On approximate solutions of certain mixed type integrodifferential equations (Q2919566)

The paper deals with the initial value problem for the Volterra-Fredholm-type integro-differential equation NEWLINE\[NEWLINEy'(t)=f(t,y(t),(Ay)(t), (By)(t)),\quad t\in [0,a];\quad y(0)=y_0,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE(Ay)(t)=\displaystyle\int_0^tg(t,\sigma,y(\sigma))\,d\sigma,NEWLINE\]NEWLINE NEWLINE\[NEWLINE(By)(t)=\displaystyle\int_0^ah(t,\sigma,y(\sigma))\,d\sigma,NEWLINE\]NEWLINE \(f\in C(I_a\times\mathbb R^3,\mathbb R)\), \(g\in C(D\times\mathbb R,\mathbb R)\), \(h\in C(I_a^2\times\mathbb R,\mathbb R)\), \(I_a=[0,a]\) (\(a>0\)), and \(D=\{(t,s)\in I_a^2: s\leq t\}\). By using some new integral inequalities with explicit estimates, the author studies the \(\varepsilon\)-approximate solutions of the above problem. The corresponding two-dimensional problem is also investigated.