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), 206212] developed monotone
iterative method for RiemannLiouville 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, 246254
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):246254
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): 246254
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 IntegroDifferential Equations with Statedependent Delay, Comp. Math. Appl., 62 (2011), 11431149.

[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), 9731033.

[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 IntegroDifferential Equations and Some Time Discretizations, Dis. Cont. Dyn. Sys., Series A, (2007), 277285.

[5]
C. Cuevas, H. Soto, A. Sepulveda , Almost Periodic and Pseudoalmost Periodic Solutions to Fractional Differential and IntegroDifferential Equations, Appl. Math. Comput., 218 (2011), 17351745.

[6]
J. V. Devi, Generalized Monotone Method for Periodic Boundary Value Problems of Caputo Fractional Differential Equations, Commun. Appl. Anal., 12 (2008), 399406.

[7]
J. V. Devi, F. A. McRae, Z. Drici, Variational Lyapunov Method for Fractional Differential Equations, Comp. Math. Appl., 64 (2012), 29822989.

[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., SeriesA :Mathematical Analysis, 19 (2012), 575584.

[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 IntegroDifferential Equations, Electron. J. Qual. Theory Differ. Equ., 58 (2010), 117.

[11]
G. M. N'Gue're'kata , A Cauchy Problem for Some Fractional Abstract Differential Equations with Nonlocal Conditions, Nonlinear Anal., 70 (2009), 18731876.

[12]
T. Jankwoski, Differential Equations with Integral Boundary Conditions, J. Comput. Appl. Math., 147 (2002), 18.

[13]
A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematical Studies, Vol.204. Elsevier(NorthHolland) 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), 102114.

[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), 395402.

[17]
V. Lakshmikantham, A. S. Vatsala, Basic Theory of Fractional Differential Equations and Applications, Nonlinear Anal., 69 (2008), 26772682.

[18]
V. Lakshmikantham, A. S. Vatsala, General Uniqueness and Monotone Iterative Technique for Fractional Differential Equations, Appl. Math. Letters, 21 (2008), 828834.

[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), 281293.

[22]
F. A. Mc Rae , Monotone Iterative Technique and Existence Results for Fractional Differential Equations , Nonlinear Anal., 71 (2009), 60936096.

[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), 3956.

[26]
J. A. Nanware, D. B. Dhaigude, Existence and Uniqueness of solution of RiemannLiouville Fractional Differential Equations with Integral Boundary Conditions, Int. J. Nonlinear Sci., 14 (2012), 410415.

[27]
J. A. Nanware, D. B. Dhaigude, Monotone Iterative Scheme for System of RiemannLiouville Fractional Differential Equations with Integral Boundary Conditions, Math. Modelling Sci. Computation, SpringerVerlag, 283 (2012), 395402.

[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), 206212.