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Coincidence points in generalized metric spaces. (English) Zbl 1328.54031

The authors continue the research (see [A. V. Arutyunov, Dokl. Math. 76, No. 2, 665–668 (2007; Zbl 1152.54351); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 416, No. 2, 151–155 (2007)]) on coincidence points for maps (single-valued and set-valued) such that one of them is \(\beta\)-Lipschitz continuous and the second one is an \(\alpha\)-covering continuous map (\(\beta<\alpha\)).
In the present paper maps in generalized metric spaces (i.e., with \(d:X\times X\to [0,\infty]\)) are studied. As an application of abstract results the existence of nontrivial solutions of the following equation in \(C([0,\infty),\mathbb{R}^n)\) is solved: \[ x(t)=\sum_{j=1}^m \theta_jx(\varphi_j(t)) \;\text{ for every } t\geq 0, \] where \(\varphi_j:[0,\infty)\to [0,\infty)\) are given continuous functions, strictly increasing for sufficiently big \(t\), and \(\theta_j\in \mathbb{R}\) are small numbers (\(\sum_{j=1}^m|\theta_j|<1\)).

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
47H04 Set-valued operators

Citations:

Zbl 1152.54351
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References:

[1] Graves, L.M.: Some mapping theorems. Duke Math. J. 17:2, 111-114 (1950) · Zbl 0037.20401 · doi:10.1215/S0012-7094-50-01713-3
[2] Mordukhovich, B.S.: Approximation methods in optimization and control problems, Nauka, Moscow. [in Russian] (1988) · Zbl 0643.49001
[3] Mordukhovich, B.S., Wang, B.: Restrictive metric regularity generalized differential calculus in Banach spaces. Inter. J. Maths. Math. Sci. 50, 2650-2683 (2004)
[4] Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. 1. Springer, Berlin (2006)
[5] Mordukhovich, B.S., Nam, N.M., Wang, B.: Metric regularity of mappings and generalized normals to set images. Set-Valued Var. Anal. 17:4, 359-387 (2009) · Zbl 1218.49023 · doi:10.1007/s11228-009-0122-3
[6] Mordukhovich, B.S., Shao, Y.: Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces. Nonlinear Anal. 25, 1401-1424 (1995) · Zbl 0863.47030 · doi:10.1016/0362-546X(94)00256-H
[7] Dontchev, A.L., Rockafellar, R.T.: Implicit functions and solution mappings, Monographs in Mathematics. Springer (2009) · Zbl 1178.26001
[8] Dontchev, A.L., Lewis, A.S.: Perturbations and metric regularity. Set-Valued Var. Anal. 13:4, 417-438 (2005) · Zbl 1086.49020 · doi:10.1007/s11228-005-4404-0
[9] Uderzo, A.: A metric version of milyutin theorem. Set-Valued and Var. Anal. 20:2, 279-306 (2011) · Zbl 1250.49020
[10] Uderzo, A.: On mappings covering at a nonlinear rate and their perturbation stability. Nonlinear Analysis, Theory, Methods and Applications 75:3, 1602-1616 (2012) · Zbl 1236.49041 · doi:10.1016/j.na.2011.03.014
[11] Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: Covering mappings and well-posedness of nonlinear Volterra equations. Nonlinear Anal. 75:3, 1026-1044 (2012) · Zbl 1237.45003 · doi:10.1016/j.na.2011.03.038
[12] Leventhal, D., Lewis, A.S.: Randomized methods for linear constraints: convergence rates and conditioning. Math. Oper. Res. 35:3, 641-654 (2010) · Zbl 1216.15006 · doi:10.1287/moor.1100.0456
[13] Mordukhovich, B.S., Nam, N.M.: Variational stability and marginal functions via generalized differentiation. Math. Oper. Res. 30, 800-816 (2005) · Zbl 1284.90083 · doi:10.1287/moor.1050.0147
[14] Pang, C.H.J.: Generalized differentiation with positively homogeneous maps: applications in set-valued analysis and metric regularity. Math. Oper. Res. 36:3, 377-397 (2011) · Zbl 1242.90250 · doi:10.1287/moor.1110.0497
[15] Arutyunov, A.V.: Covering mappings in metric spaces and fixed points. Dokl. Math. 72:2, 665-668 (2007) · Zbl 1152.54351 · doi:10.1134/S1064562407050079
[16] Arutyunov, A., Avakov, E., Gel‘man B.B., Dmitruk A.A., Obukhovskii, V.: Locally covering maps in metric spaces and coincidence points. J. Fixed Points Theory Appl. 5:1, 105-127 (2009) · Zbl 1182.54050 · doi:10.1007/s11784-008-0096-z
[17] Arutyunov, A.V.: Stability of coincidence points and properties of covering mappings. Math. Notes 86:2, 153-158 (2009) · Zbl 1186.54033 · doi:10.1134/S0001434609070177
[18] Pasynkov, B.A.: Fiberwise contraction mappings principle, to appear in Topology and its Appl. (2014) · Zbl 1305.54055
[19] Luxemburg, W.A.J.: On the convergence of successive approximations in the theory of ordinary differential equations III. Nieun. Arch. Wisk 6, 93-98 (1958) · Zbl 0085.30201
[20] Jung, C.F.K.: On generalized complete metric spaces. Bull. Amer. Math. Soc. 75:1, 113-116 (1969) · Zbl 0194.23801 · doi:10.1090/S0002-9904-1969-12165-8
[21] Kornfeld, I.P., Sinay, Y.G., Fomin, S.V.: Ergodic Theory, Moscow, Nauka. [in Russian] (1980) · Zbl 0863.47030
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