The Gronwall inequality has an important role in many differential and integral equations.
There have been many papers studying integral equations with convolution type or singular type, see, for example, Litvinchuc [8], Li [9, 10], De-Bonis [11], Du [12], Jiang [13], among which a series of valuable achievements have been obtained.
There have been many papers studying singular integral equations and a relatively complete theoretical system is almost formed (see, e.g., [1 6]).
Integral equations are used as mathematical models for many physical situations, and integral equations also occur as reformulations of other mathematical problems, such as partial differential equations and ordinary differential equations.
Due in part to the immediate application of this interesting result to differential and integral equations, many researchers tried to obtain the conclusion of [2] under weaker assumptions; for example, see [1, 5], and [10].
The integral equations have many applications in mechanics, physics, engineering, biology, economics, and so on.
Integral equations govern many topics in several disciplines, such as applied mechanics, population dynamics, and economy.
Thus it is interesting to obtain numerical solutions of Hammerstein integral equations and many methods have been proposed [8, 39 46].
As we know, the solutions of many applied problems lead to integral equations, and these equations can be reduced to Abel's integral equation.
The functional integral equations describe many physical phenomena in various areas of natural science, mathematical physics, mechanics and population dynamics [1 4].
