International Journal of Analysis The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. The de la Vallée Poussin Mean and Polynomial Approximation for Exponential Weight Mon, 23 Mar 2015 08:35:05 +0000 We study boundedness of the de la Vallée Poussin means for exponential weight on . Our main result is for every and every , where . As an application, we obtain for . K. Itoh, R. Sakai, and N. Suzuki Copyright © 2015 K. Itoh et al. All rights reserved. A New Look at Worst Case Complexity: A Statistical Approach Mon, 29 Dec 2014 07:29:17 +0000 We present a new and improved worst case complexity model for quick sort as , where the LHS gives the worst case time complexity, is the input size, is the frequency of sample elements, and is a function of both the input size and the parameter . The rest of the terms arising due to linear regression have usual meanings. We claim this to be an improvement over the conventional model; namely, , which stems from the worst case complexity for this algorithm. Niraj Kumar Singh, Soubhik Chakraborty, and Dheeresh Kumar Mallick Copyright © 2014 Niraj Kumar Singh et al. All rights reserved. -Asymptotically Lacunary Equivalent Sequences Wed, 03 Dec 2014 09:54:05 +0000 We introduce the strong -asymptotically equivalent and strong -asymptotically lacunary equivalent sequences which are some combinations of the definitions for asymptotically equivalent, statistical limit, modulus function, -convergence, and lacunary sequences. Then we use these definitions to prove strong -asymptotically equivalent and strong -asymptotically lacunary equivalent analogues of Connor’s results in Connor, 1988, Fridy and Orhan’s results in Fridy and Orhan, 1993, and Das and Patel’s results in Das and Patel, 1989. Tunay Bilgin Copyright © 2014 Tunay Bilgin. All rights reserved. Some Remarks on the Self-Exponential Function: Minimum Value, Inverse Function, and Indefinite Integral Sun, 23 Nov 2014 12:56:01 +0000 Considering the function as a real function of real variable, what is its minimum value? Surprisingly, the minimum value is reached for a negative value of . Furthermore, considering the function , and , two different expressions in closed form for the inverse function can be obtained. Also, two different series expansions for the indefinite integral of and are derived. The latter does not seem to be found in the literature. J. L. González-Santander and G. Martín Copyright © 2014 J. L. González-Santander and G. Martín. All rights reserved. Stable Numerical Evaluation of Finite Hankel Transforms and Their Application Thu, 13 Nov 2014 12:30:16 +0000 A new stable algorithm, based on hat functions for numerical evaluation of Hankel transform of order , is proposed in this paper. The hat basis functions are used as a basis to expand a part of the integrand, , appearing in the Hankel transform integral. This leads to a very simple, efficient, and stable algorithm for the numerical evaluation of Hankel transform. The novelty of our paper is that we give error and stability analysis of the algorithm and corroborate our theoretical findings by various numerical experiments. Finally, an application of the proposed algorithm is given for solving the heat equation in an infinite cylinder with a radiation condition. Manoj P. Tripathi, B. P. Singh, and Om P. Singh Copyright © 2014 Manoj P. Tripathi et al. All rights reserved. The Construction of Hilbert Spaces over the Non-Newtonian Field Tue, 21 Oct 2014 00:00:00 +0000 Although there are many excellent ways to present the principle of the classical calculus, the novel presentations probably lead most naturally to the development of the non-Newtonian calculi. In this paper we introduce vector spaces over real and complex non-Newtonian field with respect to the -calculus which is a branch of non-Newtonian calculus. Also we give the definitions of real and complex inner product spaces and study Hilbert spaces which are special type of normed space and complete inner product spaces in the sense of -calculus. Furthermore, as an example of Hilbert spaces, first we introduce the non-Cartesian plane which is a nonlinear model for plane Euclidean geometry. Secondly, we give Euclidean, unitary, and sequence spaces via corresponding norms which are induced by an inner product. Finally, by using the -norm properties of complex structures, we examine Cauchy-Schwarz and triangle inequalities. Uğur Kadak and Hakan Efe Copyright © 2014 Uğur Kadak and Hakan Efe. All rights reserved. Convergence Theorems for Fixed Points of Multivalued Mappings in Hilbert Spaces Wed, 24 Sep 2014 06:05:19 +0000 Let be a real Hilbert space and a nonempty closed convex subset of . Suppose is a multivalued Lipschitz pseudocontractive