rss_2.0Tatra Mountains Mathematical Publications FeedSciendo RSS Feed for Tatra Mountains Mathematical Publicationshttps://sciendo.com/journal/TMMPhttps://www.sciendo.comTatra Mountains Mathematical Publications 's Coverhttps://sciendo-parsed-data-feed.s3.eu-central-1.amazonaws.com/6005b97be797941b18f25131/cover-image.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20220521T213330Z&X-Amz-SignedHeaders=host&X-Amz-Expires=604800&X-Amz-Credential=AKIA6AP2G7AKDOZOEZ7H%2F20220521%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Signature=f041784aebef10a12a1cca9198cfdf7e9f0efa1ef8fd30b095f6aecdc9e394f5200300What was the River Ister in the Time of Strabo? A Mathematical Approachhttps://sciendo.com/article/10.2478/tmmp-2021-0032<abstract> <title style='display:none'>Abstract</title> <p>We introduce a novel method for map registration and apply it to transformation of the river Ister from <italic>Strabo’s map of the World</italic> to the current map in the World Geodetic System. This transformation leads to the surprising but convincing result that Strabo’s river Ister best coincides with the nowadays Tauernbach-Isel-Drava-Danube course and not with the Danube river what is commonly assumed. Such a result is supported by carefully designed mathematical measurements and it resolves all related controversies otherwise appearing in understanding and translation of Strabo’s original text. Based on this result, we also show that <italic>Strabo’s Suevi in the Hercynian Forest</italic> corresponds to the Slavic people in the Carpathian-Alpine basin and thus that the compact Slavic settlement was there already at the beginning of the first millennium AD.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Oscillation Results for Third-Order Quasi-Linear Emden-Fowler Differential Equations with Unbounded Neutral Coefficientshttps://sciendo.com/article/10.2478/tmmp-2021-0028<abstract> <title style='display:none'>Abstract</title> <p>Some new oscillation criteria are obtained for a class of thirdorder quasi-linear Emden-Fowler differential equations with unbounded neutral coefficients of the form <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0028_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo>″</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>′</mml:mo><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>λ</mml:mi></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math><tex-math>$(a(t){(z(t))^\alpha })' + f(t){y^\lambda }(g(t)) = 0,$</tex-math></alternatives></inline-formula> where <italic>z</italic>(<italic>t</italic>) = <italic>y</italic>(<italic>t</italic>) + <italic>p</italic>(<italic>t</italic>)<italic>y</italic>(<italic>σ</italic>(<italic>t</italic>)) and <italic>α, λ</italic> are ratios of odd positive integers. The established results generalize, improve and complement to known results.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00A Fractional Order Delay Differential Model for Survival of Red Blood Cells in an Animal: Stability Analysishttps://sciendo.com/article/10.2478/tmmp-2021-0034<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we analyse stability of survival of red blood cells in animal fractional order model with time delay. Results have been illustrated by numerical simulations.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Finite Volume Schemes for the Affine Morphological Scale Space (Amss) Modelhttps://sciendo.com/article/10.2478/tmmp-2021-0031<abstract> <title style='display:none'>Abstract</title> <p>Finite volume (FV) numerical schemes for the approximation of Affine Morphological Scale Space (AMSS) model are proposed. For the scheme parameter <italic>θ</italic>, 0 ≤ <italic>θ</italic> ≤ 1 the numerical schemes of Crank-Nicolson type were derived. The explicit (<italic>θ</italic> = 0), semi-implicit, fully-implicit (<italic>θ</italic> = 1) and Crank-Nicolson (<italic>θ</italic> = 0.5) schemes were studied. Stability estimates for explicit and implicit schemes were derived. On several numerical experiments the properties and comparison of the numerical schemes are presented.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Improvement and Handling of the Segmentation Model with an Inflation Termhttps://sciendo.com/article/10.2478/tmmp-2021-0033<abstract> <title style='display:none'>Abstract</title> <p>The use of balloon models to address the problems of “snakes” based models was introduced by Laurent D. Cohen. This paper presents a geodesic active contours model with a modified external force term that includes a balloon model. This balloon model makes the segmentation surface to behave like a balloon inflated by the external forces. In this paper, we show an automatic way to control the behaviour of the external force with respect to the segmentation evolution. The external forces, comprised of edge and inflation terms, push the segmentation surface to edges, while curvature regularizes the evolution. As segmentation evolves, the influence of the applied inflation force is determined by how close we are to the edges. With this setup, the initial segmentation does not need to be close to the object’s edges, instead it is inflated by the balloon model towards the edges. Closer to the edges, the influence of the inflation force is adjusted accordingly. The force’s influence is completely turned off when the evolution is stable (reached the edges), then only the curvature and edge information is used to evolve the segmentation.