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 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 EDTThe Existence of Positive Solution for Semilinear Elliptic Equations with Multiple an Inverse Square Potential and Hardy-Sobolev Critical Exponentshttps://projecteuclid.org/euclid.aaa/1557972333<strong>M. Khiddi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 10 pages.</p><p><strong>Abstract:</strong><br/>
Via the concentration compactness principle, delicate energy estimates, the strong maximum principle, and the Mountain Pass lemma, the existence of positive solutions for a nonlinear PDE with multi-singular inverse square potentials and critical Sobolev-Hardy exponent is proved. This result extends several recent results on the topic.
</p>projecteuclid.org/euclid.aaa/1557972333_20190515220602Wed, 15 May 2019 22:06 EDTInequality of Ostrowski Type for Mappings with Bounded Fourth Order Partial Derivativeshttps://projecteuclid.org/euclid.aaa/1557972334<strong>Waseem Ghazi Alshanti</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 6 pages.</p><p><strong>Abstract:</strong><br/>
A general Ostrowski’s type inequality for double integrals is given. We utilize function whose partial derivative of order four exists and is bounded.
</p>projecteuclid.org/euclid.aaa/1557972334_20190515220602Wed, 15 May 2019 22:06 EDTCommon Fixed Point Theorems for a Pair of Self-Mappings in Fuzzy Cone Metric Spaceshttps://projecteuclid.org/euclid.aaa/1557972335<strong>Saif Ur Rehman</strong>, <strong>Yongjin Li</strong>, <strong>Shamoona Jabeen</strong>, <strong>Tayyab Mahmood</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 10 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we present some common fixed point theorems for a pair of self-mappings in fuzzy cone metric spaces under the generalized fuzzy cone contraction conditions. We extend and improve some recent results given in the literature.
</p>projecteuclid.org/euclid.aaa/1557972335_20190515220602Wed, 15 May 2019 22:06 EDTCertain Subclasses of Bi-Close-to-Convex Functions Associated with Quasi-Subordinationhttps://projecteuclid.org/euclid.aaa/1557972336<strong>Gurmeet Singh</strong>, <strong>Gurcharanjit Singh</strong>, <strong>Gagandeep Singh</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 6 pages.</p><p><strong>Abstract:</strong><br/>
In the present investigation, we introduce certain new subclasses of the class of biunivalent functions in the open unit disc $U=\{z:|z|<\mathrm{1}\}$ defined by quasi-subordination. We obtained estimates on the initial coefficients $|{a}_{\mathrm{2}}|$ and $|{a}_{\mathrm{3}}|$ for the functions in these subclasses. The results present in this paper would generalize and improve those in related works of several earlier authors.
</p>projecteuclid.org/euclid.aaa/1557972336_20190515220602Wed, 15 May 2019 22:06 EDTOn the Resolution of an Inverse Problem by Shape Optimization Techniqueshttps://projecteuclid.org/euclid.aaa/1557972337<strong>Chahnaz Zakia Timimoun</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 6 pages.</p><p><strong>Abstract:</strong><br/>
In this work, we want to detect the shape and the location of an inclusion $\omega $ via some boundary measurement on $\partial \mathrm{\Omega }$ . In practice, the body $\omega $ is immersed in a fluid flowing in a greater domain $\mathrm{\Omega }$ and governed by the Stokes equations. We study the inverse problem of reconstructing $\omega $ using shape optimization methods by defining the Kohn-Vogelius cost functional. We aim to study the inverse problem with Neumann and mixed boundary conditions.
</p>projecteuclid.org/euclid.aaa/1557972337_20190515220602Wed, 15 May 2019 22:06 EDTOptimal Control against the Human Papillomavirus: Protection versus Eradication of the Infectionhttps://projecteuclid.org/euclid.aaa/1563933803<strong>Fernando Saldaña</strong>, <strong>Andrei Korobeinikov</strong>, <strong>Ignacio Barradas</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 13 pages.</p><p><strong>Abstract:</strong><br/>
We investigate the optimal vaccination and screening strategies to minimize human papillomavirus (HPV) associated morbidity and the interventions cost. We propose a two-sex compartmental model of HPV-infection with time-dependent controls (vaccination of adolescents, adults, and screening) which can act simultaneously. We formulate optimal control problems complementing our model with two different objective functionals. The first functional corresponds to the protection of the vulnerable group and the control problem consists of minimizing the cumulative level of infected females over a fixed time interval. The second functional aims to eliminate the infection, and, thus, the control problem consists of minimizing the total prevalence at the end of the time interval. We prove the existence of solutions for the control problems, characterize the optimal controls, and carry out numerical simulations using various initial conditions. The results and properties and drawbacks of the model are discussed.
