Abstract and Applied Analysis Articles (Project Euclid)
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The latest articles from Abstract and Applied Analysis on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTMon, 01 Nov 2010 10:13 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces
http://projecteuclid.org/euclid.aaa/1267538585
<strong>Siwaporn Saewan</strong>, <strong>Poom Kumam</strong>, <strong>Kriengsak Wattanawitoon</strong><p><strong>Source: </strong>Abstr. Appl. Anal., Volume 2010, 25 pages.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to introduce a new hybrid projection method for finding a common element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of
the variational inequality for an $\alpha$ -inverse-strongly monotone, and the set of solutions of the generalized equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Base on this result, we also get some new and interesting results. The results in this paper generalize, extend, and unify some well-known strong convergence results in the literature.
</p>projecteuclid.org/euclid.aaa/1267538585_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTImproving Fourier Partial Sum Approximation for Discontinuous Functions Using a Weight Functionhttps://projecteuclid.org/euclid.aaa/1513220444<strong>Beong In Yun</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 7 pages.</p><p><strong>Abstract:</strong><br/>
We introduce a generalized sigmoidal transformation ${w}_{m}(r;x)$ on a given interval $[a,b]$ with a threshold at $x=r\in (a,b)$ . Using ${w}_{m}(r;x)$ , we develop a weighted averaging method in order to improve Fourier partial sum approximation for a function having a jump-discontinuity. The method is based on the decomposition of the target function into the left-hand and the right-hand part extensions. The resultant approximate function is composed of the Fourier partial sums of each part extension. The pointwise convergence of the presented method and its availability for resolving Gibbs phenomenon are proved. The efficiency of the method is shown by some numerical examples.
</p>projecteuclid.org/euclid.aaa/1513220444_20171213220130Wed, 13 Dec 2017 22:01 ESTOn the Output Controllability of Positive Discrete Linear Delay Systemshttps://projecteuclid.org/euclid.aaa/1513220445<strong>Mouhcine Naim</strong>, <strong>Fouad Lahmidi</strong>, <strong>Abdelwahed Namir</strong>, <strong>Mostafa Rachik</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 12 pages.</p><p><strong>Abstract:</strong><br/>
Necessary and sufficient conditions for output reachability and null output controllability of positive linear discrete systems with delays in state, input, and output are established. It is also shown that output reachability and null output controllability together imply output controllability.
</p>projecteuclid.org/euclid.aaa/1513220445_20171213220130Wed, 13 Dec 2017 22:01 ESTCorrigendum to “A Three-Point Boundary Value Problem with an Integral Condition for a Third-Order Partial Differential Equation”https://projecteuclid.org/euclid.aaa/1513220446<strong>C. Latrous</strong>, <strong>A. Memou</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 1 page.</p>projecteuclid.org/euclid.aaa/1513220446_20171213220130Wed, 13 Dec 2017 22:01 ESTApproximation Properties of $q$ -Bernoulli Polynomialshttps://projecteuclid.org/euclid.aaa/1515466879<strong>M. Momenzadeh</strong>, <strong>I. Y. Kakangi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 6 pages.</p><p><strong>Abstract:</strong><br/>
We study the $q$ -analogue of Euler-Maclaurin formula and by introducing a new $q$ -operator we drive to this form. Moreover, approximation properties of $q$ -Bernoulli polynomials are discussed. We estimate the suitable functions as a combination of truncated series of $q$ -Bernoulli polynomials and the error is calculated. This paper can be helpful in two different branches: first we solve the differential equations by estimating functions and second we may apply these techniques for operator theory.
</p>projecteuclid.org/euclid.aaa/1515466879_20180108220206Mon, 08 Jan 2018 22:02 ESTFinite-Time Stability and Controller Design of Continuous-Time Polynomial Fuzzy Systemshttps://projecteuclid.org/euclid.aaa/1515466880<strong>Xiaoxing Chen</strong>, <strong>Manfeng Hu</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 12 pages.</p><p><strong>Abstract:</strong><br/>
Finite-time stability and stabilization problem is first investigated for continuous-time polynomial fuzzy systems. The concept of finite-time stability and stabilization is given for polynomial fuzzy systems based on the idea of classical references. A sum-of-squares- (SOS-) based approach is used to obtain the finite-time stability and stabilization conditions, which include some classical results as special cases. The proposed conditions can be solved with the help of powerful Matlab toolbox SOSTOOLS and a semidefinite-program (SDP) solver. Finally, two numerical examples and one practical example are employed to illustrate the validity and effectiveness of the provided conditions.
</p>projecteuclid.org/euclid.aaa/1515466880_20180108220206Mon, 08 Jan 2018 22:02 ESTContractibility of Fixed Point Sets of Mean-Type Mappingshttps://projecteuclid.org/euclid.aaa/1515466882<strong>S. Iampiboonvatana</strong>, <strong>P. Chaoha</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 7 pages.</p><p><strong>Abstract:</strong><br/>
We establish a convergence theorem and explore fixed point sets of certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces. We not only extend previous works by Matkowski to general normed linear spaces, but also obtain a new result on the structure of fixed point sets of quasi-nonexpansive mappings in a nonstrictly convex setting.
</p>projecteuclid.org/euclid.aaa/1515466882_20180108220206Mon, 08 Jan 2018 22:02 ESTNecessary and Sufficient Conditions for Set-Valued Maps with Set Optimizationhttps://projecteuclid.org/euclid.aaa/1518577260<strong>Abdessamad Oussarhan</strong>, <strong>Ikram Daidai</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 6 pages.</p><p><strong>Abstract:</strong><br/>
Optimality conditions are studied for set-valued maps with set optimization. Necessary conditions are given in terms of $S$ -derivative and contingent derivative. Sufficient conditions for the existence of solutions are shown for set-valued maps under generalized quasiconvexity assumptions.
</p>projecteuclid.org/euclid.aaa/1518577260_20180213220118Tue, 13 Feb 2018 22:01 ESTSoliton Solutions of the Coupled Schrödinger-Boussinesq Equations for Kerr Law Nonlinearityhttps://projecteuclid.org/euclid.aaa/1518577262<strong>Anwar Ja’afar Mohamad Jawad</strong>, <strong>Mahmood Jawad Abu-AlShaeer</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 8 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, the coupled Schrödinger-Boussinesq equations (SBE) will be solved by the sech, tanh, csch, and the modified simplest equation method (MSEM). We obtain exact solutions of the nonlinear for bright, dark, and singular 1-soliton solution. Kerr law nonlinearity media are studied. Results have proven that modified simple equation method does not produce the soliton solution in general case. Solutions may find practical applications and will be important for the conservation laws for dispersive optical solitons.
