Document Type : Original Article


Universidad Nacional de Educacin a Distancia (UNED), Departamento de Matemaicas. 03202 Elche (Alicante), Spain


In this paper, under suitable conditions and by using the so-called degree of nondensifiability (DND), we provide sufficient conditions for the existence of a common fixed point for two commuting self-mappings defined into a non-empty, bounded, closed and convex subset of a Banach space. Our main result generalizes a Darbo-type fixed point theorem based on the DND. To illustrate the differences between our results and a known common fixed point result for two commuting self-mappings due to Jungck or others based on the measures of noncompactness, we provide some examples.


[1] R. R. Akhmerov, M. I. Kamenski˘ı, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiı, Measures of Noncompactness and Condensing Operators, Operator Theory: Advances and Applications, vol. 55, Birkhauser Verlag, Basel, 1992.
[2] M. A. Alghamdi, S. Radenovic and N. Shahzad, On some generalizations of commuting mappings, Abstr. Appl. Anal., 2012 (2012) Article ID 952052.
[3] J. M. Ayerbe Toledano, T. Domınguez Benavides, and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory: Advances and Applications, vol. 99, Birkhauser Verlag, Basel, 1997.
[4] S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fundam. Math., 3 (1922), 133-181.
[5] N. H. Bouzara and V. Karakaya, Some common fixed point results for commuting k-set contraction
mappings and their application, Adv. Fixed Point Theory 6 (2016), 224–240.
[6] W. M. Boyce, Commuting functions with no common fixed point, Trans. Amer. Math. Soc., 137 (1969), 77-92.
[7] Y. Cherruault and G. Mora, Optimisation Globale, Th´eorie des Courbes α-denses, Economica, Paris 2005.
[8] C. Derbazi, Z. Baitiche, M. Benchohra, and Y Zhou, Boundary value problem for ψ-Caputo fractional differential equations in Banach spaces via densifiability techniques, Mathematics 10(1) (2022), 153.
[9] G. Garcıa, Existence of solutions for infinite systems of ordinary differential equations by densifiability techniques, Filomat 32 (2018), 3419-3428.
[10] G. Garcıa, Solvability of initial value problems with fractional order differential equations in Banach spaces by α-dense curves, Fract. Calc. Appl. Anal., 20 (2017), 646-661.
[11] G. Garcıa and G. Mora, Projective limits of generalized scales of Banach spaces and pplications, Ann. Funct. Anal., 13 (2022), 76.
[12] G. Garcıa and G. Mora, A fixed point result in Banach algebras based on the degree of  ondensifiability and applications to quadratic integral equations, J. Math. Anal. Appl., 472 (2019),
[13] G. Garcıa and G. Mora, The degree of convex nondensifiability in Banach spaces, J. Convex Anal., 22 (2015), 871–888.
[14] A. Hajji, A generalization of Darbo’s fixed point and common solutions of equations in Banach spaces, Fixed Point Theory Appl., 63 (2013), 2013.
[15] J. P. Huneke, On common fixed points of commuting continuous functions on an interval, Trans. Amer. Math. Soc., 139 (1969), 371–381.
[16] M. El Harrak and A. Hajji, A new common fixed point theorem for three commuting mappings, Axioms 9 (2020), 105.
[17] H. I,sik, B. Mohammadi, M. R. Haddadi and V. Parvaneh, On a new generalization of Banach contraction principle with application, Mathematics 7 (2019), 8 pages.
[18] G. Jungck, Commuting mappings and fixed points, Amer. Math. Montly 83 (1976), 261-263.
[19] N. Khodabakhshi and S. M. Vaezpour, Common fixed point theorems via measure of  noncompactness, Fixed Point Theory 17 (2016), 381-386.
[20] B. Mohammadi, V. Parvaneh, and H. Hosseinzadeh, Some Wardowski-Mizogochi-Takahashi-Type generalizations of the multi-valued version of Darbo’s fixed point theorem with applications, Int. J. Nonlinear Anal. Appl. doi: 10.22075/ijnaa.2023.23057.2470
[21] B. Mohammadi, A.A. Shole Haghighi, M. Khorshidi. M. De la Sen, and V. Parvaneh, Existence
of solutions for a system of integral equations using a generalization of Darbo’s fixed point theorem, Mathematics 8 (2020), 492.
[22] G. Mora, The Peano curves as limit of α-dense curves, Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Math. RACSAM 99 (2005), 23-28.
[23] G. Mora and Y. Cherruault, Characterization and generation of α-dense curves, Comput. Math. Appl., 33 (1997), 83-91.
[24] G. Mora and J.A. Mira, Alpha-dense curves in infinite dimensional spaces, Int. J. Pure and Appl. Math. 5 (2003), 257-266.
[25] G. Mora and D.A. Redtwitz, Densifiable metric spaces, Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Math. RACSAM 105 (2011), 71-83.
[26] H. Sagan, Space-filling Curves, Springer-Verlag, New York, 1994.
[27] J. Vujakovic, L. Ljajko, S. Radojevic, and S. Radenovic, On some new Jungck–Fisher–Wardowski type fixed point results, Symmetry 12 (2020).
[28] A. Wisnicki, An example of a nonexpansive mapping which is not 1-ball-contractive, Ann. Univ. Mariae Curie-Sk lodowska Sect. A LIX (2005), 141-146.