# Article

Full entry | PDF   (0.3 MB)
Keywords:
$\tau$-measure of noncompactness; $\tau$-sequential continuity; $\Phi_{\tau}$-condensing operator; $\Phi_{\tau}$-nonexpansive operator; nonlinear contraction; fixed point theorem; demi-$\tau$-compactness; operator $\tau$-semi-closed at origin; Lebesgue space; integral equation
Summary:
In this paper we prove a collection of new fixed point theorems for operators of the form $T+S$ on an unbounded closed convex subset of a Hausdorff topological vector space $(E,\Gamma)$. We also introduce the concept of demi-$\tau$-compact operator and $\tau$-semi-closed operator at the origin. Moreover, a series of new fixed point theorems of Krasnosel'skii type is proved for the sum $T+S$ of two operators, where $T$ is $\tau$-sequentially continuous and $\tau$-compact while $S$ is $\tau$-sequentially continuous (and $\Phi_{\tau}$-condensing, $\Phi_{\tau}$-nonexpansive or nonlinear contraction or nonexpansive). The main condition in our results is formulated in terms of axiomatic $\tau$-measures of noncompactness. Apart from that we show the applicability of some our results to the theory of integral equations in the Lebesgue space.
References:
[1] Appell J., De Pascale E.: Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi funzioni misurabili. Boll. Un. Mat. Ital. B (6) 3 (1984), 497–515. MR 0762715
[2] Appell J., Zabrejko P.P.: Nonlinear Superposition Operators. Cambridge University Press, Cambridge, 1990. MR 1066204 | Zbl 1156.47052
[3] Arino O., Gautier S., Penot J.P.: A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations. Funkcial. Ekvac. 27 (1984), no. 3, 273–279. MR 0794756 | Zbl 0599.34008
[4] Banaś J.: Demicontinuity and weak sequential continuity of operators in the Lebesgue space. Proceedings of the 1st Polish Symposium on Nonlinear Analysis, 1997, pp. 124–129.
[5] Banaś J.: Applications of measures of weak noncompcatness and some classes of operators in the theory of functional equations in the Lebesgue space. Nonlinear Anal. 30 (1997), no. 6, 3283–3293. DOI 10.1016/S0362-546X(96)00157-5 | MR 1602984
[6] Barroso C.S.: Krasnoselskii's fixed point theorem for weakly continuous maps. Nonlinear Anal. 55 (2003), 25–31. DOI 10.1016/S0362-546X(03)00208-6 | MR 2001629 | Zbl 1042.47035
[7] Barroso C.S., Teixeira E.V.: A topological and geometric approach to fixed point results for sum of operators and applications. Nonlinear Anal. 60 (2005), no. 4, 625–660. MR 2109150
[8] Barroso C.S., Kalenda O.F.K., Rebouas M.P.: Optimal approximate fixed point results in locally convex spaces. J. Math. Anal. Appl. 401 (2013), no. 1, 1–8. DOI 10.1016/j.jmaa.2012.10.026 | MR 3011241
[9] Ben Amar A., Jeribi A., Mnif M.: On a generalization of the Schauder and Krasnosel'skii fixed point theorems on Dunford-Pettis space and applications. Math. Methods Appl. Sci. 28 (2006), 1737–1756. DOI 10.1002/mma.639
[10] Ben Amar A., Jeribi A., Mnif M.: Some fixed point theorems and application to biological model. Numer. Funct. Anal. Optim. 29 (2008), no. 1–2, 1-23. DOI 10.1080/01630560701749482 | MR 2387835 | Zbl 1130.47305
[11] Ben Amar A., Mnif M.: Leray-Schauder alternatives for weakly sequentially continuous mappings and application to transport equation. Math. Methods Appl. Sci. 33 (2010), no. 1, 80–90. MR 2591226 | Zbl 1193.47056
[12] Ben Amar A., Xu S.: Measures of weak noncompactness and fixed point theory for $1$-set weakly contractive operators on unbounded domains. Anal. Theory Appl. 27 (2011), no. 3, 224–238. DOI 10.1007/s10496-011-0224-2 | MR 2844659
[13] Boyd D.W., Wong J.S.W.: On nonlinear contractions. Proc. Amer. Math. Soc. 20 (1969), 458–464. DOI 10.1090/S0002-9939-1969-0239559-9 | MR 0239559 | Zbl 0175.44903
[14] Burton T.A.: A fixed point theorem of Krasnosel'skii. Appl. Math. Lett. 11 (1998), 85–88. DOI 10.1016/S0893-9659(97)00138-9 | MR 1490385
[15] Day M.M.: Normed Linear Spaces. Academic Press, New York, 1962. MR 0145316 | Zbl 0583.00016
[16] De Blasi F.S.: On a property of the unit sphere in Banach space. Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259–262. MR 0482402
[17] Dunford N., Pettis B.J.: Linear operators on summable functions]/. Trans. Amer. Math. Soc. 47 (1940), 323–392. DOI 10.1090/S0002-9947-1940-0002020-4 | MR 0002020
[18] Dunford N., Schwartz J.T.: Linear Operators, Part I. Interscience, Leyden, 1963. Zbl 0635.47001
[19] Edwards R.E.: Functional Analysis, Theory and Applications. Holt, Reinhard and Winston, New York, 1965. MR 0221256 | Zbl 0189.12103
[20] Garcia-Falset J.: Existence of fixed points and measures of weak noncompactness. Nonlinear Anal. 71 (2009), 2625–2633. DOI 10.1016/j.na.2009.01.096 | MR 2532788 | Zbl 1194.47060
[21] Krasnosel'skii M.A.: On the continuity of the operator $Fu(x)=f(x,u(x))$. Dokl. Akad. Nauk SSSR 77 (1951), 185–188 (in Russian). MR 0041354
[22] Krasnosel'skii M.A.: Two remarks on the method of successive approximation. Uspehi Mat. Nauk 10 (1955), 123–127 (in Russian). MR 0068119
[23] Krasnosel'skii M.A., Zabrejko P.P., Pustyl'nik J.I., Sobolevskii P.J.: Integral Opertors in Spaces of Summable Functions. Noordhoff, Leyden, 1976.
[24] Kubiaczyk I.: On a fixed point theorem for weakly sequentially continuous mappings. Discuss. Math. Differential Incl. 15 (1995), 15–20. MR 1344524 | Zbl 0832.47046
[25] O'Regan D.: Fixed-point theory for weakly sequentially continuous mappings. Math. Comput. Modelling 27 (1998), no. 5, 1–14. DOI 10.1016/S0895-7177(98)00014-4 | MR 1616796 | Zbl 1185.34026
[26] O'Regan D., Taoudi M.A.: Fixed point theorems for the sum of two weakly sequentially continuous mappins. Nonlinear Anal. 73 (2010), 283–289. DOI 10.1016/j.na.2010.03.009 | MR 2650815
[27] V.I. Shragin: On the weak continuity of the Nemytskii operator. Uchen. Zap. Mosk. Obl. Ped. Inst. 57 (1957), 73–79.
[28] Taoudi M.A.: Krasnosel'skii type fixed point theorems under weak topology features. Nonlin. Anal. 72 (2010), no. 1, 478–482. DOI 10.1016/j.na.2009.06.086 | MR 2574957 | Zbl 1225.47071
[29] Zabrejko P.P., Koshelev A.I., Krasnosel'skii M.A., Mikhlin S.G., Rakovshchik L.S., Stecenko V.J.: Integral Equations. Noordhoff, Leyden, 1975.

Partner of