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 EDTOn Estimates of Deviation of Functions from Matrix Operators of Their Fourier Series by Some Expressions with $r$ -Differences of the Entrieshttp://projecteuclid.org/euclid.aaa/1465991974<strong>Włodzimierz Łenski</strong>, <strong>Bogdan Szal</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 10 pages.</p><p><strong>Abstract:</strong><br/>
We generalize the results of Krasniqi 2012 and Wei and Yu 2012 to the case of $r$ -differences.
</p>projecteuclid.org/euclid.aaa/1465991974_20160615075948Wed, 15 Jun 2016 07:59 EDTOn Certain Properties for Two Classes of Generalized Convex Functionshttp://projecteuclid.org/euclid.aaa/1465991975<strong>Mohamed S. S. Ali</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 7 pages.</p><p><strong>Abstract:</strong><br/>
Two classes of generalized convex functions in the sense of Beckenbach are considered. For both classes, we show that the existence of support curves implies their generalized convexity and obtain an extremum property of these functions. Furthermore, we establish Hadamard’s inequality for them.
</p>projecteuclid.org/euclid.aaa/1465991975_20160615075948Wed, 15 Jun 2016 07:59 EDTConsistent Approximations of the Zeno Behaviour in Affine-Type Switched Dynamic Systemshttp://projecteuclid.org/euclid.aaa/1465991976<strong>Vadim Azhmyakov</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 9 pages.</p><p><strong>Abstract:</strong><br/>
This paper proposes a new theoretic approach to a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno behaviour. We study executions of switched control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real world processes do not present the corresponding behaviour, mathematical models of many engineering systems may be Zeno due to the used formal abstraction. We propose two useful approximative approaches to the Zeno dynamics, namely, an analytic technique and a variational description of this phenomenon. A generic trajectory associated with the Zeno dynamics can finally be characterized as a result of a specific projection or/and an optimization procedure applied to the original dynamic model. The obtained analytic and variational techniques provide an effective methodology for constructive approximations of the general Zeno-type behaviour. We also discuss shortly some possible applications of the proposed approximation schemes.
</p>projecteuclid.org/euclid.aaa/1465991976_20160615075948Wed, 15 Jun 2016 07:59 EDTA Linearized Relaxing Algorithm for the Specific Nonlinear Optimization Problemhttp://projecteuclid.org/euclid.aaa/1465991977<strong>Mio Horai</strong>, <strong>Hideo Kobayashi</strong>, <strong>Takashi G. Nitta</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 8 pages.</p><p><strong>Abstract:</strong><br/>
We propose a new method for the specific nonlinear and nonconvex global optimization problem by using a linear relaxation technique. To simplify the specific nonlinear and nonconvex optimization problem, we transform the problem to the lower linear relaxation form, and we solve the linear relaxation optimization problem by the Branch and Bound Algorithm. Under some reasonable assumptions, the global convergence of the algorithm is certified for the problem. Numerical results show that this method is more efficient than the previous methods.
</p>projecteuclid.org/euclid.aaa/1465991977_20160615075948Wed, 15 Jun 2016 07:59 EDTExistence of Solutions for a Robin Problem Involving the $p(x)$ -Laplace Operatorhttp://projecteuclid.org/euclid.aaa/1471047229<strong>Mostafa Allaoui</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 8 pages.</p><p><strong>Abstract:</strong><br/>
In this article we study the nonlinear Robin boundary-value problem $-{\mathrm{\Delta }}_{p(x)}u=f(x,u) \mathrm{i}\mathrm{n} \mathrm{\Omega }$ , $|\nabla u{|}^{p(x)-\mathrm{2}}(\partial u/\partial \nu )+\beta (x){|u|}^{p(x)-\mathrm{2}}u=\mathrm{0}$ on $\partial \mathrm{\Omega }$ . Using the variational method, under appropriate assumptions
on $f$ , we obtain results on existence and multiplicity of
solutions.
</p>projecteuclid.org/euclid.aaa/1471047229_20160812201407Fri, 12 Aug 2016 20:14 EDTTwist Periodic Solutions in the Relativistic Driven Harmonic
Oscillatorhttp://projecteuclid.org/euclid.aaa/1471047230<strong>Daniel Núñez</strong>, <strong>Andrés Rivera</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 7 pages.</p><p><strong>Abstract:</strong><br/>
We study the one-dimensional forced harmonic oscillator with relativistic
effects. Under some conditions of the parameters, the existence of a
unique stable periodic solution is proved which is of twist type. The
results depend on a Twist Theorem for nonlinear Hill’s
equations which is established and proved here.
</p>projecteuclid.org/euclid.aaa/1471047230_20160812201407Fri, 12 Aug 2016 20:14 EDTExistence of General Competitive Equilibria: A Variational Approachhttp://projecteuclid.org/euclid.aaa/1471047231<strong>G. Anello</strong>, <strong>F. Rania</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 10 pages.</p><p><strong>Abstract:</strong><br/>
We study the existence of general competitive equilibria in economies
with agents and goods in a finite number. We show that there exists a
Walras competitive equilibrium in all ownership private economies such
that, for all consumers, initial endowments do not contain free goods
and utility functions are locally Lipschitz quasiconcave. The proof of
the existence of competitive equilibria is based on variational
methods by applying a theoretical existence result for Generalized
Quasi Variational Inequalities.