</p>projecteuclid.org/euclid.aaa/1518577262_20180213220118Tue, 13 Feb 2018 22:01 ESTGeneralized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusionshttps://projecteuclid.org/euclid.aaa/1518577263<strong>G. M. N’Guérékata</strong>, <strong>Marko Kostić</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 17 pages.</p><p><strong>Abstract:</strong><br/>
The main aim of this paper is to investigate generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions to a class of abstract (semilinear) multiterm fractional differential inclusions with Caputo derivatives. We illustrate our abstract results with several examples and possible applications.
</p>projecteuclid.org/euclid.aaa/1518577263_20180213220118Tue, 13 Feb 2018 22:01 ESTMultiobjective Optimization, Scalarization, and Maximal Elements of Preordershttps://projecteuclid.org/euclid.aaa/1518577264<strong>Paolo Bevilacqua</strong>, <strong>Gianni Bosi</strong>, <strong>Magalì Zuanon</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 6 pages.</p><p><strong>Abstract:</strong><br/>
We characterize the existence of (weak) Pareto optimal solutions to the classical multiobjective optimization problem by referring to the naturally associated preorders and their finite (Richter-Peleg) multiutility representation. The case of a compact design space is appropriately considered by using results concerning the existence of maximal elements of preorders. The possibility of reformulating the multiobjective optimization problem for determining the weak Pareto optimal solutions by means of a scalarization procedure is finally characterized.
</p>projecteuclid.org/euclid.aaa/1518577264_20180213220118Tue, 13 Feb 2018 22:01 ESTTwo Sufficient Conditions for Convex Ordering on Risk Aggregationhttps://projecteuclid.org/euclid.aaa/1521252060<strong>Dan Zhu</strong>, <strong>Chuancun Yin</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 5 pages.</p><p><strong>Abstract:</strong><br/>
We define new stochastic orders in higher dimensions called weak correlation orders. It is shown that weak correlation orders imply stop-loss order of sums of multivariate dependent risks with the same marginals. Moreover, some properties and relations of stochastic orders are discussed.
</p>projecteuclid.org/euclid.aaa/1521252060_20180316220113Fri, 16 Mar 2018 22:01 EDTStability for Linear Volterra Difference Equations in Banach Spaceshttps://projecteuclid.org/euclid.aaa/1523498503<strong>Rigoberto Medina</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 6 pages.</p><p><strong>Abstract:</strong><br/>
This paper is devoted to studying the existence and stability of implicit Volterra difference equations in Banach spaces. The proofs of our results are carried out by using an appropriate extension of the freezing method to Volterra difference equations in Banach spaces. Besides, sharp explicit stability conditions are derived.
</p>projecteuclid.org/euclid.aaa/1523498503_20180411220152Wed, 11 Apr 2018 22:01 EDTTime Scale Inequalities of the Ostrowski Type for Functions Differentiable on the Coordinateshttps://projecteuclid.org/euclid.aaa/1523498504<strong>Eze R. Nwaeze</strong>, <strong>Seth Kermausuor</strong>, <strong>Ana M. Tameru</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
In 2016, some inequalities of the Ostrowski type for functions (of two variables) differentiable on the coordinates were established. In this paper, we extend these results to an arbitrary time scale by means of a parameter $\lambda \in [\mathrm{0,1}]$ . The aforementioned results are regained for the case when the time scale $\mathbb{T}=\mathbb{R}$ . Besides extension, our results are employed to the continuous and discrete calculus to get some new inequalities in this direction.
</p>projecteuclid.org/euclid.aaa/1523498504_20180411220152Wed, 11 Apr 2018 22:01 EDTOn Solvability Theorems of Second-Order Ordinary Differential Equations with Delayhttps://projecteuclid.org/euclid.aaa/1525744868<strong>Nai-Sher Yeh</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 6 pages.</p><p><strong>Abstract:</strong><br/>
For each ${x}_{\mathrm{0}}\in [\mathrm{0,2}\pi )$ and $k\in \mathbf{N}$ , we obtain some existence theorems of periodic solutions to the two-point boundary value problem ${u}^{\mathrm{\prime }\mathrm{\prime }}(x)+{k}^{\mathrm{2}}u(x-{x}_{\mathrm{0}})+g(x,u(x-{x}_{\mathrm{0}}))=h(x)$ in $(\mathrm{0},\mathrm{2}\pi )$ with $u(\mathrm{0})-u(\mathrm{2}\pi )={u}^{\mathrm{\prime }}(\mathrm{0})-{u}^{\mathrm{\prime }}(\mathrm{2}\pi )=\mathrm{0}$ when $g:(\mathrm{0,2}\pi )\times\mathbf{R}\to \mathbf{R}$ is a Caratheodory function which grows linearly in $u$ as $|u|\to \mathrm{\infty }$ , and $h\in {L}^{\mathrm{1}}(\mathrm{0,2}\pi )$ may satisfy a generalized Landesman-Lazer condition $(\mathrm{1}+\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\beta )){\int }_{\mathrm{0}}^{\mathrm{2}\pi }h(x)v(x)dx<{\int }_{v(x)>\mathrm{0}}{g}_{\beta }^{+}(x){|v(x)|}^{\mathrm{1}-\beta }dx+{\int }_{v(x)<\mathrm{0}}{g}_{\beta }^{-}(x){|v(x)|}^{\mathrm{1}-\beta }dx$ for all $v\in N(L)\\{\mathrm{0}\}$ . Here $N(L)$ denotes the subspace of ${L}^{\mathrm{1}}(\mathrm{0,2}\pi )$ spanned by $\mathrm{sin}kx$ and $\mathrm{cos}kx$ , $-\mathrm{1}<\beta \le \mathrm{0}$ , ${g}_{\beta }^{+}(x)={\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}}_{u\to \mathrm{\infty }}(g(x,u)u/{|u|}^{\mathrm{1}-\beta })$ , and ${g}_{\beta }^{-}(x)={\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{f}}_{u\to -\mathrm{\infty }}(g(x,u)u/{|u|}^{\mathrm{1}-\beta })$ .
