Upper and lower bounds for solutions of nonlinear Volterra convolution integral equations with power nonlinearity (Q5939768)

The nonlinear Volterra integral equation NEWLINE\[NEWLINE\varphi^m(x)= a(x) \int_0^x k(x-t) b(t) \varphi(t) dt+ f(x) \quad (0< x< d\leq \infty) \tag{1}NEWLINE\]NEWLINE is studied. Here it is assumed that \(m>1\) and \(a(x)\), \(k(u)\), \(b(t)\) and \(f(x)\) are real nonnegative functions. The main result is concerned with some upper bounds of the average \((1/x) \int_0^x\varphi(t) dt\) of the solution \(\varphi(t)\) of (1). A lot of particular cases of \(a(x)\), \(k(u)\), \(b(t)\) and \(f(x)\) are considered in detail.