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Fixed point theorems for monotone mappings in ordered Banach spaces under weak topology features

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We present several fixed point theorems for monotone nonlinear operators in ordered Banach spaces. The main assumptions of our results are formulated in terms of the weak topology. As an application, we study the existence of solutions to a class of first-order vectorvalued ordinary differential equations. Our conclusions generalize many well-known results.
Rocznik
Tom
Strony
5--19
Opis fizyczny
Bibliogr. 35 poz.
Twórcy
  • Department of Mathematics, College of Applied Sciences, P.O. Box 715, Makkah 21955 KSA
  • Department of Mathematics, College of Applied Sciences, P.O. Box 715, Makkah 21955 KSA
  • Department of Mathematics, Faculty of Sciences of Gabès, University of Gabès, Cité Erriadh, 6072 Zrig, Gabès, TUNISIA
  • National School of Applied Sciences, Cadi Ayyad University, Marrakech, MOROCCO
Bibliografia
  • [1] R.P. Agarwal, D. O’Regan, M.-A. Taoudi, Fixed point theorems for ws-compact mappings in Banach spaces, Fixed Point Theory Appl. 2010, Article ID 183596 (2010) 13 pages.
  • [2] R.P. Agarwal, D. O’Regan, M.-A. Taoudi, Fixed point theorems for convex-power condensing operators relative to the weak topology and applications to Volterra integral equations, J. Int. Eq. Appl. 24 (2) (2012) 167–181.
  • [3] O. Arino, S. Gautier, J.P. Penot, A fixed point theorem for sequentially continuous mappings with applications to ordinary differential equations, Funkc. Ekvac. 27 (1984) 273–279.
  • [4] J. Appell, The superposition operator in function spaces – a survey, Expo. Math. 6 (1988) 209–270.
  • [5] J. Banaś, J. Rivero, On measures of weak noncompactness, Ann. Mat. Pura Appl. 151 (1988) 213–224.
  • [6] J. Bana´s, Z. Knap, Measure of weak noncompactness and nonlinear integral equations of convolution type, J. Math. Anal. Appl. 146 (1990) 353–362.
  • [7] J. Banaś, Z. Knap, Integrable solutions of a functional-integral equation, Rev. Mat. Univ. Complut. Madrid 2 (1) (1989) 31–38.
  • [8] J. Banaś, Integrable solutions of Hammerstein and Urysohn integral equations, J. Austral. Math. Soc. Ser. A 46 (1) (1989 points and solutions) 61–68.
  • [9] J. Banaś, M.-A. Taoudi, Fixed of operator equations for the weak topology in Banach algebras, Taiwanese Journal of Mathematics 18 (2014) 871–893.
  • [10] C.S. Barroso, Krasnosel’skii’s fixed point theorem for weakly continuous maps, Nonlinear Analysis 55 (1) (2003) 25–31.
  • [11] C.S. Barroso, E.V. Teixeira, A topological and geometric approach to fixed points results for sum of operators and applications, Nonlin. Anal. 60 (4) (2005) 625– 650.
  • [12] A. Bellour, D. O’Regan, M.-A. Taoudi, On the existence of integrable solutions for a nonlinear quadratic integral equation, J. Appl. Math. Comput. 46 (1-2) (2014) 67–77.
  • [13] A. Bellour, M. Bousselsal, M.-A. Taoudi, Integrable solutions of a nonlinear integral equation related to some epidemic models, Glasnik Matematicki 49 (69) (2014) 395–406.
  • [14] A. Chlebowicz, M-A. Taoudi, Measures of weak noncompactness and fixed points, in: Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, Springer, Singapore, 2017, 247–296.
  • [15] F.S. De Blasi, On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Sci. Math. Roum. 21 (1977) 259–262.
  • [16] S. Djebali, Z. Sahnoun, Nonlinear alternatives of Schauder and Krasnosel’skij types with applications to Hammerstein integral equations in L 1 spaces, J. Differential Equations 249 (9) (2010) 2061–2075.
  • [17] Y. Du, Fixed points of increasing operators in ordered Banach spaces and applications, Applicable Analysis 38 (1990) 1–20.
  • [18] S.W. Du, V. Lakshmikantham, Monotone iterative technique for differential equations in a Banach space, J. Math. Anal. Appl. 87 (2) (1982) 454–459.
  • [19] N. Dunford, J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, New York, 1958.
  • [20] J. Garcia-Falset, Existence of fixed points and measure of weak noncompactness, Nonlin. Anal. 71 (2009) 2625–2633.
  • [21] J. Garcia-Falset, Existence of fixed points for the sum of two operators, Math. Nachr. 283 (12) (2010) 1736–1757.
  • [22] J. Garcia-Falset, K. Latrach, E. Moreno-Galvez, M.-A. Taoudi, Schaefer-Krasnoselskii fixed point theorems using a usual measure of weak noncompactness, J. Differential Equations 252 (5) (2012) 3436–3452.
  • [23] D. Guo, Y.J. Chow, J. Zhu, Partial Ordering Methods in Nonlinear Problems, Nova Publishers, 2004.
  • [24] D.J. Guo, J.X. Sun, Z.L. Liu, The functional methods in nonlinear differential equation, Shandong Technical and Science Press (in chinese) (2006) 1–6.
  • [25] S. Heikkila, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, CRC Press, 1994.
  • [26] N. Hussain, M.-A. Taoudi, Fixed point theorems for multivalued mappings in ordered Banach spaces with application to integral inclusions, Fixed Point Theory Appl. (2016) 2016:65.
  • [27] Y. Li, Z. Liu, Monotone iterative technique for addressing impulsive integrodifferential equations in Banach spaces, Nonlinear Anal. 66 (1) (2007) 83–92.
  • [28] E. Liz, Monotone iterative techniques in ordered Banach spaces, Proceedings of the Second World Congress of Nonlinear Analysts, Part 8 (Athens, 1996), Nonlinear Anal. 30 (8) (1997) 5179–5190.
  • [29] K. Latrach, M.-A. Taoudi, A. Zeghal, Some fixed point theorems of the Schauder and Krasnosel’skii type and application to nonlinear transport equations, J. Differential Equations 221 (1) (2006) 256–271.
  • [30] K. Latrach, M.-A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L 1 - spaces, Nonlin. Anal. 66 (2007) 2325–2333.
  • [31] A.R. Mitchell, C.K.L. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, in: Nonlinear Equations in Abstract Spaces, (edited by V. Lakshmikantham), Academic Press, 1978, 387–404.
  • [32] J. Sun, X, Zhang, The fixed point theorem of convex-power condensing operator and applications to abstract semilinear evolution equations, Acta Math. Sinica (in chinese) 48 (2005) 339–446.
  • [33] M.-A. Taoudi, Integrable solutions of a nonlinear functional integral equation on an unbounded interval, Nonlin. Anal. 71 (2009) 4131–4136.
  • [34] M.-A. Taoudi, Krasnosel’skii type fixed point theorems under weak topology features, Nonlinear Anal. 72 (1) (2010) 478–482.
  • [35] K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1965.
Uwagi
PL
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-87cdcab0-1264-4b27-a987-13b64bf17897
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