A fixed point theorem for nonautonomous type superposition operators and integrable solutions of a general nonlinear functional integral equation. (English) Zbl 1472.47048
Summary: We first establish a new fixed point theorem for nonautonomous type superposition operators. After that, we prove the existence of integrable solutions for a general nonlinear functional integral equation in an \(L^1\) space on an unbounded interval by using our theorem. Our main tool is the measure of weak noncompactness.
MSC:
| 47H10 | Fixed-point theorems |
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
| 47H08 | Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc. |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
superposition operators; fixed points; measure of weak noncompactness; nonlinear functional integral equationsReferences:
| [1] | Appell, J.; Zabrejko, PP, Cambridge Tracts in Math. 95 (1990), Cambridge · Zbl 0701.47041 · doi:10.1017/CBO9780511897450 |
| [2] | Krasnosel’skii MA:On the continuity of the operator[InlineEquation not available: see fulltext.]. Dokl. Akad. Nauk SSSR 1951, 77:185-188. (in Russian) · Zbl 0042.12101 |
| [3] | Krasnosel’skii MA: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, New York; 1964. · Zbl 0111.30303 |
| [4] | Gaidukevich OI, Maslyuchenko VK: New generalizations of the Scorza-Dragoni theorem.Ukr. Math. J. 2000,52(7):1010-1017. · Zbl 0964.28007 · doi:10.1023/A:1005217430845 |
| [5] | Banas J, Rivero J: On measures of weak noncompactness.Ann. Math. Pures Appl. 1988, 151:213-224. · Zbl 0653.47035 · doi:10.1007/BF01762795 |
| [6] | De Blasi FS: On a property of the unit sphere in Banach spaces.Bull. Math. Soc. Sci. Math. Roum. 1977, 21:259-262. · Zbl 0365.46015 |
| [7] | Appell J, De Pascale E: Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili.Boll. Unione Mat. Ital., B 1984, 3:497-515. · Zbl 0507.46025 |
| [8] | Banas J, Knap K: Measures of weak noncompactness and nonlinear integral equations of convolution type.J. Math. Anal. Appl. 1990, 146:353-362. · Zbl 0699.45002 · doi:10.1016/0022-247X(90)90307-2 |
| [9] | Dieudonné J: Sur les espaces de Köthe.J. Anal. Math. 1951, 1:81-115. · Zbl 0044.11703 · doi:10.1007/BF02790084 |
| [10] | Boyd D, Wong JSW: On nonlinear contractions.Proc. Am. Math. Soc. 1969, 20:458-464. · Zbl 0175.44903 · doi:10.1090/S0002-9939-1969-0239559-9 |
| [11] | Smart DR: Fixed Point Theorems. Cambridge University Press, Cambridge; 1980. · Zbl 0427.47036 |
| [12] | Dhage BC: Local fixed point theory involving three operators in Banach algebras.Topol. Methods Nonlinear Anal. 2004, 24:377-386. · Zbl 1083.47039 |
| [13] | Joshi M: Existence theorems for a variant of Hammerstein integral equations.Comment. Math. Univ. Carol. 1975,16(2):255-271. · Zbl 0316.45014 |
| [14] | Lucchetti R, Patrone F: On Nemytskii’s operator and its application to the lower semicontinuity of integral functional.Indiana Univ. Math. J. 1980,29(5):703-735. · Zbl 0476.47049 · doi:10.1512/iumj.1980.29.29051 |
| [15] | Moreira DR, Teixeira EV: Weak convergence under nonlinearities.An. Acad. Bras. Ciênc. 2003,75(1):9-19. · Zbl 1042.47040 |
| [16] | Banas J, Chlebowicz A: On existence of integrable solutions of a functional integral equation under Carathéodory conditions.Nonlinear Anal. 2009, 70:3172-3179. · Zbl 1168.45005 · doi:10.1016/j.na.2008.04.020 |
| [17] | Banas J, Chlebowicz A: On integrable solutions of a nonlinear Volterra integral equation under Carathéodory conditions.Bull. Lond. Math. Soc. 2009,41(6):1073-1084. · Zbl 1191.47087 · doi:10.1112/blms/bdp088 |
| [18] | Taoudi MA: Integrable solutions of a mixed type operator equation.Nonlinear Anal. 2009, 71:4131-4136. · Zbl 1203.45004 · doi:10.1016/j.na.2009.02.072 |
| [19] | Liang J, Yan SH, Agarwal RP, Huang TW: Integral solution of a class of nonlinear integral equations.Appl. Math. Comput. 2013,219(10):4950-4957. · Zbl 1285.45004 · doi:10.1016/j.amc.2012.10.099 |
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.
