UPPER AND LOWER BOUNDS FOR SOLUTIONS OF NONLINEAR VOLTERRA CONVOLUTION INTEGRAL EQUATIONS WITH POWER NONLINEARITY

The Volterra nonlinear integral equation ϕm(x) = a(x) ∫ x 0 k(x − t)b(t)ϕ(t)dt+ f(x), 0 < x < d ≤ ∞, with m> 1 and real nonnegative functions a(x), k(u), b(t) and f(x) is studied. In the general case some upper bounds are given for the averages 1 x ∫ x 0 ϕ(t)dt. In the case when a(x), k(u), b(t) and f(x) have power lower estimates near the origin, lower power type bounds directly for solutions ϕ(x) are given. Conditions for the uniqueness of the solution in weighted space of continuous functions are proved. Particular cases of the above equation are specially considered