Krasnoselskii's unification, Volterra's integrodifferential equation, and the method of Aizerman (Q2925595)

The authors consider the nonlinear integro-differential equation NEWLINE\[NEWLINEx'(t)= -\int^t_0 D(t- s) g(x(s))\,ds+ f(t),\quad t\geq 0,\tag{1}NEWLINE\]NEWLINE where \(f:[0,+\infty)\to \mathbb{R}\), \(D;[0,+\infty)\to \mathbb{R}\), \(g:\mathbb{R}\to \mathbb{R}\) are continuous functions with NEWLINE\[NEWLINED(t)> 0,\quad \int^{+\infty}_0 D(t)\,dt<+\infty,\quad xg(x)> 0,\;x\neq 0.NEWLINE\]NEWLINE Assume that there exist \(G_1\geq 0\), \(G_2> 0\) with \(G_1\leq x^{-1}g(x)\leq G_2\), \(x\neq 0\). If \(G_1= 1= G_2\), (1) reduces to the forced linear equation NEWLINE\[NEWLINEx'(t)= -\int^t_0 D(t- s)x(s)\,ds+ f(t),\quad t\geq 0,\tag{2}NEWLINE\]NEWLINE and the authors prove that (2) has a bounded solution. Denote NEWLINE\[NEWLINEF(t)= e^{-Jt} \int^t_0 e^{Js}f(s)\,ds,\quad t\geq 0,\;J>0,NEWLINE\]NEWLINE \((BC,\|.\|)\) the Banach space of bounded continuous functions \(\Phi: [0,+\infty)\to\mathbb{R}\) with the supremum norm, \(M= \{\Phi\in BC;\,\|\Phi\|\leq 1\}\), \(A:M\to M\) be defined by NEWLINE\[NEWLINE(Ax)(t)= z(t)+ F(t)+ \int^t_0 \Biggl(R(t- u) x(u)- \Biggl(\int^t_0 R(t- s) J^{-1} D(s- u)\,ds\Biggr) g(x(u))\Biggr)\,du,\;t\geq 0,NEWLINE\]NEWLINE where NEWLINE\[NEWLINEz(t)= x(0) e^{-Jt},\quad R(t)= Je^{-Jt},\quad t\geq 0,\quad J> 0.NEWLINE\]NEWLINE Consider NEWLINE\[NEWLINEm(u,t)= \max_{i\in \{1,2\}} \Biggl\{\Biggl| Je^{-J(t-u)}- \int^t_0 e^{-J(t-s)} D_i(s- u)\,ds\Biggr|\Biggr\},\quad 0\leq u\leq t,NEWLINE\]NEWLINE where \(D_i(t)= G_i D(t)\), \(t\geq 0\), \(i\in \{1,2\}\). -- The authors prove that if \(F\) is uniformly continuous and NEWLINE\[NEWLINE\Biggl(|x(0)|+ \Biggl|\int^t_0 e^{js} f(s)\,ds\Biggr|\Biggr) e^{-Jt}+ \int^t_0 m(u,t)\,du\leq 1,\quad t\geq 0,\quad J>0,NEWLINE\]NEWLINE then the natural mapping NEWLINE\[NEWLINE\begin{multlined} x(t)= z(t)+\\ F(t)+ \int^t_0 \Biggl(R(t- u)- \Biggl(\int^t_u R(t-s) J^{-1} D(s-u)\,ds\Biggr)(x(u))^{-1} g(x(u))\Biggr) x(u)\,du\end{multlined}NEWLINE\]NEWLINE for \(t\geq 0\), \(J>0\), on the set \(M\) maps \(M\) into \(M\) and a theorem of Brouwer-Schauder type gives a fixed point in \(M= \{\Phi\in BC;\,a\leq\Phi\leq b\}\), \(a< b\), for \(A: M\to M\), if \(A\) is continuous.