An extension of Gronwall's inequality (Q2735880)

One of the results in this paper is the extended Gronwall's inequality. NEWLINENEWLINENEWLINETheorem: Let \(k\geq 1\) be an integer. Suppose that NEWLINE\[NEWLINE\varphi^k(t)\leq c_0^k(t)+ \int_0^t [kc_1(s)+ kc_2(s) \varphi^2(s)+\dots+ kc_k(s) \varphi^k(s)] dsNEWLINE\]NEWLINE for a.e. \(t\in[0,T]\), where \(c_0\geq 0\) is nonincreasing, \(\varphi\in L^\infty\) and \(\varphi\geq 0\) for a.e. \(t\in [0,T]\), \(c_i\geq 0\) and \(c_i\in L^1\) for \(i\geq 1\). Then NEWLINE\[NEWLINE\varphi(t)\leq W_k (c_0, c_1,\dots, c_{k-1}) \exp \Biggl( \int_0^t c_k(s) ds \Biggr) \quad\text{a.e.},NEWLINE\]NEWLINE where the function \(W_k\) is defined recursively by NEWLINE\[NEWLINEw_1(c_0)= c_0, \quad W_{k+1} (c_0,c_1,\dots, c_k)= W_k \left( \Biggl[ c_0^k+k \int_0^t c_1(s) ds\Biggr]^{1/k}, c_2,\dots, c_k\right),\;k\geq 1.NEWLINE\]NEWLINE The classical Gronwall's inequality is the consequence of the Theorem for \(k=1\). The results of this paper are very useful in the theory of differential and integral equations.