Abstract:
Focus of this study is to cover up some gaps and further build up the theory of discrete
fractional calculus. This dissertation starts with brief introduction and de nitions to discrete
fractional calculus. Two new de nitions of generalized fractional di erence operator are introduced
namely Hilfer fractional di erence operator and substantial fractional di erence operator.
A missing property in the literature for delta Laplace transform i.e. delta exponential shift
is established. The delta Laplace transform is presented for the newly introduced Hilfer and
substantial fractional di erences. The double Laplace transform in a delta discrete setting is
introduced, and its existence, uniqueness and basic properties are discussed. The delta double
Laplace transform is presented for integer and non-integer order partial di erences.
Another goal of this study is to establish the existence and UHR stability for various classes
of fractional di erence equations. Conditions are acquired for RL, Caputo, Hilfer and substantialtype
fractional di erence equations. Moreover we establish a technique to transforming arbitrary
real order delta di erence equations with impulses to corresponding summation equations.
Existence results are built up for impulsive delta fractional di erence equation with nonlocal
initial condition and two-point and four-point boundary conditions. The conditions for existence
and UHR stability of the solution to multi-point summation boundary value problem are
established.
