On approximation properties of non-convolution type nonlinear integral operators

Abstract

In the present paper we state some approximation theorems concerning pointwise convergence and its rate for a class of non-convolution type nonlinear integral operators of the form: $$ T_\lambda \left( {f;x} \right) = \int\limits_A^B {K_\lambda \left( {t,x,f\left( t \right)} \right)dt , x \in } \left\langle {a,b} \right\rangle ,\lambda \in \Lambda $$. In particular, we obtain the pointwise convergence and its rate at some characteristic points x 0 of f as (x, λ) ar(x 0, λ0) in L 1 〈 A,B 〉, where 〈 a,b 〉 and 〈 A,B 〉 are is an arbitrary intervals in R, Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology.The results of the present paper generalize several ones obtained previously in the papers [19]–[23].