Abstract:
Fractional calculus, the study of of integration and differentiation of fractional order,
has recently been extended to include its discrete analogues of fractional difference
calculus and fractional quantum calculus. Due to this, there was a question whether
there exist a single theory pertaining the above theory. Answer to this task was
proposed by great scientist Stephen Hilger (1988), P. A. Williams and Bastos (2012),
Jiang Zhu et al. (2013). This field is diverse, having a lot of applications in different
fields of sciences such as: differential equations, probability theory, Mathematical
and Economical models, optimization theory, signal processing, chaotic dynamics,
atomic Bose-Einstein condensation, theory of inequalities etc.
Inequalities play a significant role in many branches of Sciences as well as to discuss
the abstract analysis of the solutions of differential, difference equations and
Cauchy type problems. Among others inequalities, Gronwall-Bellman type integral
inequalities have a significant part in this direction. We propose −Delta integral,
−multi-time scale integral, Itˆo−Isometry, generalized fractional dynamical
Gronwall-Bellman type integral inequalities to analyze some qualitative and quantitative
properties of solutions of integro-differential equations, cauchy type problems,
nonlinear fractional stochastic differential equation and fractional −stochastic differential
equation.