</p>projecteuclid.org/euclid.aaa/1525744868_20180507220114Mon, 07 May 2018 22:01 EDTOptimal Rational Approximations by the Modified Fourier Basishttps://projecteuclid.org/euclid.aaa/1525744869<strong>Arnak V. Poghosyan</strong>, <strong>Tigran K. Bakaryan</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 21 pages.</p><p><strong>Abstract:</strong><br/>
We consider convergence acceleration of the modified Fourier expansions by rational trigonometric corrections which lead to modified-trigonometric-rational approximations. The rational corrections contain some unknown parameters and determination of their optimal values for improved pointwise convergence is the main goal of this paper. The goal was accomplished by deriving the exact constants of the asymptotic errors of the approximations with further elimination of the corresponding main terms by appropriate selection of those parameters. Numerical experiments outline the convergence improvement of the optimal rational approximations compared to the expansions by the modified Fourier basis.
</p>projecteuclid.org/euclid.aaa/1525744869_20180507220114Mon, 07 May 2018 22:01 EDTA Version of Uncertainty Principle for Quaternion Linear Canonical Transformhttps://projecteuclid.org/euclid.aaa/1528855378<strong>Mawardi Bahri</strong>, <strong> Resnawati</strong>, <strong>Selvy Musdalifah</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 7 pages.</p><p><strong>Abstract:</strong><br/>
In recent years, the two-dimensional (2D) quaternion Fourier and quaternion linear canonical transforms have been the focus of many research papers. In the present paper, based on the relationship between the quaternion Fourier transform (QFT) and the quaternion linear canonical transform (QLCT), we derive a version of the uncertainty principle associated with the QLCT. We also discuss the generalization of the Hausdorff-Young inequality in the QLCT domain.
</p>projecteuclid.org/euclid.aaa/1528855378_20180612220317Tue, 12 Jun 2018 22:03 EDTA Deposition Model: Riemann Problem and Flux-Function Limits of Solutionshttps://projecteuclid.org/euclid.aaa/1528855379<strong>Hongjun Cheng</strong>, <strong>Shiwei Li</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 14 pages.</p><p><strong>Abstract:</strong><br/>
The Riemann solutions of a deposition model are shown. A singular flux-function limit of the obtained Riemann solutions is considered. As a result, it is shown that the Riemann solutions of the deposition model just converge to the Riemann solutions of the limit system, the scalar conservation law with a linear flux function involving discontinuous coefficient. Especially, for some initial data, the two-shock Riemann solution of the deposition model tends to the delta-shock Riemann solution of the limit system; by contrast, for some initial data, the two-rarefaction-wave Riemann solution of the deposition model tends to the vacuum Riemann solution of the limit system. Some numerical results exhibiting the formation processes of delta-shocks and vacuum states are presented.
</p>projecteuclid.org/euclid.aaa/1528855379_20180612220317Tue, 12 Jun 2018 22:03 EDT${C}^{\mathrm{1}}$ Hermite Interpolation with PH Curves Using the Enneper Surfacehttps://projecteuclid.org/euclid.aaa/1528855380<strong>Hyun Chol Lee</strong>, <strong>Jae Hoon Kong</strong>, <strong>Gwangil Kim</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
We show that the geometric and PH-preserving properties of the Enneper surface allow us to find PH interpolants for all regular ${C}^{\mathrm{1}}$ Hermite data-sets. Each such data-set is satisfied by two scaled Enneper surfaces, and we can obtain four interpolants on each surface. Examples of these interpolants were found to be better, in terms of bending energy and arc-length, than those obtained using a previous PH-preserving mapping.
</p>projecteuclid.org/euclid.aaa/1528855380_20180612220317Tue, 12 Jun 2018 22:03 EDTThe Implementation of Milstein Scheme in Two-Dimensional SDEs Using the Fourier Methodhttps://projecteuclid.org/euclid.aaa/1528855381<strong>Yousef Alnafisah</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 7 pages.</p><p><strong>Abstract:</strong><br/>
Multiple stochastic integrals of higher multiplicity cannot always be expressed in terms of simpler stochastic integrals, especially when the Wiener process is multidimensional. In this paper we describe how the Fourier series expansion of Wiener process can be used to simulate a two-dimensional stochastic differential equation (SDE) using Matlab program. Our numerical experiments use Matlab to show how our truncation of Itô’-Taylor expansion at an appropriate point produces Milstein method for the SDE.
</p>projecteuclid.org/euclid.aaa/1528855381_20180612220317Tue, 12 Jun 2018 22:03 EDTControllability and Observability of Nonautonomous Riesz-Spectral Systemshttps://projecteuclid.org/euclid.aaa/1528855382<strong>Sutrima Sutrima</strong>, <strong>Christiana Rini Indrati</strong>, <strong>Lina Aryati</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
There are many industrial and biological reaction diffusion systems which involve the time-varying features where certain parameters of the system change during the process. A part of the transport-reaction phenomena is often modelled as an abstract nonautonomous equation generated by a (generalized) Riesz-spectral operator on a Hilbert space. The basic problems related to the equations are existence of solutions of the equations and how to control dynamical behaviour of the equations. In contrast to the autonomous control problems, theory of controllability and observability for the nonautonomous systems is less well established. In this paper, we consider some relevant aspects regarding the controllability and observability for the nonautonomous Riesz-spectral systems including the Sturm-Liouville systems using a ${C}_{\mathrm{0}}$ -quasi-semigroup approach. Three examples are provided. The first is related to sufficient conditions for the existence of solutions and the others are to confirm the approximate controllability and observability of the nonautonomous Riesz-spectral systems and Sturm-Liouville systems, respectively.
</p>projecteuclid.org/euclid.aaa/1528855382_20180612220317Tue, 12 Jun 2018 22:03 EDTGeneralized Fractional Integral Operators Involving Mittag-Leffler Functionhttps://projecteuclid.org/euclid.aaa/1531274540<strong>Hafte Amsalu</strong>, <strong>D. L. Suthar</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 8 pages.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to study various properties of Mittag-Leffler (M-L) function. Here we establish two theorems which give the image of this M-L function under the generalized fractional integral operators involving Fox’s $H$ -function as kernel. Corresponding assertions in terms of Euler, Mellin, Laplace, Whittaker, and $K$ -transforms are also presented. On account of general nature of M-L function a number of results involving special functions can be obtained merely by giving particular values for the parameters.
