@article {104722,
	title = {Revisiting Krasnoselskii\&rsquo;s fixed point theorem: Extensions and applications to operator equations},
	journal = {International Journal of Optimization and Control: Theories \& Applications},
	volume = {16},
	number = {1},
	year = {2026},
	pages = {191-204},
	abstract = {
	Fixed point theory stands as a fundamental pillar in nonlinear functional anal ysis, being essential for proving the existence of solutions for nonlinear dif ferential and integral equations. Krasnoselskii{\textquoteright}s hybrid fixed point theorem, which combines the Banach contraction principle with Schauder{\textquoteright}s theorem, is a pivotal contribution. Recent efforts have focused on refining and relaxing the conditions of this theorem. This study aims to extend the theoretical frame work of Krasnoselskii-type fixed point theorems to address a broader and more general class of nonlinear operator equations within a Banach algebra setting. It also seeks to establish rigorous conditions for the existence (and uniqueness, where possible) of solutions. The approach involves developing local variants of the classic Krasnoselskii fixed point theorems. We performed a comparative analysis of previous studies, introduced modifications to the operator equations to relax restrictive assumptions, and theoretically generalized the theorems to accommodate a complex structure involving four operators. To validate the results, they were applied to a nonlinear functional integral equation within the Banach space C[0,1]. We successfully generalized existing results by in corporating the H{\textasciidieresis}older continuity condition, which is less restrictive than the standard Lipschitz condition. The unified theoretical framework led to the es tablishment of a comprehensive set of theorems and corollaries covering a wide class of operator equations such as : Ax(ρ2)Bxρ1 + Cx(ρ3)Dxρ1 = x. Our re sults provide less restrictive local existence conditions and wider applicability in the analysis of complex mathematical systems.

},
	url = {https://article.accscience.com/journal/IJOCTA/16/1/ijocta025350147/ijocta025350147.pdf},
	author = {Yakoub, Abir and MENNOUNI, ABDELAZIZ and P. Agarwal,  Ravi}
}
