Properties of convergence of a class of iterative processes generated by sequences of self-mappings with applications to switched dynamic systems. (English) Zbl 1346.47081
Summary: This article investigates the convergence properties of iterative processes involving sequences of self-mappings of metric or Banach spaces. Such sequences are built from a set of primary self-mappings which are either expansive or non-expansive self-mappings and some of the non-expansive ones can be contractive including the case of strict contractions. The sequences are built subject to switching laws which select each active self-mapping on a certain activation interval in such a way that essential properties of boundedness and convergence of distances and iterated sequences are guaranteed. Applications to the important problem of stability of dynamic switched systems are also given.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 54E40 | Special maps on metric spaces |
| 47N70 | Applications of operator theory in systems, signals, circuits, and control theory |
Keywords:
expansive; non-expansive; contractive and strictly contractive self-mappings; switched dynamic systems; convergence; fixed point; stabilityReferences:
| [1] | Berinde, V., Lecture Notes in Mathematics 1912 (2002), Heidelberg |
| [2] | Farnum NR: A fixed point method for finding percentage points.Appl. Stat. 1991,40(1):123-126. · Zbl 0825.62373 · doi:10.2307/2347910 |
| [3] | Antia HM: Numerical Methods for Scientists and Engineers. Birkhäuser, Boston; 2002. · Zbl 1014.65001 |
| [4] | Miñambres JJ, De la Sen M: Application of numerical-methods to the acceleration of the convergence of the adaptive-control algorithms: the one dimensional case.Comput. Math. Appl. 1986,12(1):1049-1056. · Zbl 0617.65045 · doi:10.1016/0898-1221(86)90010-6 |
| [5] | Soleymani, F.; Sharifi, M.; Sateyi, S.; Khaksar Haghani, F., A class of Steffensen-type iterative methods for nonlinear systems, No. 2014 (2014) · Zbl 1474.65123 |
| [6] | Samreen, M.; Kamram, T.; Shahzad, N., Some fixed point theorems in b-metric space endowed with graph, No. 2013 (2013) · Zbl 1470.54109 |
| [7] | Yao, Y.; Liou, YC; Yao, JC, Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, No. 2007 (2007) · Zbl 1153.54024 |
| [8] | Nashine HK, Khan MS: An application of fixed point theorem to best approximation in locally convex space.Appl. Math. Lett. 2010, 23:121-127. 10.1016/j.aml.2009.06.025 · Zbl 1200.47076 · doi:10.1016/j.aml.2009.06.025 |
| [9] | Yao, YH; Liou, YC; Kang, SM, An iterative approach to mixed equilibrium problems and fixed point problems, No. 2013 (2013) |
| [10] | Cho, YJ; Kadelburg, Z.; Saadati, R.; Shatanawi, W., Coupled fixed point theorems under weak contractions, No. 2012 (2012) · Zbl 1250.54046 |
| [11] | Cho, SY; Li, WL; Kang, SM, Convergence analysis of an iterative algorithm for monotone operators, No. 2013 (2013) · Zbl 1364.47017 |
| [12] | Hussain, N.; Cho, YJ, Weak contractions, common fixed points, and invariant approximations, No. 2009 (2009) · Zbl 1183.47049 |
| [13] | Jleli, M.; Karapınar, E.; Samet, B., Best proximity point results for MK-proximal contractions, No. 2012 (2012) · Zbl 1296.54066 |
| [14] | Jleli, M.; Samet, B., Best proximity point results for MK-proximal contractions on ordered sets (2013) · Zbl 1325.54033 |
| [15] | De la Sen, M.; Agarwal, RP, Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type, No. 2011 (2011) · Zbl 1312.54019 |
