Twin Fixed Point Theorem for Operator Sums with Application to a Discrete ϕ-Laplacian Problem
Keywords:
Boundary value problems, Cone, Difference equations , Fixed point theorem, Sum of operators, ϕ-LaplacianAbstract
Fixed point theorems on cones of functional type provide powerful tools for studying positive solutions to nonlinear problems, particularly in settings where standard methods fail to capture multiplicity or positivity. In this work, we establish an extension of the Avery-Henderson twin fixed point theorem to the setting of operator sums by employing fixed point index theory on cones. This new variant significantly broadens the applicability of fixed point techniques to problems involving composite operator structures. As an application, we establish the existence of at least two positive solutions for a discrete boundary value problem involving the ϕ-Laplacian. The result not only generalizes existing theory but also addresses the challenge of establishing multiplicity in nonlinear difference equations, which arise in models of population dynamics, mechanical systems, and network flows.
MSC 2020 Classifications: 47H10, 39A27.
References
[1] Krasnoselskii MA. Positive Solutions of Operator Equations. Groningen: Noordhoff; 1964
[2] Deimling K. Nonlinear Functional Analysis. Berlin: Springer-Verlag; 1985.
[3] Guo D, Lakshmikantham V. Nonlinear Problems in Abstract Cones. Boston: Academic Press; 1988.
[4] Guo D, Cho YJ, Zhu J. Partial Ordering Methods in Nonlinear Problems. Jinan: Shandong Sci. Technol. Publ. Press; 1985
[5] Leggett RW, Williams LR. Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana Univ Math J. 1979;28:673-688.
[6] Anderson DR, Avery RI, Henderson J. Functional expansion-compression fixed point theorem of Leggett-Williams type. Electron J Differ Equ. 2010;2010(63):1-9.
[7] Anderson DR, Avery RI. Fixed point theorem of cone expansion and compression of functional type. J Differ Equ Appl. 2002;8:1073-1083.
[8] Avery RI, Henderson J, ORegan D. Dual of the compression-expansion fixed point theorems. Fixed Point Theory Appl. 2007;2007:Article ID 90715, 11 pages.
[9] Guo D. A new fixed point theorem. Acta Math Sinica. 1981;24:444-450.
[10] Guo D. Some fixed point theorems on cone maps. Kexue Tongbao. 1984;29:575-578.
[11] Sun J, Zhang G. A generalization of the cone expansion and compression fixed point theorem and applications. Nonlinear Anal. 2007;67:579-586.
[12] Avery RI, Henderson J. Two positive fixed points of nonlinear operators on ordered Banach spaces. Commun Appl Nonlinear Anal. 2001;8(1):27-36.
[13] Djebali S, Mebarki K. Fixed point index theory for perturbation of expansive mappings by k-set contractions. Topol Methods Nonlinear Anal. 2019;54(2):613-640.
[14] Benslimane S, Djebali S, Mebarki K. On the fixed point index for sums of operators. Fixed Point Theory. 2022;23(1):143-162.
[15] Benslimane S, Georgiev SG, Mebarki K. Multiple nonnegative solutions for a class of fourth-order BVPs via a new topological approach. Adv Theory Nonlinear Anal Appl. 2022;6(3):390-404.
[16] Benzenati L, Mebarki K. Multiple positive fixed points for the sum of expansive mappings and k-set contractions. Math Methods Appl Sci. 2019;42(13):4412-4426.
[17] Benzenati L, Mebarki K, Precup R. A vector version of the fixed point theorem of cone compression and expansion for a sum of two operators. Nonlinear Stud. 2020;27(3):563-575.
[18] Georgiev SG, Mebarki K. On fixed point index theory for the sum of operators and applications in a class of ODEs and PDEs. Gen Topol. 2021;22(2):259-294.
[19] Georgiev SG, Mebarki K. A new multiple fixed point theorem with applications. Adv Theory Nonlinear Anal Appl. 2021;5(3):393411.
[20] Georgiev SG, Mebarki K. Leggett-Williams fixed point theorem type for sums of two operators and applications in PDEs. Differ Equ Appl. 2021;13(3):321-344.
[21] Georgiev SG, Mebarki K. Existence of positive solutions for a class of ODEs, FDEs and PDEs via fixed point index theory for the sum of operators. Commun Appl Nonlinear Anal. 2019;26(4):16-40.
[22] Banas J, Goebel K. Measures of noncompactness in Banach spaces. vol. 60 of Lecture Notes in Pure and Appl. Math. New York: Marcel Dekker; 1980.
[23] Xiang T, Georgiev SG. Noncompact-type Krasnoselskii fixed-point theorems and their applications. Math Methods Appl Sci. 2016;39(4):833-863.
[24] Mebarki K, Georgiev SG, Djebali S, Zennir K. Fixed Point Theorems with Applications. Chapman and Hall/CRC; 2023.
[25] Liu Y, Ge W. Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator. J Math Anal Appl. 2003;278(2):551-561.
