×

Existence of solutions of nonlinear Fredholm-type integral equations in Hölder space. (English) Zbl 07714666

Summary: Using Darbo’s fixed point theorem, we present an existence result for the solution of a nonlinear integral equation of Fredholm type in the space of functions satisfying the Hölder condition. We also give an example to illustrate the mentioned existence result.

MSC:

26B35 Special properties of functions of several variables, Hölder conditions, etc.
45B05 Fredholm integral equations
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] R. P. Agarwal, J. Banaś, K. Banaś, and D. O’Regan, “Solvability of a quadratic Hammerstein integral equation in the class of functions having limits at infinity”, J. Integral Equations Appl. 23:2 (2011), 157-181. · Zbl 1223.45006 · doi:10.1216/JIE-2011-23-2-157
[2] I. K. Argyros, “On a class of quadratic integral equations with perturbation”, Funct. Approx. Comment. Math. 20 (1992), 51-63. · Zbl 0780.45005
[3] K. Balachandran and S. Ilamaran, “An existence theorem for a Volterra integral equation with deviating arguments”, J. Appl. Math. Stochastic Anal. 3:3 (1990), 155-162. · Zbl 0703.45005 · doi:10.1155/S1048953390000144
[4] J. Banaś and I. J. Cabrera, “On existence and asymptotic behaviour of solutions of a functional integral equation”, Nonlinear Anal. 66:10 (2007), 2246-2254. · Zbl 1128.45004 · doi:10.1016/j.na.2006.03.015
[5] J. Banaś and A. Chlebowicz, “On an elementary inequality and its application in the theory of integral equations”, J. Math. Inequal. 11:2 (2017), 595-605. · Zbl 1368.26012 · doi:10.7153/jmi-11-48
[6] J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics 60, Marcel Dekker, New York, 1980. · Zbl 0441.47056
[7] J. Banaś and R. Nalepa, “On the space of functions with growths tempered by a modulus of continuity and its applications”, J. Funct. Spaces Appl. (2013), art. id. 820437. · Zbl 1275.46014 · doi:10.1155/2013/820437
[8] J. Banaś and R. Nalepa, “On a measure of noncompactness in the space of functions with tempered increments”, J. Math. Anal. Appl. 435:2 (2016), 1634-1651. · Zbl 1355.46030 · doi:10.1016/j.jmaa.2015.11.033
[9] J. Banaś and B. Rzepka, “On existence and asymptotic stability of solutions of a nonlinear integral equation”, J. Math. Anal. Appl. 284:1 (2003), 165-173. · Zbl 1029.45003 · doi:10.1016/S0022-247X(03)00300-7
[10] J. Banaś and K. Sadarangani, “On some measures of noncompactness in the space of continuous functions”, Nonlinear Anal. 68:2 (2008), 377-383. · Zbl 1134.46012 · doi:10.1016/j.na.2006.11.003
[11] J. Caballero, M. A. Darwish, and K. Sadarangani, “Solvability of a quadratic integral equation of Fredholm type in Hölder spaces”, Electron. J. Differential Equations (2014), art. id. 31. · Zbl 1287.45003
[12] J. Caballero Mena, R. Nalepa, and K. Sadarangani, “Solvability of a quadratic integral equation of Fredholm type with supremum in Hölder spaces”, J. Funct. Spaces (2014), art. id. 856183. · Zbl 1293.45001 · doi:10.1155/2014/856183
[13] S. Chandrasekhar, Radiative transfer, Oxford University Press, 1950. · Zbl 0037.43201
[14] E. T. Copson, “On an integral equation arising in the theory of diffraction”, Quart. J. Math. Oxford Ser. 17 (1946), 19-34. · Zbl 0063.00965 · doi:10.1093/qmath/os-17.1.19
[15] G. Darbo, “Punti uniti in trasformazioni a codominio non compatto”, Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92. · Zbl 0064.35704
[16] K. Deimling, Nonlinear functional analysis, Springer, 1985. · Zbl 0559.47040 · doi:10.1007/978-3-662-00547-7
[17] M. T. Ersoy and H. Furkan, “On the existence of the solutions of a Fredholm integral equation with a modified argument in Hölder spaces”, Symmetry 10:10 (2018), art. id. 522. · doi:10.3390/sym10100522
[18] S. Hu, M. Khavanin, and W. Zhuang, “Integral equations arising in the kinetic theory of gases”, Appl. Anal. 34:3-4 (1989), 261-266. · Zbl 0697.45004 · doi:10.1080/00036818908839899
[19] C. Kuratowski, “Sur les spaces complets”, Fund. Math. 15:1 (1930), 301-309. · doi:10.4064/fm-15-1-301-309
[20] H. Okrasińska-Płociniczak, Ł. Płociniczak, J. Rocha, and K. Sadarangani, “Solvability in Hölder spaces of an integral equation which models dynamics of the capillary rise”, J. Math. Anal. Appl. 490:1 (2020), art. id. 124237. · Zbl 1467.45004 · doi:10.1016/j.jmaa.2020.124237
[21] İ. Özdemir, “An existence theorem for some nonlinear Volterra-Fredholm integral equations in the space of continuous tempered functions”, Numer. Funct. Anal. Optim. 42:11 (2021), 1287-1307. · Zbl 1477.45003 · doi:10.1080/01630563.2021.1954659
[22] İ. Özdemir, “On the solvability of a class of nonlinear integral equations in Hölder spaces”, Numer. Funct. Anal. Optim. 43:4 (2022), 365-393. · Zbl 1489.45003 · doi:10.1080/01630563.2022.2032148
[23] S. Peng, J. Wang, and F. Chen, “A quadratic integral equation in the space of functions with tempered moduli of continuity”, J. Appl. Math. Inform. 33:3-4 (2015), 351-363. · Zbl 1333.26006 · doi:10.14317/jami.2015.351
[24] M. Rabbani, “An iterative algorithm to find a closed form of solution for Hammerstein nonlinear integral equation constructed by the concept of cosm-rs”, Math. Sci. 13:3 (2019), 299-305. · Zbl 1447.45007 · doi:10.1007/s40096-019-00299-4
[25] M. Rabbani and S. Kiasoltani, “Solving of nonlinear system of Fredholm-Volterra integro-differential equations by using discrete collocation method”, J. Math. Comput. Sci. 3:4 (2011), 382-389. · doi:10.22436/jmcs.03.04.03
[26] D. Saha, M. Sen, N. Sarkar, and S. Saha, “Existence of a solution in the Holder space for a nonlinear functional integral equation”, Armen. J. Math. 12 (2020), art. id. 7. · Zbl 1454.45002
[27] S. Saiedinezhad, “On a measure of noncompactness in the Holder space \[C^{k,\gamma}(\Omega)\] and its application”, J. Comput. Appl. Math. 346 (2019), 566-571. · Zbl 1452.47002 · doi:10.1016/j.cam.2018.07.030
[28] M. Temizer Ersoy, “Solutions of Fredholm type integral equations via the classical Schauder fixed point theorem”, J. Integral Equations Appl. 33:2 (2021), 259-270. · Zbl 1480.45003 · doi:10.1216/jie.2021.33.259
[29] M. Temizer Ersoy, “On the existence of the solutions of a nonlinear Fredholm integral equation in Hölder spaces”, Math. Sci. Appl. E-Notes 10:1 (2022), 16-26 · doi:10.36753/mathenot.756916
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.