</p>projecteuclid.org/euclid.aaa/1563933803_20190723220407Tue, 23 Jul 2019 22:04 EDTOn a Parametric Mulholland-Type Inequality and Applicationshttps://projecteuclid.org/euclid.aaa/1563933804<strong>Bicheng Yang</strong>, <strong>Meifa Huang</strong>, <strong>Yanru Zhong</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 8 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, by the use of the weight functions, and the idea of introducing parameters, a discrete Mulholland-type inequality with the general homogeneous kernel and the equivalent form are given. The equivalent statements of the best possible constant factor related to a few parameters are provided. As applications, the operator expressions and a few particular examples are considered.
</p>projecteuclid.org/euclid.aaa/1563933804_20190723220407Tue, 23 Jul 2019 22:04 EDTA Convolution Theorem Related to Quaternion Linear Canonical Transformhttps://projecteuclid.org/euclid.aaa/1563933807<strong>Mawardi Bahri</strong>, <strong>Ryuichi Ashino</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 9 pages.</p><p><strong>Abstract:</strong><br/>
We introduce the two-dimensional quaternion linear canonical transform (QLCT), which is a generalization of the classical linear canonical transform (LCT) in quaternion algebra setting. Based on the definition of quaternion convolution in the QLCT domain we derive the convolution theorem associated with the QLCT and obtain a few consequences.
</p>projecteuclid.org/euclid.aaa/1563933807_20190723220407Tue, 23 Jul 2019 22:04 EDTMultiple of Solutions for Nonlocal Elliptic Equations with Critical Exponent Driven by the Fractional $p$ -Laplacian of Order $s$https://projecteuclid.org/euclid.aaa/1563933808<strong>M. Khiddi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 6 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the existence of infinitely many weak solutions for nonlocal elliptic equations with critical exponent driven by the fractional $p$ -Laplacian of order $s$ . We show the above result when $\lambda >\mathrm{0}$ is small enough. We achieve our goal by making use of variational methods, more specifically, the Nehari Manifold and Lusternik-Schnirelmann theory.
</p>projecteuclid.org/euclid.aaa/1563933808_20190723220407Tue, 23 Jul 2019 22:04 EDTBlow-Up of Solutions with High Energies of a Coupled System of Hyperbolic Equationshttps://projecteuclid.org/euclid.aaa/1563933809<strong>Jorge A. Esquivel-Avila</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 11 pages.</p><p><strong>Abstract:</strong><br/>
We consider an abstract coupled evolution system of second order in time. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We compare our results with those in the literature and show how we improve them.
</p>projecteuclid.org/euclid.aaa/1563933809_20190723220407Tue, 23 Jul 2019 22:04 EDTFractional Integral and Derivative Formulas by Using Marichev-Saigo-Maeda Operators Involving the S-Functionhttps://projecteuclid.org/euclid.aaa/1563933810<strong>D. L. Suthar</strong>, <strong>Hafte Amsalu</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 19 pages.</p><p><strong>Abstract:</strong><br/>
We establish fractional integral and derivative formulas by using Marichev-Saigo-Maeda operators involving the S-function. The results are expressed in terms of the generalized Gauss hypergeometric functions. Corresponding assertions in terms of Saigo, Erdélyi-Kober, Riemann-Liouville, and Weyl type of fractional integrals and derivatives are presented. Also we develop their composition formula by applying the Beta and Laplace transforms. Further, we point out also their relevance.
</p>projecteuclid.org/euclid.aaa/1563933810_20190723220407Tue, 23 Jul 2019 22:04 EDTThe Modified Coupled Hirota Equation: Riemann-Hilbert Approach and N-Soliton Solutionshttps://projecteuclid.org/euclid.aaa/1563933811<strong>Siqi Xu</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 10 pages.</p><p><strong>Abstract:</strong><br/>
The Cauchy initial value problem of the modified coupled Hirota equation is studied in the framework of Riemann-Hilbert approach. The N-soliton solutions are given in a compact form as a ratio of $(N+\mathrm{1})\times(N+\mathrm{1})$ determinant and $N\timesN$ determinant, and the dynamical behaviors of the single-soliton solution are displayed graphically.
