Journal of the Nigerian Mathematical Society <p>Journal of the Nigerian Mathematical Society (JNMS)</p> <p>JNMS provides a means of communication and exchange of ideas among workers in mathematical sciences (mathematics, mathematical physics, statistics, computer science), and offers an effective method of bringing new results quickly to the public. By doing so the journal establishes an informal vehicle enabling the field of mathematical sciences to be understood in a broad sense. It will include theoretical and experimental results, and fundamental and practical research.<br /><br /></p> en-US (Professor Samuel S. Okoya) (ICTS) Thu, 08 Apr 2021 15:08:23 +0000 OJS 60 The Influence Of J. O. C. Ezeilo On Mathematics (1930-2013) <p>Emeritus Professor J.O.C. Ezeilo (popularly called JOC) was born on January 17, 1930 at Nanka in Anambra State, Nigeria. He was married with four children. He had the following degrees: B.Sc. (First Class Hons.) in Mathematics (1953); B.Sc. Advanced (Passed with Distinction) (1954); M.Sc. (1955) all from the University of London; Ph.D. (1958) Cambridge University, England; D.Sc. (h.c.) (1989) University of Maiduguri; D. Tech. (h.c.) (1995) Federal University of Technology, Akure; D.Sc. (h.c.) (1996) University of Nigeria, Nsukka; D.Sc. (h.c) (2008) Anambra State University, Uli.</p> F. I. Ochor Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Review Of Selected Publications Of Professor J. O. C. Ezeilo <p>Any attempt to classify the research works of Professor J.O.C. Ezeilo is a big job. Why? He had spent all his research life, (spanning over fifty-one years), in studying the qualitative properties of higher order nonlinear differential equation of orders 2,3,4,5,6, Nth order ( N odd and even) and systems in general. The qualitative properties studied covered many types of properties which include: Boundedness, Ultimate boundedness, Existence and Uniqueness results, Stability, Instability, periodicity, Oscillations, Resonance and Non-resonance, etc. While most of his life-long work was on the construction and use of Lyapunov functions, he spent the latter part of his life occasionally going into the use of topological degree methods and Leray Schauder techniques. These results initiated into higher order differential equations some well known results of the 2nd order, generalizing works of initiators in the subject, such as Cartwright (his Ph.D. supervisor), Barbashin, Loud, Erugin, Malkin, Krasovskii, Ogurcov, Reissig, Tejumola (his Ph.D. student), Pliss, Swick, Chow, Dunninger, among others.</p> A. U. Afuwape, F. I. Ochor Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 What Some Colleagues Say Of Ezeilo <p>We begin this article, which deals largely with what J. O. C. Ezeilo’s colleagues have to say about his significant influence on Mathematics and impact upon lives as attested to by those that worked with him. Some comments from his colleagues are as presented below</p> Colleagues Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Picard Iteration Process For A General Class Of Contractive Mappings <p>Let (E, ρ) be a metric space. Let T : E → E be a map with F(T) := {x ∈ E : T x = x} 6= ∅ such that ρ(T x, p) ≤ aρ(x, p), ∀x ∈ E, p ∈ F(T) and some a ∈ [0, 1). It is shown that the class of mappings satisfying this condition is more general than the class of contraction mappings with fixed points. Several classes of nonlinear operators studied by various authors are shown to belong to this class. Finally, it is shown that the Picard iteration process converges to the unique fixed point of T. Our theorem improves a recent result of Akewe and Olaleru (British Journal of Mathematics and Computer Science, 3(3): 437-447, 2013) and a host of other results.</p> C. E. Chidume, J. O> Olaleru Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Pendulum-Like Systems Of A Certain Class Of Nonlinear Differential Equations - A Review∗ <p>From the ideal pendulum equation of second order non-linear differential equation</p> <p>x¨ + b sin x = 0</p> <p>to the damped and forced equations, we look at the generalized mathematical pendulum and systems associated with it. The works of M.L.Cartwright &amp; J.E. Littlewood, J.O.C. Ezeilo and H.O. Tejumola are indicated. The generalization to systems with multiple-equilibria, canonical forms, reduction methods and generalizations to higher-order differential equations are highlighted. Pendulum-like systems of third order are given as examples, for dichotomy and gradient-like solutions. Challenges to Nigerian Mathematical Society are highly emphasised in conclusion.