Abstract
The purpose of this work is to present two new notions of \(\mu \)-set contraction of a bounded subset of a Banach space and establish some fixed point and coupled fixed point results in the direction of Darbo (Rend Sem Math Univ Padova 4:84–92, 1995). We apply our work to get existence of solutions to nonlinear functional-integral equations followed by an illustration. Our work generalizes many existing results in the literature.
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References
Aghajani, A., Banaś, J., Sabzali, N.: Existence of solution for a class of nonlinear Volterra singular integral equations. Comput. Math. Appl. 62, 1215–1227 (2011)
Aghajani, A., Banaś, J., Sabzali, N.: Some generalizations of Darbo fixed point theorem and applications. Bull. Belg. Math. Soc. Simon Stevin. 20(2), 345–358 (2013)
Aghajani, A., Sabzali, N.: Existence of coupled fixed points via measure of noncompactness and applications. J. Nonlinear Convex Anal. 15(5), 941–952 (2014)
Aghajani, A., Allahyari, R., Mursaleen, M.: A generalization of Darbos theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 260, 68–77 (2014)
Altun, I., Turkoglu, D.: A fixed point theorem for mappings satisfying a general contractive condition of operator type. J. Comput. Anal. Appl. 9(1), 9–14 (2007)
Arab, R.: Some fixed point theorems in generalized darbo fixed point theorem and the existence of solutions for system of integral equations. J. Korean Math. Soc. 52(1), 125–139 (2015)
Arab, R.: The existence of fixed points via the measure of noncompactness and its application to functional-integral equations. Mediterr. J. Math. 13, 759–773 (2016)
Banaś, J.: Measures of noncompactness in the space of continuous tempered functions. Demonstr. Math. 14, 127–133 (1981)
Banaś, J., Goebel, K.: Measures of noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Dekker, New York (1980)
Burton, T.A.: Krasnoselskii’s inversion principle and fixed points. Nonlinear Anal. 30, 3975–3986 (1997)
Burton, T.A., Kirk, C.: A fixed point theorem of Krasnoselskii–Schaefer type. Math. Nachrichten 189, 23–31 (1998)
Darbo, G.: Punti uniti in transformazioni a condominio non compatto. Rend. Sem. Math. Univ. Padova 4, 84–92 (1995)
Geraghty, M.: On contractive mappings. Proc. Am. Math. Soc. 40, 604–608 (1973)
Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces, Mathematics and Its Applications, vol. 373. Kluwer Academic Publishers, Dordrecht (1996)
Kuratowski, K.: Sur les espaces completes. Fund. Math. 15, 301–309 (1930)
Mizoguchi, N., Takahashi, W.: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141(1), 177–188 (1989)
Mursaleen, M., Mohiuddine, S.A.: Applications of measures of noncompactness to the infinite system of differential equations in \(l_p\) spaces. Nonlinear Anal. TMA 75, 2111–2115 (2012)
Mursaleen, M., Rizvi, S.M.H.: Solvability of infinite systems of second order differential equations in \(c_0\) and \(\ell _1\) by Meir-Keeler condensing operators. Proc. Am. Math. Soc. 144(10), 4279–4289 (2016)
Reich, S.: Fixed points in locally convex spaces. Math. Z. 125(1), 17–31 (1972)
Reich, S.: Fixed points of condensing functions. J. Math. Anal. Appl. 41, 460–467 (1973)
Acknowledgements
The authors express their gratitude to the referees for careful reading of the manuscript. The first author is thankful to the United States-India Education Foundation, New Delhi, India, and IIE/CIES, Washington, DC, USA, for Fulbright-Nehru PDF Award (no. 2052/FNPDR/2015).
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Nashine, H.K., Arab, R. Existence of solutions to nonlinear functional-integral equations via the measure of noncompactness. J. Fixed Point Theory Appl. 20, 66 (2018). https://doi.org/10.1007/s11784-018-0546-1
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DOI: https://doi.org/10.1007/s11784-018-0546-1