</p>projecteuclid.org/euclid.aaa/1563933811_20190723220407Tue, 23 Jul 2019 22:04 EDTConstructive Existence of (1,1)-Solutions to Two-Point Value Problems for Fuzzy Linear Multiterm Fractional Differential Equationshttps://projecteuclid.org/euclid.aaa/1563933812<strong>HuiChol Choi</strong>, <strong>SungHyok Kwon</strong>, <strong>Kinam Sin</strong>, <strong>Sunae Pak</strong>, <strong>Sungryol So</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 18 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider the following two-point boundary value problems of fuzzy linear fractional differential equations: $({{}^{c}D}_{\mathrm{1,1}}^{\alpha }y)(t)\oplus b(t)\otimes ({{}^{c}D}_{\mathrm{1,1}}^{\beta }y)(t)\oplus c(t)\otimes y(t)=f(t)$ , $t\in (\mathrm{0,1})$ , $y(\mathrm{0})={y}_{\mathrm{0}}$ and $y(\mathrm{1})={y}_{\mathrm{1}}$ , where $b,c\in C(I)$ , $b(t),c(t)\ge \mathrm{0}$ , $y,f\in C(I,{\mathbf{R}}_{\mathrm{F}})$ , $I=[\mathrm{0,1}]$ , ${y}_{\mathrm{0}},{y}_{\mathrm{1}}\in {\mathbf{R}}_{\mathrm{F}}$ and $\mathrm{1}<\beta <\alpha \le \mathrm{2}$ . Our existence result is based on Banach fixed point theorem and the approximate solution of our problem is obtained by applying the Haar wavelet operational matrix.
</p>projecteuclid.org/euclid.aaa/1563933812_20190723220407Tue, 23 Jul 2019 22:04 EDTStable Numerical Solutions Preserving Qualitative Properties of Nonlocal Biological Dynamic Problemshttps://projecteuclid.org/euclid.aaa/1566439711<strong>M.-A. Piqueras</strong>, <strong>R. Company</strong>, <strong>L. Jódar</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 7 pages.</p><p><strong>Abstract:</strong><br/>
This paper deals with solving numerically partial integrodifferential equations appearing in biological dynamics models when nonlocal interaction phenomenon is considered. An explicit finite difference scheme is proposed to get a numerical solution preserving qualitative properties of the solution. Gauss quadrature rules are used for the computation of the integral part of the equation taking advantage of its accuracy and low computational cost. Numerical analysis including consistency, stability, and positivity is included as well as numerical examples illustrating the efficiency of the proposed method.
</p>projecteuclid.org/euclid.aaa/1566439711_20190821220859Wed, 21 Aug 2019 22:08 EDTLeast-Norm of the General Solution to Some System of Quaternion Matrix Equations and Its Determinantal Representationshttps://projecteuclid.org/euclid.aaa/1568858819<strong>Abdur Rehman</strong>, <strong>Ivan Kyrchei</strong>, <strong>Muhammad Akram</strong>, <strong>Ilyas Ali</strong>, <strong>Abdul Shakoor</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 18 pages.</p><p><strong>Abstract:</strong><br/>
We constitute some necessary and sufficient conditions for the system ${A}_{\mathrm{1}}{X}_{\mathrm{1}}={C}_{\mathrm{1}}$ , ${X}_{\mathrm{1}}{B}_{\mathrm{1}}={C}_{\mathrm{2}}$ , ${A}_{\mathrm{2}}{X}_{\mathrm{2}}={C}_{\mathrm{3}}$ , ${X}_{\mathrm{2}}{B}_{\mathrm{2}}={C}_{\mathrm{4}}$ , ${A}_{\mathrm{3}}{X}_{\mathrm{1}}{B}_{\mathrm{3}}+{A}_{\mathrm{4}}{X}_{\mathrm{2}}{B}_{\mathrm{4}}={C}_{c}$ , to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.
</p>projecteuclid.org/euclid.aaa/1568858819_20190918220727Wed, 18 Sep 2019 22:07 EDTThe Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Functionhttps://projecteuclid.org/euclid.aaa/1589335219<strong>Pierpaolo Soravia</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2019, 9 pages.</p><p><strong>Abstract:</strong><br/>
We define and study ${C}^{1}$ -solutions of the Aronsson equation (AE), a second order quasi linear equation. We show that such super/subsolutions make the Hamiltonian monotone on the trajectories of the closed loop Hamiltonian dynamics. We give a short, general proof that ${C}^{1}$ -solutions are absolutely minimizing functions. We discuss how ${C}^{1}$ -supersolutions of (AE) become special Lyapunov functions of symmetric control systems, and allow to find continuous feedbacks driving the system to a target in finite time, except on a singular manifold. A consequence is a simple proof that the corresponding minimum time function is locally Lipschitz continuous away from the singular manifold, despite classical results showing that it should only be Hölder continuous unless appropriate conditions hold. We provide two examples for Hörmander and Grushin families of vector fields where we construct ${C}^{1}$ -solutions (even classical) explicitly.
</p>projecteuclid.org/euclid.aaa/1589335219_20200512220032Tue, 12 May 2020 22:00 EDT