Authors:David Aron, Santosh Kumar Pages: 3 - 12 Abstract: In this paper, we present some fixed point results for multivalued non-self mappings. We generalize the fixed point theorem due to Altun and Minak [2] by using Jleli and Sameti [9] \(\vartheta\)-contraction. To validate the results proved here, we provide an appropriate application of our main result. PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.001 Issue No:Vol. 8, No. 1 (2022)

Authors:Chiranjib Choudhury, Shyamal Debnath Pages: 13 - 22 Abstract: In this paper, we introduce and investigate the concept of \(A^{\mathcal{I^{K}}}\)-summability as an extension of \(A^{\mathcal{I^{*}}}\)-summability which was recently (2021) introduced by O.H.H.~Edely, where \(A=(a_{nk})_{n,k=1}^{\infty}\) is a non-negative regular matrix and \(\mathcal{I}\) and \(\mathcal{K}\) represent two non-trivial admissible ideals in \(\mathbb{N}\). We study some of its fundamental properties as well as a few inclusion relationships with some other known summability methods. We prove that \(A^{\mathcal{K}}\)-summability always implies \(A^{\mathcal{I^{K}}}\)-summability whereas \(A^{\mathcal{I}}\)-summability not necessarily implies \(A^{\mathcal{I^{K}}}\)-summability. Finally, we give a condition namely \(AP(\mathcal{I},\mathcal{K})\) (which is a natural generalization of the condition \(AP\)) under which \(A^{\mathcal{I}}\)-summability implies \(A^{\mathcal{I^{K}}}\)-summability.

Authors:Dinesan Deepthy, Joseph Varghese Kureethara Pages: 23 - 33 Abstract: The induced \(nK_2\) decomposition of infinite square grids and hexagonal grids are described here. We use the multi-level distance edge labeling as an effective technique in the decomposition of square grids. If the edges are adjacent, then their color difference is at least 2 and if they are separated by exactly a single edge, then their colors must be distinct. Only non-negative integers are used for labeling. The proposed partitioning technique per the edge labels to get the induced \(nK_2\) decomposition of the ladder graph is the square grid and the hexagonal grid. PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.003 Issue No:Vol. 8, No. 1 (2022)

Authors:Tatiana F. Filippova Pages: 34 - 42 Abstract: Using the technique of generalized inequalities of the Hamilton--Jacobi--Bellman type, we study here the state estimation problem for a control system which operates under conditions of uncertainty and nonlinearity of a special kind, when the dynamic equations describing the studied system simultaneously contain the different forms of nonlinearity in state velocities. Namely, quadratic functions and uncertain matrices of state elocity coefficients are presented therein. The external ellipsoidal bounds for reachable sets are found, some approaches which may produce internal estimates for such sets are also mentioned. The example is included to illustrate the result. PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.004 Issue No:Vol. 8, No. 1 (2022)

Authors:Ivan A. Finogenko, Alexander N. Sesekin Pages: 43 - 54 Abstract: Nonlinear control systems presented as differential inclusions with positional impulse controls are investigated. By such a control we mean some abstract operator with the Dirac function concentrated at each time. Such a control ("running impulse"), as a generalized function, has no meaning and is formalized as a sequence of correcting impulse actions on the system corresponding to a directed set of partitions of the control interval. The system responds to such control by discontinuous trajectories, which form a network of so-called "Euler's broken lines." If, as a result of each such correction, the phase point of the object under study is on some given manifold (hypersurface), then a slip-type effect is introduced into the motion of the system, and then the network of "Euler's broken lines" is called an impulse-sliding mode. The paper deals with the problem of approximating impulse-sliding modes using sequences of continuous delta-like functions. The research is based on Yosida's approximation of set-valued mappings and some well-known facts for ordinary differential equations with impulses. PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.005 Issue No:Vol. 8, No. 1 (2022)

Authors:Elena K. Kostousova Pages: 55 - 63 Abstract: We consider the problem of an enhanced evasion for linear discrete-time systems, where there are two conflicting bounded controls and the aim of one of them is to be guaranteed to avoid the trajectory hitting a given target set at a given final time and also at intermediate instants. First we outline a common solution scheme based on the construction of so called solvability tubes or repulsive tubes. Then a much more quick and simple for realization method based on the construction of the tubes with parallelepiped-valued cross-sections is presented under assumptions that the target set is a parallelepiped and parallelotope-valued constraints on controls are imposed. An example illustrating this method is considered. PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.006 Issue No:Vol. 8, No. 1 (2022)

