On α-Fractional Bregman Divergence to study α-Fractional Minty’s Lemma

In this paper fractional variational inequality problems (FVIP) and dual fractional variational inequality problems (DFVIP), Fractional minimization problems are defined with the help of fractional Caputo differential operator and Reimann-Liouville differential operator. The equivalence theorem of Caputo FVIP and fractional minimization problem is established. The equivalence between FVIP and DFVIP is studied with the help of fractional Bregman divergence to establish the fractional Minty’s lemma. Introduction: The theory of variational inequality problems is one of the best tool to solve the equilibrium and non-equilibrium problems arises in the branches of Applied Mathematics, Applied Physics, Engineering, Medical, Financial and many more. On the other hand, fractional calculus manages to study various types of dynamic problems arises in real and complex analysis It is certain for the development of fractional calculus because the fractional derivative  has global correlation, which can reflect the historical process of the systematic function.With regard to the definition of fractional derivatives, many mathematicians studied fractional derivatives from different aspects and advanced different definitions for them, for example  Riemann-Liouville (RL) derivative, Caputo derivative and other derivatives. Conclusions: The concept of -fractionally differential inequality problem is defined and studied its existence theorem with the help of -fractionally convex function. The concept of -fractionally Bregman divergence is developed and using it the existence of -fractionally minimization problem and -fractionally differential inequality problem are studied. Equivalence theorem in between -fractionally differential inequality problem and -fractionally dual differential inequality problem is studied to establish the Minty’s lemma in fractional calculus.