On the application of measure of noncompactness to existence theorems. (English) Zbl 0589.45007

Szufla, Stanisław

Rend. Sem. Mat. Univ. Padova 75, 1-14 (1986).

The measure of noncompactness \(\alpha\) is used to prove existence theorems for Hammerstein integral equations, boundary value problems for nonlinear ordinary differential equations of second order and quasilinear evolution equations. In the case of Hammerstein integral equation \((*)\quad x(t)=p(t)+\lambda \int_{I}K(t,s)f(s,x(s))ds\) assume: 1) p is a continuous function from \(I:=[0,a]\) into the Banach space E; 2) let f be a continuous function from \(I\times E\) into the Banach space F satisfying: (i) f is continuous in x and strongly measurable in s; (ii) for any \(r>0\) there exists an integrable function \(m_ r:I\to {\mathbb{R}}_+\) such that \(\| f(s,x)\| \leq m_ r(s)\) for all \(s\in I\) and \(\| x\| \leq r\); 3) K is a continuous function from \(I^ 2\) into the space of bounded linear mappings \(F\to E\). In addition assume that there exists an integrable function \(h:I\to {\mathbb{R}}_+\) such that for any bounded subset X of E there exists a closed subset \(I_{\epsilon}\) of I such that \(\mu (I\setminus I_{\epsilon})<\epsilon\) and \(\alpha (f(T\times X))\leq \sup_{s\in T}h(s)\alpha (X)\) for each closed subset T of \(I_{\epsilon}\). Then there exists \(\rho >0\) such that for any \(\lambda\in {\mathbb{R}}\) with \(| \lambda | <\rho\) the equation (*) has at least one continuous solution.

Reviewer: W.Petry

MSC:

45N05	Abstract integral equations, integral equations in abstract spaces
45G10	Other nonlinear integral equations
47H10	Fixed-point theorems

Keywords:

measure of noncompactness; Hammerstein; quasilinear evolution equations; Banach space; continuous solution

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