Revisiting Krasnoselskii’s fixed point theorem: Extensions and applications to operator equations

Abstract

Fixed point theory stands as a fundamental pillar in nonlinear functional analysis, being essential for proving the existence of solutions for nonlinear differential and integral equations. Krasnoselskii’s hybrid fixed point theorem, which combines the Banach contraction principle with Schauder’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 framework 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 incorporating the Hölder continuity condition, which is less restrictive than the standard Lipschitz condition. The unified theoretical framework led to the establishment of a comprehensive set of theorems and corollaries covering a wide class of operator equations such as: $A x^{(\rho 2)} B x^{\rho 1}+C x^{(\rho 3)} D x^{\rho 1}=x$. Our results provide less restrictive local existence conditions and wider applicability in the analysis of complex mathematical systems.

Keywords

Fixed point / Operator equations / Hölderian / Nonlinear functional / Lipschitzian / Integral equations

References

[1] Zguaid K, El Alaoui FZ. On the regional boundary observability of semilinear time-fractional systems with Caputo derivative. Int J Optimiz Control. 2023; 13(2): 161-170. https://doi.org/10.11121/ijocta.2023.1286

[2] Kalimuthu K, Muthuvel K. A study on the approximate controllability results of fractional stochastic integro-differential inclusion systems via sectorial operators. IJOCTA. 2023; 13(2): 193-204. https://doi.org/10.11121/ijocta.2023.1348

[3] Sakthivel R, Anandhi ER, Mahmudov NI. Approximate controllability of second-order systems with state-dependent delay. Numer Funct Anal Optim. 2008; 29(11-12): 1347-1362. https://doi.org/10.1080/01630560802580901

[4] Beyene MT, Firdi MD, Dufera TT. Existence and Ulam-Hyers stability results for Caputo-Hadamard fractional differential equations with non-instantaneous impulses. Bound Value Probl. 2025; 2025(1): 6. https://doi.org/10.1186/s13661-024-01958-9

[5] Al-Dosari AAM. Controllability of mild solution to Hilfer fuzzy fractional differential inclusion with infinite continuous delay. Fractal Fract. 2024; 8(4): 235. https://doi.org/10.3390/fractalfract8040235

[6] Almalahi M, Panchal SK. Existence and stability results of relaxation fractional differential equations with Hilfer-Katugampola fractional derivative. Adv Theory Nonlinear Anal Appl. 2020; 4(4): 299-315. https://doi.org/10.31197/atnaa.686693

[7] Kamsrisuk N, Ntouyas SK, Ahmad B, Samadi A, Tariboon J. Existence results for a coupled system of (k, φ)-Hilfer fractional differential equations with nonlocal integro-multi-point boundary conditions . AIMS Math. 2023; 8: 4079-4097. https://doi.org/10.3934/math.2023203

[8] Acar C, Öztürk V. Fixed point theorems for almost α-ψ-contractive mappings in F-metric spaces. Fund J Math Appl. 2024; 7(4): 203-211. https://doi.org/10.33401/fujma.1400093

[9] Yamaod O, Sintunavarat W, Cho YJ. Common fixed point theorems for generalized cyclic contraction pairs in b-metric spaces with applications. Fixed Point Theory Appl. 2015; 2015(1): 164. https://doi.org/10.1186/s13663-015-0409-z

[10] Huang H, Xu S. Fixed point theorems of contractive mappings in cone b-metric spaces and applications. Fixed Point Theory Appl. 2013; 2013(1): 112. https://doi.org/10.1186/1687-1812-2013-112

[11] Raj JCP, Xavier AA, Joseph JM. Common fixed point theorems under rational contractions in complex valued extended b-metric spaces. Int J Nonlinear Anal Appl. 2022; 13(1): 3479-3490. https://doi.org/10.22075/IJNAA.2020.20025.2118

[12] Majid NA, Jumaili AMA, Ng ZC, Lee SK. New results of fixed point theorems and their applications in complete complex D*-Metric spaces. J Funct Spaces. 2024; 2024(1): 5532624. https://doi.org/10.1155/2024/5532624

[13] Zada MB, Sarwar M, Tunc C. Fixed point theorems in b-metric spaces and their applications to non-linear fractional differential and integral equations. J Fixed Point Theor Appl. 2018; 20(1): 25. https://doi.org/10.1007/s11784-018-0510-0

[14] Kasinathan D, Kasinathan R, Kasinathan R, Chalishajar D. Exponential stability for higher-order impulsive fractional neutral stochastic integro-delay differential equations with mixed Brownian motions and non-local conditions. Int J Optimiz Control: Theor Appl. 2025; 15(1): 103-122. https://doi.org/10.36922/ijocta.1524