</p>projecteuclid.org/euclid.aaa/1471047231_20160812201407Fri, 12 Aug 2016 20:14 EDTStudying Radiation and Reaction Effects on Unsteady MHD Non-Newtonian
(Walter’s B) Fluid in Porous Mediumhttp://projecteuclid.org/euclid.aaa/1471047232<strong>Gamal M. Abdel-Rahman Rashed</strong>, <strong>Faiza M. N. El-fayez</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 7 pages.</p><p><strong>Abstract:</strong><br/>
This paper describes the studied effects of thermal radiation and
chemical reaction on unsteady MHD non-Newtonian (obeying
Walter’s B model) fluid in porous medium. The resulting
problems are solved numerically. Graphical results for various
interesting parameters are presented. Also the effects of the
different parameters on the skin-friction and the heat fluxes are
obtained and discussed numerically.
</p>projecteuclid.org/euclid.aaa/1471047232_20160812201407Fri, 12 Aug 2016 20:14 EDTCertain Subclasses of Bistarlike and Biconvex Functions Based on Quasi-Subordinationhttp://projecteuclid.org/euclid.aaa/1463662652<strong>Nanjundan Magesh</strong>, <strong>Vitalrao Kupparao Balaji</strong>, <strong>Jagadesan Yamini</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 6 pages.</p><p><strong>Abstract:</strong><br/> We introduce the unified biunivalent function class ${\mathcal{M}}_{q,\sigma }^{\delta ,\lambda }(\gamma ,\phi )$ defined based on quasi-subordination and obtained the coefficient estimates for Taylor-Maclaurin coefficients $|{a}_{\mathrm{2}}|$ and $|{a}_{\mathrm{3}}|$ . Several related classes of functions are also considered and connections to earlier known and new results are established. </p>projecteuclid.org/euclid.aaa/1463662652_20161003085512Mon, 03 Oct 2016 08:55 EDTMaximality Theorems on the Sum of Two Maximal Monotone Operators and Application to Variational Inequality Problemshttp://projecteuclid.org/euclid.aaa/1475499297<strong>Teffera M. Asfaw</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 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 $T:X\supseteq D(T)\to {\mathrm{2}}^{{X}^{⁎}}$ and $A:X\supseteq D(A)\to {\mathrm{2}}^{{X}^{⁎}}$ be maximal monotone operators. The maximality of the sum of two maximal monotone operators has been an open problem for many years. In this paper, new maximality theorems are proved for $T+A$ under weaker sufficient conditions. These theorems improved the well-known maximality results of Rockafellar who used condition $\stackrel{\circ }{D(T)}\cap D(A)\ne \mathrm{\varnothing }$ and Browder and Hess who used the quasiboundedness of $T$ and condition $\mathrm{0}\in D(T)\cap D(A)$ . In particular, the maximality of $T+\partial \varphi $ is proved provided that $\stackrel{\circ }{D(T)}\cap D(\varphi )\ne \mathrm{\varnothing }$ , where $\varphi :X\to (-\mathrm{\infty },\mathrm{\infty }]$ is a proper, convex, and lower semicontinuous function. Consequently, an existence theorem is proved addressing solvability of evolution type variational inequality problem for pseudomonotone perturbation of maximal monotone operator. </p>projecteuclid.org/euclid.aaa/1475499297_20161003085512Mon, 03 Oct 2016 08:55 EDTDiscrete Approaches to Continuous Boundary Value Problems: Existence and Convergence of Solutionshttp://projecteuclid.org/euclid.aaa/1475499298<strong>Douglas R. Anderson</strong>, <strong>Christopher C. Tisdell</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 6 pages.</p><p><strong>Abstract:</strong><br/> We investigate two types of first-order, two-point boundary value problems (BVPs). Firstly, we study BVPs that involve nonlinear difference equations (the “discrete” BVP); and secondly, we study BVPs involving nonlinear ordinary differential equations (the “continuous” BVP). We formulate some sufficient conditions under which the discrete BVP will admit solutions. For this, our choice of methods involves a monotone iterative technique and the method of successive approximations (a.k.a. Picard iterations) in the absence of Lipschitz conditions. Our existence results for the discrete BVP are of a constructive nature and are of independent interest in their own right. We then turn our attention to applying our existence results for the discrete BVP to the continuous BVP. We form new existence results for solutions to the continuous BVP with our methods involving linear interpolation of the data from the discrete BVP, combined with a priori bounds and the convergence Arzela-Ascoli theorem. Thus, our use of discrete BVPs to yield results for the continuous BVP may be considered as a discrete approach to continuous BVPs. </p>projecteuclid.org/euclid.aaa/1475499298_20161003085512Mon, 03 Oct 2016 08:55 EDTVariational Approaches to Characterize Weak Solutions for Some Problems of Mathematical Physics Equationshttp://projecteuclid.org/euclid.aaa/1475499299<strong>Irina Meghea</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 10 pages.