</p> <p>This approach solves the issues associated with inclusion of balloon model. These issues are that the inflation force can overpower forces from weak edges, or they can cause the contour to be slightly larger than the actual minima. We present examples of the improved model for segmentation of human bladder images. Weak edges are more prevalent in medical images, and the automated handling of the inflation forces gives promising results for this kind of images.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00A Quintic Spline Collocation Method for Solving Time-Dependent Convection-Diffusion Problemshttps://sciendo.com/article/10.2478/tmmp-2021-0029<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we develop a new numerical algorithm for solving a time dependent convection-diffusion equation with Dirichlet’s type boundary conditions. The method comprises the horizontal method of lines for time integration and (<italic>θ</italic>-method, <italic>θ</italic> ∈ [1/2, 1] (<italic>θ</italic> = 1 corresponds to the backward Euler method and <italic>θ</italic> = 1/2 corresponds to the Crank-Nicolson method) to discretize in temporal direction and the quintic spline collocation method. The convergence analysis of proposed method is discussed in detail, and it justified that the approximate solution converges to the exact solution of orders <italic>O</italic>(Δ<italic>t</italic> + <italic>h</italic><sup>3</sup>) for the backward Euler method and <italic>O</italic>(Δ<italic>t</italic><sup>2</sup> + <italic>h</italic><sup>3</sup>) for the Crank-Nicolson method, where Δ<italic>t</italic> and <italic>h</italic> are mesh sizes in the time and space directions, respectively. It is also shown that the proposed method is unconditionally stable. This scheme is applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behaviour of the rates of convergence. Results shown by this method are in good agreement with the known exact solutions. The produced results are also more accurate than some available results given in the literature.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Solving Nonlinear Volterra-Fredholm Integral Equations using an Accurate Spectral Collocation Methodhttps://sciendo.com/article/10.2478/tmmp-2021-0030<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we present a Jacobi spectral collocation method to solve nonlinear Volterra-Fredholm integral equations with smooth kernels. The main idea in this approach is to convert the original problem into an equivalent one through appropriate variable transformations so that the resulting equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. The convergence and error analysis are discussed for both <italic>L</italic><sup>∞</sup> and weighted <italic>L</italic><sup>2</sup> norms. We confirm the theoretical prediction of the exponential rate of convergence by the numerical results which are compared with well-known methods.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Application of the Extended Fan Sub-Equation Method to Time Fractional Burgers-Fisher Equationhttps://sciendo.com/article/10.2478/tmmp-2021-0016<abstract> <title style='display:none'>Abstract</title> <p>In this paper, the extended Fan sub-equation method to obtain the exact solutions of the generalized time fractional Burgers-Fisher equation is applied. By applying this method, we obtain different solutions that are benefit to further comprise the concepts of complex nonlinear physical phenomena. This method is simple and can be applied to several nonlinear equations. Fractional derivatives are taken in the sense of Jumarie’s modified Riemann-Liouville derivative. A comparative study with the other methods approves the validity and effectiveness of the technique, and on the other hand, for suitable parameter values, we plot 2D and 3D graphics of the exact solutions by using the extended Fan sub-equation method. In this work, we use Mathematica for computations and programming.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Oscillation Tests for Linear Difference Equations with Non-Monotone Argumentshttps://sciendo.com/article/10.2478/tmmp-2021-0021<abstract> <title style='display:none'>Abstract</title> <p>This paper presents sufficient conditions involving limsup for the oscillation of all solutions of linear difference equations with general deviating argument of the form <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0021_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Δ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>ℕ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mtext> </mml:mtext><mml:mo stretchy="false">[</mml:mo><mml:mo>∇</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℕ</mml:mi><mml:mo stretchy="false">]</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math><tex-math>$\Delta x(n) + p(n)x(\tau (n)) = 0,\,n \in {_0}\quad [\nabla x(n) - q(n)x(\sigma (n)) = 0,\,n \in ],\ </tex-math></alternatives></inline-formula>, where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0021_eq_002.