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A generalization of Darbo's theorem with application to the solvability of systems of integral-differential equations in Sobolev spaces | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 23، دوره 12، شماره 1، مرداد 2021، صفحه 287-300 اصل مقاله (499.33 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2021.4784 | ||
| نویسندگان | ||
| Hojjatollah Amiri Kayvanloo1؛ Mahnaz Khanehgir1؛ Reza Allahyari* 2 | ||
| 1Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran | ||
| 2Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran. | ||
| تاریخ دریافت: 29 تیر 1397، تاریخ بازنگری: 17 شهریور 1398، تاریخ پذیرش: 26 شهریور 1399 | ||
| چکیده | ||
| In this article, we introduce the notion of $(\alpha,\beta)$-generalized Meir-Keeler condensing operator in a Banach space, a characterization using strictly L-functions and provide an extension of Darbo's fixed point theorem associated with measures of noncompactness. Then, we establish some results on the existence of coupled fixed points for a class of condensing operators in Banach spaces. As an application, we study the problem of existence of entire solutions for a general system of nonlinear integral-differential equations in a Sobolev space. Further, an example is presented to verify the effectiveness and applicability of our main results. | ||
| کلیدواژه‌ها | ||
| Coupled fixed points؛ Measure of noncompactness؛ Meir-Keleer condensing operator؛ Sobolev space؛ System of integral equations | ||
| مراجع | ||
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[1] R. Agarwal, M. Meehan and D. O’Regan, Fixed Point Theory and Applications, Cambridge University Press, 2004. [2] A. Aghajani, R. Allahyari, and M. Mursaleen, A generalization of Darbo’s theorem with application to the solvability of systems of integral equations, J. Comput. Appl. Math. 260 (2014) 68–77. [3] A. Aghajani and Y. Jalilian, Existence of nondecreasing positive solution for a system of singular integral equations, Mediter. J. Math. 8 (2011) 563–576. [4] A. Aghajani, M. Mursaleen and A. Shole Haghighi, Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness, Acta Math. Science. 35B (2015) 552–566. [5] A. Aghajani, D. O’Regan and A. Shole Haghighi, Measure of noncompactness on Lp(Rn) and applications, Cubo A Math. J. 17 (2015) 85–97. [6] A. Aghajani, and N. Sabzali, Existence of coupled fixed points via measure of noncompactness and applications, J. Nonlinear Convex Anal. 15(5) (2014) 941–952. [7] R.R. Akhmerov, M.I. Kamenski, A.S. Potapov, A.E. Rodkina and B.N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhauser Verlag. Basel, 1992. [8] J. Bana´s, and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes In Pure And Applied Mathematics, 60, Dekker, New York, 1980. [9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Science. Business Media, LLC, 2011. [10] S.S. Chang, Y.J. Cho and N.J. Huang, Coupled fixed point theorems with applications, J. Korean Math. Soc. 33 (1996) 575–585. [11] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Uni. Padova. 24 (1955) 84–92. [12] P. Drabek, and J. Milota, Methods of Nonlinear Analysis, Birkhauser Velgar AG, 2007. [13] H. Hanche-Olsen and H. Holden, The Kolmogrov-Riesz compactness theorem, Expo. Math. 28 (2010) 385–394. [14] K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930) 301-309. [15] T. C. Lim, On characterizations of Meir-Keeler contractive maps, Nonlinear Anal. 46 (2001) 113–120. [16] A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969) 326–329. | ||
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