</p> A. U. Afuwape Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Mathematical Analysis Of An Age-Structured Vaccination Model For Measles <p>An age-structured vaccination model for the transmission dynamics of measles in a population is designed and rigorously analysed. In the absence of vaccination, the model exhibits the phenomenon of backward bifurcation (where an asymptotically-stable disease-free equilibrium (DFE) co-exists with an asymptotically-stable endemic equilibrium whenever the associated reproduction number is less than unity). This phenomenon is shown to arise due to the imperfect nature of children’s natural immunity against infection or measles induced mortality. For the case when the measles-induced mortality is negligible, it is shown, using a linear Lyapunov function, that the DFE of the model without vaccination is globally-asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the vaccination free model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a non-linear Lyapunov function of Goh-Volterra type, to be globally-asymptotically stable for a special case. Numerical simulations of the vaccination model show that the use of an imperfect anti-measles vaccine can result in the effective control of measles in the community provided the vaccine efficacy and coverage rate are high enough.</p> S. M. Garba, A. B. Gumel, N. Hussaini Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Strong Convergence Theorems For Equilibrium Problems And Fixed Points Of Asymptotically Nonexpansive Maps <p>Zhenhua He and Wei-Shih Du, Fixed Point Theory and Applications 2011, 2011:33 introduced a new method of finding a common element in the intersection of the set of solutions of a finite family of equilibrium problems and the set of fixed points of a nonexpansive mapping in real Hilbert spaces. In this paper we modify the algorithm of He and Du and prove strong convergence results for finding a common element in the intersection of the set of solutions of a finite family of equilibrium problems and the set of fixed points of an asymptotically nonexpansive mapping in real Hilbert spaces.</p> M. O. Osilike, C. I. Ugwuogor Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Iterative Procedures For Finite Family Of Total Asymptotically Nonexpansive Mappings <p>It is our aim in this paper to introduce an explicit iterative scheme for a finite family of total asymptotically nonexpansive mappings and prove its strong convergence to a common fixed point of these mappings in smooth reflexive real Banach spaces which admits weakly sequentially continuous duality mappings. In addition, we proved path existence theorem for finite family of asymptotically nonexpansive mappings; and further showed that the convergence of the path guarantees that the set of common fixed points of finite family of asymptotically nonexpansive mappings is a sunny nonexpansive retract. Our theorems improve, generalize and unify several recently announced results in the literature.</p> E. U. Ofoedu, L. O. Madu Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Modified Iterative Algorithm For Family Of Asymptotically Nonexpansive Mappings In Banach Spaces <p>In this paper we introduce a new modified iterative scheme for approximation of common fixed points of countably infinite family of asymptotically nonexpansive mappings and solutions of some variational inequality problems. We prove strong convergence theorem that extend and generalize some recent results. Our Theorem particularly, is applicable in lp spaces (1 &lt;p&lt; ∞).</p> B. Ali, G. C. Ogwunnadi Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Weak And Strong Convergence Theorems For Approximating Fixed Points Of Nonexpansive Mappings Using A Composite Hybrid Iteration Method <p>We prove that the recent results of Miao and Li [Applicable Analysis and Discrete Mathematics 2(2008), 197- 204,] concerning the iterative approximation of fixed points of nonexpansive mappings in Hilbert spaces using a composite hybrid iteration method can be extended to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operators.</p> D. I. Igbokwe, U. S. Jim Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Some Fixed Point Theorems In Cone Rectangular Metric Spaces <p>We establish new fixed point theorems in cone rectangular metric spaces. The presented theorems generalize, extend and improve some existing results in the literature including the results of M. Jleli and B. Samet (2009), A. Azam and M. Arshad (2008), S. Moradi (2009), L.G. Huang and X. Zhang (2007), Sh. Rezapour and R. Hamlbarani (2008), I. Sahin and M. Telci (2009), and others. In all our results, we dispense with the the normality assumption which is a characteristic of most of the previous results.</p> J. O. Olaleru, B. Samet Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 A Generalization Of Afuwape-Barbashin-Ezeilo Problem For Certain Third Order Nonlinear Vibrations <p>Using frequency-domain results of Afuwape [2], a theorem of Corduneanu on abstract Volterra operators [4] we have given conditions under which the third order system<br>μ (x) ˙x = y − x<br>αy˙ = g(z − y)<br>z˙ = f(y − x) − gy<br>has a periodic solution - where the nonlinearity f satisfies a sector condition, α and g are constants. Our result constitute an application of [4] (pp. 124) and a generalization of [2].</p> X. Udo-Utun Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 On Span Of Flag Manifolds Rf(1, 1, 1, N − 3) <p>We obtain bounds for the span of the incomplete flag manifold of length 3, RF(1, 1, 1, n−3), n ≥ 4 using suitable fiberings where RF(1, 1, 1, n − 3) is either a total space or base space. We obtain exact values for n = 5 and 6 using non vanishing Stiefel-Whitney classes.