Author details

Céleste" by Poincaré and Fichot [9], and to the Hilbert works on contour and

operators of subalgebras of this special Banach algebra (e.g., the algebra of the bounded operators B Xð Þ, and K Xð Þ,the ideal of compact operators) [10]. For example, consider the bounded operators in a Banach space with closed range and with kernel and co-kernel of finite dimension. These are called Fredholm operators and are the operators that give invertible elements of the Calkin algebra. The operators of the Calkin algebra radical are called Reisz operators and can be char-

research. However, also the integral equations research has developed more the functional analysis, considering the function theory, integral transforms and the

For other side, a general resolution method to the singular integral equations cannot be given in detail on the effective resolution of these equations, because is followed the research on a general methods to this integral equations class through certain special functions and integral transforms, which are of diverse and varied nature [5, 11]. In fact, the resolution of singular integrals considering the Hilbert transform and the Fourier transform [11] has been in the last years strongly researched. Here we only consider the intimate relation between this singular integral equations theory with the analytic functions theory and special functions related with the regularity and completeness of the solutions required.

One of the new developments on nonlinear integral equations are followed to

where K tð Þ ; s and f tð Þ ; s are given functions, while ωð Þt is the unknown function. Hammerstein considered for K tð Þ ; s , a symmetric and positive Fredholm kernel. This last condition establishes that all their eigenvalues are positive. Thus, the function f tð Þ ; s is continuous and satisfies j j f tð Þ ; s ≤ C1j js þ C2, where C<sup>1</sup> and C<sup>2</sup> are

kernel K tð Þ ; s ;then, the Hammerstein integral equation has at least one continuous solution. Also are considered certain observations on the no decreasing of the function f tð Þ ; s , on s, considering fix t, from the interval ð Þ a; b :The Hammerstein's

where the positive constant C is smaller than the first eigenvalue of the kernel K tð Þ ; s : A solution of the Hammerstein equation may be constructed by the method of successive approximation. In regard to this point, many approximation methods are designed to solve these integral equations and other nonlinear integral equations. Also of interest are the recent developments on Hammerstein-Volterra integral equations:

K tð Þ ; s f sð Þ ;ωð Þs ds ¼ 0, a≤ t≤ b (2)

j j f tð Þ� ; s<sup>1</sup> f tð Þ ; s<sup>2</sup> ≤ C sj j <sup>1</sup> � s<sup>2</sup> , (3)

K tð Þ ; s f sð Þ ;ωð Þs ds, 0≤ t≤ 1 (4)

the Hammerstein integral equations [12], which is written as

positive constants and C<sup>1</sup> is smaller than the first eigenvalue of the

equation cannot have more than one solution. This property holds also if

ðb a

ωðÞþt

f tðÞ¼ ωð Þþt

ðt 0

acterized spectrally and in terms of the dimensions of Rec ð Þ <sup>λ</sup><sup>I</sup> � <sup>T</sup> <sup>k</sup> � �

A possible treatment, bringing the Cauchy ideas together with Banach algebras,

, etc. Very questions on these algebras are motive of modern

K Xð Þ , on a Banach space X, likewise as the

,

boundary problems of the analytic functions theory.

is the consideration of the Calkin algebra B Xð Þ

Recent Advances in Integral Equations

ker ð Þ <sup>λ</sup><sup>I</sup> � <sup>T</sup> <sup>k</sup> � �

Kernels study in a wide form.

f tð Þ ; s satisfies the condition

4

Francisco Bulnes1,2

1 Research Department in Mathematics and Engineering, TESCHA, Mexico

2 IINAMEI, Mexico

\*Address all correspondence to: francisco.bulnes@tesch.edu.mx

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