mapping such that . An Ishikawa-type iterative algorithm is constructed and it is shown that, for the corresponding sequence , under appropriate conditions on the iteration parameters, holds. Finally, convergence theorems are proved under approximate additional conditions. Our theorems are significant improvement on important recent results of Panyanak (2007) and Sastry and Babu (2005). N. Djitte and M. Sene Copyright © 2014 N. Djitte and M. Sene. All rights reserved. Best Proximity Point Results in Complex Valued Metric Spaces Wed, 27 Aug 2014 00:00:00 +0000 We introduce the concept of proximity points for nonself-mappings between two subsets of a complex valued metric space which is a recently introduced extension of metric spaces obtained by allowing the metric function to assume values from the field of complex numbers. We apply this concept to obtain the minimum distance between two subsets of the complex valued metric spaces. We treat the problem as that of finding the global optimal solution of a fixed point equation although the exact solution does not in general exist. We also define and use the concept of P-property in such spaces. Our results are illustrated with examples. Binayak S. Choudhury, Nikhilesh Metiya, and Pranati Maity Copyright © 2014 Binayak S. Choudhury et al. All rights reserved. System of Operator Quasi Equilibrium Problems Thu, 19 Jun 2014 06:43:32 +0000 We consider a system of operator quasi equilibrium problems and system of generalized quasi operator equilibrium problems in topological vector spaces. Using a maximal element theorem for a family of set-valued mappings as basic tool, we derive some existence theorems for solutions to these problems with and without involving -condensing mappings. Suhel Ahmad Khan Copyright © 2014 Suhel Ahmad Khan. All rights reserved. On the Spectral Properties of the Weighted Mean Difference Operator over the Sequence Space Mon, 16 Jun 2014 08:56:02 +0000 In the present work the generalized weighted mean difference operator has been introduced by combining the generalized weighted mean and difference operator under certain special cases of sequences and . For any two sequences and of either constant or strictly decreasing real numbers satisfying certain conditions the difference operator is defined by with for all . Furthermore, we compute the spectrum and the fine spectrum of the operator over the sequence space . In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this operator on the sequence space . P. Baliarsingh and S. Dutta Copyright © 2014 P. Baliarsingh and S. Dutta. All rights reserved. Integration over an Infinite-Dimensional Banach Space and Probabilistic Applications Sun, 15 Jun 2014 05:34:37 +0000 We study, for some subsets of , the Banach space of bounded real sequences . For any integer , we introduce a measure over that generalizes the -dimensional Lebesgue measure; consequently, also a theory of integration is defined. The main result of our paper is a change of variables' formula for the integration. Claudio Asci Copyright © 2014 Claudio Asci. All rights reserved. Frames of Eigenfunctions Associated with a Boundary Value Problem Thu, 05 Jun 2014 09:53:36 +0000 We introduce and study a redundant system of retro Banach frames consisting of eigenfunctions associated with a given boundary value problem. L. K. Vashisht and Shalu Sharma Copyright © 2014 L. K. Vashisht and Shalu Sharma. All rights reserved. On Nonautonomous Discrete Dynamical Systems Mon, 02 Jun 2014 13:14:48 +0000 We define and study expansiveness, shadowing, and topological stability for a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a metric space. Dhaval Thakkar and Ruchi Das Copyright © 2014 Dhaval Thakkar and Ruchi Das. All rights reserved. Some Properties of Generalized Gegenbauer Matrix Polynomials Thu, 29 May 2014 09:23:32 +0000 Various new generalized forms of the Gegenbauer matrix polynomials are introduced using the integral representation method, which allows us to express them in terms of Hermite matrix polynomials. Certain properties for these new generalized Gegenbauer matrix polynomials such as recurrence relations and expansion in terms of Hermite matrix polynomials are derived. Further, several families of bilinear and bilateral generating matrix relations for these polynomials are established and their applications are presented. Ghazala Yasmin Copyright © 2014 Ghazala Yasmin. All rights reserved. Some New Difference Sequence Spaces of Invariant Means Defined by Ideal and Modulus Function Wed, 28 May 2014 11:29:25 +0000 The main objective of this paper is to introduce a new kind of sequence spaces by combining the concepts of modulus function, invariant means, difference sequences, and