</p>projecteuclid.org/euclid.aaa/1531274540_20180710220243Tue, 10 Jul 2018 22:02 EDTThe Existence and Structure of Rotational Systems in the Circlehttps://projecteuclid.org/euclid.aaa/1531274541<strong>Jayakumar Ramanathan</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 11 pages.</p><p><strong>Abstract:</strong><br/>
By a rotational system, we mean a closed subset $X$ of the circle, $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ , together with a continuous transformation $f:X\to X$ with the requirements that the dynamical system $(X,f)$ be minimal and that $f$ respect the standard orientation of $\mathbb{T}$ . We show that infinite rotational systems $(X,f)$ , with the property that map $f$ has finite preimages, are extensions of irrational rotations of the circle. Such systems have been studied when they arise as invariant subsets of certain specific mappings, $F:\mathbb{T}\to \mathbb{T}$ . Because our main result makes no explicit mention of a global transformation on $\mathbb{T}$ , we show that such a structure theorem holds for rotational systems that arise as invariant sets of any continuous transformation $F:\mathbb{T}\to \mathbb{T}$ with finite preimages. In particular, there are no explicit conditions on the degree of $F$ . We then give a development of known results in the case where $F(\theta )=d·\theta \mathrm{mod}\mathrm{1}$ for an integer $d>\mathrm{1}$ . The paper concludes with a construction of infinite rotational sets for mappings of the unit circle of degree larger than one whose lift to the universal cover is monotonic.
</p>projecteuclid.org/euclid.aaa/1531274541_20180710220243Tue, 10 Jul 2018 22:02 EDTMultiresolution Analysis Applied to the Monge-Kantorovich Problemhttps://projecteuclid.org/euclid.aaa/1531274542<strong>Armando Sánchez-Nungaray</strong>, <strong>Carlos González-Flores</strong>, <strong>Raquiel R. López-Martínez</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 9 pages.</p><p><strong>Abstract:</strong><br/>
We give a scheme of approximation of the MK problem based on the symmetries of the underlying spaces. We take a Haar type MRA constructed according to the geometry of our spaces. Thus, applying the Haar type MRA based on symmetries to the MK problem, we obtain a sequence of transportation problem that approximates the original MK problem for each of MRA. Moreover, the optimal solutions of each level solution converge in the weak sense to the optimal solution of original problem.
</p>projecteuclid.org/euclid.aaa/1531274542_20180710220243Tue, 10 Jul 2018 22:02 EDTA Novel Method for Solving Nonlinear Volterra Integro-Differential Equation Systemshttps://projecteuclid.org/euclid.aaa/1531274543<strong>Mohammad Hossein Daliri Birjandi</strong>, <strong>Jafar Saberi-Nadjafi</strong>, <strong>Asghar Ghorbani</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 6 pages.</p><p><strong>Abstract:</strong><br/>
An efficient iteration method is introduced and used for solving a type of system of nonlinear Volterra integro-differential equations. The scheme is based on a combination of the spectral collocation technique and the parametric iteration method. This method is easy to implement and requires no tedious computational work. Some numerical examples are presented to show the validity and efficiency of the proposed method in comparison with the corresponding exact solutions.
</p>projecteuclid.org/euclid.aaa/1531274543_20180710220243Tue, 10 Jul 2018 22:02 EDTNumerical Simulation of a One-Dimensional Water-Quality Model in a Stream Using a Saulyev Technique with Quadratic Interpolated Initial-Boundary Conditionshttps://projecteuclid.org/euclid.aaa/1521252087<strong>Pawarisa Samalerk</strong>, <strong>Nopparat Pochai</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 7 pages.</p><p><strong>Abstract:</strong><br/>
The one-dimensional advection-diffusion-reaction equation is a mathematical model describing transport and diffusion problems such as pollutants and suspended matter in a stream or canal. If the pollutant concentration at the discharge point is not uniform, then numerical methods and data analysis techniques were introduced. In this research, a numerical simulation of the one-dimensional water-quality model in a stream is proposed. The governing equation is advection-diffusion-reaction equation with nonuniform boundary condition functions. The approximated pollutant concentrations are obtained by a Saulyev finite difference technique. The boundary condition functions due to nonuniform pollutant concentrations at the discharge point are defined by the quadratic interpolation technique. The approximated solutions to the model are verified by a comparison with the analytical solution. The proposed numerical technique worked very well to give dependable and accurate solutions to these kinds of several real-world applications.
</p>projecteuclid.org/euclid.aaa/1521252087_20180918220240Tue, 18 Sep 2018 22:02 EDTSolution of Sequential Hadamard Fractional Differential Equations by Variation of Parameter Techniquehttps://projecteuclid.org/euclid.aaa/1521252086<strong>Mohammed M. Matar</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 7 pages.</p><p><strong>Abstract:</strong><br/>
We obtain in this article a solution of sequential differential equation involving the Hadamard fractional derivative and focusing the orders in the intervals $(\mathrm{1,2})$ and $(\mathrm{2,3})$ . Firstly, we obtain the solution of the linear equations using variation of parameter technique, and next we investigate the existence theorems of the corresponding nonlinear types using some fixed-point theorems. Finally, some examples are given to explain the theorems.
</p>projecteuclid.org/euclid.aaa/1521252086_20180918220240Tue, 18 Sep 2018 22:02 EDTNumerical Simulation for a Three-Dimensional Air Pollution Measurement Model in a Heavy Traffic Area under the Bangkok Sky Train Platformhttps://projecteuclid.org/euclid.aaa/1523498522<strong>Kewalee Suebyat</strong>, <strong>Nopparat Pochai</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
Air pollutant levels in Bangkok are generally high in street tunnels. They are particularly elevated in almost closed street tunnels such as an area under the Bangkok sky train platform with high traffic volume where dispersion is limited. There are no air quality measurement stations in the vicinity, while the human population is high. In this research, the numerical simulation is used to measure the air pollutant levels. The three-dimensional air pollution measurement model in a heavy traffic area under the Bangkok sky train platform is proposed. The finite difference techniques are employed to approximate the modelled solutions. The vehicle air pollutant emission due to the high traffic volume is mathematically assumed by the pollutant sources term. The simulation is also considered in averaged and moving pollutant sources due to manner vehicle emission. The proposed approximated air pollutant concentration indicators can be replaced by user required gaseous pollutants indices such as NOx, SO2, CO, and PM2.5.
