Locally Lipschitz composition operators and applications to nonlinear integral equations (Q388705)

The authors study the nonlinear composition (Nemytskij, superposition) operator on the closed balls NEWLINE\[NEWLINEB_r (X) = \left\{ f \in X : \left\| f \right\| _{X} \leq r\right\} , \quad \quad r>0,NEWLINE\]NEWLINE in several function spaces, with particular emphasis on two spaces of functions of bounded variation and Hölder spaces.NEWLINENEWLINEIt is well known that imposing a global Lipschitz condition on composition operators leads to a strong degeneracy phenomenon in many function spaces (the generating function has to be an affine function with respect to the second variable). In contrast to this, the authors show that a local version of Banach's contraction mapping principle is less restrictive and applies to a large variety of nonlinear problems. As examples, they obtain existence and uniqueness results of solutions of the nonlinear integral equations of Hammerstein type NEWLINE\[NEWLINE f\left( s\right) =\int_{0}^{1}k\left( s,t\right) h\left( t,f\left( t\right) \right) dt+b\left( s\right) \quad \quad ( 0\leq s\leq 1 ) , NEWLINE\]NEWLINE where \(k: [ 0,1] \times [ 0,1 ] \rightarrow \mathbb{R}\) is a given kernel function, and of the nonlinear weakly singular Abel-Volterra equations NEWLINE\[NEWLINE f\left( s\right) -\int_{0}^{s} {{k\left( s,t\right) h\left( f\left( t\right) \right) } \over {\left| s-t\right| ^{\nu }}}\, dt = b\left( s\right) \quad \quad ( 0\leq s\leq 1 ) , NEWLINE\]NEWLINE where \(k:[ 0,1] \times [ 0,1 ] \rightarrow \mathbb{R}\) is continuous and \(0<\nu <1\).