Measures of noncompactness in the study of asymptotically stable and ultimately nondecreasing solutions of integral equations
The authors introduce a new class of measures of noncompactness related to asymptotic stability and ultimate monotonicity in the space \(BC(\mathbb R)^+\). By using those measures of noncompactness and a fixed point theorem of Darbo type, they prove that the following functional integral equation \[ x(t)=m(t)+f(t,x(t))\int^t_0 g(t,\tau)h(\tau,x(\tau))\,d\tau,\quad t\geq 0, \] has at least one solution \(x\in BC(\mathbb R)^+\), which is nonnegative, asymptotically stable and ultimately nondecreasing. A concrete example is given to illustrate the abstract results.
scientific article; zbMATH DE number 5770967
scientific article