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math><tex-math><?CDATA \[{(p(n))_{n \ge 0}}$$?></tex-math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0021_eq_003.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math><tex-math><?CDATA$${(q(n))_{n \ge 1}}$\$?></tex-math></alternatives></inline-formula> are sequences of nonnegative real numbers and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0021_eq_004.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math><tex-math>${(\tau (n))_{n \ge 0}},\quad {(\sigma (n))_{n \ge 1}}$</tex-math></alternatives></inline-formula> are (not necessarily monotone) sequences of integers. The results obtained improve all well-known results existing in the literature and an example, numerically solved in MATLAB, illustrating the significance of these results is provided.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Explicit Non Algebraic Limit Cycle for a Discontinuous Piecewise Differential Systems Separated by One Straight Line and Formed by Linear Center and Linear System Without Equilibriahttps://sciendo.com/article/10.2478/tmmp-2021-0019<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we deal with the discontinuous piecewise differential linear systems formed by two differential systems separated by a straight line when one of these two differential systems is a linear without equilibria and the other is a linear center. We are going to show that the maximum number of crossing limit cycles is one, and if exists, it is non algebraic and analytically given.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Oscillation Behaviour of Solutions for a Class of a Discrete Nonlinear Fractional-Order Derivativeshttps://sciendo.com/article/10.2478/tmmp-2021-0022<abstract> <title style='display:none'>Abstract</title> <p>Based on the generalized Riccati transformation technique and some inequality, we study some oscillation behaviour of solutions for a class of a discrete nonlinear fractional-order derivative equation <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0022_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Δ</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mi>Δ</mml:mi><mml:mi>μ</mml:mi></mml:msup><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mi>η</mml:mi></mml:msup><mml:mo stretchy="false">]</mml:mo><mml:mo>+</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math><tex-math> $\Delta [\gamma (\ell ){[\alpha (\ell ) + \beta (\ell ){\Delta ^\mu }u(\ell )]^\eta }] + \phi (\ell )f[G(\ell )] = 0,\ell \in {N_{{\ell _0} + 1 - \mu }},$ </tex-math></alternatives></inline-formula> where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0022_eq_002.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>−</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:math><tex-math>${\ell _0} &gt; 0,\quad G(\ell ) = \sum\limits_{j = {\ell _0}}^{\ell - 1 + \mu } {{{(\ell - j - 1)}^{( - \mu )}}u(j)}$</tex-math></alternatives></inline-formula> and Δ<sup><italic>μ</italic></sup> is the Riemann-Liouville (R-L) difference operator of the derivative of order <italic>μ</italic>, 0 &lt; <italic>μ</italic> ≤ 1 and <italic>η</italic> is a quotient of odd positive integers. Illustrative examples are given to show the validity of the theoretical results.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Oscillatory Behaviour of Second-Order Nonlinear Differential Equations with Mixed Neutral Termshttps://sciendo.com/article/10.2478/tmmp-2021-0023<abstract> <title style='display:none'>Abstract</title> <p>The authors examine the oscillation of second-order nonlinear differential equations with mixed nonlinear neutral terms. They present new oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are illustrated by some examples.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Some Results Involving the Airy Functions and Airy Transformshttps://sciendo.com/article/10.2478/tmmp-2021-0017<abstract> <title style='display:none'>Abstract</title> <p>In the present work, the author studied some properties of the modified Bessel’s functions and Airy functions. It is worth mentioning that the Airy functions are used in many fields of physics. They are applied in many branches of classical and quantum physics. The author also studied certain properties of the Airy transform and derived some new integral relations involving the Airy functions. Non-trivial illustrative examples are provided as well. All the results are presented in lucid and comprehensible language.