Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions
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Authors
J. A. Nanware
- Department of Mathematics, Shrikrishna Mahavidyalaya, Gunjoti - 413 606, Dist. Osmanabad (M.S), India.
D. B. Dhaigude
- Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad - 431 004, India.
Abstract
Recently, Wang and Xie [T. Wang, F. Xie, J. Nonlinear Sci. Appl., 1 (2009), 206-212] developed monotone
iterative method for Riemann-Liouville fractional differential equations with integral boundary conditions
with the strong hypothesis of locally Hölder continuity and obtained existence and uniqueness of a solution
for the problem. In this paper, we apply the comparison result without locally Hölder continuity due to
Vasundhara Devi to develop monotone iterative method for the problem and obtain existence and uniqueness
of a solution of the problem.
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ISRP Style
J. A. Nanware, D. B. Dhaigude, Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions, Journal of Nonlinear Sciences and Applications, 7 (2014), no. 4, 246--254
AMA Style
Nanware J. A., Dhaigude D. B., Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions. J. Nonlinear Sci. Appl. (2014); 7(4):246--254
Chicago/Turabian Style
Nanware, J. A., Dhaigude, D. B.. "Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions." Journal of Nonlinear Sciences and Applications, 7, no. 4 (2014): 246--254
Keywords
- Fractional differential equations
- existence and uniqueness
- lower and upper solutions
- integral boundary conditions.
MSC
References
-
[1]
R. P. Agarwal, B. de Andrade, G. Siracusa, On Fractional Integro-Differential Equations with State-dependent Delay, Comp. Math. Appl., 62 (2011), 1143-1149.
-
[2]
R. P. Agarwal, M. Benchohra, S. Mamani , A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions , Acta Appl. Math., 109 (2010), 973-1033.
-
[3]
R. P. Agarwal, B. de Andrade, C. Cuevas , On type of periodicity and Ergodicity to a class of Fractional Order Differential Equations, Adv. Differen. Eqs., Hindawi Publl.Corp., NY , USA, Article ID 179750, (2010), 25 pages.
-
[4]
E. Cuesta, Asymptotic Behaviour of the Solutions of Fractional Integro-Differential Equations and Some Time Discretizations, Dis. Cont. Dyn. Sys., Series A, (2007), 277-285.
-
[5]
C. Cuevas, H. Soto, A. Sepulveda , Almost Periodic and Pseudo-almost Periodic Solutions to Fractional Differential and Integro-Differential Equations, Appl. Math. Comput., 218 (2011), 1735-1745.
-
[6]
J. V. Devi, Generalized Monotone Method for Periodic Boundary Value Problems of Caputo Fractional Differential Equations, Commun. Appl. Anal., 12 (2008), 399-406.
-
[7]
J. V. Devi, F. A. McRae, Z. Drici, Variational Lyapunov Method for Fractional Differential Equations, Comp. Math. Appl., 64 (2012), 2982-2989.
-
[8]
D. B. Dhaigude, J. A. Nanware, V. R. Nikam, Monotone Technique for System of Caputo Fractional Differential Equations with Periodic Boundary Conditions, Dyn. Conti. Dis. Impul. Sys., Series-A :Mathematical Analysis, 19 (2012), 575-584.
-
[9]
D. B. Dhaigude, J. A. Nanware, Monotone Technique for Finite System of Caputo Fractional Differential Equations with Periodic Boundary Conditions, , (To appear)
-
[10]
T. Diagana, G. M. Mophou, G. M. N'Gue're'kata , On the Existence of Mild Solutions to Some Semilinear Fractional Integro-Differential Equations, Electron. J. Qual. Theory Differ. Equ., 58 (2010), 1-17.
-
[11]
G. M. N'Gue're'kata , A Cauchy Problem for Some Fractional Abstract Differential Equations with Non-local Conditions, Nonlinear Anal., 70 (2009), 1873-1876.
-
[12]
T. Jankwoski, Differential Equations with Integral Boundary Conditions, J. Comput. Appl. Math., 147 (2002), 1-8.
-
[13]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematical Studies, Vol.204. Elsevier(North-Holland) Sciences Publishers, Amsterdam (2006)
-
[14]
P. Kumar, D. N. Pandey, D. Bahuguna, On a new class of abstract impulsive functional differential equations of fractional order, J. Nonlinear Sci. Appl., 7 (2014), 102-114.
-
[15]
G. S. Ladde, V. Lakshmikantham, A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations , Pitman Advanced Publishing Program, London (1985)
-
[16]
V. Lakshmikantham, A. S. Vatsala, Theory of Fractional Differential Equations and Applications , Communications in Applied Analysis, 11 (2007), 395-402.
-
[17]
V. Lakshmikantham, A. S. Vatsala, Basic Theory of Fractional Differential Equations and Applications, Nonlinear Anal., 69 (2008), 2677-2682.
-
[18]
V. Lakshmikantham, A. S. Vatsala, General Uniqueness and Monotone Iterative Technique for Fractional Differential Equations, Appl. Math. Letters, 21 (2008), 828-834.
-
[19]
V. Lakshmikantham, S. Leela, Differential and Integral Inequalities Vol.I., Academic Press, Newyork (1969)
-
[20]
V. Lakshmikantham, S. Leela, J. V. Devi, Theory and Applications of Fractional Dynamic Systems, Cambridge Scientific Publishers Ltd., (2009)
-
[21]
Y. Liu, H. Shi , Existence of unbounded positive solutions for BVPs of singular fractional differential equations , J. Nonlinear Sci. Appl., 5 (2012), 281-293.
-
[22]
F. A. Mc Rae , Monotone Iterative Technique and Existence Results for Fractional Differential Equations , Nonlinear Anal., 71 (2009), 6093-6096.
-
[23]
J. A. Nanware, Monotone Method In Fractional Differential Equations and Applications, Ph.D Thesis, Dr. Babasaheb Ambedkar Marathwada University (2013)
-
[24]
J. A. Nanware, D. B. Dhaigude, Boundary Value Problems for Differential Equations of Noninteger Order Involving Caputo Fractional Derivative, Proceedings of Jangjeon Mathematical Society, South Korea (To appear)
-
[25]
J. A. Nanware , Existence and Uniqueness Results for Fractional Differential Equations Via Monotone Method, Bull. Marathwada Math. Soc., 14 (2013), 39-56.
-
[26]
J. A. Nanware, D. B. Dhaigude, Existence and Uniqueness of solution of Riemann-Liouville Fractional Differential Equations with Integral Boundary Conditions, Int. J. Nonlinear Sci., 14 (2012), 410-415.
-
[27]
J. A. Nanware, D. B. Dhaigude, Monotone Iterative Scheme for System of Riemann-Liouville Fractional Differential Equations with Integral Boundary Conditions, Math. Modelling Sci. Computation, Springer-Verlag, 283 (2012), 395-402.
-
[28]
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999)
-
[29]
T. Wang, F. Xie, Existence and Uniqueness of Fractional Differential Equations with Integral Boundary Conditions, J. Nonlinear Sci. Appl., 1 (2009), 206-212.