ideal convergence. We also examine some topological properties of the resulting sequence spaces. Further, we introduce a new concept of -convergence and obtain a condition under which this convergence coincides with above-mentioned sequence spaces. Sudhir Kumar, Vijay Kumar, and S. S. Bhatia Copyright © 2014 Sudhir Kumar et al. All rights reserved. Solvability of a Third-Order Singular Generalized Left Focal Problem in Banach Spaces Mon, 26 May 2014 08:11:58 +0000 We consider the existence of positive solution for a third-order singular generalized left focal boundary value problem with full derivatives in Banach spaces. Green’s function and its properties, explicit a priori, estimates will be presented. By means of the theories of the fixed point in cones, we establish some new and general results on the existence of single and multiple positive solutions to the third-order singular generalized left focal boundary value problem. Our results are generalizations and extensions of the results of the focal boundary value problem. An example is included to illustrate the results obtained. Youwei Zhang Copyright © 2014 Youwei Zhang. All rights reserved. Hermite-Hadamard and Simpson Type Inequalities for Differentiable P-GA-Functions Thu, 22 May 2014 08:26:35 +0000 The author introduces the concept of the -GA-functions, gives Hermite-Hadamard's inequalities for -GA-functions, and defines a new identity. By using this identity, the author obtains new estimates on generalization of Hadamard and Simpson type inequalities for -GA-functions. Some applications to special means of real numbers are also given. İmdat İşcan Copyright © 2014 İmdat İşcan. All rights reserved. New Relations Involving an Extended Multiparameter Hurwitz-Lerch Zeta Function with Applications Tue, 13 May 2014 07:33:01 +0000 We derive several new expansion formulas involving an extended multiparameter Hurwitz-Lerch zeta function introduced and studied recently by Srivastava et al. (2011). These expansions are obtained by using some fractional calculus methods such as the generalized Leibniz rules, the Taylor-like expansions in terms of different functions, and the generalized chain rule. Several (known or new) special cases are also given. H. M. Srivastava, Sébastien Gaboury, and Richard Tremblay Copyright © 2014 H. M. Srivastava et al. All rights reserved. The Hopf Bifurcation Analysis and Optimal Control of a Delayed SIR Epidemic Model Wed, 07 May 2014 11:34:29 +0000 We propose a delayed SIR model with saturated incidence rate. The delay is incorporated into the model in order to model the latent period. The basic reproductive number is obtained. Furthermore, using time delay as a bifurcation parameter, it is proven that there exists a critical value of delay for the stability of diseases prevalence. When the delay exceeds the critical value, the system loses its stability and a Hopf bifurcation occurs. The model is extended to assess the impact of some control measures, by reformulating the model as an optimal control problem with vaccination and treatment. The existence of the optimal control is also proved. Finally, some numerical simulations are performed to verify the theoretical analysis. Abdelhadi Abta, Hassan Laarabi, and Hamad Talibi Alaoui Copyright © 2014 Abdelhadi Abta et al. All rights reserved. An Estimate of the Rate of Convergence of the Fourier Series in the Generalized Hölder Metric by Delayed Arithmetic Mean Wed, 07 May 2014 08:27:36 +0000 We study the rate of convergence problem of the Fourier series by Delayed Arithmetic Mean in the generalized Hölder metric space which was earlier introduced by Das, Nath, and Ray and obtain a sharper estimate of Jackson's order. L. Nayak, G. Das, and B. K. Ray Copyright © 2014 L. Nayak et al. All rights reserved. Existence and Nonexistence of a Solution for a Nonlinear -Elliptic Problem with Right-Hand Side Measure Sun, 04 May 2014 00:00:00 +0000 We discuss the existence and nonexistence of solution of a nonlinear problem -elliptic-, where is a Radon measure with bounded total variation, by considering the Sobolev spaces with variable exponents. This study is done in two cases: (i) is absolutely continuous with respect to -capacity. and (ii) is concentrated on a Borel set of null -capacity. Elhoussine Azroul, Abdelkrim Barbara, and Hicham Redwane Copyright © 2014 Elhoussine Azroul et al. All rights reserved. On Convergence with respect to an Ideal and a Family of Matrices Thu, 24 Apr 2014 10:13:45 +0000 P. Das et al. recently introduced and studied the notions of strong -summability with respect to an Orlicz function and -statistical convergence, where is a nonnegative regular matrix and is an ideal on the set of natural numbers. In