Page 197 - Mathematical Techniques of Fractional Order Systems

186 Mathematical Techniques of Fractional Order Systems

the study of differential inclusions to the world of fractional calculus is
essential based on the widespread applications of multivalued analysis in
science and engineering. The study of fractional differential inclusions was
initiated by El-Sayed and Ibrahim (1995) and much interest has been given
along this line (see, Henderson and Ouahab, 2010; Vijayakumar et al., 2014;
Zhou, 2016; Balasubramaniam and Tamilalagan, 2015).
In the theory of abstract differential equations of integer order,
C 0 -semigroup, also known as a strongly continuous one-parameter family of
semigroup, is a generalization of the exponential function, which provides
solutions of integer order differential equations in Banach spaces, such as
delay differential equations and partial differential equations etc. It is worth-
while to mention that the Riemann Liouville and Caputo fractional opera-
tors do not possess semigroup and commutative properties, which are
inherent to the derivatives of integer order (Balachandran and Kiruthika,
2011). In order to study the existence of solutions, controllability, and other
qualitative properties of FDEs, various approaches are proposed, namely by
employing the method of solution operators, ða; kÞ-regularized families
of bounded linear operators and α-resolvent operators, etc. (see Kexue
and Jigen, 2012; Li et al., 2012; Lizama and N’Gue ´re ´kata, 2013; Peng and
Li, 2012).
In particular, Herna ´ndez et al. (2013, 2010) noted that the concept of a
solution is not realistic, while using variation of constant formula, also they
pointed out that similar problem arises, where the operator A is taken as a
Hille Yosida type operator. Further, they have studied the existence result
for a general class of abstract FDEs by using the well-developed theory of
resolvent operators for integral equations.
In the case of FDEs of order 1 , α , 2, Dos Santos et al. (2013) estab-
lished the existence of mild solutions for the nonlocal Cauchy problem of
abstract fractional neutral integro-differential equations with unbounded
delay by using the theory of resolvent operators. The functional equation
associated with general ða; kÞ regularized families of bounded linear operator,
which can replace the property of semigroups has been studied by Lizama
and Poblete (2012). The variation of constants formula obtained for FDEs
through ða; kÞ-regularized families of bounded linear operators is considered
to be an unified functional analytic approach, which covers the theories of
C 0 -semigroups and cosine families as particular cases. Lizama and
N’Gue ´re ´kata (2013) proved the existence of solutions for some classes of
FDEs with nonlocal conditions of order 1 , α , 2 by using ða; kÞ-regularized
families.
Further, during the last decades fractional optimal control problems gained
the attention of the researchers (see Agrawal, 2004; Agrawal et al., 2010;
Balasubramaniam and Tamilalagan, 2016). The solvability and optimal con-
trols of a class of fractional integro-differential evolution systems with infinite
delay in Banach spaces has been investigated by Wang et al. (2012). Wang