### Nonlinear Implicit Hadamard's Fractional Differential Equa- tions with Retarded and Advanced Arguments

#### Abstract

In this paper, we establish the existence and uniqueness of solutions for a class of

problem for nonlinear implicit fractional dierential equations (NIFDE for short) of Hadamard type involving both retarded and advanced arguments. The proofs of our main results are based upon the Banach contraction principle and the Schauder xed point theorem. We present two examples to show the applicability of our results.

#### Keywords

#### References

S. Abbas, M. Benchohra and G M. N'Guerekata, Topics in Fractional Differential Equations, Springer-Verlag, New York, 2012.

S. Abbas, M. Benchohra and G M. N'Guerekata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.

B. Ahmad and S.K. Ntouyas, A fully Hadamard type integral boundary value problem of a coupled system of fractional dierential equations, Fract. Calc. Appl. Anal. 17 (2014), 348-360.

B. Ahmad and S.K. Ntouyas, Initial-value problem for hybrid Hadamard fractional differential equations, Electron. J. Dierential Equations 2014 (2014), No 161, pp.1-8.

B. Ahmad and S. Sivasundaram, Existence results and monotone iterative technique for impulsive hybrid functional dierential systems with anticipation and retardation, Appl. Math. Comput. 197 (2008), no. 2, 515-524.

M. Ammi, E. El Kinani and D. Torres, Existence and uniqueness of solutions to functional integro-diferential fractional equations, Electron. J. Dierential Equations 2012 (2012), No. 103, pp. 1-9.

M. Benchohra, S. Bouriah and J. Henderson, Existence and stability results for non-

linear implicit neutral fractional dierential equations with nite delay and impulses,

Comm. Appl. Nonlinear Anal. 22 (1) (2015), 46-67.

M. Benchohra and S. Bouriah, Existence and stability results for nonlinear boundary

value problem for implicit dierential equations of fractional order, Moroccan J. Pure.

Appl. Anal. 1 (1) (2015), 22-36.

P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Compositions of Hadamard-type frac-

tional integration operators and the semigroup property, J. Math. Anal. Appl. 269

(2002), 387-400.

P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Fractional calculus in the Mellin setting

and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), 1-27.

P. L. Butzer, A. A. Kilbas and J. J. Trujillo, Mellin transform analysis and integration

by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002),

-15.

K. Diethelm, The Analysis of Fractional Dierential Equations. An Application-

oriented Exposition Using Dierential Operators of Caputo Type. Lecture Notes in

Mathematics, 2004. Springer-Verlag, Berlin, 2010.

M. El Borai and M. Abbas, On some integro-dierential equations of fractional orders

involving Carathodory nonlinearities, Int. J. Mod. Math. 2 (2007), 41-52.

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.

T. Gnana Bhaskar, V. Lakshmikantham and J. Vasundhara Devi, Monotone itera-

tive technique for functional dierential equations with retardation and anticipation,

Nonlinear Anal. 66 (2007), no. 10, 2237-2242.

A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001),

-1204.

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Frac-

tional Dierential Equations. North-Holland Mathematics Studies, 204. Elsevier Sci-

ence B.V., Amsterdam, 2006.

S. Sun, Y. Zhao, Z. Hand and Y. Li, The existence of solutions for boundary value

problem of fractional hybrid dierential equations, Commun. Nonlinear Sci. Numer.

Simul. 17 (2012), 4961-4967.

Y. Sun and P. Wang, Iterative methods for a fourth-order dierential equations with

retardation and anticipation, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algo-

rithms 17 (2010), no. 4, 487-500.

V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics

of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing,

P. Thiramanus, S. K. Ntouyas and J. Tariboon, Existence and uniqueness result

for Hadamard type fractional dierential equations with nonlocal fractional integral

boundary conditions, Abstr. Appl. Anal. (2014), Art. ID 902054, 9 pp.

J. Vasundhara Devi and Ch. V. Sreedhar, Euler solutions for integrodierential equa-

tions with retardation and anticipation, Nonlinear Dyn. Syst. Theory 12 (2012), no.

, 237-250.

J. Vasundhara Devi and Ch V. Sreedhar, Quasilinearization for integrodierential

equations with retardation and anticipation, Nonlinear Stud. 19 (2012), no. 2, 303-

J. Vasundhara Devi, Ch. V. Sreedhar and S. Nagamani, Monotone iterative technique

for integrodierential equations with retardation and anticipation, Commun. Appl.

Anal. 14 (2010), no. 3-4, 325-335.

M. Yao, A. Zhao and J. Yan, Monotone method for rst order functional dierential

equations with retardation and anticipation., Nonlinear Anal. 71 (2009), no. 9, 4223-

M. Yao, L. Wen and X. Hu, Monotone method for rst-order impulsive dierential

equations with retardation and anticipation, Dyn. Contin. Discrete Impuls. Syst. Ser.

A Math. Anal. 15 (2008), suppl. S1, 12-14.

J. Wang, M. Feckan and Y. Zhou, Ulam's type stability of impulsive ordinary dier-

ential equations, J. Math. Anal. Appl. 395 (2012), 258-264.

Y. Zhao, S. Sun, Z. Han and Q. Li, Theory of fractional hybrid dierential equations,

Comput. Math. Appl. 62 (2011), 1312-1324.

Y. Zhou, Basic Theory of Fractional Dierential Equations, World Scientic, Singa-

pore, 2014.

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