss, Adv. Difference Equ., 2014 (2014), 16 pages
-
[3]
I. Bachar, H. Mâagli, V. D. Radulescu, Fractional Navier boundary value problems, Bound. Value Probl., 2016 (2016), 14 pages
-
[4]
I. Bachar, H. Mâagli, F. Toumi, Z. Zine El Abidine, Existence and global asymptotic behavior of positive solutions for sublinear and superlinear fractional boundary value problems, Chin. Ann. Math. Ser. B, 37 (2016), 1--28
-
[5]
Z. Bai, H. Lü , Positive solutions for boundary value problem of nonlinear fractional differential equation , J. Math. Anal. Appl., 311 (2005), 495--505
-
[6]
L. Del Pezzo, J. Rossi, N. Saintier, A. Salort, An optimal mass transport approach for limits of eigenvalue problems for the fractional p-Laplacian, Adv. Nonlinear Anal., 4 (2015), 235--249
-
[7]
K. Diethelm, A. D. Freed , On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, Sci. Comput. Chem. Eng. II, Springer, Heidelberg (1999)
-
[8]
L. Gaul, P. Klein, S. Kempe , Damping description involving fractional operators, Mech. Syst. Signal Process, 5 (1991), 81--88
-
[9]
J. Giacomoni, P. K. Mishra, K. Sreenadh, Fractional elliptic equations with critical exponential nonlinearity, Adv. Nonlinear Anal., 5 (2016), 57--74
-
[10]
W. G. Glöckle, T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68 (1995), 46--53
-
[11]
S. Goyal, K. Sreenadh, Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function, Adv. Nonlinear Anal., 4 (2015), 37--58
-
[12]
J. R. Graef, L. Kong, Q. Kong, M. Wang, Existence and uniqueness of solutions for a fractional boundary value problem with Dirichlet boundary condition, Electron. J. Qual. Theory Differ. Equ., 2013 (2013), 11 pages
-
[13]
R. Hilfer, Applications of fractional calculus in physics, World Scientific Publishing Co., Inc., River Edge (2000)
-
[14]
M. Jleli, B. Samet, Existence of positive solutions to a coupled system of fractional differential equations, Math. Methods Appl. Sci., 38 (2015), 1014--1031
-
[15]
E. R. Kaufmann, E. Mboumi , Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equ., 2008 (2008), 11 pages
-
[16]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North- Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006)
-
[17]
S. Liang, J. Zhang, Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Anal., 71 (2009), 5545--5550
-
[18]
H. Mâagli, N. Mhadhebi, N. Zeddini , Existence and estimates of positive solutions for some singular fractional boundary value problems, Abstr. Appl. Anal., 2014 (2014), 7 pages
-
[19]
F. Mainardi , Fractional calculus: some basic problems in continuum and statical mechanics, Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Vienna and New York, 1997 (1997), 291--348
-
[20]
R. Metzler, J. Klafter , Boundary value problems for fractional diffusion equations, Phys. A, 278 (2000), 107--125
-
[21]
K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley- Inter-science Publication, John Wiley & Sons, Inc., New York (1993)
-
[22]
G. Molica Bisci, V. D. Radulescu, R. Servadei, Variational methods for nonlocal fractional problems, With a foreword by Jean Mawhin, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (2016)
-
[23]
G. Molica Bisci, D. Repovs, Existence and localization of solutions for nonlocal fractional equations, Asymptot. Anal., 90 (2014), 367--378
-
[24]
G. Molica Bisci, D. Repovs, Higher nonlocal problems with bounded potential, J. Math. Anal. Appl., 420 (2014), 167--176
-
[25]
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 (1999)
-
[26]
S. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikolʹskiĭ, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon (1993)
-
[27]
H. Scher, E. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B., 12 (1975), 2455--2477
-
[28]
V. E. Tarasov , Fractional dynamics, Applications of fractional calculus to dynamics of particles, fields and media, Nonlinear Physical Science, Springer, Heidelberg, Higher Education Press, Beijing (2011)
-
[29]
S. P. Timoshenko, J. M. Gere, Theory of elastic stability, McGraw-Hill, New York (1961)
-
[30]
X. Zhang, L. Liu, Y. Wu, Multiple positive solutions of a singular fractional differential equation with negatively perturbed term, Math. Comput. Modelling, 55 (2012), 1263--1274