</p>projecteuclid.org/euclid.aaa/1523498522_20180918220240Tue, 18 Sep 2018 22:02 EDTAn Extended Generalized $q$ -Extensions for the Apostol Type Polynomialshttps://projecteuclid.org/euclid.aaa/1537322513<strong>Letelier Castilla</strong>, <strong>William Ramírez</strong>, <strong>Alejandro Urieles</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 13 pages.</p><p><strong>Abstract:</strong><br/>
Through a modification on the parameters associated with generating function of the $q$ -extensions for the Apostol type polynomials of order $\alpha $ and level $m$ , we obtain some new results related to a unified presentation of the $q$ -analog of the generalized Apostol type polynomials of order $\alpha $ and level $m$ . In addition, we introduce some algebraic and differential properties for the $q$ -analog of the generalized Apostol type polynomials of order $\alpha $ and level $m$ and the relation of these with the $q$ -Stirling numbers of the second kind, the generalized $q$ -Bernoulli polynomials of level $m$ , the generalized $q$ -Apostol type Bernoulli polynomials, the generalized $q$ -Apostol type Euler polynomials, the generalized $q$ -Apostol type Genocchi polynomials of order $\alpha $ and level $m$ , and the $q$ -Bernstein polynomials.
</p>projecteuclid.org/euclid.aaa/1537322513_20180918220240Tue, 18 Sep 2018 22:02 EDTOn the Rate of Convergence of P-Iteration, SP-Iteration, and D-Iteration Methods for Continuous Nondecreasing Functions on Closed Intervalshttps://projecteuclid.org/euclid.aaa/1537322514<strong>Jukkrit Daengsaen</strong>, <strong>Anchalee Khemphet</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 6 pages.</p><p><strong>Abstract:</strong><br/>
We introduce a new iterative method called D-iteration to approximate a fixed point of continuous nondecreasing functions on arbitrary closed intervals. The purpose is to improve the rate of convergence compared to previous work. Specifically, our main result shows that D-iteration converges faster than P-iteration and SP-iteration to the fixed point. Consequently, we have that D-iteration converges faster than the others under the same computational cost. Moreover, the analogue of their convergence theorem holds for D-iteration.
</p>projecteuclid.org/euclid.aaa/1537322514_20180918220240Tue, 18 Sep 2018 22:02 EDTExistence Theorems on Solvability of Constrained Inclusion Problems and Applicationshttps://projecteuclid.org/euclid.aaa/1537322515<strong>Teffera M. Asfaw</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space ${X}^*$ . Let $T:X\supseteq D(T)\to {\mathrm{2}}^{{X}^*}$ be a maximal monotone operator and $C:X\supseteq D(C)\to {X}^*$ be bounded and continuous with $D(T)\subseteq D(C)$ . The paper provides new existence theorems concerning solvability of inclusion problems involving operators of the type $T+C$ provided that $C$ is compact or $T$ is of compact resolvents under weak boundary condition. The Nagumo degree mapping and homotopy invariance results are employed. The paper presents existence results under the weakest coercivity condition on $T+C$ . The operator $C$ is neither required to be defined everywhere nor required to be pseudomonotone type. The results are applied to prove existence of solution for nonlinear variational inequality problems.
</p>projecteuclid.org/euclid.aaa/1537322515_20180918220240Tue, 18 Sep 2018 22:02 EDTEstimates on the Bergman Kernels in a Tangential Direction on Pseudoconvex Domains in ${\mathbb{C}}^{3}$https://projecteuclid.org/euclid.aaa/1537322516<strong>Sanghyun Cho</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 13 pages.</p><p><strong>Abstract:</strong><br/>
Let $\mathrm{\Omega }$ be a smoothly bounded pseudoconvex domain in ${\mathbb{C}}^{\mathrm{3}}$ and assume that ${T}_{\mathrm{\Omega }}^{reg}({z}_{\mathrm{0}})<\mathrm{\infty }$ where ${z}_{\mathrm{0}}\in b\mathrm{\Omega }$ , the boundary of $\mathrm{\Omega }$ . Then we get optimal estimates of the Bergman kernel function along some “almost tangential curve” ${C}_{b}({z}_{\mathrm{0}},{\mathrm{\delta }}_{\mathrm{0}})\subset \mathrm{\Omega }\cup \{{z}_{\mathrm{0}}\}$ .
</p>projecteuclid.org/euclid.aaa/1537322516_20180918220240Tue, 18 Sep 2018 22:02 EDTGlobal Dynamics of an SVEIR Model with Age-Dependent Vaccination, Infection, and Latencyhttps://projecteuclid.org/euclid.aaa/1537322517<strong>Rodrigue Yves M’pika Massoukou</strong>, <strong>Suares Clovis Oukouomi Noutchie</strong>, <strong>Richard Guiem</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 21 pages.</p><p><strong>Abstract:</strong><br/>
Vaccine-induced protection is substantial to control, prevent, and reduce the spread of infectious diseases and to get rid of infectious diseases. In this paper, we propose an SVEIR epidemic model with age-dependent vaccination, latency, and infection. The model also considers that the waning vaccine-induced immunity depends on vaccination age and the vaccinated individuals fall back to the susceptible class after losing immunity. The model is a coupled system of (hyperbolic) partial differential equations with ordinary differential equations. The global dynamics of the model is established through construction of appropriate Lyapunov functionals and application of Lasalle’s invariance principle. As a result, the global stability of the infection-free equilibrium and endemic equilibrium is obtained and is fully determined by the basic reproduction number ${\mathfrak{R}}_{\mathrm{0}}$ .