</p> D. O. A. Ajayi Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Coefficient Estimates For Certain Classes Of Analytic Functions <p>Bounds on early coefficients of analytic functions normalized by f(0) = f(0) − 1 = 0 which satisfy</p> <p>Re f(z) α−1f(z)zα−11 + zf(z)f(z)&gt; 0</p> <p>in the unit disk U = {z ∈ C : |z| &lt; 1} are obtained using known properties of functions with positive real part.</p> M. A. Ganiyu, F. M. Jimoh, C. N. Ejieji, K. O. Babalola Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Symmetric Two-Step Runge-Kutta Collocation Methods For Stiff Systems Of Ordinary Differential Equations <p>Symmetric two-step Runge-Kutta collocation methods have been derived for solution of stiff and oscillatory differential equations. The methods are of orders six and eight with five and seven stages respectively and hence substantial improvements in efficiency and flexibility are achieved when using them. They are shown to be A-stable, self-starting, convergent and cope effectively with stable systems of initial value problems with large Lipschitz constants. These methods as compared, for example, with some other recently derived methods of the same order, provide approximations of high accuracy to solutions of systems of initial value problems in ordinary differential equations over the entire interval of integration. The analytic discussion is confirmed by numerical examples</p> D. G. Yakubu, U. I. Huoma, A. M. Kwami Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 A Comparison Of The Implicit Determinant Method And Inverse Iteration <p>It is well known that if the largest or smallest eigenvalue of a matrix has been computed by some numerical algorithms and one is interested in computing the corresponding eigenvector, one method that is known to give such good approximations to the eigenvector is inverse iteration with a shift. However, in a situation where the desired eigenvalue is defective, inverse iteration converges harmonically to the eigenvalue close to the shift. In this paper, we extend the implicit determinant method of Spence and Poulton [13] to compute a defective eigenvalue given a shift close to the eigenvalue of interest. For a defective eigenvalue, the proposed approach gives quadratic convergence and this is verified by some numerical experiments.</p> R. O. Akinola, A. Spence Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 A Five-Step Extended Block Backward Differentiation Formula For Solutions Of Semi-Explicit Index-1 Dae Systems <p>A five-step Extended Block Backward Differentiation Formulae (EBBDF) for the solutions of semi-explicit index-1 systems of Differential Algebraic Equations (DAEs) is presented. The processes compute the solutions of DAEs in a block by block fashion by some discrete schemes obtained from the associated continuous scheme which are combined and implemented as a set of block formulae. Numerical results revealed this method to be efficient and very accurate, and particularly suitable for semi implicit index one DAEs.</p> O. A. Akinfenwa Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Network Coding For Wireless Security <p>In this paper, a Random Linear Network Coding (RLNC) scheme which takes advantage of redundant network capacity for improved success probability and robustness of wireless communication networks is proposed. We provide sufficient literature and illustrate vital potential benefits of this scheme over the normal Hierarchical Routing Scheme (HRS). Our model derivation yields a new bound on the required field size with useful transition probabilities. Specifically, the system overheads of both schemes are compared through extensive computer simulations in an ideal network setting. Simulation results reveal that optimum field size for effective throughput and overhead management are achievable with the RLNC scheme, and the delivery ratio can be controlled by varying the number of<br>subnets (k) in both schemes</p> M. E. Ekpenyong, B. L. Ibanga Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Steady Arrhenius Laminar Free Convective Mhd Flow And Heat Transfer Past A Vertical Stretching Sheet With Viscous Dissipation <p>An analysis of the effects of Arrhenius kinetics on hydromagnetic free convective flow(set up due to temperature) of an electrically conducting fluid past a vertical stretching sheet kept at constant temperature with viscous dissipation is presented. A similarity transformation is used to reduce the governing partial differential equations into a system of ordinary differential equations, which is solved numerically . The effect of various parameters on the velocity and temperature profiles as well as the skin friction and Nusselt number are presented in graphs and tables. It was shown that the velocity and temperature increases as local Eckert number (or viscous dissipation parameter) increases.