The stability of periodic solutions of discontinuous systems that intersect several surfaces of discontinuity

International audienceSystems of differential equations with discontinuous right-hand sides are considered, specifically investigating periodic solutions which simultaneously intersect two or more surfaces of discontinuity. It is shown that the Poincare mapping along phase trajectories of the system in the neighbourhood of a fixed point, corresponding to periodic motion, is in general piecewise-differentiable: this neighbourhood divides into several sectors in which the Jacobians are different. For such mappings, theorems of stability in the first approximation are not applicable, and one has to devise new stability cn1eria. Several necessary conditions for stability are obtained, as well as sufficient conditions. The results are used to investigate symmetric modes of motion of a vibro-impact system with two impact pairs. The method of investigating stability in the first approximation was previously applied to discontinuous systems for solutions that intersect one surface of discontinuity [2]. It turned out that under such conditions the Poincare mapping is differentiable, so that Lyapunov's theorems could be used

Discontinuity at fixed point and metric completeness

[EN] In this paper, we prove some new fixed point theorems for a generalized class of Meir-Keeler type mappings, which give some new solutions to the Rhoades open problem regarding the existence of contractive mappings that admit discontinuity at the fixed point. In addition to it, we prove that our theorems characterize completeness of the metric space as well as Cantor's intersection property.Bisht, RK.; Rakocevic, V. (2020). Discontinuity at fixed point and metric completeness. Applied General Topology. 21(2):349-362. https://doi.org/10.4995/agt.2020.13943OJS349362212R. M. T. Bianchini, Su un problema di S. Reich riguardante la teoria dei puntifissi, Boll. Un. Mat. Ital. 5 (1972), 103-108.R. K. Bisht and N. Özgür, Geometric properties of discontinuous fixed point set of $(epsilon-delta)$ contractions and applications to neural networks, Aequat. Math. 94 (2020), 847-863. https://doi.org/10.1007/s00010-019-00680-7R. K. Bisht and R. P. 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Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968), 71-76. https://doi.org/10.2307/2316437R. Kannan, Some results on fixed points-II, Amer. Math. Monthly 76 (1969), 405-408. https://doi.org/10.1080/00029890.1969.12000228M. Maiti and T. K. Pal, Generalizations of two fixed point theorems, Bull. Calcutta Math. Soc. 70 (1978), 57-61.A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329. https://doi.org/10.1016/0022-247X(69)90031-6L. V. Nguyen, On fixed points of asymptotically regular mappings, Rend. Circ. Mat. Palermo, II. Ser., to appear.X. Nie and W. X. Zheng, On multistability of competitive neural networks with discontinuous activation functions. In: 4th Australian Control Conference (AUCC), (2014) 245-250. https://doi.org/10.1109/AUCC.2014.7358690X. Nie and W. X. Zheng, Multistability of neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays, Neural Networks 65 (2015), 65-79. https://doi.org/10.1016/j.neunet.2015.01.007X. Nie and W. X. Zheng, Dynamical behaviors of multiple equilibria in competitive neural networks with discontinuous nonmonotonic piecewise linear activation functions, IEEE Transactions On Cybernatics 46, no. 3 (2015), 679-693.https://doi.org/10.1109/TCYB.2015.2413212N. Y. Özgür and N. Tas, Some fixed-circle theorems and discontinuity at fixed circle, AIP Conference Proceedings 1926 (2018), 020048. https://doi.org/10.1063/1.5020497N. Y. Özgür and N. Tas, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc. 42, no. 4 (2019), 1433-1449. https://doi.org/10.1007/s40840-017-0555-zA. Pant and R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31, no. 11 (2017), 3501-3506. https://doi.org/10.2298/FIL1711501PA. Pant, R. P. Pant and M. C. Joshi, Caristi type and Meir-Keeler type fixed point theorems, Filomat 33, no. 12 (2019), 3711-3721. https://doi.org/10.2298/FIL1912711PR. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240 (1999), 284-289. https://doi.org/10.1006/jmaa.1999.6560R. P. Pant, Fixed points of Lipschitz type mappings, preprint.R. P. Pant, N. Özgür, N. Tas, A. Pant and M. C. Joshi, New results on discontinuity at fixed point, J. Fixed Point Theory Appl. (2020) 22:39. https://doi.org/10.1007/s11784-020-0765-0R. P. Pant, N. Y. Özgür and N. Tas, On discontinuity problem at fixed point, Bull. Malays. Math. Sci. Soc. 43 (2020), 499-517. https://doi.org/10.1007/s40840-018-0698-6R. P. Pant, N. Y. Özgür and N. Tas}, Discontinuity at fixed points with applications, Bulletin of the Belgian Mathematical Society-Simon Stevin 25, no. 4 (2019), 571-589.M. Rashid, I. Batool and N. Mehmood, Discontinuous mappings at their fixed points and common fixed points with applications, J. Math. 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Effective Choice and Boundedness Principles in Computable Analysis

In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles which are cornerstones among Weihrauch degrees and it turns out that certain core theorems in analysis can be classified naturally in this structure. In particular, we study theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem and others. We also explore how existing classifications of the Hahn-Banach Theorem and Weak K"onig's Lemma fit into this picture. We compare the results of our classification with existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective theorems. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metatheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many theorems and we discuss the Brouwer Fixed Point Theorem as an example

Some fixed point theorems for discontinuous mappings

This paper provides a fixed point theorem à la Schauder, where the mappings considered are possibly discontinuous. Our main result generalizes and unifies several well-known results.Schauder fixed point theorem, Brouwer fixed point theorem, discontinuity.

Typicalness of chaotic fractal behaviour of integral vortexes in Hamiltonian systems with discontinuous right hand side

We consider a linear-quadratic deterministic optimal control problem where the control takes values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely the chaotic behaviour of bounded pieces of optimal trajectories. We find the hyperbolic domains in the neighbourhood of a homoclinic point and estimate the corresponding contraction-extension coefficients. This gives us the possibility to calculate the entropy and the Hausdorff dimension of the non-wandering set which appears to have a Cantor-like structure as in Smale's Horseshoe. The dynamics of the system is described by a topological Markov chain. In the second part it is shown that this behaviour is generic for piece-wise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hyper-surface strata.Comment: 113 pages, 22 figure