The paper deals with the second order integro-differential equation \[ u_{tt}=c u_t+bu(t,\xi)+u(t,\xi)\int_\Omega k(\xi,\eta)u(t,\eta)d\eta +h(t,u(t,\xi)), \] where \(\Omega\subset\mathbb{R}^n,\) \(n\geq2,\) is a \(C^1\)-smooth and bounded domain, \(b\) and \(c\) are constants and \(k: \Omega\times \Omega\to\mathbb{R},\) \(h: [0,T]\times\mathbb{R}\to\mathbb{R}\) are given functions. Under suitable conditions on the data, the authors prove existence of solutions to the above equation under various conditions such as the periodic ones \[ u(0,\xi)=u(T,\xi),\;u_t(0,\xi)=u_t(T,\xi), \] the Cauchy multipoint \[ u(0,\xi)=\sum_{i=1}^k \alpha_i u(t_i,\xi),\;u(T,\xi)=\sum_{i=1}^k \beta_i u(t_i,\xi), \] and the weighted mean value conditions \[ u(0,\xi)=\dfrac{1}{T}\int_0^T p_1(t)u(t,\xi)dt,\;u(T,\xi)=\dfrac{1}{T}\int_0^T p_2(t)u(t,\xi)dt. \] The approach relies on reducing the problem to an abstract setting and application of suitable approximation solvability methods, based on Schauder degree arguments, Hartman-type inequality and Scorza-Dragoni results. The solutions are located in bounded sets and result limits of functions with values in finite dimensional spaces.