Applying the Fourier Method to Solve One Class of Third Order Differential Equations in Banach Spaces

Mixed problem for one class of third order differential equations with non-linear operator on the right-hand side is considered in the Banach space B-2+alpha p/p,B- 1 + alpha p/p, 1 < p < +infinity, alpha(p) = max {p - 2; 0}. The concept of generalized solution in B-p,p,T(2+alpha p/p, 1 + alpha p/p) is introduced, and the existence and uniqueness theorems for generalized solution of considered problem are proved. Note that for p >= 2 this problem has been treated in ([1, 2]). The results obtained in this work are the generalizations of previously known corresponding results for p >= 2.