[15] Ibrahim AG, Elmandouh AA. Existence and stability of solutions of ψ-Hilfer fractional functional differential inclusions with non-instantaneous impulses. AIMS Math. 2021; 6(10): 10802-10832. https://doi.org/10.3934/math.2021628

[16] Arab R, Zare K. New fixed point results for rational type contractions in partially ordered b-metric spaces. Int J Anal Appl. 2016; 10(2): 64-70. https://www.etamaths.com/index.php/ijaa/article/view/626

[17] Burton TA, Kirk C. A fixed point theorem of Krasnoselskii-Schaefer type. Math Nachr. 1998; 189(1): 23-31. https://doi.org/10.1002/mana.19981890103

[18] Granas A, Dugundji J. Fixed Point Theory. Springer-Verlag; 2003.

[19] Lau ATM, Mah PF. Fixed point property for Banach algebras associated to locally compact groups. J Funct Anal. 2010; 258(2): 357-372. https://doi.org/10.1016/j.jfa.2009.07.011

[20] Dhage BC. Multivalued operators and fixed-point theorems in Banach algebras II. Comput Math Appl. 2004; 48(9-10): 1461-1476. https://doi.org/10.1016/j.camwa.2004.07.004

[21] Dhage BC. Local fixed point theory for the sum of two operators in Banach spaces. Fixed Point Theor. 2003; 4(1): 49-60. https://scispace.com/pdf/local-fixed-point-theory-for-the-sum-of-two-operators-in-2blkt5vl10.pdf

[22] Dhage BC. On a fixed point theorem in Banach algebras with applications. Appl Math Lett. 2005; 18(3): 273-280. https://doi.org/10.1016/j.aml.2003.10.014

[23] Xiang T, Georgiev SG. Noncompact-type Krasnoselskii fixed point theorems and their applications. Math Methods Appl Sci. 2016; 39(4): 833-863. https://doi.org/10.1002/mma.3525

[24] Burton TA, Zhang B. Fractional equations and generalizations of Schaefer’s and Krasnoselskii’s fixed point theorems. Nonlinear Anal Theory Method Appl. 2012; 75(18): 6485-6495. https://doi.org/10.1016/j.na.2012.07.022

[25] Julius H. Fixed product preserving mappings on Banach algebras. J Math Anal Appl. 2023; 517(1): 126615. https://doi.org/10.1016/j.jmaa.2022.126615

[26] Agarwal RP, Metwali MMA, O’Regan D. On existence and uniqueness of L₁-solutions for quadratic integral equations via a Krasnoselskii type fixed point theorem. Rocky Mount J Math. 2018; 48: 1743-1762. https://doi.org/10.1216/RMJ-2018-48-6-1743

[27] Nashine HK, Ibrahim RW, Agarwal RP. Moments solution of fractional evolution equation found by new Krasnoselskii type fixed point theorems. Fixed Point Theor. 2021; 22: 263-278. https://doi.org/10.24193/fpt-ro.2021.1.19

[28] Park S. Generalizations of the Krasnoselskii fixed point theorem. Nonlinear Anal Theor Methods Appl. 2007; 67(12): 3401-3410. https://doi.org/10.1016/j.na.2006.10.024

[29] Mennouni A. Two projection methods for skew-hermitian operator equations. Math Comput Model. 2012; 55(3-4): 1649-1654. https://doi.org/10.1016/j.mcm.2011.10.078

[30] Xu S, Radenovi S. Fixed point theorems of generalized Lipschitz mappings on cone metric spaces over Banach algebras without assumption of normality. Fixed Point Theor Appl. 2014; 2014(1): 102. https://doi.org/10.1186/1687-1812-2014-102

[31] Arab R, Rabbani M, Mollapourasl R. The solution of a nonlinear integral equation with deviating argument based the on fixed point technique. Appl Comput Math. 2015; 14(1): 38-49. http://acmij.az/view.php?lang=az&menu=journal&id=374

[32] Hussain N, Taoudi MA. Krasnosel’skii-type fixed point theorems with applications to Volterra integral equations. Fixed Point Theor Appl. 2013; 2013(1): 196. https://doi.org/10.1186/1687-1812-2013-196

[33] Alsaadi A, Kazemi M, Metwali MM. On generalization of Petryshyn’s fixed point theorem and its application to the product of n-nonlinear integral equations. AIMS Math. 2023; 8(12): 30562-30573. https://doi.org/10.3934/math.20231562