</p><p><strong>Abstract:</strong><br/> This paper is aimed at providing three versions to solve and characterize weak solutions for Dirichlet problems involving the $p$ -Laplacian and the $p$ -pseudo-Laplacian. In this way generalized versions for some results which use Ekeland variational principle, critical points for nondifferentiable functionals, and Ghoussoub-Maurey linear principle have been proposed. Three sequences of generalized statements have been developed starting from the most abstract assertions until their applications in characterizing weak solutions for some mathematical physics problems involving the abovementioned operators. </p>projecteuclid.org/euclid.aaa/1475499299_20161003085512Mon, 03 Oct 2016 08:55 EDTOn the Existence of Infinitely Many Solutions for Nonlocal Systems with Critical Exponentshttp://projecteuclid.org/euclid.aaa/1475499300<strong>M. Khiddi</strong>, <strong>R. Echarghaoui</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 10 pages.</p><p><strong>Abstract:</strong><br/> We study a class of semilinear nonlocal elliptic systems posed on settings without compact Sobolev embedding. By employing critical point theory and concentration estimates, we prove the existence of infinitely many solutions for values of the dimension $N$ , where $N>\mathrm{6}s,$ provided $\mathrm{0}<s<\mathrm{1}.$ </p>projecteuclid.org/euclid.aaa/1475499300_20161003085512Mon, 03 Oct 2016 08:55 EDTOptimality Conditions for Nondifferentiable Multiobjective Semi-Infinite Programming Problemshttp://projecteuclid.org/euclid.aaa/1475499301<strong>D. Barilla</strong>, <strong>G. Caristi</strong>, <strong>A. Puglisi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 6 pages.</p><p><strong>Abstract:</strong><br/> We have considered a multiobjective semi-infinite programming problem with a feasible set defined by inequality constraints. First we studied a Fritz-John type necessary condition. Then, we introduced two constraint qualifications and derive the weak and strong Karush-Kuhn-Tucker (KKT in brief) types necessary conditions for an efficient solution of the considered problem. Finally an extension of a Caristi-Ferrara-Stefanescu result for the ( $\mathrm{\Phi },\rho $ )-invexity is proved, and some sufficient conditions are presented under this weak assumption. All results are given in terms of Clark subdifferential. </p>projecteuclid.org/euclid.aaa/1475499301_20161003085512Mon, 03 Oct 2016 08:55 EDTOn Some Inequalities Involving Three or More Meanshttp://projecteuclid.org/euclid.aaa/1475499339<strong>Mustapha Raïssouli</strong>, <strong>Mohamed Chergui</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 8 pages.</p><p><strong>Abstract:</strong><br/> We investigate some results about mean-inequalities involving a large number of bivariate means. As application, we derive a lot of inequalities between four or more means among the standard means known in the literature. </p>projecteuclid.org/euclid.aaa/1475499339_20161216220306Fri, 16 Dec 2016 22:03 ESTGeneralized Jensen-Mercer Inequality for Functions with Nondecreasing Incrementshttp://projecteuclid.org/euclid.aaa/1481943735<strong>Asif R. Khan</strong>, <strong>Sumayyah Saadi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 12 pages.</p><p><strong>Abstract:</strong><br/> In the year 2003, McD Mercer established an interesting variation of Jensen’s inequality and later in 2009 Mercer’s result was generalized to higher dimensions by M. Niezgoda. Recently, Asif et al. has stated an integral version of Niezgoda’s result for convex functions. We further generalize Niezgoda’s integral result for functions with nondecreasing increments and give some refinements with applications. In the way, we generalize an important result, Jensen-Boas inequality, using functions with nondecreasing increments. These results would constitute a valuable addition to Jensen-type inequalities in the literature. </p>projecteuclid.org/euclid.aaa/1481943735_20161216220306Fri, 16 Dec 2016 22:03 ESTExistence of Mild Solutions to Nonlocal Fractional Cauchy Problems via Compactnesshttp://projecteuclid.org/euclid.aaa/1481943736<strong>Rodrigo Ponce</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 15 pages.</p><p><strong>Abstract:</strong><br/> We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases $\mathrm{0}<\alpha <\mathrm{1}$ and $\mathrm{1}<\alpha <\mathrm{2}.$ </p>projecteuclid.org/euclid.aaa/1481943736_20161216220306Fri, 16 Dec 2016 22:03 ESTCertain Properties of Some Families of Generalized Starlike Functions with respect to $q$ -Calculushttp://projecteuclid.org/euclid.aaa/1481943737<strong>Ben Wongsaijai</strong>, <strong>Nattakorn Sukantamala</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 8 pages.</p><p><strong>Abstract:</strong><br/> By making use of the concept of $q$ -calculus, various types of generalized starlike functions of order $\alpha $ were introduced and studied from different viewpoints. In this paper, we investigate the relation between various former types of $q$ -starlike functions of order $\alpha $ . We also introduce and study a new subclass of $q$ -starlike functions of order $\alpha $ . Moreover, we give some properties of those $q$ -starlike functions with negative coefficient including the radius of univalency and starlikeness. Some illustrative examples are provided to verify the theoretical results in case of negative coefficient functions class. </p>projecteuclid.org/euclid.aaa/1481943737_20161216220306Fri, 16 Dec 2016 22:03 ESTAntinormal Weighted Composition Operatorshttp://projecteuclid.org/euclid.aaa/1481943738<strong>Dilip Kumar</strong>, <strong>Harish Chandra</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 5 pages.