| [16] | Karapınar, E., On best proximity point of ψ-Geraghty contractions, No. 2013 (2013) · Zbl 1295.41037 |
| [17] | Gupta A, Rajput SS, Kaurav PS: Coupled best proximity point theorem in metric spaces.Int. J. Anal. Appl. 2014,4(2):201-215. · Zbl 1399.54110 |
| [18] | De la Sen, M., Fixed point and best proximity theorems under two classes of integral-type contractive conditions in uniform metric spaces, No. 2010 (2010) · Zbl 1206.54041 |
| [19] | Jleli, M.; Karapınar, E.; Samet, B., A best proximity point result in modular spaces with the Fatou property, No. 2013 (2013) · Zbl 1325.41019 |
| [20] | Pathak HK, Shahzad N: Some results on best proximity points for cyclic mappings.Bull. Belg. Math. Soc. Simon Stevin 2013,20(3):559-572. · Zbl 1278.54043 |
| [21] | De la Sen, M.; Karapınar, E., Best proximity points of generalized semicyclic impulsive self-mappings: applications to impulsive differential and difference equations, No. 2013 (2013) · Zbl 1357.54032 |
| [22] | Dey, D.; Kumar Laha, A.; Saha, M., Approximate coincidence point of two nonlinear mappings, No. 2013 (2013) · Zbl 1268.54022 |
| [23] | Dey D, Saha M: Approximate fixed point of Reich operator.Acta Math. Univ. Comen. 2013,LXXXII(1):119-123. · Zbl 1324.54067 |
| [24] | De la Sen M, Ibeas A: Asymptotically non-expansive self-maps and global stability with ultimate boundedness of dynamic systems.Appl. Math. Comput. 2013,219(22):10655-10667. · Zbl 1298.47063 · doi:10.1016/j.amc.2013.04.009 |
| [25] | De la Sen M, Agarwal RP, Nistal N: Non-expansive and potentially expansive properties of two modifiedp-cyclic self-maps in metric spaces.J. Nonlinear Convex Anal. 2013,14(4):661-686. · Zbl 1325.54027 |
| [26] | Karapınar E: Best proximity points of Kannan type cyclic weakϕ-contractions in ordered metric spaces.An. Univ. ‘Ovidius’ Constanţa, Ser. Mat. 2012,20(3):51-64. · Zbl 1299.54095 |
| [27] | Karapınar E: Best proximity points of cyclic mappings.Appl. Math. Lett. 2012,25(11):1761-1766. · Zbl 1269.47040 · doi:10.1016/j.aml.2012.02.008 |
| [28] | Jleli, M.; Karapınar, E.; Samet, B., A short note on the equivalence between ‘best proximity’ points and ‘fixed point’ results, No. 2014 (2014) · Zbl 1308.41034 |
| [29] | Ashyralyev, A.; Koksal, ME, Stability of a second order of accuracy difference scheme for hyperbolic equation in a Hilbert space, No. 2007 (2007) · Zbl 1156.65079 |
| [30] | Ashyralyev, A.; Sharifov, YA, Optimal control problem for impulsive systems with integral boundary conditions, 8-11 (2012) · Zbl 1286.49035 |
| [31] | Li XD, Akca H, Fu XL: Uniform stability of impulsive infinite delay differential equations with applications to systems with integral impulsive conditions.Appl. Math. Comput. 2013,219(14):7329-7337. · Zbl 1297.34082 · doi:10.1016/j.amc.2012.12.033 |
| [32] | Stamov, G.; Akca, H.; Stamova, I., Uncertain dynamic systems: analysis and applications, No. 2013 (2013) |
| [33] | Li, H.; Sun, X.; Karimi, HR; Niu, B., Dynamic ouput-feedback passivity control for fuzzy systems under variable sampling, No. 2013 (2013) · Zbl 1299.93157 |
| [34] | Xiang, Z.; Liu, S.; Mahmoud, MS, Robust [InlineEquation not available: see fulltext.] reliable control for uncertain switched neutral systems with distributed delays (2013) |
