Theorem on integral inequalities with functional argument (Q1332030)

Let \(u_ 0\in C[0,T]\), \(u_ 0\geq 0\). Let \(L_ i\geq 0\) \((i= 1,2)\) be an integrable function on \([0,T]\) and let \(\alpha\in C^ 1[0,T]\) be such that \(\alpha'(t)\geq 0\) and \(\alpha'(t)\leq t\), \(t\in [0,T]\). Put \(\ell_ 1(t)= \exp \int^ t_ 0 L_ 1(s)ds\), \(t\in [0,T]\), and suppose that \(L_ 0= 1-\int^ T_ 0 L_ 2(t)\ell_ 1(\alpha(t))dt> 0\). If \(v\in C[0,T]\) is a positive function which satisfies \[ v(t)\leq u_ 0(t)+ \int^{\alpha(s)}_ 0 L_ 1(s) v(s)ds+ \int^ T_ 0 L_ 2(s) v(s)ds,\quad t\in [0,T], \] then \[ v(t)\leq u_ 0(t)+ U_ 0\cdot [\ell_ 1(\alpha(t))- 1]+ L^{-1}_ 0 \ell_ 1(T) \int^ T_ 0 L_ 2(s)\{u_ 0(s)+ U_ 0\cdot [\ell_ 1(\alpha(s))- 1]\}ds, \] where \(U_ 0= \max\{u_ 0(t); t\in [0,T]\}\). This is one of the main results; the others are its modifications.