</p><p><strong>Abstract:</strong><br/> Let ${l}^{\mathrm{2}}={L}^{\mathrm{2}}(\mathbb{N},\mu )$ , where $\mathbb{N}$ is set of all positive integers and $\mu $ is the counting measure whose $\sigma $ -algebra is the power set of $\mathbb{N}$ . In this paper, we obtain necessary and sufficient conditions for a weighted composition operator to be antinormal on the Hilbert space ${l}^{\mathrm{2}}$ . We also determine a class of antinormal weighted composition operators on Hardy space ${H}^{\mathrm{2}}(\mathbb{D})$ . </p>projecteuclid.org/euclid.aaa/1481943738_20161216220306Fri, 16 Dec 2016 22:03 ESTHyperplanes That Intersect Each Ray of a Cone Once and a Banach Space Counterexamplehttp://projecteuclid.org/euclid.aaa/1481943739<strong>Chris McCarthy</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 7 pages.</p><p><strong>Abstract:</strong><br/> Suppose $C$ is a cone contained in real vector space $V$ . When does $V$ contain a hyperplane $H$ that intersects each of the 0-rays in $C\setminus \{\mathrm{0}\}$ exactly once? We build on results found in Aliprantis, Tourky, and Klee Jr.’s work to give a partial answer to this question. We also present an example of a salient, closed Banach space cone $C$ for which there does not exist a hyperplane that intersects each 0-ray in $C\setminus \{\mathrm{0}\}$ exactly once. </p>projecteuclid.org/euclid.aaa/1481943739_20161216220306Fri, 16 Dec 2016 22:03 ESTA Note on First Passage Functionals for Lévy Processes with Jumps of Rational Laplace Transformshttp://projecteuclid.org/euclid.aaa/1481943740<strong>Djilali Ait-Aoudia</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 8 pages.</p><p><strong>Abstract:</strong><br/> This paper investigates the two-sided first exit problem for a jump process having jumps with rational Laplace transform. The corresponding boundary value problem is solved to obtain an explicit formula for the first passage functional. Also, we derive the distribution of the first passage time to two-sided barriers and the value at the first passage time. </p>projecteuclid.org/euclid.aaa/1481943740_20161216220306Fri, 16 Dec 2016 22:03 ESTQuasi-Hyperbolicity and Delay Semigroupshttp://projecteuclid.org/euclid.aaa/1481943741<strong>Shard Rastogi</strong>, <strong>Sachi Srivastava</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 6 pages.</p><p><strong>Abstract:</strong><br/> We study quasi-hyperbolicity of the delay semigroup associated with the equation ${u}^{\mathrm{\prime }}(t)=Bu(t)+\mathrm{\Phi }{u}_{t}$ , where ${u}_{t}$ is the history function and $(B,D(B))$ is the generator of a quasi-hyperbolic semigroup. We give conditions under which the associated solution semigroup of this equation generates a quasi-hyperbolic semigroup. </p>projecteuclid.org/euclid.aaa/1481943741_20161216220306Fri, 16 Dec 2016 22:03 ESTThe Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operatorhttp://projecteuclid.org/euclid.aaa/1481943742<strong>Hassan Kamil Jassim</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 5 pages.</p><p><strong>Abstract:</strong><br/> We used the local fractional variational iteration transform method (LFVITM) coupled by the local fractional Laplace transform and variational iteration method to solve three-dimensional diffusion and wave equations with local fractional derivative operator. This method has Lagrange multiplier equal to minus one, which makes the calculations more easily. The obtained results show that the presented method is efficient and yields a solution in a closed form. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new method. </p>projecteuclid.org/euclid.aaa/1481943742_20161216220306Fri, 16 Dec 2016 22:03 ESTCompleteness of Ordered Fields and a Trio of Classical Series Testshttp://projecteuclid.org/euclid.aaa/1481943743<strong>Robert Kantrowitz</strong>, <strong>Michael M. Neumann</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 6 pages.</p><p><strong>Abstract:</strong><br/> This article explores the fate of the infinite series tests of Dirichlet, Dedekind, and Abel in the context of an arbitrary ordered field. It is shown that each of these three tests characterizes the Dedekind completeness of an Archimedean ordered field; specifically, none of the three is valid in any proper subfield of $\mathbb{R}$ . The argument hinges on a contractive-type property for sequences in Archimedean ordered fields that are bounded and strictly increasing. For an arbitrary ordered field, it turns out that each of the tests of Dirichlet and Dedekind is equivalent to the sequential completeness of the field. </p>projecteuclid.org/euclid.aaa/1481943743_20161216220306Fri, 16 Dec 2016 22:03 ESTFréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaceshttp://projecteuclid.org/euclid.aaa/1481943744<strong>Joaquín Motos</strong>, <strong>María Jesús Planells</strong>, <strong>César F. Talavera</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 9 pages.