</p>projecteuclid.org/euclid.aaa/1537322517_20180918220240Tue, 18 Sep 2018 22:02 EDTA New Method of Hypothesis Test for Truncated Spline Nonparametric Regression Influenced by Spatial Heterogeneity and Applicationhttps://projecteuclid.org/euclid.aaa/1539137013<strong> Sifriyani</strong>, <strong>I. N. Budiantara</strong>, <strong>S. H. Kartiko</strong>, <strong> Gunardi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 13 pages.</p><p><strong>Abstract:</strong><br/>
This study developed a new method of hypothesis testing of model conformity between truncated spline nonparametric regression influenced by spatial heterogeneity and truncated spline nonparametric regression. This hypothesis test aims to determine the most appropriate model used in the analysis of spatial data. The test statistic for model conformity hypothesis testing was constructed based on the likelihood ratio of the parameter set under H 0 whose components consisted of parameters that were not influenced by the geographical factor and the set under the population parameter whose components consisted of parameters influenced by the geographical factor. We have proven the distribution of test statistics $V$ and verified that each of the numerators and denominators in the statistic test $V$ followed a distribution of ${\chi }^{\mathrm{2}}$ . Since there was a symmetric and idempotent matrix S, it could be proved that ${\stackrel{~}{\mathrm{Y}}}^{\mathrm{T}}S \stackrel{~}{\mathrm{Y}}/{\sigma }^{\mathrm{2}}~{\chi }_{(n-lm-\mathrm{1})}^{\mathrm{2}}$ . Matrix $D({u}_{i},{v}_{i})$ was positive semidefinite and contained weighting matrix $\mathbf{W}({u}_{i},{v}_{i})$ which had different values in every location; therefore matrix $D({u}_{i},{v}_{i})$ was not idempotent. If ${\stackrel{~}{\mathrm{Y}}}^{\mathrm{T}}D({u}_{i},{v}_{i})\stackrel{~}{\mathrm{Y}}\ge \mathrm{0}$ and $D({u}_{i},{v}_{i})$ was not idempotent and also $\stackrel{~}{\mathrm{Y}}$ was a $N(\mathbf{0},\mathbf{I})$ distributed random vector, then there were constants $k$ and $r$ ; hence ${\stackrel{~}{\mathrm{Y}}}^{\mathrm{T}}D({u}_{i},{v}_{i})\stackrel{~}{\mathrm{Y}}~k{\chi }_{r}^{\mathrm{2}}$ ; therefore it was concluded that test statistic $V$ followed an F distribution. The modeling is implemented to find factors that influence the unemployment rate in 38 areas in Java in Indonesia.
</p>projecteuclid.org/euclid.aaa/1539137013_20181009220356Tue, 09 Oct 2018 22:03 EDTSystems with Local and Nonlocal Diffusions, Mixed Boundary Conditions, and Reaction Termshttps://projecteuclid.org/euclid.aaa/1542337406<strong>Mauricio Bogoya</strong>, <strong>Julio D. Rossi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
We study systems with different diffusions (local and nonlocal), mixed boundary conditions, and reaction terms. We prove existence and uniqueness of the solutions and then analyze global existence vs blow up in finite time. For blowing up solutions, we find asymptotic bounds for the blow-up rate.
</p>projecteuclid.org/euclid.aaa/1542337406_20181115220401Thu, 15 Nov 2018 22:04 ESTOn Computability and Applicability of Mann-Reich-Sabach-Type Algorithms for Approximating the Solutions of Equilibrium Problems in Hilbert Spaceshttps://projecteuclid.org/euclid.aaa/1542337407<strong>F. O. Isiogugu</strong>, <strong>P. Pillay</strong>, <strong>P. U. Nwokoro</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 9 pages.</p><p><strong>Abstract:</strong><br/>
We establish the existence of a strong convergent selection of a modified Mann-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points $F(T)$ of a multivalued (or single-valued) $k-$ strictly pseudocontractive-type mapping $T$ and the set of solutions $EP(F)$ of an equilibrium problem for a bifunction $F$ in a real Hilbert space $H$ . This work is a continuation of the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of a sequence $\{{K}_{n}{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ of closed convex subsets of $H$ from an arbitrary ${x}_{\mathrm{0}}\in H$ and a sequence $\{{x}_{n}{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$ of the metric projections of ${x}_{\mathrm{0}}$ into ${K}_{n}$ . The obtained result is a partial resolution of the controversy over the computability of such algorithms in the contemporary literature.
</p>projecteuclid.org/euclid.aaa/1542337407_20181115220401Thu, 15 Nov 2018 22:04 ESTA Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigrouphttps://projecteuclid.org/euclid.aaa/1542337408<strong>Maxim J. Goldberg</strong>, <strong>Seonja Kim</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 9 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider a general symmetric diffusion semigroup ${\{{T}_{t}f\}}_{t\ge \mathrm{0}}$ on a topological space $X$ with a positive $\sigma $ -finite measure, given, for $t>\mathrm{0}$ , by an integral kernel operator: ${T}_{t}f(x)\triangleq {\int }_{X}\mathrm{}{\rho }_{t}(x,y)f(y)dy$ . As one of the contributions of our paper, we define a diffusion distance whose specification follows naturally from imposing a reasonable Lipschitz condition on diffused versions of arbitrary bounded functions. We next show that the mild assumption we make, that balls of positive radius have positive measure, is equivalent to a similar, and an even milder looking, geometric demand. In the main part of the paper, we establish that local convergence of ${T}_{t}f$ to $f$ is equivalent to local equicontinuity (in $t$ ) of the family ${\{{T}_{t}f\}}_{t\ge \mathrm{0}}$ . As a corollary of our main result, we show that, for ${t}_{\mathrm{0}}>\mathrm{0}$ , ${T}_{t+{t}_{\mathrm{0}}}f$ converges locally to ${T}_{{t}_{\mathrm{0}}}f$ , as $t$ converges to ${\mathrm{0}}^{+}$ . In the Appendix, we show that for very general metrics $\mathcal{D}$ on $X$ , not necessarily arising from diffusion, ${\int }_{X}\mathrm{}{\rho }_{t}(x,y)\mathcal{D}(x,y)dy\to \mathrm{0}\text{\hspace\{0.17em\}\hspace\{0.17em\}a.e.}$ , as $t\to {\mathrm{0}}^{+}.$ R. Coifman and W. Leeb have assumed a quantitative version of this convergence, uniformly in $x$ , in their recent work introducing a family of multiscale diffusion distances and establishing quantitative results about the equivalence of a bounded function $f$ being Lipschitz, and the rate of convergence of ${T}_{t}f$ to $f$ , as $t\to {\mathrm{0}}^{+}$ . We do not make such an assumption in the present work.
