Monotonic solutions of a quadratic integral equation of Volterra type
The paper concerns the existence of nondecreasing solutions of the nonlinear quadratic Volterra integral equation of the form \[ x(t)=a(t)+x(t)\int_0^tv(t,\tau,x(\tau))d\tau, \quad t\in I=[0,T] \] in the space \(C(I)\) consisting of all real functions defined and continuous on the interval \(I\), where \(a(t)\) and \(v(t,\tau,x)\) are given while \(x=x(t)\) is an unknown function. The proof of the main result is based on the Darbo type fixed point theorem formulated in terms of axiomatic measures of noncompactness introduced by \textit{J. Banás} and \textit{K. Goebel} [Measures of noncompactness in Banach spaces, Lect. Notes Pure Appl. Math. 60 (1980; Zbl 0441.47056)]. The authors illustrate the main result and compare its assumptions by giving suitable interesting examples.
nonlinear quadratic Volterra integral equation
scientific article; zbMATH DE number 2075459
scientific article