</p><p><strong>Abstract:</strong><br/> We show that the dual ${({B}_{p(·)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega }))}^{\mathrm{\prime }}$ of the variable exponent Hörmander space ${B}_{p(·)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ is isomorphic to the Hörmander space ${B}_{\mathrm{\infty }}^{c}(\mathrm{\Omega })$ (when the exponent $p(·)$ satisfies the conditions $\mathrm{0}<{p}^{-}\le {p}^{+}\le \mathrm{1}$ , the Hardy-Littlewood maximal operator $M$ is bounded on ${L}_{p(·)/{p}_{\mathrm{0}}}$ for some $\mathrm{0}<{p}_{\mathrm{0}}<{p}^{-}$ and $\mathrm{\Omega }$ is an open set in ${\mathbb{R}}^{n}$ ) and that the Fréchet envelope of ${B}_{p(·)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ is the space ${B}_{\mathrm{1}}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ . Our proofs rely heavily on the properties of the Banach envelopes of the ${p}_{\mathrm{0}}$ -Banach local spaces of ${B}_{p(·)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for $p(·)\equiv p$ , $\mathrm{0}<p<\mathrm{1}$ , are also given (e.g., all quasi-Banach subspace of ${B}_{p}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ is isomorphic to a subspace of ${l}_{p}$ , or ${l}_{\mathrm{\infty }}$ is not isomorphic to a complemented subspace of the Shapiro space ${h}_{{p}^{-}}$ ). Finally, some questions are proposed. </p>projecteuclid.org/euclid.aaa/1481943744_20161216220306Fri, 16 Dec 2016 22:03 ESTIntegrodifferential Inequalities Arising in the Theory of Differential Equationshttp://projecteuclid.org/euclid.aaa/1481943745<strong>Zareen A. Khan</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 6 pages.</p><p><strong>Abstract:</strong><br/> The goal of this paper is to achieve some new results related to integrodifferential inequalities of one independent variable which can be applied as a study of qualitative and quantitative properties of solutions of some nonlinear integral equations. </p>projecteuclid.org/euclid.aaa/1481943745_20161216220306Fri, 16 Dec 2016 22:03 ESTExistence and Boundedness of Solutions for Nonlinear Volterra Difference Equations in Banach Spaceshttp://projecteuclid.org/euclid.aaa/1481943746<strong>Rigoberto Medina</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 6 pages.</p><p><strong>Abstract:</strong><br/> We consider a class of nonlinear discrete-time Volterra equations in Banach spaces. Estimates for the norm of operator-valued functions and the resolvents of quasi-nilpotent operators are used to find sufficient conditions that all solutions of such equations are elements of an appropriate Banach space. These estimates give us explicit boundedness conditions. The boundedness of solutions to Volterra equations with infinite delay is also investigated. </p>projecteuclid.org/euclid.aaa/1481943746_20161216220306Fri, 16 Dec 2016 22:03 ESTA Variational Approach to Perturbed Discrete Anisotropic Equationshttp://projecteuclid.org/euclid.aaa/1481943747<strong>Amjad Salari</strong>, <strong>Giuseppe Caristi</strong>, <strong>David Barilla</strong>, <strong>Alfio Puglisi</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 12 pages.</p><p><strong>Abstract:</strong><br/> We continue the study of discrete anisotropic equations and we will provide new multiplicity results of the solutions for a discrete anisotropic equation. We investigate the existence of infinitely many solutions for a perturbed discrete anisotropic boundary value problem. The approach is based on variational methods and critical point theory. </p>projecteuclid.org/euclid.aaa/1481943747_20161216220306Fri, 16 Dec 2016 22:03 ESTA Simplified Proof of Uncertainty Principle for Quaternion Linear Canonical Transformhttp://projecteuclid.org/euclid.aaa/1455115144<strong>Mawardi Bahri</strong>, <strong>Ryuichi Ashino</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 11 pages.</p><p><strong>Abstract:</strong><br/> We provide a short and simple proof of an uncertainty principle associated with the quaternion linear canonical transform (QLCT) by considering the fundamental relationship between the QLCT and the quaternion Fourier transform (QFT). We show how this relation allows us to derive the inverse transform and Parseval and Plancherel formulas associated with the QLCT. Some other properties of the QLCT are also studied. </p>projecteuclid.org/euclid.aaa/1455115144_20170124220641Tue, 24 Jan 2017 22:06 ESTLocal Hypoellipticity by Lyapunov Functionhttp://projecteuclid.org/euclid.aaa/1455115145<strong>E. R. Aragão-Costa</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 8 pages.</p><p><strong>Abstract:</strong><br/> We treat the local hypoellipticity, in the first degree, for a class of abstract differential operators complexes; the ones are given by the following differential operators: ${L}_{j}=\partial /\partial {t}_{j}+(\partial \varphi /\partial {t}_{j})(t,A)A$ , $j=\mathrm{1,2},\dots ,n$ , where $A:D(A)\subset H\to H$ is a self-adjoint linear operator, positive with $\mathrm{0}\in \rho (A)$ , in a Hilbert space $H$ , and $\varphi =\varphi (t,A)$ is a series of nonnegative powers of ${A}^{-\mathrm{1}}$ with coefficients in ${C}^{\mathrm{\infty }}(\mathrm{\Omega })$ , $\mathrm{\Omega }$ being an open set of ${\mathbb{R}}^{n}$ , for any $n\in \mathbb{N}$ , different from what happens in the work of Hounie (1979) who studies the problem only in the case $n=\mathrm{1}$ . We provide sufficient condition to get the local hypoellipticity for that complex in the