Authors:X. Lenin Xaviour, S. Ancy Mary Pages: 64 - 75 Abstract: A set \(S\) of vertices in a connected graph \(G=(V,E)\) is called a signal set if every vertex not in \(S\) lies on a signal path between two vertices from \(S\). A set \(S\) is called a double signal set of \(G\) if \(S\) if for each pair of vertices \(x,y \in G\) there exist \(u,v \in S\) such that \(x,y \in L[u,v]\). The double signal number \(\mathrm{dsn}\,(G)\) of \(G\) is the minimum cardinality of a double signal set. Any double signal set of cardinality \(\mathrm{dsn}\,(G)\) is called \(\mathrm{dsn}\)-set of \(G\). In this paper we introduce and initiate some properties on double signal number of a graph. We have also given relation between geodetic number, signal number and double signal number for some classes of graphs. PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.007 Issue No:Vol. 8, No. 1 (2022)

Authors:Alena I. Machtakova, Nikolai N. Petrov Pages: 76 - 89 Abstract: In finite-dimensional Euclidean space, we analyze the problem of pursuit of a single evader by a group of pursuers, which is described by a system of differential equations with Caputo fractional derivatives of order \(alpha.\) The goal of the group of pursuers is the capture of the evader by at least \(m\) different pursuers (the instants of capture may or may not coincide). As a mathematical basis, we use matrix resolving functions that are generalizations of scalar resolving functions. We obtain sufficient conditions for multiple capture of a single evader in the class of quasi-strategies. We give examples illustrating the results obtained. PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.008 Issue No:Vol. 8, No. 1 (2022)

Authors:M. Nithya, C. Sugapriya, S. Selvakumar, K. Jeganathan, T. Harikrishnan Pages: 90 - 116 Abstract: This paper explores the two-commodity (TC) inventory system in which commodities are classified as major and complementary items. The system allows a customer who has purchased a free product to conduct Bernoulli trials at will. Under the Bernoulli schedule, any entering customer will quickly enter an orbit of infinite capability during the stock-out time of the major item. The arrival of a retrial customer in the system follows a classical retrial policy. These two products' re-ordering process occurs under the \((s, Q)\) and instantaneous ordering policies for the major and complimentary items, respectively. A comprehensive analysis of the retrial queue, including the system's stability and the steady-state distribution of the retrial queue with the stock levels of two commodities, is carried out. The various system operations are measured under the stability condition. Finally, numerical evidence has shown the benefits of the proposed model under different random situations. PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.009 Issue No:Vol. 8, No. 1 (2022)

Authors:Olga A. Tilzo Pages: 117 - 127 Abstract: We study the monopolistic competition model with producer-retailer-consumers two-level interaction. The industry is organized according to the Dixit–Stiglitz model. The retailer is the only monopolist. A quadratic utility function represents consumer preferences. We consider the case of the retailer's leadership; namely, we study two types of behavior: with and without the free entry condition. Earlier, we obtained the result: to increase social welfare and/or consumer surplus, the government needs to subsidize (not tax!) retailers. In the presented paper, we develop these results for the situation when the producer imposes an entrance fee for retailers. PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.010 Issue No:Vol. 8, No. 1 (2022)

Authors:D. Vamshee Krishna, D. Shalini Pages: 128 - 135 Abstract: In this paper, we are introducing certain subfamilies of holomorphic functions and making an attempt to obtain an upper bound (UB) to the second and third order Hankel determinants by applying certain lemmas, Toeplitz determinants, for the normalized analytic functions belong to these classes, defined on the open unit disc in the complex plane. For one of the inequality, we have obtained sharp bound. PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.011 Issue No:Vol. 8, No. 1 (2022)

Authors:Sergey V. Zakharov Pages: 136 - 144 Abstract: The solution of the Cauchy problem for the vector Burgers equation with a small parameter of dissipation \(\varepsilon\) in the \(4\)-dimensional space-time is studied: $$ \mathbf{u}_t + (\mathbf{u}\nabla) \mathbf{u} = \varepsilon \triangle \mathbf{u}, \quad u_{\nu} (\mathbf{x}, -1, \varepsilon) = - x_{\nu} + 4^{-\nu}(\nu + 1) x_{\nu}^{2\nu + 1}, $$ With the help of the Cole–Hopf transform \(\mathbf{u} = - 2 \varepsilon \nabla \ln H,\) the exact solution and its leading asymptotic approximation, depending on six space-time scales, near a singular point are found. A formula for the growth of partial derivatives of the components of the vector field \(\mathbf{u}\) on the time interval from the initial moment to the singular point, called the formula of the gradient catastrophe, is established: $$ \frac{\partial u_{\nu} (0, t, \varepsilon)}{\partial x_{\nu}} = \frac{1}{t} \left[ 1 + O \left( \varepsilon t ^{- 1 - 1/\nu} \right) \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to -\infty, \quad t \to -0. $$ The asymptotics of the solution far from the singular point, involving a multistep reconstruction of the space-time scales, is also obtained: $$ u_{\nu} (\mathbf{x}, t, \varepsilon) \approx - 2 \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \tanh \left[ \frac{x_{\nu}}{\varepsilon} \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to +\infty. $$ PubDate: 2022-07-29 DOI: 10.15826/umj.2022.1.012 Issue No:Vol. 8, No. 1 (2022)