</p>projecteuclid.org/euclid.aaa/1542337408_20181115220401Thu, 15 Nov 2018 22:04 ESTBest Proximity Point Theorems for Cyclic Relatively $\rho $ -Nonexpansive Mappings in Modular Spaceshttps://projecteuclid.org/euclid.aaa/1542337409<strong>Karim Chaira</strong>, <strong>Samih Lazaiz</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 8 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we introduce the notion of proximal $\rho $ -normal structure of pair of $\rho $ -admissible sets in modular spaces. We prove some results of best proximity points in this setting without recourse to Zorn’s lemma. We provide some examples to support our conclusions.
</p>projecteuclid.org/euclid.aaa/1542337409_20181115220401Thu, 15 Nov 2018 22:04 ESTExistence and Attractivity Results for Coupled Systems of Nonlinear Volterra–Stieltjes Multidelay Fractional Partial Integral Equationshttps://projecteuclid.org/euclid.aaa/1542337410<strong>Saïd Abbas</strong>, <strong>Mouffak Benchohra</strong>, <strong>Naima Hamidi</strong>, <strong>Gaston N’Guérékata</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
We are concerned with some existence and attractivity results of a coupled fractional Riemann–Liouville–Volterra–Stieltjes multidelay partial integral system. We prove the existence of solutions using Schauder’s fixed point theorem; then we show that the solutions are uniformly globally attractive.
</p>projecteuclid.org/euclid.aaa/1542337410_20181115220401Thu, 15 Nov 2018 22:04 ESTExact Null Controllability, Stabilizability, and Detectability of Linear Nonautonomous Control Systems: A Quasisemigroup Approachhttps://projecteuclid.org/euclid.aaa/1544756625<strong>Sutrima Sutrima</strong>, <strong>Christiana Rini Indrati</strong>, <strong>Lina Aryati</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 12 pages.</p><p><strong>Abstract:</strong><br/>
In the theory control systems, there are many various qualitative control problems that can be considered. In our previous work, we have analyzed the approximate controllability and observability of the nonautonomous Riesz-spectral systems including the nonautonomous Sturm-Liouville systems. As a continuation of the work, we are concerned with the analysis of stability, stabilizability, detectability, exact null controllability, and complete stabilizability of linear non-autonomous control systems in Banach spaces. The used analysis is a quasisemigroup approach. In this paper, the stability is identified by uniform exponential stability of the associated ${C}_{\mathrm{0}}$ -quasisemigroup. The results show that, in the linear nonautonomous control systems, there are equivalences among internal stability, stabizability, detectability, and input-output stability. Moreover, in the systems, exact null controllability implies complete stabilizability.
</p>projecteuclid.org/euclid.aaa/1544756625_20181213220430Thu, 13 Dec 2018 22:04 ESTOn the Convex and Convex-Concave Solutions of Opposing Mixed Convection Boundary Layer Flow in a Porous Mediumhttps://projecteuclid.org/euclid.aaa/1544756626<strong>M. Aïboudi</strong>, <strong>K. Boudjema Djeffal</strong>, <strong>B. Brighi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 5 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we are concerned with the solution of the third-order nonlinear differential equation ${f}^{″\mathrm{\prime }}+f{f}^{″}+\beta {f}^{\mathrm{\prime }}({f}^{\mathrm{\prime }}-\mathrm{1})=\mathrm{0}$ , satisfying the boundary conditions $f(\mathrm{0})=a\in \mathbb{R}$ , ${f}^{\mathrm{\prime }}(\mathrm{0})=b<\mathrm{0}$ , and ${f}^{\mathrm{\prime }}(t)\to \lambda $ , as $t\to +\mathrm{\infty }$ , where $\lambda \in \{\mathrm{0,1}\}$ and $\mathrm{0}<\beta <\mathrm{1}.$ The problem arises in the study of the opposing mixed convection approximation in a porous medium. We prove the existence, nonexistence, and the sign of convex and convex-concave solutions of the problem above according to the mixed convection parameter $b<\mathrm{0}$ and the temperature parameter $\mathrm{0}<\beta <\mathrm{1}$ .
</p>projecteuclid.org/euclid.aaa/1544756626_20181213220430Thu, 13 Dec 2018 22:04 ESTQualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponentshttps://projecteuclid.org/euclid.aaa/1544756627<strong>Zakariya Chaouai</strong>, <strong>Abderrahmane El Hachimi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 14 pages.</p><p><strong>Abstract:</strong><br/>
We consider the Dirichlet initial boundary value problem ${\partial }_{t}{u}^{m(x)}-\mathrm{div}({|\nabla u|}^{p(x,t)-\mathrm{2}}\nabla u)=a(x,t){u}^{q(x,t)}$ , where the exponents $p(x,t)>\mathrm{1}$ , $q(x,t)>\mathrm{0}$ , and $m(x)>\mathrm{0}$ are given functions. We assume that $a(x,t)$ is a bounded function. The aim of this paper is to deal with some qualitative properties of the solutions. Firstly, we prove that if $\mathrm{ess}\mathrm{sup}p(x,t)-\mathrm{1}<\mathrm{ess}\mathrm{inf}m(x)$ , then any weak solution will be extinct in finite time when the initial data is small enough. Otherwise, when $\mathrm{ess}\mathrm{sup}m(x)<\mathrm{ess}\mathrm{inf}p(x,t)-\mathrm{1}$ , we get the positivity of solutions for large $t$ . In the second part, we investigate the property of propagation from the initial data. For this purpose, we give a precise estimation of the support of the solution under the conditions that $\mathrm{ess}\mathrm{sup}m(x)<\mathrm{ess}\mathrm{inf}p(x,t)-\mathrm{1}$ and either $q(x,t)=m(x)$ or $a(x,t)\le \mathrm{0}$ a.e. Finally, we give a uniform localization of the support of solutions for all $t>\mathrm{0}$ , in the case where $a(x,t)<{a}_{\mathrm{1}}<\mathrm{0}$ a.e. and $\mathrm{ess}\mathrm{sup}q(x,t)<\mathrm{ess}\mathrm{inf}p(x,t)-\mathrm{1}$ .
