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APPROXIMATIONS OF SOLUTIONS FOR A NONLOCAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION WITH DEVIATED ARGUMENT
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 Title & Authors
APPROXIMATIONS OF SOLUTIONS FOR A NONLOCAL FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION WITH DEVIATED ARGUMENT
CHADHA, ALKA; PANDEY, DWIJENDRA N.;
 
 Abstract
This paper investigates the existence of mild solution for a fractional integro-differential equations with a deviating argument and nonlocal initial condition in an arbitrary separable Hilbert space H via technique of approximations. We obtain an associated integral equation and then consider a sequence of approximate integral equations obtained by the projection of considered associated nonlocal fractional integral equation onto finite dimensional space. The existence and uniqueness of solutions to each approximate integral equation is obtained by virtue of the analytic semigroup theory via Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. We consider the Faedo-Galerkin approximation of the solution and demonstrate some convergenceresults. An example is also given to illustrate the abstract theory.
 Keywords
Analytic Semigroup;Banach fixed point Theorem;Caputo derivative;Neutral integro-differential equation;Faedo-Galerkin approximation;
 Language
English
 Cited by
 References
1.
K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, 1993.

2.
S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishe, Yverdon, 1993.

3.
I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Vol. 198. Academic Press, San Diego, 1999.

4.
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

5.
M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals, 14 (2002), 433-440. crossref(new window)

6.
L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494-505. crossref(new window)

7.
L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Applicable Anal. 40 (1991), 11-19. crossref(new window)

8.
D. Bahuguna and M. Muslim, Approximation of solutions to non-local history-valued retarded differential equations, Appl. Math. Comp. 174 (2006), 165-179. crossref(new window)

9.
D. Bahuguna and S. Agarwal, Approximations of solutions to neutral functional differential equations with nonlocal history conditions, J. Math. Anal. Appl. 317 (2006), 583-602. crossref(new window)

10.
K. Balachandran and M. Chandrasekaran, Existence of solutions of a delay differential equations with nonlocal conditions, Indian J. Pure. Appl. Math. 27 (1996), 443-449.

11.
Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal.: Real World Appl. 11 (2010), 4465-4475. crossref(new window)

12.
F. Li, J. Liang and H.-K. Xu, Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions, J. Math. Anal. Appl. 391 (2012), 510-525. crossref(new window)

13.
X.-B. Shu and Q. Wang, The existence and uniqueness of mild solutions for fractional differential equations with nonlocal conditions of order 1 < α < 2, Comput. Math. Appl. 64 (2012), 2100-2110. crossref(new window)

14.
R.S. Jain and M.B. Dhakne, On mild solutions of nonlocal semilinear impulsive functional integro-differential equations, Appl. Math. E-notes, 13 (2013), 109-119.

15.
H.M. Ahmed, Fractional neutral evolution equations with nonlocal conditions, Adv. Diff. Equ. 2013:117. crossref(new window)

16.
S. Liang and R. Mei, Existence of mild solutions for fractional impulsive neutral evolution equations with nonlocal conditions, Adv. Diff. Equ. 2014:101.

17.
C.G. Gal, Nonlinear abstract differential equations with deviated argument, J. Math. Anal. Appl. 333 (2007), 971-983. crossref(new window)

18.
C.G. Gal, Semilinear abstract differential equations with deviated argument, Int. J. Evol. Equ. 2 (2008), 381-386.

19.
R. Göthel, Faedo-Galerkin approximation in equations of evolution, Math. Meth. Appl. Sci. 6 (1984), 41-54. crossref(new window)

20.
P.D. Miletta, Approximation of solutions to evolution equations, Math. Methods Appl. Sci. 17 (1994), 753-763. crossref(new window)

21.
D. Bahuguna and S.K. Srivastava, Approximation of solutions to evolution integrodifferential equations, J. Appl. Math. Stoch. Anal. 9 (1996), 315-322. crossref(new window)

22.
D. Bahuguna, S.K. Srivastava and S. Singh, Approximations of solutions to semilinear integrodifferential equations, Numer. Funct. Anal. Optimiz. 22 (2001), 487-504. crossref(new window)

23.
D. Bahuguna and R. Shukla, Approximations of solutions to nonlinear Sobolev type evolution equations, Elect. J. Diff. Equ. 2003 (2003), 1-16.

24.
D. Bahuguna and M. Muslim, Approximation of solutions to retarded differential equations with applications to populations dynamics, J. Appl. Math. Stoch. Anal. 2005 (2005), 1-11. crossref(new window)

25.
M. Muslim and D. Bahuguna, Existence of solutions to neutral differential equations with deviated argument, Elect. J. Qualit. The. Diff. Equ. 2008 (2008), 1-12. crossref(new window)

26.
P. Kumar, D.N. Pandey and D. Bahuguna, Approximations of solutions to a fractional differential equations with a deviating argument, Diff. Equ. Dyn. Syst. 2013 (2013), pp-20.

27.
M. Muslim, R.P. Agarwal and A.K. Nandakumaran, Existence, uniqueness and convergence of approximate solutions of impulsive neutral differential equations, Funct. Diff. Equ. 16 (2009), 529-544.

28.
M. Muslim and A.K. Nandakumaran, Existence and approximations of solutions to some fractional order functional integral equations, J. Int. Equ. Appl. 22 (2010), 95-114. crossref(new window)

29.
M. Muslim, C. Conca and R.P. Agarwal, Existence of local and global solutions of fractional-order differential equations, Nonlinear Oscillations, 14 (2011), 77-85. crossref(new window)

30.
A. Chadha and D.N. Pandey, Existence, uniqueness and Aapproximation of solution for the fractional semilinear integro-differential equation, Int. J. Applied Math. Stat. 52 (2014), 73-89.

31.
A. Chadha and D.N. Pandey, Approximations of solutions for a Sobolev type fractional order differential equation, Nonlinear Dyn. Sys. The. 14 (2014), 11-29.

32.
F. Mainardi, On a special function arising in the time fractional diffusion-wave equation: Transform methods and special functions, Science Culture Technology, pp. 171-83, Singopore, 1994.

33.
H. Pollard, The representation of e-xλ as a Laplace integral, Bull. Am. Math. Soc. 52 (1946), 908-910. crossref(new window)

34.
Mahmoud M. El-Borai and Hamza A.S. Abujabal, On the Cauchy problem for some abstract nonlinear differential equations, Korean J. Comput. Appl. Math. 3 (1996), 279-290.

35.
A. Pazy, Semigroups of linear operators and aplications to partial differential equations, Springer, New York, 1983.

36.
I.M. Gelfand and G.E. Shilov, Generalized Functions. Vol. 1. Nauka, Moscow, 1959.