O równaniach funkcyjnych związanych z rozdzielnością implikacji rozmytych

In classical logic conjunction distributes over disjunction and disjunction distributes over conjunction. Moreover, implication is left-distributive over conjunction and disjunction: p ! (q ^ r) (p ! q) ^ (p ! r); p ! (q _ r) (p ! q) _ (p ! r): At the same time it is neither right-distributive over conjunction nor over disjunction. However, the following two equalities, that are kind of right-distributivity of implications, hold: (p ^ q) ! r (p ! r) _ (q ! r); (p _ q) ! r (p ! r) ^ (q ! r): We can rewrite the above four classical tautologies in fuzzy logic and obtain the following distributivity equations: I(x;C1(y; z)) = C2(I(x; y); I(x; z)); (D1) I(x;D1(y; z)) = D2(I(x; y); I(x; z)); (D2) I(C(x; y); z) = D(I(x; z); I(y; z)); (D3) I(D(x; y); z) = C(I(x; z); I(y; z)); (D4) that are satisfied for all x; y; z 2 [0; 1], where I is some generalization of classical implication, C, C1, C2 are some generalizations of classical conjunction and D, D1, D2 are some generalizations of classical disjunction. We can define and study those equations in any lattice L = (L;6L) instead of the unit interval [0; 1] with regular order „6” on the real line, as well. From the functional equation’s point of view J. Aczél was probably the one that studied rightdistributivity first. He characterized solutions of the functional equation (D3) in the case of C = D, among functions I there are bounded below and functions C that are continuous, increasing, associative and have a neutral element. Part of the results presented in this thesis may be seen as a generalization of J. Aczél’s theorem but with fewer assumptions on the functions F and G. As a generalization of classical implication we consider here a fuzzy implication and as a generalization of classical conjunction and disjunction - t-norms and t-conorms, respectively (or more general conjunctive and disjunctive uninorms). We study the distributivity equations (D1) - (D4) for such operators defined on different lattices with special focus on various functional equations that appear. In the first two sections necessary fuzzy logic concepts are introduced. The background and history of studies on distributivity of fuzzy implications are outlined, as well. In Sections 3, 4 and 5 new results are presented and among them solutions to the following functional equations (with different assumptions): f(m1(x + y)) = m2(f(x) + f(y)); x; y 2 [0; r1]; g(u1 + v1; u2 + v2) = g(u1; u2) + g(v1; v2); (u1; u2); (v1; v2) 2 L1; h(xc(y)) = h(x) + h(xy); x; y 2 (0;1); k(min(j(y); 1)) = min(k(x) + k(xy); 1); x 2 [0; 1]; y 2 (0; 1]; where: f : [0; r1] ! [0; r2], for some constants r1; r2 that may be finite or infinite, and for functions m2 that may be injective or not; g : L1 ! [1;1], for L1 = f(u1; u2) 2 [1;1]2 j u1 u2g (function g satisfies two-dimensional Cauchy equation extended to the infinities); h; c : (0;1) ! (0;1) and function h is continuous or is a bijection; k : [0; 1] ! [0; 1], g : (0; 1] ! [1;1) and function k is continuous. Most of the results in Sections 3, 4 and 5 are new and obtained by the author in collaboration with M. Baczynski, R. Ger, M. E. Kuczma or T. Szostok. Part of them have been already published either in scientific journals (see [5]) or in refereed papers in proceedings (see [4, 1, 2, 3])

Lp-calculus approach to the random autonomous linear differential equation with discrete delay

[EN] In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay , with initial condition x(t)=g(t), -t0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. Using Lp-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an Lp-solution too. An analysis of Lp-convergence when the delay tends to 0 is also performed in detail.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). 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Mech. 6(4), 403–418 (2014). https://doi.org/10.4208/aamm.2012.m38Shi, W., Zhang, C.: Generalized polynomial chaos for nonlinear random delay differential equations. Appl. Numer. Math. 115, 16–31 (2017). https://doi.org/10.1016/j.apnum.2016.12.004Lupulescu, V., Abbas, U.: Fuzzy delay differential equations. Fuzzy Optim. Decis. Mak. 11(1), 91–111 (2012). https://doi.org/10.1007/s10700-011-9112-7Liu, S., Debbouche, A., Wang, J.R.: Fuzzy delay differential equations. On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths. J. Comput. Appl. Math. 312, 47–57 (2017). https://doi.org/10.1016/j.cam.2015.10.028Krapivsky, P.L., Luck, J.L., Mallick, K.: On stochastic differential equations with random delay. J. Stat. Mech. Theory Exp. (2011). https://doi.org/10.1088/1742-5468/2011/10/P10008Garrido-Atienza, M.J., Ogrowsky, A., Schmalfuss, B.: Random differential equations with random delays. Stoch. 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Existence results of delay and fractional differential equations via fuzzy weakly contraction mapping principle

[EN] The purpose of this article is to extend the results derived through former articles with respect to the notion of weak contraction into intuitionistic fuzzy weak contraction in the context of (T,N,∝) -cut set of an intuitionistic fuzzy set. We intend to prove common fixed point theorem for a pair of intuitionistic fuzzy mappings satisfying weakly contractive condition in a complete metric space which generalizes many results existing in the literature. Moreover, concrete results on existence of the solution of a delay differential equation and a system of Riemann-Liouville Cauchy type problems have been derived. In addition, we also present illustrative examples to substantiate the usability of our main result.Tabassum, R.; Azam, A.; Mohammed, SS. (2019). Existence results of delay and fractional differential equations via fuzzy weakly contraction mapping principle. Applied General Topology. 20(2):449-469. https://doi.org/10.4995/agt.2019.11683SWORD449469202H. M. 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Fuzzy Topology, Quantization and Gauge Invariance

Dodson-Zeeman fuzzy topology considered as the possible mathematical framework of quantum geometric formalism. In such formalism the states of massive particle m correspond to elements of fuzzy manifold called fuzzy points. Due to their weak (partial) ordering, m space coordinate x acquires principal uncertainty dx. It's shown that m evolution with minimal number of additional assumptions obeys to schroedinger and dirac formalisms in norelativistic and relativistic cases correspondingly. It's argued that particle's interactions on such fuzzy manifold should be gauge invariant.Comment: 12 pages, Talk given on 'Geometry and Field Theory' conference, Porto, July 2012. To be published in Int. J. Theor. Phys. (2015

General conditioned and aimed information on fuzzy setting

In this paper our investigation on aimed information, started in 2011, will be completed on fuzzy setting. Here will be given a form of information for fuzzy setting, when it is conditioned and aimed. This information is called "general", because it is defined without using probability or fuzzy measure

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515