International Journal of Analysis The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. Singular Differential Equations and -Drazin Invertible Operators Tue, 15 Mar 2016 11:38:07 +0000 We extend results of Favini, Nashed, and Zhao on singular differential equations using the -Drazin inverse and the order of a quasinilpotent operator in the sense of Miekka and Nevanlinna. Two classes of singularly perturbed differential equations are studied using the continuity properties of the -Drazin inverse obtained by Koliha and Rakočević. Alrazi Abdeljabbar and Trung Dinh Tran Copyright © 2016 Alrazi Abdeljabbar and Trung Dinh Tran. All rights reserved. Cyclicity of Special Operators on a BK with AK Space Mon, 21 Dec 2015 11:59:14 +0000 Let be a complex domain and let be a reflexive BK space with AK such that and the functional of evaluation at is bounded for all . We will investigate the cyclicity for the adjoint of a weighted composition operator acting on . Leila Bagheri and Bahmann Yousefi Copyright © 2015 Leila Bagheri and Bahmann Yousefi. All rights reserved. A Survey on Operator Monotonicity, Operator Convexity, and Operator Means Wed, 11 Nov 2015 09:13:36 +0000 This paper is an expository devoted to an important class of real-valued functions introduced by Löwner, namely, operator monotone functions. This concept is closely related to operator convex/concave functions. Various characterizations for such functions are given from the viewpoint of differential analysis in terms of matrix of divided differences. From the viewpoint of operator inequalities, various characterizations and the relationship between operator monotonicity and operator convexity are given by Hansen and Pedersen. In the viewpoint of measure theory, operator monotone functions on the nonnegative reals admit meaningful integral representations with respect to Borel measures on the unit interval. Furthermore, Kubo-Ando theory asserts the correspondence between operator monotone functions and operator means. Pattrawut Chansangiam Copyright © 2015 Pattrawut Chansangiam. All rights reserved. Some Ostrowski Type Inequalities for Harmonically -Convex Functions in Second Sense Mon, 26 Oct 2015 12:34:27 +0000 The authors introduce the concept of harmonically -convex functions in second sense and establish some Ostrowski type inequalities of these classes of functions. Imran Abbas Baloch and İmdat İşcan Copyright © 2015 Imran Abbas Baloch and İmdat İşcan. All rights reserved. On Equalities Involving Integrals of the Logarithm of the Riemann -Function with Exponential Weight Which Are Equivalent to the Riemann Hypothesis Sun, 18 Oct 2015 16:22:19 +0000 Integral equalities involving integrals of the logarithm of the Riemann -function with exponential weight functions are introduced, and it is shown that an infinite number of them are equivalent to the Riemann hypothesis. Some of these equalities are tested numerically. The possible contribution of the Riemann function zeroes nonlying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than nine milliards of decimals for the maximal possible weight function exp(). We also show how certain Fourier transforms of the logarithm of the Riemann zeta-function taken along the real (demi)axis are expressible via elementary functions plus logarithm of the gamma-function and definite integrals thereof, as well as certain sums over trivial and nontrivial Riemann function zeroes. Sergey K. Sekatskii, Stefano Beltraminelli, and Danilo Merlini Copyright © 2015 Sergey K. Sekatskii et al. All rights reserved. On Simultaneous Approximation of Modified Baskakov-Durrmeyer Operators Sun, 18 Oct 2015 13:04:37 +0000 We discuss properties of modified Baskakov-Durrmeyer-Stancu (BDS) operators with parameter . We compute the moments of these modified operators. Also, we establish pointwise convergence, Voronovskaja type asymptotic formula, and an error estimation in terms of second order modification of continuity of the function for the operators . Prashantkumar G. Patel and Vishnu Narayan Mishra Copyright © 2015 Prashantkumar G. Patel and Vishnu Narayan Mishra. All rights reserved. Geometric Properties of a Class of Analytic Functions Defined by a Differential Inequality Sun, 04 Oct 2015 17:06:01 +0000 Let be the class of analytic functions defined in the open unit disk and normalized by . For in , let , where and . In the present paper, we find conditions under which functions in the class are starlike of order , . Manpreet Kaur, Sushma Gupta, and Sukhjit Singh Copyright © 2015 Manpreet Kaur et al. All rights reserved. An Integral Mean Value Theorem concerning Two Continuous Functions and Its Stability Sun, 27 Sep 2015 13:17:24 +0000 The aim of this paper is to investigate an integral mean value theorem proposed by one of the references of this paper. Unfortunately, the proof contains a gap. First, we present a counterexample which shows that this theorem fails in this form. Then, we present two improved versions of this theorem. The stability of the mean point arising from the second result concludes this paper. Monea Mihai Copyright © 2015 Monea Mihai. All rights reserved. Error Estimation of Functions by Fourier-Laguerre Polynomials Using Matrix-Euler Operators Tue, 01 Sep 2015 11:44:27 +0000 Various investigators have studied the degree of approximation of a function using different summability (Cesáro means of order : , Euler , and Nörlund ) means of its Fourier-Laguerre series at the point after replacing the continuity condition in Szegö theorem by much lighter conditions. The product summability methods are more powerful than the individual summability methods and thus give an approximation for wider class of functions than the individual methods. This has motivated us to investigate the error estimation of a function by -transform of its Fourier-Laguerre series at frontier point , where is a general lower triangular regular matrix. A particular case, when is a Cesáro matrix of order 1, that is, , has also been discussed as a corollary of main result. M. L. Mittal and Mradul Veer Singh Copyright © 2015 M. L. Mittal and Mradul Veer Singh. 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.