The Hadamard product and recursively defined sequences

ODAM-Vol. 3 (2020), Issue 1, pp. 20 – 24 Open Access Full-Text PDF
Sergei Dmitrievich Kazenas
Abstract: In this paper, the approach to obtaining nontrivial formulas for some recursively defined sequences is illustrated. The most interesting result in the paper is the formula for the solution of quadratic map-like recurrence. Also, some formulas for the solutions of linear difference equations with variable coefficients are obtained. At the end of the paper, some integer sequences associated with a quadratic map are considered.
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Dynamic analysis of non-homogenous varying thickness rectangular plates resting on Pasternak and Winkler foundations

EASL-Vol. 3 (2020), Issue 1, pp. 1 – 20 Open Access Full-Text PDF
S. A. Salawu, M. G. Sobamowo, O. M. Sadiq
Abstract: Modern day technological advancement has resulted in manufacturing industries intensify the use and application of thin plates in their productions thereby, resulting in increased research awareness in the study of dynamic behavior of thin plates. This research analyzes the free vibration dynamic behavior of thin rectangular plates resting on elastic Winkler and Pasternak foundations using two-dimensional differential transformation method. The reliability of the obtained analytical solutions are validated with results presented in cited literature and confirmed very precise. However, the analytical solutions obtained are used to investigate the influence of elastic foundations, homogeneity and thickness variation on the dynamic behavior of the plates under clamped and condition. From the results obtained, it is realized that increase in non-homogenous material results in corresponding increase in natural frequency of the plates. Also, increase in Winkler, Pasternak and combine Winkler and Pasternak foundations stiffness leads to increase in natural frequency of the plates. Increase in thickness results to natural frequency increases. The findings will serve as benchmark for further study of plate vibration research.
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Walk counting and Nikiforov’s problem

ODAM-Vol. 3 (2020), Issue 1, pp. 11 – 19 Open Access Full-Text PDF
Lihua Feng, Lu Lu, Dragan Stevanović
Abstract: For a given graph, let \(w_k\) denote the number of its walks with \(k\) vertices and let \(\lambda_1\) denote the spectral radius of its adjacency matrix. Nikiforov asked in [Linear Algebra Appl 418 (2006), 257–268] whether it is true in a connected bipartite graph that \(\lambda_1^r\geq\frac{w_{s+r}}{w_s}\) for every even \(s\geq 2\) and even \(r\geq 2\)? We construct here several infinite sequences of connected bipartite graphs with two main eigenvalues for which the ratio \(\frac{w_{s+r}}{\lambda_1^r w_s}\) is larger than~1 for every even \(s,r\geq 2\), and thus provide a negative answer to the above problem.
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Extremal total eccentricity of \(k\)-apex trees

ODAM-Vol. 3 (2020), Issue 1, pp. 8 – 10 Open Access Full-Text PDF
Naveed Akhter, Hafiza Iqra Yasin
Abstract: In a simple connected graph \(G\), eccentricity of a vertex is one of the first, distance-based invariants. The eccentricity of a vertex \(v\) in a connected graph \(G\) is the maximum distance of the vertex \(v\) to any other vertex \(u\). The total eccentricity of the graph \(G\) is the sum of the all vertex eccentricities. A graph \(G\) is called an apex tree if it has a vertex \(x\) such that \(G-x\) is a tree. In this work we have found the graph having extremal total eccentricity of \(k\)-apex trees.
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Antimagicness of subdivided fans

OMS-Vol. 4 (2020), Issue 1, pp. 18 – 22 Open Access Full-Text PDF
Afshan Tabassum, Muhammad Awais Umar, Muzamil Perveen, Abdul Raheem
Abstract: A graph \(\Gamma\) (simple, finite, undirected) with an \(\Omega\)-covering has an \((\alpha,\delta)\)-\(\Omega\)-antimagic labeling if the weights of all subgraphs \(\Omega\) of graph \(\Gamma\) constitute an arithmetic progression with the common difference \(\delta\). Such a~graph is called super \((\alpha,\delta)\)-\(\Omega\)-antimagic if \(\nu(V(\Gamma))= \{ 1,2,3,\dots,|V(\Gamma)|\}\). In the present paper, the cycle coverings of subdivision of fan graphs has been considered and results are proved for several differences.
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A unified integral operator and further its consequences

OMA-Vol. 4 (2020), Issue 1, pp. 1 – 7 Open Access Full-Text PDF
Ghulam Farid
Abstract: The aim of this paper is to construct left sided and right sided integral operators in a unified form. These integral operators produce various well known integral operators in the theory of fractional calculus. Formulated integral operators of this study include generalized fractional integral operators of Riemann-Liouville type and operators containing Mittag-Leffler functions in their kernels. Also boundedness of all these fractional integral operators is derived from the boundedness of unified integral operators. The existence of new integral operators may have useful consequences in applied sciences besides in fractional calculus.
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Positive solutions for boundary value problem of sixth-order elastic beam equation

OMS-Vol. 4 (2020), Issue 1, pp. 9 – 17 Open Access Full-Text PDF
Zouaoui Bekri, Slimane Benaicha
Abstract: In this paper, we study the existence of positive solutions for boundary value problem of sixth-order elastic beam equation of the form \(-u^{(6)}(t)=q(t)f(t,u(t),u^{‘}(t),u^{”}(t),u^{”’}(t),u^{(4)}(t),u^{(5)}(t)),~~0<t<1,\) with conditions \(u(0)=u^{‘}(1)=u^{”}(0)=u^{”’}(1)=u^{(4)}(0)=u^{(5)}(1)=0,\) where \(f\in C([0,1]\times[0,\infty)\times[0,\infty)\times(-\infty,0]\times(-\infty,0]\times[0,\infty)\times[0,\infty)\rightarrow [0,\infty))\). The boundary conditions describe the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. We give sufficient conditions that allow us to obtain the existence of positive solution. The main tool used in the proof is the Leray-Schauder nonlinear alternative and Leray-Schauder fixed point theorem. As an application, we also give example to illustrate the results obtained.
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A fixed point theorem for generalized weakly contractive mappings in \(b\)-metric spaces

OMS-Vol. 4 (2020), Issue 1, pp. 1 – 8 Open Access Full-Text PDF
Eliyas Zinab, Kidane Koyas, Aynalem Girma
Abstract: In this paper we establish a fixed point theorem for generalized weakly contractive mappings in the setting of \(b\)-metric spaces and prove the existence and uniqueness of a fixed point for a self-mappings satisfying the established theorem. Our result extends and generalizes the result of Cho [1]. Finally, we provided an example in the support of our main result.
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Existence of solution for a nonlinear fifth-order three-point boundary value problem

OMA-Vol. 3 (2019), Issue 2, pp. 125 – 136 Open Access Full-Text PDF
Zouaoui Bekri, Slimane Benaicha
Abstract: In this paper, we explore the existence of nontrivial solution for the fifth-order three-point boundary value problem of the form \(u^{(5)}(t)+f(t,u(t))=0,\quad\text 0<t<1,\) with boundary conditions \(u(0)=0,\quad u^{‘}(0)=u^{”}(0)=u^{”’}(0)=0,\quad u(1)=\alpha u(\eta),\) where \(0<\eta<1\), \(\alpha\in\mathbb{R}\), \(\alpha\eta^{4}\neq1\), \(f\in C([0,1]\times\mathbb{R},\mathbb{R})\). Under certain growth conditions on the non-linearity \(f\) and using Leray-Schauder nonlinear alternative, we prove the existence of at least one solution of the posed problem. Furthermore, the obtained results are further illustrated by mean of some examples.
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