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Properties of the Katugampola Fractional Operatorshttps://sciendo.com/article/10.2478/tmmp-2021-0024<abstract> <title style='display:none'>Abstract</title> <p>In this work, there are considered higher order fractional operators defined in the sense of Katugampola. There are proved some fundamental properties of the Katugampola fractional operators of any arbitrary real order. Moreover, there are given conditions ensuring existence of the higher order Katugampola fractional derivative in space of the absolutely continuous functions.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Certain Singular Distributions and Fractalshttps://sciendo.com/article/10.2478/tmmp-2021-0026<abstract> <title style='display:none'>Abstract</title> <p>In the presented paper, the main attention is given to fractal sets whose elements have certain restrictions on using digits or combinations of digits in their own nega-P-representation. Topological, metric, and fractal properties of images of certain self-similar fractals under the action of some singular distributions, are investigated.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Two Non Algebraic Limit Cycles of a Class of Polynomial Differential Systems with Non-Elementary Equilibrium Pointhttps://sciendo.com/article/10.2478/tmmp-2021-0018<abstract> <title style='display:none'>Abstract</title> <p>The problems of existence of limit cycles and their numbers are the most difficult problems in the dynamical planar systems. In this paper, we study the limit cycles for a family of polynomial differential systems of degree 6<italic>k</italic> + 1, <italic>k</italic> ∈ ℕ*, with the non-elementary singular point. Under some suitable conditions, we show our system exhibiting two non algebraic or two algebraic limit cycles explicitly given. To illustrate our results we present some examples.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Controllability of Nonlocal Impulsive Functional Differential Equations with Measure of Noncompactness in Banach Spaceshttps://sciendo.com/article/10.2478/tmmp-2021-0020<abstract> <title style='display:none'>Abstract</title> <p>This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii’s Fixed Point Theorem, we get the controllability of nonlocal impulsive differential equations under compactness conditions, Lipschitz conditions and mixed-type conditions, respectively.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Existence and Multiplicity of Positive Solutions for a Third-Order Two-Point Boundary Value Problemhttps://sciendo.com/article/10.2478/tmmp-2021-0027<abstract> <title style='display:none'>Abstract</title> <p>We study the existence and multiplicity of positive solutions for a third-order two-point boundary value problem by applying Krasnosel’skii’s fixed point theorem. To illustrate the applicability of the obtained results, we consider some examples.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00Existence of The Asymptotically Periodic Solution to the System of Nonlinear Neutral Difference Equationshttps://sciendo.com/article/10.2478/tmmp-2021-0025<abstract> <title style='display:none'>Abstract</title> <p>The system of nonlinear neutral difference equations with delays in the form <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2021-0025_eq_001.png"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>{</mml:mo> <mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>Δ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mi>Δ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mi>τ</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:mrow></mml:math><tex-math>$\left\{ \begin{array}{l} \Delta ({y_i}(n) + {p_i}(n){y_i}(n - {\tau _i})) = {a_i}(n){f_i}({y_{i + 1}}(n)) + {g_i}(n),\\ \Delta ({y_m}(n) + {p_m}(n){y_m}(n - {\tau _m})) = {a_m}(n){f_m}({y_1}(n)) + {g_m}(n), \end{array} \right.$</tex-math></alternatives></inline-formula> for <italic>i</italic> = 1, . . . , <italic>m</italic> − 1, <italic>m</italic> ≥ 2, is studied. The sufficient conditions for the existence of an asymptotically periodic solution of the above system, are established. Here sequences (<italic>p<sub>i</sub></italic>(<italic>n</italic>)), <italic>i</italic> = 1,..., <italic>m</italic>, are bounded away from -1. The presented results are illustrated by theoretical and numerical examples.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00A Fixed Point Approach to the Hyers-Ulam-Rassias Stability Problem of Pexiderized Functional Equation in Modular Spaceshttps://sciendo.com/article/10.2478/tmmp-2021-0005<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we consider pexiderized functional equations for studying their Hyers-Ulam-Rassias stability. This stability has been studied for a variety of mathematical structures. Our framework of discussion is a modular space. We adopt a fixed-point approach to the problem in which we use a generalized contraction mapping principle in modular spaces. The result is illustrated with an example.</p> </abstract>ARTICLE2022-01-01T00:00:00.000+00:00en-us-1