On a fixed point theorem for the product of operators

Abstract

Abstract. In this paper, we study the existence of fixed points for the product of nonlinear operators. This kind of fixed point theorems is necessary in consideration of quadratic differential and integral problems. We emphasize a possible extension of the applicability of obtained theorems and consequently we prove the existence of fixed points for operators acting on some function spaces that are not necessarily Banach algebras. This result can be successfully applied to many quadratic problems.Mathematics Subject Classification. 46E30, 45G10, 47H30, 47N20.

Solvability of Chandrasekhar’s Quadratic Integral Equations in Banach Algebra

“…be three operators such that: 1) A and C are Lipschitzian with constants α and β respectively, 2) B is completely continuous, and, 3) x AxBy Cx x S = + ⇒ ∈ , for all y S ∈ .…”

“…Functional integral and differential equations of different types play an im-portant and a fascinating role in nonlinear analysis and finding various ap-plications in describing of several real world problems [2] [3] [4] [5] [6] [7] [8] [9].…”

Fixed point theorems in Fréchet algebras and Fréchet spaces and applications to nonlinear integral equations

“…A lot of these equations were considered in Banach algebra (cf. [1,5,6,9,10,14,15]) but only few were investigated in Fréchet algebra [8]. However, it seems that convenient environment for integral equations on unbounded interval R + are various Fréchet function spaces, which in the case of some types of the product integral equations, naturally lead to Fréchet algebras.…”

On Some Qualitative Properties of Integrable Solutions for Cauchy-type Problem of Fractional Order

“…We need to emphasize that the following Cauchy-type problem is also strictly related to quadratic equations (cf. [8])…”

“…Let us recall that the quadratic integral equations were discussed in many different functions spaces (see [3,6,7,8,20]) and have numerous applications in the theories of radiative transfer, neutron transport and in the kinetic theory of gases [3,6,7].…”