We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption...

In this paper, we aim to study the global solvability of the following system of third order nonlinear neutral delay differential equations $$\frac{d}{dt}\left\{r_i\left(t\right)\frac{d}{dt}\left[{\lambda }_i\left(t\right)\frac{d}{dt}\left(x_i\left(t\right)-f_i(t,x_1(t-{\sigma }_{i1}),x_2(t-{\sigma }_{i2}),x_3(t-{\sigma }_{i3}))\right)\right]\right\}+\frac{d}{dt}\left[r_i\left(t\right)\frac{d}{dt}{g}_i(t,x_1\left(p_{i1}\left(t\right)\right),x_2\left(p_{i2}\left(t\right)\right),x_3\left(p_{i3}\left(t\right)\right))\right]+\frac{d}{dt}{h}_i(t,x_1\left(q_{i1}\left(t\right)\right),x_2\left(q_{i2}\left(t\right)\right),x_3\left(q_{i3}\left(t\right)\right))=l_i(t,x_1\left({\eta }_{i1}\left(t\right)\right),x_2\left({\eta }_{i2}\left(t\right)\right),x_3\left({\eta }_{i3}\left(t\right)\right)),t\ge t_0,i\in \{1,2,3\}$$ in the following bounded closed and convex set $$\Omega (a,b)=\left\{x\left(t\right)=(x_1\left(t\right),x_2\left(t\right),x_3\left(t\right))\in C([t_0,+\infty ),{ℝ}^3):a\left(t\right)\le x_i\left(t\right)\le b\left(t\right),\forall t\ge t_0,i\in \{1,2,3\}\right\},$$ where ${\sigma }_{ij}>0$, $r_i,{\lambda }_i,a,b\in C([t_0,+\infty ),{ℝ}^{+})$, $f_i,g_i,h_i,l_i\in C([t_0,+\infty )\times {ℝ}^3,ℝ)$, $p_{ij},q_{ij},{\eta }_{ij}\in C([t_0,+\infty ),ℝ)$ for $i,j\in \{1,2,3\}$. By applying the Krasnoselskii fixed point theorem, the Schauder fixed point theorem, the Sadovskii fixed point theorem and the Banach contraction principle, four existence results of uncountably many bounded positive solutions of the system are established.