elliptic region, using a Lyapunov function and the dynamics properties of solutions of the Cauchy problem $t$ ′ $(s)=-\nabla \mathrm{R}\mathrm{e}\mathrm{}{\varphi }_{\mathrm{0}}(t(s))$ , $s\ge \mathrm{0}$ , $t(\mathrm{0})={t}_{\mathrm{0}}\in \mathrm{\Omega },{\varphi }_{\mathrm{0}}:\mathrm{\Omega }\to \mathbb{C}$ being the first coefficient of $\varphi (t,A)$ . Besides, to get over the problem out of the elliptic region, that is, in the points $t$ ∗ $\in \mathrm{\Omega }$ such that $\nabla \mathrm{R}\mathrm{e}{\varphi }_{\mathrm{0}}(t$ ∗ $)$ = 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operator $A=\mathrm{1}-\mathrm{\Delta }:{H}^{\mathrm{2}}({\mathbb{R}}^{N})\subset {L}^{\mathrm{2}}({\mathbb{R}}^{N})\to {L}^{\mathrm{2}}({\mathbb{R}}^{N})$ . </p>projecteuclid.org/euclid.aaa/1455115145_20170124220641Tue, 24 Jan 2017 22:06 ESTGeneration and Identification of Ordinary Differential Equations of Maximal Symmetry Algebrahttp://projecteuclid.org/euclid.aaa/1485313541<strong>J. C. Ndogmo</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2016, 9 pages.</p><p><strong>Abstract:</strong><br/> An effective method for generating linear ordinary differential equations of maximal symmetry in their most general form is found, and an explicit expression for the point transformation reducing the equation to its canonical form is obtained. New expressions for the general solution are also found, as well as several identification and other results and a direct proof of the fact that a linear ordinary differential equation is iterative if and only if it is reducible to the canonical form by a point transformation. New classes of solvable equations parameterized by an arbitrary function are also found, together with simple algebraic expressions for the corresponding general solution. </p>projecteuclid.org/euclid.aaa/1485313541_20170124220641Tue, 24 Jan 2017 22:06 ESTA Variation on Uncertainty Principle and Logarithmic Uncertainty Principle
for Continuous Quaternion Wavelet Transformshttp://projecteuclid.org/euclid.aaa/1488423778<strong>Mawardi Bahri</strong>, <strong>Ryuichi Ashino</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 11 pages.</p><p><strong>Abstract:</strong><br/>
The continuous quaternion wavelet transform(CQWT) is a generalization of the classical continuous wavelet transformwithin the
context of quaternion algebra. First of all, we show that the directional quaternion Fourier transform (QFT) uncertainty principle
can be obtained using the component-wise QFT uncertainty principle. Based on this method, the directional QFT uncertainty
principle using representation of polar coordinate form is easily derived. We derive a variation on uncertainty principle related
to the QFT. We state that the CQWT of a quaternion function can be written in terms of the QFT and obtain a variation on
uncertainty principle related to the CQWT. Finally, we apply the extended uncertainty principles and properties of the CQWT to
establish logarithmic uncertainty principles related to generalized transform.
</p>projecteuclid.org/euclid.aaa/1488423778_20170301220315Wed, 01 Mar 2017 22:03 ESTBoundedness Criteria and Norm of Some Multilinear Hilbert-Type Operatorshttp://projecteuclid.org/euclid.aaa/1491962535<strong>Justice S. Bansah</strong>, <strong>Benoît F. Sehba</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 13 pages.</p><p><strong>Abstract:</strong><br/>
We consider two families of multilinear Hilbert-type operators for which we give exact relations between the parameters so that they are bounded. We also find the exact norm of these operators.
</p>projecteuclid.org/euclid.aaa/1491962535_20170411220232Tue, 11 Apr 2017 22:02 EDTModification of Nonlinear Conjugate Gradient Method with Weak Wolfe-Powell Line Searchhttp://projecteuclid.org/euclid.aaa/1491962536<strong>Ahmad Alhawarat</strong>, <strong>Zabidin Salleh</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 6 pages.</p><p><strong>Abstract:</strong><br/>
Conjugate gradient (CG) method is used to find the optimum solution for the large scale unconstrained optimization problems. Based on its simple algorithm, low memory requirement, and the speed of obtaining the solution, this method is widely used in many fields, such as engineering, computer science, and medical science. In this paper, we modified CG method to achieve the global convergence with various line searches. In addition, it passes the sufficient descent condition without any line search. The numerical computations under weak Wolfe-Powell line search shows that the efficiency of the new method is superior to other conventional methods.
</p>projecteuclid.org/euclid.aaa/1491962536_20170411220232Tue, 11 Apr 2017 22:02 EDTSome Notes about the Continuous-in-Time Financial Modelhttp://projecteuclid.org/euclid.aaa/1491962537<strong>Tarik Chakkour</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 11 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we investigate the properties of operators in the continuous-in-time model which is designed to be used for the finances of public institutions. These operators are involved in the inverse problem of this model. We discuss this inverse problem in Schwartz space that we prove the uniqueness theorem.