this paper, we will generalise these notions by replacing with a family of matrices and with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' sup-limsup-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal has a countable base), continuing some of the author's previous work. Jan-David Hardtke Copyright © 2014 Jan-David Hardtke. All rights reserved. On the Logarithmic Regularity Conditions for the Variable Exponent Hardy Type Inequality Wed, 23 Apr 2014 00:00:00 +0000 We discuss a logarithmic regularity condition in a neighborhood of the origin and infinity on the exponent functions and for the variable exponent Hardy inequality to hold. Aziz Harman and Mustafa Özgür Keleş Copyright © 2014 Aziz Harman and Mustafa Özgür Keleş. All rights reserved. New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals Tue, 22 Apr 2014 12:12:05 +0000 The author obtains new estimates on generalization of Hadamard, Ostrowski, and Simpson type inequalities for Lipschitzian functions via Hadamard fractional integrals. Some applications to special means of positive real numbers are also given. İmdat İşcan Copyright © 2014 İmdat İşcan. All rights reserved. Initial Coefficient Bounds for a General Class of Biunivalent Functions Wed, 16 Apr 2014 11:53:26 +0000 We find estimates on the coefficients and for functions in the function class . The results presented in this paper improve or generalize the recent work of Srutha Keerthi and Raja (2013). Şahsene Altınkaya and Sibel Yalçın Copyright © 2014 Şahsene Altınkaya and Sibel Yalçın. All rights reserved. Unexpected Solutions of the Nehari Problem Thu, 10 Apr 2014 09:40:17 +0000 The Nehari characteristic numbers are the minimal values of an integral functional associated with a boundary value problem (BVP) for nonlinear ordinary differential equation. In case of multiple solutions of the BVP, the problem of identifying of minimizers arises. It was observed earlier that for nonoscillatory (positive) solutions of BVP those with asymmetric shape can provide the minimal value to a functional. At the same time, an even solution with regular shape is not a minimizer. We show by constructing the example that the same phenomenon can be observed in the Nehari problem for the fifth characteristic number which is associated with oscillatory solutions of BVP (namely, with those having exactly four zeros in . A. Gritsans and F. Sadyrbaev Copyright © 2014 A. Gritsans and F. Sadyrbaev. All rights reserved. Jensen Functionals on Time Scales for Several Variables Thu, 10 Apr 2014 07:43:15 +0000 We define Jensen functionals and concerned generalized means for several variables on time scales. We derive properties of Jensen functionals and apply them to generalized means. In this setting, we obtain generalizations, refinements, and conversions of many remarkable inequalities. Matloob Anwar, Rabia Bibi, Martin Bohner, and Josip Pečarić Copyright © 2014 Matloob Anwar et al. All rights reserved. On the Inequalities for the Generalized Trigonometric Functions Mon, 07 Apr 2014 07:34:14 +0000 This paper deals with Huygens-type and Wilker-type inequalities for the generalized trigonometric functions of P. Lindqvist. A major mathematical tool used in this work is a generalized version of the Schwab-Borchardt mean introduced recently by the author of this work. Edward Neuman Copyright © 2014 Edward Neuman. All rights reserved. Pascu-Type Harmonic Functions with Positive Coefficients Involving Salagean Operator Sun, 06 Apr 2014 06:29:21 +0000 Making use of a Salagean operator, we introduce a new class of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc. Among the results presented in this paper including the coeffcient bounds, distortion inequality, and covering property, extreme points, certain inclusion results, convolution properties, and partial sums for this generalized class of functions are discussed. K. Vijaya, G. Murugusundaramoorthy, and M. Kasthuri Copyright © 2014 K. Vijaya et al. All rights reserved. On a Pointwise Convergence of Quasi-Periodic-Rational Trigonometric Interpolation Thu, 03 Apr 2014 13:37:13 +0000 We introduce a procedure for convergence acceleration of the quasi-periodic trigonometric interpolation by application of rational corrections which leads to quasi-periodic-rational trigonometric interpolation. Rational corrections contain unknown parameters whose determination is important for realization of interpolation. We investigate the pointwise convergence of the resultant interpolation for special choice of the unknown parameters and derive the exact constants of the main terms of asymptotic errors. Arnak Poghosyan and Lusine Poghosyan Copyright © 2014 Arnak Poghosyan and Lusine Poghosyan. All rights reserved.