Articles

FRACTIONAL INTEGRAL INEQUALITIES AND THEIR APPLICATIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS

  • Yaghoub JALILIAN
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  • Department of Mathematics, Razi University, Kermanshah, Iran

Received date: 2015-06-15

  Revised date: 2015-11-18

  Online published: 2016-10-25

Abstract

In this paper, first we obtain some new fractional integral inequalities. Then using these inequalities and fixed point theorems, we prove the existence of solutions for two different classes of functional fractional differential equations.

Cite this article

Yaghoub JALILIAN . FRACTIONAL INTEGRAL INEQUALITIES AND THEIR APPLICATIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia, Series B, 2016 , 36(5) : 1317 -1330 . DOI: 10.1016/S0252-9602(16)30071-6

References

[1] Magin R, Ortigueira M, Podlubny I, Trujillo J J. On the fractional signals and systems. Signal Processing, 2011, 91:350-371
[2] Magin R. Fractional calculus models of complex dynamics in biological tissues. Comput Math Appl, 2010, 59:1586-1593
[3] Merala F C, Roystona T J, Magin R. Fractional calculus in viscoelasticity:an experimental study. Commun Nonlinear Sci Numer Simul, 2010, 15:939-945
[4] Sabatier J, Nguyen H C, Farges C, Deletage J Y, Moreau X, Guillemard F, Bavoux B. Fractional models for thermal modeling and temperature estimation of a transistor junction. Adv Difference Equations, 2011, Article 687363
[5] Tenreiro Machado J A, Kiryakova V, Mainardi F. Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul, 2011, 16:1140-1153
[6] Das S. Functional Fractional Calculus for System Identification and Controls. Berlin, Heidelberg:SpringerVerlag, 2008
[7] Das S. Functional Fractional Calculus. Berlin, Heidelberg:Springer-Verlag, 2011
[8] Tarasov V E. Fractional Dynamics:Applications of Fractional Calculus to Dynamics of Particles. Fields and Media. Berlin, Heidelberg:Springer-Verlag, 2010
[9] Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity:An Introduction to Mathematical Models. London:Imperial College Press, 2010
[10] Kiryakova V. Generalized Fractional Calculus and Applications. Harlow, New York:Longman & Wiley, 1994
[11] Kiryakova V. All the special functions are fractional differintegrals of elementary functions. J Physics A:Math & Gen, 1997, 30:5085-5103
[12] Kiryakova V. Some special functions related to fractional calculus and fractional (non-integer) order control systems and equations. Facta Universitatis (Sci J of University of Nis), Series:Autom Control Robot, 2008, 7:79-98
[13] Machado J T, Kiryakova V, Mainardi F. Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul, 2011, 16:1140-1153
[14] Klafter J, Lim S C, Metzler R. Fractional Dynamics in Physics:Recent Advances. Singapore:World Scientific, 2011
[15] Agarwal R P, Ahmad B. Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput Math Appl, 2011, 62:1200-1214
[16] Agarwal R P, Benchohra M, Hamani S. A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl Math, 2010, 109:973-1033
[17] Ahmad B, Ntouyas S K. Nonlocal fractional boundary value problems with slit-strips boundary conditions. Fractional Calculus Appl Anal, 2015, 18(1):261-280
[18] Balachandran K, Kiruthika S, Trujillo J J. Existence results for fractional impulsive integrodifferential equations in Banach spaces. Commun Nonlinear Sci Numer Simul, 2011, 16:1970-1977
[19] Balachandran K, Trujillo J J. The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces. Nonlinear Analysis:TMA, 2010, 72:4587-4593
[20] Balachandran K, Kiruthika S. Existence results for fractional integrodifferential equations with nonlocal condition via resolvent operators. Comput Math Appl, 2011, 62:1350-1358
[21] Benchohra M, Berhoun F, Nguerekata G. Bounded solutions for fractional order differential equations on the half-line. Bull Math Anal Appls, 2012, 4:62-71
[22] Caballero J, Harjani J, Sadarangani K. On existence and uniqueness of positive solutions to a class of fractional boundary value problems. Boundary Value Problems, 2011, 2011:25
[23] Hu Z, Liu W. Solvability for fractional order boundary value problems at resonance. Boundary Value Problems, 2011, 2011:20
[24] Rui W. Existence of solutions of nonlinear fractional differential equations at resonance. Elect J Qual Theory Differ Equ, 2011, 66:1-12
[25] Sun W, Wang Y. Multiple positive solutions of nonlinear fractional differential equations with integral boundary value conditions. Fractional Calculus and Applied Analysis, 2014, 17(3):605-616
[26] Kilbas A A, Srivastava H, Trujillo J J. Theory and Applications of Fractional Differential Equations. Amsterdam:Elsevier, 2006
[27] Podlubny I. Fractional Differential Equations. New York:Academic Press, 1999
[28] Diethelm K. The Analysis of Fractional Differential Equations. Berlin, Heidelberg:Springer-Verlag, 2010
[29] Samko S G, Kilbas A A, Marichev O I. Fractional Integrals and Derivatives:Theory and Applications. Yverdon:Gordon and Breach, 1993
[30] Bainov D D, Simeonov P. Integral Inequalities and Applications. Kluwer Academic Publishers, 1992
[31] Lakshiliikantham V, Leela S. Differential and Integral Inequalities:Theory and Applications:Ordinary Differential Equations, Vol 1. New York:Academic Press, 1969
[32] Aghajani A, Jalilian Y, Trujillo J J. On the existence of solutions of fractional integro-differential equations. Fract Calculus Appl Anal, 2012, 15(1):44-69
[33] Jalilian Y, Jalilian R. Existence of solution for delay fractional differential equations. Mediterr J Math, 2013, 10:1731-1747
[34] Abbas S. Existence of solutions to fractional order ordinary and delay differential equations and applications. Elect J Differ Equ, 2011, 9:1-11
[35] Balachandran K, Kiruthika S, Trujillo J J. Existence of solutions of nonlinear fractional pantograph equations. Acta Math Sci, 2013, 33B:712-720
[36] Zeidler E. Nonlinear Functional Analysis and Applications, I:Fixed Point Theorems. New York:SpringerVerlag, 1986
[37] Pachpatte B G. Inequalities for Differential and Integral Equations. New York:Academic Press, 1998

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