Annals of Communications in Mathematics

AbstractEmploying sequential generalized Caputo fractional left and right vectorial Taylor formulae we establish mixed sequential generalized fractional Ostrowski and Gruss ¨ type inequalities for several Banach algebra valued functions. The estimates are with respect to all norms k·kp , 1 ≤ p ≤ ∞. We finish with applications.

ABSTRACT.In this paper, we determine some stability results concerning the quartic functional equation as of the form\[\sum_{b=1}^{s} \phi\!\left(-v_b + \sum_{a=1,\,a\ne b}^{s} v_a\right)- 4 \sum_{1 \le a < b < c \le s} \phi(v_a + v_b + v_c)- \sum_{b=1}^{s} \phi(2v_b)\]\[= (-4s+14)\sum_{a=1,\,a\ne b}^{s} \phi(v_a + v_b)+ (s-8)\phi\!\left(\sum_{a=1}^{s} v_a\right)\]\[\quad + 2 \left[\sum_{a=1,\,a\ne b}^{s} \phi(v_a - v_b)+ (s^{2}-7s+7)\sum_{a=1}^{s} \phi(v_a)\right],\]where any positive integers \( s \ge 3 \), in Banach algebra via direct and fixed point approaches.