Abstract
In this paper, we prove some fixed point theorems for the nonlinear operator A · B + C in Banach algebra. Our fixed point results are obtained under a weak topology and measure of weak noncompactness; and we give an example of the application of our results to a nonlinear integral equation in Banach algebra.
Highlights
Integral equations are involved in various scientific problems such as transport theory, the theory of radiative transfer, biomathematics, etc
In 2014, Banas et al [8] proved some existence results of operator equations under the weak topology using the measure of weak noncompactness
Let Ω be a nonempty, bounded, closed, and convex subset of a Banach algebra X and ω be a subadditive MWNC on X
Summary
Integral equations are involved in various scientific problems such as transport theory, the theory of radiative transfer, biomathematics, etc (see [1,2,3,4,5,6]). The problems of the existence of solutions for an integral equation can be resolved by searching fixed points for nonlinear operators in a Banach algebra. The history of fixed point theory in Banach algebra started in 1977 with R.W. Legget [12], who considered the existence of solutions for the equation:. The study of nonlinear integral equations in Banach algebra via fixed point theory was in initiated by B.C. Dhage [15]. In 2014, Banas et al [8] proved some existence results of operator equations under the weak topology using the measure of weak noncompactness. We use the measure of noncompactness to prove some fixed point results for a nonlinear operator of type AB + C in a Banach algebra. We discuss the existence of solutions for an abstract nonlinear integral equation in the Banach algebra C ([0, 1], X ); and an example of a nonlinear integral equation in the Banach algebra C ([0, 1], R)
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