</p> A. J. Omowaye, O. K. Koriko Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Initial-Boundary-Value Problem Of Hyperbolic Equations For Viscous Blood Flow Through A Tapered Vessel <p>In this paper, the effect of viscosity on blood flow through a tapered artery is studied. Approximate solutions of the coupled nonlinear partial differential equations that model the viscous blood a complaint artery are obtained using Adomian decomposition method (ADM). The convergence and parametric study of the solution are presented and discussed including shock development. The result of the computation shows that viscosity has significant influence on blood flow.</p> S. O. Adesanya Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Dynamic Behaviour Of Non-Prismatic Rayleigh Beam On Pasternak Foundation And Under Partially Distributed Masses Moving At Varying Velocities <p>The dynamic analysis of the behaviour of Non-prismatic Rayleigh beam on Pasternak foundation under partially distributed masses moving at varying velocities is investigated in this paper. The solution technique is based on the expansion of Heaviside function in series form, the use of the generalized Galerkin method and a modification of Struble’s asymptotic method which reduces the governing fourth order partial differential equation to a coupled second order ordinary differential equation. Closed form solutions are obtained and numerical results in plotted curves are presented. The results show that as the value of rotatory inertia correction factor r0 increases, the response amplitude of the Rayleigh beam decreases. Similarly, higher values of the foundation stiffness K, shear modulus G and axial force N decrease the transverse deflection of the beam. The results further show that for fixed r0,K, G and N, the transverse deflection of the non-uniform Rayleigh beam resting on Pasternak foundation and under partially distributed masses moving at varying velocities are higher than those when only the force effects of the moving load are considered indicating clearly that resonance is reached earlier in moving distributed mass problem. This further confirms results in literature stressing the need to always consider the inertia terms when heavy loads traverse any form of structural members.</p> S. T. Oni, O. K. Ogunbamike Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Optimization Of Investment Returns With N-Step Utility Functions <p>In this paper, we examine different ways of allocating investments, maximizing and generating optimal wealth of investment returns with N-step utility functions; in an N period setting where the investor maximizes the expected utility of the terminal wealth in a stochastic market with different utility functions. The specific utility functions considered are negative exponential, logarithm, square root and power structures as the market state changes according to a Markov chain. The states of the market describe the prevailing economic, financial, social and other conditions that affect the deterministic parameters of the models using martingale approach to obtain the optimal solution. Thus, we determine the optimization strategies for investment returns in situations where investors at different utility functions could end up doubling or halving their stake. The performance of any utility function is determined by the ratio q : q'&nbsp;of the probability of rising to falling as well as the ratio p : p' of the risk neutral probability measure of rising to the falling.</p> J. T. Eghwerido, T. O. Obilade Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 A Note On Just-In-Time Scheduling On Flow Shop Machines <p>In this paper, the scheduling to maximize the weighted number of Just-In-Time jobs is considered. This problem is known to be NP Complete for when the due date is at a point in time indicating no efficient optimal solution is feasible in reliable time. Due dates with interval in time are considered in this work. The problem formulation is suggested, two greedy heuristics are proposed for solving the problem. A numerical example to illustrate its use and extensive computational experiments performed with promising results are presented. Likely areas of extensions are provided.</p> M. O. Adamu, N. Budlender, G. A. Idowu Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000 Modelling Of Quality Enhancement System For Queuing Situation In Wireless Communication Network <p>In this study, we considered queuing in wireless communication network system with two different arrival points S1 and S2 linked with a service point S3 which locates two different departure points S4 and S5. The stochastic processes of the arrival follow a Poisson distribution while the stochastic process of departure follows exponential distribution. Various probability functions derived were reduced to steady state equations. By using the generation functions techniques, we obtained the operating characteristics like the expected number of calls in the system, in queue and the expected time spent.</p> F. O. Ikpotokin, O. Aselinmen Copyright (c) 2021 Thu, 08 Apr 2021 00:00:00 +0000