| [35] | Tang J, Huang C: Impulsive control and synchronization analysis of complex dynamical networks with non-delayed and delayed coupling.Int. J. Innov. Comput. Inf. Control 2013,9(11):4555-4564. |
| [36] | Li S, Xiang Z, Karimi HR:Stability and[InlineEquation not available: see fulltext.]-gain controller design for positive switched systems with mixed time-varying delays.Appl. Math. Comput. 2013,222(1):507-518. · Zbl 1328.93215 · doi:10.1016/j.amc.2013.07.070 |
| [37] | De la Sen M: On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays.Appl. Math. Comput. 2007,190(1):382-401. · Zbl 1117.93034 · doi:10.1016/j.amc.2007.01.053 |
| [38] | De la Sen M: About the positivity of a class of hybrid dynamic linear systems.Appl. Math. Comput. 2007,189(1):853-868. · Zbl 1119.93038 · doi:10.1016/j.amc.2006.11.182 |
| [39] | De la Sen, M., Total stability properties based on fixed point theory for a class of hybrid dynamic systems, No. 2009 (2009) · Zbl 1189.34023 |
| [40] | Marchenko VM: Hybrid discrete-continuous systems: I. Stability and stabilizability.Differ. Equ. 2012,48(12):1623-1638. · Zbl 1258.93092 · doi:10.1134/S0012266112120087 |
| [41] | Marchenko VM: Observability of hybrid discrete-continuous time system.Differ. Equ. 2013,49(11):1389-1404. · Zbl 1290.93027 · doi:10.1134/S0012266113110074 |
| [42] | Marchenko VM: Hybrid discrete-continuous systems: II. Controllability and reachability.Differ. Equ. 2013,49(1):112-125. · Zbl 1267.93023 · doi:10.1134/S0012266113010114 |
| [43] | Bilbao-Guillerna A, De la Sen M, Ibeas A, Alonso-Quesada S: Robustly stable multiestimation scheme for adaptive control and identification with model reduction issues.Discrete Dyn. Nat. Soc. 2005,2005(1):31-67. 10.1155/DDNS.2005.31 · Zbl 1102.93027 · doi:10.1155/DDNS.2005.31 |
| [44] | De la Sen M: On some structures of stabilizing control laws for linear and time-invariant systems with bounded point delays and unmensurable states.Int. J. Control 1994,59(2):529-541. · Zbl 0799.93048 · doi:10.1080/00207179408923091 |
| [45] | De la Sen, M.; Ibeas, A., Stability results for switched linear systems with constant discrete delays, No. 2008 (2008) · Zbl 1162.93393 |
| [46] | De la Sen, M.; Ibeas, A., On the global asymptotic stability of switched linear time-varying systems with constant point delays, No. 2008 (2008) · Zbl 1166.34040 |
| [47] | De la Sen, M.; Ibeas, A.; Alonso-Quesada, S., Asymptotic hyperstability of a class of linear systems under impulsive controls subject to an integral Popovian constraint, No. 2013 (2013) · Zbl 1291.93263 |
| [48] | Niamsup, P.; Rojsiraphisa, T.; Rajchakit, M., Robust stability and stabilization of uncertain switched discrete-time systems, No. 2012 (2012) · Zbl 1388.93065 |
| [49] | Niamsup, P.; Rajchakit, G., New results on robust stability and stabilization of linear discrete-time stochastic systems with convex polytopic uncertainties, No. 2013 (2013) · Zbl 1266.93124 |
| [50] | Darwish MA, Henderson J: Existence and asymptotic stability of solutions of a perturbed quadratic fractional integral equation.Fract. Calc. Appl. Anal. 2009,12(1):71-86. · Zbl 1181.45014 |
| [51] | De la Sen M: Robust stability of a class of linear time-varying systems.IMA J. Math. Control Inf. 2002,19(4):399-418. · Zbl 1016.93052 · doi:10.1093/imamci/19.4.399 |
| [52] | Tsakalis KS, Ioannou PA: Linear Time-Varying Systems. Prentice Hall, New York; 1993. · Zbl 0795.93001 |
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.
