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The paper deals with the existence of periodic solutions of the boundary value problem for nonlinear heat equation, where various types of nonlinearities are considered. The proofs are based on the investigation of Liapunov-Schmidt bifurcation system via Leray-Schauder degree theory.

One investigates the existence of an $\omega$-periodic solution of the problem $u_t=u_{xx}+cu+g(t,x)+ϵf(t,x,u,u_x,ϵ),u(t,0)=h_0\left(t\right)+ϵ{\chi }_0(t,u(t,0),u(t,\pi )),u(t,\pi )=h_1\left(t\right)+ϵ{\chi }_1(t,u(t,0),u(t,\pi ))$, provided the functions $g,f,h_0,h_1,{\chi }_0,{\chi }_1$ are sufficiently smooth and $\omega$-periodic in $t$. If $c\ne k^2$, $k$ natural, such a solution always exists for sufficiently small $ϵ>0$. On the other hand, if $c=l^2$, $l$ natural, some additional conditions have to be satisfied.