International Journal of Nonlinear Analysis and Applications

Fixed point theorem on functional intervals for sum of two operators and application in ODEs

Document Type : Research Paper

Authors

Laboratory of Applied Mathematics, Faculty of Exact Sciences, University of Bejaia, 06000 Bejaia, Algeria

Abstract

In this paper, we present a generalization of the functional expansion-compression fixed point theorem developed by Avery et al. in [5] to the case of a k-set contraction perturbed by an operator T, where I -T is Lipschitz invertible. The arguments are based upon recent fixed point index theory in cones of Banach spaces. Next, we apply the obtained result to discuss the existence of a nontrivial positive solution to a nonautonomous second order boundary value problem.

Keywords

[1] D.R. Anderson and R.I. Avery, A Topological proof and extension of the Leggett-Williams fixed point theorem, Comm. Appl. Nonlinear Anal. 16 (2009), no. 4, 39–44.
[2] D.R. Anderson, R.I. Avery, and J. Henderson, Some fixed point theorems of Leggett-Williams type, Rocky Mt. J. Math. 41 (2011), 371–386.
[3] D.R. Anderson, R.I. Avery, J. Henderson, and  X. Liu, Operator compression-expansion fixed point theorem, Electron. J. Differ. Equ. 2011 (2011), 1–9.
[4] D.R. Anderson, R.I. Avery, and J. Henderson, Functional compression-expansion fixed point theorem of Leggett-Williams type, Electron. J. Differ. Equ. 2010 (2010), 1–9.
[5] R.I. Avery, D.R. Anderson, and J. Henderson, An Extension of the compression-expansion fixed point theorem of functional type, Electron. J. Differ. Equ. 2016 (2016), no. 253, 1–9.
[6] R.I. Avery, D.R. Anderson, and J. Henderson, Generalization of the functional omitted ray fixed point theorem, Comm. Appl. Nonlinear Anal. 25 (2018), no. 1, 39–51.
[7] J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60. Marcel Dekker, Inc., New York, 1980.
[8] J. Banas and M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, New Delhi:, Springer, 2014.
[9] S. Benslimane, S. Djebali, and K. Mebarki, On the fixed point index for sums of operators, Fixed Point Theory, 23 (2022), no. 1, 143-162.
[10] L. Benzenati, K. Mebarki, and R. Precup, A vector version of the fixed point theorem of cone compression and expansion for a sum of two operators, Nonlinear Stud. 27 (2020), no. 3, 563–575.
[11] A. Das, B. Hazarika, N.K. Mahato, and V. Parvaneh, Application of measure of noncompactness on integral equations involving generalized proportional fractional and Caputo-Fabrizio fractional integrals, Filomat 36 (2022), no. 17, 5885–5893.
[12] A. Das, B. Hazarika, V. Parvaneh, and M. Mursaleen, Solvability of generalized fractional order integral equations via measures of noncompactness, Math. Sci. 15 (2021), 241–251.
[13] A. Das, M. Paunovi´c, V. Parvaneh, M. Mursaleen, and Z. Bagheri, Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness, Demonstr. Math. 56 (2023), no. 1, 20220192.
[14] S. Djebali and K. Mebarki, Fixed point index theory for perturbation of expansive mappings by k-set contractions, Top. Meth. Nonli. Anal. 54 (2019), no. 2, 613–640.
[15] S.G. Georgiev and K. Mebarki, Existence of positive solutions for a class ODEs, FDEs and PDEs via fixed point index theory for the sum of operators, Comm. Appl. Nonlinear Anal. 23 (2019), no. 4, 16–40.
[16] S.G. Georgiev and K. Mebarki, Leggett-Williams fixed point theorem type for sums of two operators and application in PDEs, Differ. Equ. Appl. 13 (2021), no. 3, 321–344.
[17] S.G. Georgiev and K. Mebarki, A new multiple fixed point theorem with applications, Adv. Theory Nonlinear Anal. 5 (2021), no. 3, 393–411.
[18] S.G. Georgiev and K. Mebarki, On fixed point index theory for the sum of operators and applications in a class ODEs and PDEs, Gen. Topol. 22 (2021), no. 2, 259–294.
[19] K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1934), 301–335.
[20] A. Mouhous and K. Mebarki, Existence of fixed points in conical shells of a Banach space for sum of two operators and application in ODEs, Turk. J. Math. 46 (2022), no. 7, 2556–2571.
[21] R.W. Leggett and L.R. Williams Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), no. 4, 673–688.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 10
October 2023
Pages 127-137
  • Receive Date: 22 September 2022
  • Revise Date: 17 August 2023
  • Accept Date: 19 August 2023