Solvability and existence of asymptotically stable solutions for a Volterra-Hammerstein integral equation on an infinite interval (Q380226)

Using a Krasnoselskii type fixed point theorem (for the sum of a contraction and compact map) in a Fréchet space, the existence of a solution of the Volterra-Urysohn functional integral equation NEWLINE\[NEWLINE\begin{multlined} x(t)=Q(t)+f(t,x(t),x(\pi(t)))\\ + \int_0^{\mu(t)}V(t,s,x(\sigma_1(s)),\dots,V(\sigma_p(s)))\,ds+ \int_0^\infty G(t,s,x(\chi_1(s)),\dots,x(\chi_q(s)))\,ds\end{multlined}NEWLINE\]NEWLINE of vector-valued continuous functions (with values in Banach spaces) on the unbounded interval \([0,\infty)\) is shown; here, \(\sigma_k(t),\pi(t),\mu(t)\in[0,t]\). The main hypotheses are a contraction type assumption for \(f\), a Lipschitz type assumption for \(V\), and a compactness and uniform boundedness (by an integrable function) of \(G\).NEWLINENEWLINEAlso, under additional hypotheses, the unique asymptotic behaviour of all solutions is shown (for some strange reason this is called asymptotic stability). The results are illustrated by some examples.