## Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations.(English)Zbl 1252.47044

Summary: We present some new common fixed point theorems for a pair of nonlinear mappings defined on an ordered Banach space. Our results extend several earlier works. An application is given to show the usefulness and the applicability of the obtained results.

### MSC:

 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47H08 Measures of noncompactness and condensing mappings, $$K$$-set contractions, etc. 47N20 Applications of operator theory to differential and integral equations 45G10 Other nonlinear integral equations
Full Text:

### References:

 [1] N. Hussain and M. H. Shah, “KKM mappings in cone b-metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1677-1684, 2011. · Zbl 1231.54022 [2] S. Janković, Z. Kadelburg, and S. Radenović, “On cone metric spaces: a survey,” Nonlinear Analysis A, vol. 74, no. 7, pp. 2591-2601, 2011. · Zbl 1221.54059 [3] B. C. Dhage, “Condensing mappings and applications to existence theorems for common solution of differential equations,” Bulletin of the Korean Mathematical Society, vol. 36, no. 3, pp. 565-578, 1999. · Zbl 0940.47043 [4] N. Hussain, A. R. Khan, and R. P. Agarwal, “Krasnoselskii and Ky Fan type fixed point theorems in ordered Banach spaces,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 3, pp. 475-489, 2010. · Zbl 1219.47079 [5] J. Banaś and J. Rivero, “On measures of weak noncompactness,” Annali di Matematica Pura ed Applicata, vol. 151, pp. 213-224, 1988. · Zbl 0653.47035 [6] F. S. De Blasi, “On a property of the unit sphere in a Banach space,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 21, no. 3-4, pp. 259-262, 1977. · Zbl 0365.46015 [7] K. Kuratowski, “Sur les espaces complets,” Fundamenta Mathematicae, vol. 15, pp. 301-309, 1930. · JFM 56.1124.04 [8] G. Darbo, “Punti uniti in trasformazioni a codominio non compatto,” Rendiconti del Seminario Matematico della Università di Padova, vol. 24, pp. 84-92, 1955. · Zbl 0064.35704 [9] M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, John Wiley & Sons, New York, NY, USA, 2001. · Zbl 1318.47001 [10] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. · Zbl 0441.47056 [11] J. Jachymski, “On Isac’s fixed point theorem for selfmaps of a Galerkin cone,” Annales des Sciences Mathématiques du Québec, vol. 18, no. 2, pp. 169-171, 1994. · Zbl 0823.47057 [12] J. García-Falset, “Existence of fixed points and measures of weak noncompactness,” Nonlinear Analysis A, vol. 71, no. 7-8, pp. 2625-2633, 2009. · Zbl 1194.47060 [13] K. Latrach and M. A. Taoudi, “Existence results for a generalized nonlinear Hammerstein equation on L1 spaces,” Nonlinear Analysis A, vol. 66, no. 10, pp. 2325-2333, 2007. · Zbl 1128.45006 [14] K. Latrach, M. A. Taoudi, and A. Zeghal, “Some fixed point theorems of the Schauder and the Krasnosel’skii type and application to nonlinear transport equations,” Journal of Differential Equations, vol. 221, no. 1, pp. 256-271, 2006. · Zbl 1091.47046 [15] M. A. Taoudi, “Integrable solutions of a nonlinear functional integral equation on an unbounded interval,” Nonlinear Analysis A, vol. 71, no. 9, pp. 4131-4136, 2009. · Zbl 1203.45004 [16] R. P. Agarwal, D. O’Regan, and M.-A. Taoudi, “Fixed point theorems for ws-compact mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 183596, 13 pages, 2010. · Zbl 1206.47047 [17] M. A. Taoudi, N. Salhi, and B. Ghribi, “Integrable solutions of a mixed type operator equation,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1150-1157, 2010. · Zbl 1195.45017 [18] N. Salhi and M. A. Taoudi, “Existence of integrable solutions of an integral equation of Hammerstein type on an unbounded interval,” Mediterranean Journal of Mathematics. In press. · Zbl 1276.47078 [19] S. Djebali and Z. Sahnoun, “Nonlinear alternatives of Schauder and Krasnosel’skij types with applications to Hammerstein integral equations in L1 spaces,” Journal of Differential Equations, vol. 249, no. 9, pp. 2061-2075, 2010. · Zbl 1208.47044 [20] M. Fe\vckan, “Nonnegative solutions of nonlinear integral equations,” Commentationes Mathematicae Universitatis Carolinae, vol. 36, no. 4, pp. 615-627, 1995. · Zbl 0840.45007 [21] M. A. Krasnosel’skii, “Some problems of nonlinear analysis,” in American Mathematical Society Translations, Series 2, vol. 10, pp. 345-409, 1958. · Zbl 0080.10403 [22] M. A. Krasnosel’skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York, NY, USA, 1964. · Zbl 0111.30303 [23] J. Appell and E. De Pascale, “Some parameters associated with the Hausdorff measure of noncompactness in spaces of measurable functions,” Unione Matematica Italiana Bollettino B, vol. 3, no. 2, pp. 497-515, 1984. · Zbl 0507.46025 [24] N. Dunford and J. T. Schwartz, Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, New York, NY, USA, 1958. · Zbl 0084.10402
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.