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Existence and Uniqueness of Mittag-Leffler-Ulam Stable Solution for Fractional Integrodifferential Equations with Nonlocal Initial Conditions

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Abstract (2. Language): 
In this paper, the existence and uniqueness of mild solution for fractional integrodifferential equations with nonlocal initial conditions are investigated by using Hölder’s inequality, p−mean continuity and Schauder’s fixed point theorem in Banach spaces. The Mittag-Leffler-Ulam stability results are also obtained by using generalized singular Gronwall’s inequality.
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REFERENCES

References: 

[1] M. I. Abbas. Existence for fractional order impulsive integrodifferential inclusions with
nonlocal conditions, International Journal of Mathematical Analysis, 6(37), 1813-1828.
2012.
[2] M. I. Abbas. On the existence of mild solutions for a class of fractional differential equations
with nonlocal conditions in the α-norm, Studia Scientiarum Mathematicarum Hungarica,
51(2), 141-154. 2014.
[3] L. Cãdariu. Stabilitatea Ulam-Hyers-Bourgin pentru ecuatii functionale, Ed. Univ. Vest Timi
¸soara, Timi¸sara, 2007.
[4] M. M. El-Borai. The fundamental solutions for fractional evolution equations of parabolic
type, Journal of Applied Mathematics and Stochastic Analysis, 3, 197-211. 2004.
[5] M. M. El-Borai. On some fractional evolution equations with nonlocal conditions, International
Journal of Pure and Applied Mathematics, 24, 405-413. 2005.
[6] E. Hernández, D. O’Regan, and K. Balachandran. On recent developments in the theory of
abstract differential equations with fractional derivatives, Nonlinear Analysis, 73, 3462-
3471. 2010.
[7] L. Hu, Y. Ren, and R. Sakthivel. Existence and uniqueness of mild solutions for semilinear
integro-differential equations of fractional order with nonlocal initial conditions and delays,
Semigroup Forum, 79, 507-514. 2009.
[8] D. H. Hyers, G. Isac, and Th.M. Rassias. Stability of Functional Equations in Several Vari-
ables, Birkhäuser, 1998.
[9] S.M. Jung. Hyers-Ulam-Rassias Stability of Functional Equations inMathematical Analysis,
Hadronic Press, Palm Harbor, 2001.
[10] A. A. Kilbas, Hari M. Srivastava, and J. Juan Trujillo. Theory and applications of fractional
differential equations, North-HollandMathematics Studies, vol. 204, Elsevier Science B.V.,
Amsterdam, 2006.
[11] V. Lakshmikantham, S. Leela, and J. Vasundhara Devi. Theory of Fractional Dynamic Sys-
tems, Cambridge Scientific Publishers, 2009.
REFERENCES 498
[12] H. Liu and J. C. Chang. Existence for a class of partial differential equations with nonlocal
conditions, Nonlinear Analysis, 70, 3076-3083. 2009.
[13] M. M. Meerschaert, D.A. Benson, H. Scheffler, and B. Baeumer, Stochastic solution of
space-time fractional diffusion equations, Physical Review E, 65, 1103-1106. 2002.
[14] K. S. Miller and B. Ross. An Introduction to the Fractional Calculus and Differential Equa-
tions, John Wiley, New York, 1993.
[15] I. Podlubny. Fractional Differential Equations, Academic Press, San Diego, 1999.
[16] Shi-you Lin. Generalized Gronwall inequalities and their applications to fractional differ-
ential equations, Journal of Inequalities and Applications 2013, 549. 2013.
[17] J. Wang, Y. Zhou, W. Wei, and H. Xu. Nonlocal problems for fractional integrodifferential
equations via fractional operators and optimal controls, Computers and Mathematics with
Applications, 62, 1427-1441. 2011.
[18] J. Wang, Y. Zhou. Mittag-Leffler-Ulam stabilities of fractional evolution equations, Applied
Mathematics Letters, 25, 723-728. 2012.
[19] J. Wang and Yuruo Zhang. Ulam-Hyers-Mittag-Leffler stability of fractional-order delay
differential equations, Optimization, 63(8), 1181-1190. 2014.
[20] J. Wang and Y. Zhou. A class of fractional evolution equations and optimal controls, Nonlinear
Analysis RWA, 12, 262-272. 2011.
[21] E. Zeidler. Nonlinear Functional Analysis and its Application II/A, Springer-Verlag, New
York, 1990.
[22] Y. Zhou and F. Jiao. Existence of mild solutions for fractional neutral evolution equations,
Computer and Mathematics with Applications, 59, 1063-1077. 2010.
[23] Y. Zhou and F. Jiao. Nonlocal Cauchy problem for fractional evolution equations, Nonlinear
Analysis, 11, 4465-4475. 2010.
[24] Y. Zhou and F. Jiao. Existence of mild solutions for fractional neutral evolution equations,
Computer and Mathematics with Applications, 59, 1063-1077, 2010.

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