consists of the "mixed sum-product of powers of norms," introduced by Rassias (in 2007) [28] and applied afterwards by Ravi et al. (2007-2008).
We started with two platters of mixed dim sum: two rose champagne shrimp; two scallop and pumpkin; two crystal crabmeat; two bamboo pith with assorted vegetables.
"Mixed feelings sums it up quite well," added Cook.
In addition, J.M. Rassias considered the mixed product-sum of powers of norms as the controlfunction.
Furthermore, as far as we know, the theory of discrete fractional mixed type sum-difference equations boundary value problems in Banach spaces is still a new research area.
This paper is concerned with the existence of a unique solution to a nonlinear discrete fractional mixed type sum-difference equation boundary value problem in a Banach space.
In 2008, J. M. Rassias [26] generalized even further the above two stabilities via a new stability involving a mixed product-sum of powers of norms, called JMRassias stability by several authors [27 30].
In this paper, by means of Darbo's fixed point theorem, we establish the existence of solutions to a nonlinear discrete fractional mixed type sum-difference equation boundary value problem in a Banach space.
We extend the results concerning Hyers-Ulam stability, Hyers-Ulam-Rassias stability and Ulam-Găvruta-Rassias stability controlled by the mixed product-sum of powers of norms for equations (1.7) and (1.8).
In addition, J. M. Rassias et al. [13 16] generalized the Hyers stability result by introducing two weaker conditions controlled by the Ulam-Gavruta-Rassias (or UGR) product of different powers of norms and the JM Rassias (or JMR) mixed product-sum of powers of norms, respectively.
From Theorem 3.1, we obtain the following corollary concerning the stability of (1.5) in the sense of the JMRassias stability of functional equations controlled by the mixed product-sum of powers of norms introduced by J. M. Rassias [26] and called JMRassias stability by several authors [27 30].
