Summary: 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 \(B_{p,p,T}^{2+ \frac{\alpha_p}{p},1+ \frac{\alpha_p}{p}}\), \(1< p <+ \infty\), \(\alpha_p= \max\{p-2; 0\}\). The concept of generalized solution in \(B_{p,p,T}^{2+ \frac{\alpha_p}{p},1+ \frac{\alpha_p}{p}}\) is introduced, and the existence and uniqueness theorems for generalized solution of considered problem are proved. Note that for \(p\geq 2\) this problem has been treated in 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 \(B_{p,p,T}^{2+ \frac{\alpha_p}{p},1+ \frac{\alpha_p}{p}}\), \(1< p <+ \infty\), \(\alpha_p= \max\{p-2; 0\}\). The concept of generalized solution in \(B_{p,p,T}^{2+ \frac{\alpha_p}{p},1+ \frac{\alpha_p}{p}}\) is introduced, and the existence and uniqueness theorems for generalized solution of considered problem are proved. Note that for \(p\geq 2\) this problem has been treated in [K. I. Khudaverdiev and A. A. Veliev, Study of one-dimensional mixed problem for one class of third order pseudohyperbolic equations with nonlinear operator right-hand side. (Issledovanie odnomernoǐ smeshannoǐ zadachi dlya odnogo psevdogiperbolicheskikh uravneniǐ tret’ego poryadka s nelineǐnoǐ operatornoǐ pravoǐ chast’yu) (Russian). Baku: Chashioglu (2010); B. T. Bilalov et al., Ukr. Math. J. 73, No. 3, 367–383 (2021; Zbl 1479.35240)]. The results obtained in this work are the generalizations of previously known corresponding results for \(p\geq 2\).. The results obtained in this work are the generalizations of previously known corresponding results for \(p\geq 2\).