Abstract
In this paper, we prove some generalizations of Darbo’s fixed point theorem for multivalued mappings by considering a measure of noncompactness which does not necessarily have the maximum property. Moreover, we prove some coupled fixed point theorems for multivalued mappings. Our results generalize, prove and extend well-known results in the subject. An application to solve a nonlinear system of integral inclusions is given.
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References
Aghajani, A., Banas, J., Sabzali, N.: Some generalizations of Darbo fixed point theorem and applications. Bull. Belg. Math. Soc. 20(2), 345–358 (2013)
Aghajani, A., Allahyari, R., Mursaleen, M.: A generalization of Darbo’s theorem with application to the solvability of system of integral equation. J. Comput. App. Math. 260, 68–77 (2014)
Akutsah, F., Mebawondu, A.A., Narain, O.K.: Existence of solution for a volterra type integral equation using darbo-typef-contraction. Adv. Math. Sci. J. 10(6), 2687–2710 (2021)
Aghajani, A., Mursaleen, M., Shole Haghighi, A.: Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness. Acta. Math. Sci. 35B(3), 552–566 (2015)
Altun, I., Minak, G., Daǧ, H.: Multivalued \(F\)-contractions on complete metric space. J. Nonlinear Convex Anal. 16(4), 659–666 (2015)
Banas, J., Goebel, K.: Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Math, vol. 60. Marcel Dekker, New York (1980)
Darbo, G.: Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Univ. Padova 24, 84–92 (1955)
Deimling, K.: Multivalued Differential Equations. W. de Gruyter, Berlin (1992)
Dhage, B.C.: Some generalizations of multivalued version of Schauder’s fixed point theorem with applications. CUBO Math. J. 12(03), 139–151 (2010)
Dhompongsa, S., Yingtaweesittikul, H.: Diametrically contractive multivalued mapping. Fixed Point Theory Appl. (2007)
Garcia-Falset, J., Latrach, K.: On Darbo–Sadovskii’s fixed point theorems type for abstract measures of (weak) noncompactness. Bull. Belg. Math. Soc. 22, 797–812 (2015)
Gabeleh, M., Malkowsky, E., Mursaleen, M., Rakočević, V.: A new survey of measures of noncompactness and their applications. Axioms 11(6), 299 (2022)
Ghiocel, M., Petrusel, A., Petrusel, G.: Topic in Nonlinear Analysis and Applications to Mathematical Economics. Cluj-Napoca (2007)
Kakutani, S.: A generalization of Brouwer’s fixed point theorems. Duke Math. J. 8, 457–459 (1941)
Kandilakis, D., Papageorgiou, N.S.: On the properties of the Aumann integral with applications to differential inclusions and control systems. Czechoslov. Math. J. 39, 1–15 (1989)
Kelley, J.L.: General Topology. Van Nostrand, New York (1955)
Kuratowski, K.: Sur les espaces completes. Fund. Math. 1(15), 301–309 (1930)
Karakaya, V., Sekman, D.: A new type of contraction via measure of non-compactness with an application to Volterra integral equation. Publ. Inst. Math. (Beograd) (N.S.) 111(125), 111–121 (2022)
Nasiri, H., Roshan, J.R., Mursaleen, M.: Solvability of system of Volterra integral equations via measure of noncompactness. Comput. Appl. Math. 40(5), 166, 25 (2021)
O’Regan, D.: Fixed point theory for closed multifunctions. Arch. Math. (Brno) 34, 191–197 (1998)
Piri, H., Kumam, P.: Some fixed point theorems concerning F-contraction in complete metric spaces. Fixed Point Theory Appl. (2014)
Roshan, J.R., Nashine, H.K.: Fixed point theorem via measure of noncompactness and application to Volterra integral equations in Banach algebras. J. Math. Ext. 13, 91–116 (2019)
Sadovskii, B.N.: On a fixed point principle. Funkt. Anal. 1(2), 151–153 (1967)
Secelean, N.A.: Iterated function system consisting of F-contraction. Fixed Point Theory Appl. (2013) 1(2), 151–153 (1967)
Vetro, C., Vetro, F.: The Class of F-contraction Mappings with a Measure of Noncompactness. Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, pp. 297–331. Springer, Singapore (2017)
Xu, H.K.: Diametrically contractive mappings. Bull. Aust. Math. Soc. 70, 463–468 (2004)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Part I. Springer, Berlin (1985)
Zada, M.B., Sarwar, M., Abdeljawad, T., Jarad, F.: Generalized Darbo-type F-contraction and F-expansion and its applications to a nonlinear fractional-order differential equation. J. Funct. Spaces 2020, 4581035 (2020)
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Belhadj, M., Boumaiza, M. Fixed point theory for F-set-contraction multimaps and application to a system of integral inclusions. Afr. Mat. 35, 3 (2024). https://doi.org/10.1007/s13370-023-01146-5
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DOI: https://doi.org/10.1007/s13370-023-01146-5