</p>projecteuclid.org/euclid.aaa/1491962537_20170411220232Tue, 11 Apr 2017 22:02 EDTOn Approximations by Trigonometric Polynomials of Classes of Functions Defined by Moduli of Smoothnesshttp://projecteuclid.org/euclid.aaa/1491962538<strong>Nimete Sh. Berisha</strong>, <strong>Faton M. Berisha</strong>, <strong>Mikhail K. Potapov</strong>, <strong>Marjan Dema</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 11 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we give a characterization of Nikol’skiĭ-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to such a class are given. In order to prove our results, we make use of certain recent reverse Copson-type and Leindler-type inequalities.
</p>projecteuclid.org/euclid.aaa/1491962538_20170411220232Tue, 11 Apr 2017 22:02 EDTNew Conditions for the Exponential Stability of Nonlinear Differential Equationshttp://projecteuclid.org/euclid.aaa/1494468086<strong>Rigoberto Medina</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 7 pages.</p><p><strong>Abstract:</strong><br/>
We develop a method for proving local exponential stability of nonlinear nonautonomous differential equations as well as pseudo-linear differential systems. The logarithmic norm technique combined with the “freezing” method is used to study stability of differential systems with slowly varying coefficients and nonlinear perturbations. Testable conditions for local exponential stability of pseudo-linear differential systems are given. Besides, we establish the robustness of the exponential stability in finite-dimensional spaces, in the sense that the exponential stability for a given linear equation persists under sufficiently small perturbations. We illustrate the application of this test to linear approximations of the differential systems under consideration.
</p>projecteuclid.org/euclid.aaa/1494468086_20170510220138Wed, 10 May 2017 22:01 EDTItô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scaleshttp://projecteuclid.org/euclid.aaa/1494468087<strong>Wenqing Hu</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 13 pages.</p><p><strong>Abstract:</strong><br/>
We provide an Itô formula for stochastic dynamical equation on general time scales. Based on this Itô’s formula we give a closed-form expression for stochastic exponential on general time scales. We then demonstrate Girsanov’s change of measure formula in the case of general time scales. Our result is being applied to a Brownian motion on the quantum time scale ( $q$ -time scale).
</p>projecteuclid.org/euclid.aaa/1494468087_20170510220138Wed, 10 May 2017 22:01 EDTNonnegative Infinite Matrices that Preserve $(p,q)$ -Convexity of Sequenceshttp://projecteuclid.org/euclid.aaa/1497578540<strong>Chikkanna R. Selvaraj</strong>, <strong>Suguna Selvaraj</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 9 pages.</p><p><strong>Abstract:</strong><br/>
This paper deals with matrix transformations that preserve the $(p,q)$ -convexity of sequences. The main result gives the necessary
and sufficient conditions for a nonnegative infinite matrix $A$ to preserve the $(p,q)$ -convexity of sequences. Further, we give examples of such
matrices for different values of $p$ and $q$ .
</p>projecteuclid.org/euclid.aaa/1497578540_20170615220243Thu, 15 Jun 2017 22:02 EDTOn the Boundedness of the Fractional Bergman Operatorshttp://projecteuclid.org/euclid.aaa/1497578541<strong>Benoît F. Sehba</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 3 pages.</p><p><strong>Abstract:</strong><br/>
We give a necessary and sufficient condition for the boundedness of the
Bergman fractional operators.
</p>projecteuclid.org/euclid.aaa/1497578541_20170615220243Thu, 15 Jun 2017 22:02 EDTCorrigendum to “Existence of Solutions for a Coupled System of
Second and Fourth Order Elliptic Equations”http://projecteuclid.org/euclid.aaa/1497578542<strong>Fanglei Wang</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 1 page.</p>projecteuclid.org/euclid.aaa/1497578542_20170615220243Thu, 15 Jun 2017 22:02 EDTOn the Convergence of the Uniform Attractor for the 2D Leray- α
Modelhttp://projecteuclid.org/euclid.aaa/1497578543<strong>Gabriel Deugoué</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 11 pages.</p><p><strong>Abstract:</strong><br/>
We consider a nonautonomous 2D Leray- $\alpha $ model of fluid turbulence. We prove the existence of the
uniform attractor ${\mathcal{A}}^{\alpha }$ . We also study the convergence of ${\mathcal{A}}^{\alpha }$ as $\alpha $ goes to zero. More precisely, we prove that the uniform
attractor ${\mathcal{A}}^{\alpha }$ converges to the uniform attractor of the 2D Navier-Stokes
system as $\alpha $ tends to zero.
</p>projecteuclid.org/euclid.aaa/1497578543_20170615220243Thu, 15 Jun 2017 22:02 EDTApproximation of Durrmeyer Type Operators Depending on Certain
Parametershttp://projecteuclid.org/euclid.aaa/1497578544<strong>Neha Malik</strong>, <strong>Serkan Araci</strong>, <strong>Man Singh Beniwal</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 9 pages.</p><p><strong>Abstract:</strong><br/>
Motivated by a number of recent investigations, we consider a new
analogue of Bernstein-Durrmeyer operators based on certain variants.
We derive some approximation properties of these operators. We also
compute local approximation and Voronovskaja type asymptotic formula.
We illustrate the convergence of aforementioned operators by making
use of the software MATLAB which we stated in the paper.
