Existence and uniqueness of solutions for random fuzzy fractional integral and differential equations

In this paper the random fuzzy fractional integral and differential equations are introduced. Under Lipschitz condition we obtain the existence and uniqueness theorems of solutions for two general forms of random fuzzy fractional integral equations. To prove this assertion we use an idea of successive approximations. Moreover, the approach is followed to prove the existence and uniqueness of solutions for random fuzzy fractional initial value problem under Caputo-type fuzzy fractional derivatives. The method is illustrated by solving an example.

[1]  Tofigh Allahviranloo,et al.  Existence and uniqueness results for fractional differential equations with uncertainty , 2012, Advances in Difference Equations.

[2]  I. Podlubny.  Fractional differential equations , 1998 .

[3]  Ali Vahidian Kamyad,et al.  Modified fractional Euler method for solving Fuzzy Fractional Initial Value Problem , 2013, Commun. Nonlinear Sci. Numer. Simul..

[4]  Tofigh Allahviranloo,et al.  Fuzzy fractional differential equations under generalized fuzzy Caputo derivative , 2014, J. Intell. Fuzzy Syst..

[5]  A. Kilbas,et al.  The cauchy problem for differential equations with fractional caputo derivative , 2004 .

[6]  Tofigh Allahviranloo,et al.  Explicit solutions of fractional differential equations with uncertainty , 2012, Soft Comput..

[7]  Ravi P. Agarwal,et al.  On the concept of solution for fractional differential equations with uncertainty , 2010 .

[8]  Tofigh Allahviranloo,et al.  SOLVING FUZZY FRACTIONAL DIFFERENTIAL EQUATIONS BY FUZZY LAPLACE TRANSFORMS , 2012 .

[9]  Vasile Lupulescu,et al.  On the fractional differential equations with uncertainty , 2011 .

[10]  Vasile Lupulescu,et al.  On a class of fuzzy functional differential equations , 2009, Fuzzy Sets Syst..

[11]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[12]  R. Alikhani,et al.  Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations , 2013, Commun. Nonlinear Sci. Numer. Simul..

[13]  Zaid M. Odibat,et al.  Approximations of fractional integrals and Caputo fractional derivatives , 2006, Appl. Math. Comput..

[14]  Vasile Lupulescu,et al.  Fractional calculus for interval-valued functions , 2015, Fuzzy Sets Syst..

[15]  V. Lakshmikantham,et al.  Theory of fractional functional differential equations , 2008 .

[16]  Ravi P. Agarwal,et al.  Fuzzy fractional integral equations under compactness type condition , 2012 .

[17]  V. Lakshmikantham,et al.  Theory of Fuzzy Differential Equations and Inclusions , 2003 .

[18]  V. Lakshmikantham,et al.  Fractional differential equations with a Krasnoselskii–Krein type condition , 2009 .

[19]  Shiji Song,et al.  Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations , 2000, Fuzzy Sets Syst..

[20]  Marek T. Malinowski,et al.  On random fuzzy differential equations , 2009, Fuzzy Sets Syst..

[21]  Barnabás Bede,et al.  Generalized differentiability of fuzzy-valued functions , 2013, Fuzzy Sets Syst..

[22]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[23]  V. Lakshmikantham,et al.  A Krasnoselskii–Krein-type uniqueness result for fractional differential equations , 2009 .

[24]  M. T. Malinowski.  Random fuzzy differential equations under generalized Lipschitz condition , 2012 .

[25]  M. T. Malinowski.  Existence theorems for solutions to random fuzzy differential equations , 2010 .

[26]  Zaid M. Odibat,et al.  Generalized Taylor's formula , 2007, Appl. Math. Comput..

[27]  Hung T. Nguyen,et al.  A note on the extension principle for fuzzy sets , 1978 .

[28]  V. Lakshmikantham,et al.  PERSPECTIVES OF FUZZY INITIAL VALUE PROBLEMS , 2007 .