Measures of noncompactness in the study of solutions of nonlinear differential and integral equations
The paper presents a survey on some existence results for nonlinear differential and integral equations obtained by the author and his co-workers during last years. No proofs are given. First of all, the axiomatic measures of noncompactness in a Banach space \(E\) are defined and a Darbo-type fixed point theorem is recalled. Then, after the definition of measure of noncompactness in the space \(BC(\mathbb R_+)\), it is illustrated how the technique of measures of noncompactness can be applied to provide existence theorems for nonlinear functional integral equations in the space \(BC(\mathbb R_+)\). Finally, the author considers an existence result for an infinite system of ordinary differential equations in the classical sequence space \(\ell^\infty\). The system is handled as a special case of a Cauchy problem of the form \(x'=f(t,x)\), \(x(0)=x_0\), and it is treated by using the sublinear measure of noncompactness in the space \(\ell^\infty\).
infinite system of ordinary differential equations
scientific article; zbMATH DE number 6137125