</p>projecteuclid.org/euclid.aaa/1544756627_20181213220430Thu, 13 Dec 2018 22:04 ESTSome Oscillation Results for Even Order Delay Difference Equations with a Sublinear Neutral Termhttps://projecteuclid.org/euclid.aaa/1544756628<strong>Govindasamy Ayyappan</strong>, <strong>Gunasekaran Nithyakala</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 6 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, some new results are obtained for the even order neutral delay difference equation $\mathrm{\Delta }({a}_{n}{\mathrm{\Delta }}^{m-\mathrm{1}}({x}_{n}+{p}_{n}{x}_{n-k}^{\alpha }))+{q}_{n}{x}_{n-\mathcal{l}}^{\beta }=\mathrm{0}$ , where $m\ge \mathrm{2}$ is an even integer, which ensure that all solutions of the studied equation are oscillatory. Our results extend, include, and correct some of the existing results. Examples are provided to illustrate the importance of the main results.
</p>projecteuclid.org/euclid.aaa/1544756628_20181213220430Thu, 13 Dec 2018 22:04 ESTThe Second Kummer Function with Matrix Parameters and Its Asymptotic Behaviourhttps://projecteuclid.org/euclid.aaa/1547089410<strong>Georg Wehowar</strong>, <strong>Erika Hausenblas</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 8 pages.</p><p><strong>Abstract:</strong><br/>
In the present article, we introduce the second Kummer function with matrix parameters and examine its asymptotic behaviour relying on the residue theorem. Further, we provide a closed form of a solution of a Weber matrix differential equation and give a representation using the second Kummer function.
</p>projecteuclid.org/euclid.aaa/1547089410_20190109220412Wed, 09 Jan 2019 22:04 ESTFixed Point Theorems for $\mathcal{L}$ -Contractions in Generalized Metric Spaceshttps://projecteuclid.org/euclid.aaa/1547089411<strong>Seong-Hoon Cho</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 6 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, the notion of $\mathcal{L}$ -contractions is introduced and a new fixed point theorem for such contractions is established.
</p>projecteuclid.org/euclid.aaa/1547089411_20190109220412Wed, 09 Jan 2019 22:04 ESTAn Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracyhttps://projecteuclid.org/euclid.aaa/1547089412<strong>Khalid Atifi</strong>, <strong>Idriss Boutaayamou</strong>, <strong>Hamed Ould Sidi</strong>, <strong>Jawad Salhi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 16 pages.</p><p><strong>Abstract:</strong><br/>
The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.
</p>projecteuclid.org/euclid.aaa/1547089412_20190109220412Wed, 09 Jan 2019 22:04 ESTGeneralized Fractional-Order Bernoulli Functions via Riemann-Liouville Operator and Their Applications in the Evaluation of Dirichlet Serieshttps://projecteuclid.org/euclid.aaa/1547089413<strong>Jorge Sanchez-Ortiz</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2018, 5 pages.</p><p><strong>Abstract:</strong><br/>
In this work, we define a new class of functions of the Bernoulli type using the Riemann-Liouville fractional integral operator and derive a generating function for these class generalized functions. Then, these functions are employed to derive formulas for certain Dirichlet series.
</p>projecteuclid.org/euclid.aaa/1547089413_20190109220412Wed, 09 Jan 2019 22:04 ESTThe Numbers of Positive Solutions by the Lusternik-Schnirelmann Category for a Quasilinear Elliptic System Critical with Hardy Termshttps://projecteuclid.org/euclid.aaa/1551150388<strong>Mustapha Khiddi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 9 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the quasilinear elliptic system with Sobolev critical exponent involving both concave-convex and Hardy terms in bounded domains. By employing the technique introduced by Benci and Cerami (1991), we obtain at least $\mathrm{c}\mathrm{a}\mathrm{t}(\mathrm{\Omega })+\mathrm{1}$ distinct positive solutions.
</p>projecteuclid.org/euclid.aaa/1551150388_20190225220640Mon, 25 Feb 2019 22:06 ESTDeterminantal Representations of General and (Skew-)Hermitian Solutions to the Generalized Sylvester-Type Quaternion Matrix Equationhttps://projecteuclid.org/euclid.aaa/1551150389<strong>Ivan I. Kyrchei</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 14 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we derive explicit determinantal representation formulas of general, Hermitian, and skew-Hermitian solutions to the generalized Sylvester matrix equation involving $⁎$ -Hermicity $\mathbf{A}\mathbf{X}{\mathbf{A}}^{⁎}+\mathbf{B}\mathbf{Y}{\mathbf{B}}^{⁎}=\mathbf{C}$ over the quaternion skew field within the framework of the theory of noncommutative column-row determinants.
</p>projecteuclid.org/euclid.aaa/1551150389_20190225220640Mon, 25 Feb 2019 22:06 ESTHopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delayshttps://projecteuclid.org/euclid.aaa/1552615229<strong>Fulgensia Kamugisha Mbabazi</strong>, <strong>Joseph Y. T. Mugisha</strong>, <strong>Mark Kimathi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 21 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. The stability theory of delay differential equations is used to analyze the model. The results show that the disease-free equilibrium is asymptotically stable if the control reproduction ratio ${R}_{\mathrm{0}}$ is less than unity and unstable otherwise. The stability of equilibria with delays shows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. The existence of Hopf-bifurcation is investigated and transversality conditions are proved. The model results suggest that, as the respective delays exceed some critical value past the endemic equilibrium, the system loses stability through the process of local birth or death of oscillations. Further, a decrease or an increase in the delays leads to asymptotic stability or instability of the endemic equilibrium, respectively. The analytical results are supported by numerical simulations.
</p>projecteuclid.org/euclid.aaa/1552615229_20190314220049Thu, 14 Mar 2019 22:00 EDTOn the Relationship between the Inhomogeneous Wave and Helmholtz Equations in a Fractional Settinghttps://projecteuclid.org/euclid.aaa/1552615230<strong>Mateo Dulce</strong>, <strong>Alexander Getmanenko</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 9 pages.</p><p><strong>Abstract:</strong><br/>
We study convergence of solutions of a space and time inhomogeneous fractional wave equation on the quarter-plane to the stationary regime described by solutions of the Helmholtz equation.
</p>projecteuclid.org/euclid.aaa/1552615230_20190314220049Thu, 14 Mar 2019 22:00 EDT