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

[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.