</p>projecteuclid.org/euclid.aaa/1497578544_20170615220243Thu, 15 Jun 2017 22:02 EDTA New Class of Contraction in $b$ -Metric Spaces and Applicationshttp://projecteuclid.org/euclid.aaa/1500429779<strong>Preeti Kaushik</strong>, <strong>Sanjay Kumar</strong>, <strong>Kenan Tas</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 10 pages.</p><p><strong>Abstract:</strong><br/>
A novel class of $\alpha $ - $\beta $ -contraction for a pair of mappings is introduced in the setting of $b$ -metric spaces. Existence and uniqueness of coincidence and common fixed
points for such kind of mappings are investigated. Results are supported with
relevant examples. At the end, results are applied to find the solution of an
integral equation.
</p>projecteuclid.org/euclid.aaa/1500429779_20170718220321Tue, 18 Jul 2017 22:03 EDTWeak and Strong Solutions for a Strongly Damped Quasilinear Membrane
Equationhttp://projecteuclid.org/euclid.aaa/1500429780<strong>Jin-soo Hwang</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 9 pages.</p><p><strong>Abstract:</strong><br/>
We consider a strongly damped quasilinear membrane equation with Dirichlet
boundary condition. The goal is to prove the well-posedness of the equation in
weak and strong senses. By setting suitable function spaces and making use of
the properties of the quasilinear term in the equation, we have proved the
fundamental results on existence, uniqueness, and continuous dependence on data
including bilinear term of weak and strong solutions.
</p>projecteuclid.org/euclid.aaa/1500429780_20170718220321Tue, 18 Jul 2017 22:03 EDTGeneralized Hölder’s and Minkowski’s Inequalities for
Jackson’s $q$ -Integral and Some Applications to the Incomplete $q$ -Gamma Functionhttp://projecteuclid.org/euclid.aaa/1502762543<strong>Kwara Nantomah</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 6 pages.</p><p><strong>Abstract:</strong><br/>
We establish some generalized Hölder’s and Minkowski’s
inequalities for Jackson’s $q$ -integral. As applications, we derive some inequalities involving the
incomplete $q$ -Gamma function.
</p>projecteuclid.org/euclid.aaa/1502762543_20170814220252Mon, 14 Aug 2017 22:02 EDTThe Approximation Szász-Chlodowsky Type Operators Involving Gould-Hopper
Type Polynomialshttp://projecteuclid.org/euclid.aaa/1502762544<strong>Behar Baxhaku</strong>, <strong>Artan Berisha</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 8 pages.</p><p><strong>Abstract:</strong><br/>
We introduce the Szász and Chlodowsky operators based on Gould-Hopper
polynomials and study the statistical convergence of these operators in a
weighted space of functions on a positive semiaxis. Further, a Voronovskaja type
result is obtained for the operators containing Gould-Hopper polynomials.
Finally, some graphical examples for the convergence of this type of operator
are given.
</p>projecteuclid.org/euclid.aaa/1502762544_20170814220252Mon, 14 Aug 2017 22:02 EDTOn Weighted Montgomery Identity for $k$ Points and Its Associates on Time Scaleshttp://projecteuclid.org/euclid.aaa/1502762545<strong>Eze R. Nwaeze</strong>, <strong>Ana M. Tameru</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 7 pages.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to establish a weighted Montgomery identity for
$k$ points and then use this identity to prove a new weighted Ostrowski type
inequality. Our results boil down to the results of Liu and Ngô if we take
the weight function to be the identity map. In addition, we also generalize an
inequality of Ostrowski-Grüss type on time scales for $k$ points. For $k=\mathrm{2},$ we recapture a result of Tuna and Daghan. Finally, we apply our results
to the continuous, discrete, and quantum calculus to obtain more results in this
direction.
</p>projecteuclid.org/euclid.aaa/1502762545_20170814220252Mon, 14 Aug 2017 22:02 EDTBifurcation and Global Dynamics of a Leslie-Gower Type Competitive System of Rational Difference Equations with Quadratic Termshttps://projecteuclid.org/euclid.aaa/1505786568<strong>V. Hadžiabdić</strong>, <strong>M. R. S. Kulenović</strong>, <strong>E. Pilav</strong>. <p><strong>Source: </strong>Abstract and Applied Analysis, Volume 2017, 19 pages.</p><p><strong>Abstract:</strong><br/>
We investigate global dynamics of the following systems of difference equations ${x}_{n+\mathrm{1}}={x}_{n}/({A}_{\mathrm{1}}+{B}_{\mathrm{1}}{x}_{n}+{C}_{\mathrm{1}}{y}_{n})$ , ${y}_{n+\mathrm{1}}={y}_{n}^{\mathrm{2}}/({A}_{\mathrm{2}}+{B}_{\mathrm{2}}{x}_{n}+{C}_{\mathrm{2}}{y}_{n}^{\mathrm{2}})$ , $n=\mathrm{0,1},\dots $ , where the parameters ${A}_{\mathrm{1}}$ , ${A}_{\mathrm{2}}$ , ${B}_{\mathrm{1}}$ , ${B}_{\mathrm{2}}$ , ${C}_{\mathrm{1}}$ , and ${C}_{\mathrm{2}}$ are positive numbers and the initial conditions ${x}_{\mathrm{0}}$ and ${y}_{\mathrm{0}}$ are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.
</p>projecteuclid.org/euclid.aaa/1505786568_20170918220316Mon, 18 Sep 2017 22:03 EDT