Mathematical Analysis and Applications

Inhaltsverzeichnis

1. Frontmatter

2. A Note on Isolated Removable Singularities of Harmonic Functions

   Gopala Krishna Srinivasan

   Abstract

   A proof of the removable singularities theorem for harmonic functions is presented which seems to be different from existing proofs in the literature. This is an important result in analysis with applications to many areas of mathematics. Weyl’s lemma which is used in the course of the argument is also proved in a special case to make the note self-contained.

3. Nonlinear Evolution Equations by a Ky Fan Minimax Inequality Approach

   Ouayl Chadli, Ram N. Mohapatra, G. Pany

   Abstract

   Nonlinear evolution equations appear in various fields of sciences including mechanics, physics, engineering, and material sciences. Essential functional methods for the treatment of both the linear as well as nonlinear evolution equations are based on the theory of spectral methods, maximal monotone operators, fixed point theorems, and concept of \(C_0\)-semigroups of linear mappings along with the Leray-Schauder degree theory. Recently, the solvability for the evolution equations of nonlinear type has been considered using the Ky Fan minimax inequality. This approach is quite new and different as compared to the traditional approaches. In 1972, Ky Fan (Inequalities. Academic, New York, pp. 103–113, 1972) put forward his pioneering result concerning the existence of solutions for an inequality of minimax type, which is nowadays called as “equilibrium problem” in literature. This kind of model has shown to be a cornerstone result of nonlinear analysis and has gained much interest in the past because it has been used in several contexts such as physics, chemistry, economics, engineering, and so on. This work aims to present a review of recently obtained results on the use of the equilibrium problem theory in the study of nonlinear (implicit) evolution equations. Along with that, we discuss the problem with initial value condition as well as periodic and anti-periodic solutions.

4. Sufficient Conditions Concerning the Unified Class of Starlike and Convex Functions

   Lateef Ahmad Wani, A. Swaminathan

   Abstract

   Let \(\mathcal {A}_n\) be the family of analytic functions \(f(\xi )=\xi +\sum _{j=n+1}^\infty a_j\xi ^j\), defined in the open unit disk \(\mathbb {D}\). We use differential subordinations to establish sufficient conditions involving third-order differential inequalities for to be in the unified class of starlike and convex functions

   $$ \mathcal {S^*C}_n(\alpha ,\beta ): =\left\{ f\in \mathcal {A}_n:{\Re }\left( \frac{\xi f'(\xi )+\beta \xi ^2f''(\xi )}{\beta \xi f'(\xi )+(1-\beta )f(\xi )}\right) >\alpha \right\} , $$

   where \(\alpha \in [0,1)\) and \(\beta \in [0,1]\). As applications, we construct certain members of \(\mathcal {S^*C}_n(\alpha ,\beta )\) involving triple-integrals and also derive conditions for the Pascu class of functions. Apart from obtaining new results, some of the already known results concerning starlikeness of \(f\in \mathcal {A}_n\) are obtained as special cases.

5. One Dimensional Parametrized Test Functions Space of Entire Functions

   Sheila M. Menchavez, Irene Mae Y. Antabo

   Abstract

   Inspired by the construction of Kondratiev test functions in infinite dimensional analysis, this paper constructs a nuclear space of entire test functions of minimal type, endowed with the projective limit topology.

6. Extremal Mild Solutions of Hilfer Fractional Impulsive Systems

   Divya Raghavan, N. Sukavanam

   Abstract

   The well-established monotone iterative technique that is used to study the existence and uniqueness of fractional impulsive system is extended to Hilfer fractional order in this paper. The results are derived by using the method of upper and lower solution and Gronwall inequality. Also, conditions on non-compactness of measure are used effectively to prove the main result.

7. On Integral Solutions for a Class of Mixed Volterra-Fredholm Integro Differential Equations with Caputo Fractional Derivatives

   Bandita Roy, Swaroop Nandan Bora

   Abstract

   This work studies the existence of integral solution for a class of neutral integro-differential equation of mixed type involving Caputo fractional derivative under the assumption that the associated operator A is not dense. Utilizing semigroup theory, fractional calculus, Darbo-Sadovskii’s fixed point theorem and measure of noncompactness, we have established some sufficient conditions which ensure the existence of integral solutions of our problem.

8. Trajectory Controllability of Integro-Differential Systems of Fractional Order in a Banach Space with Deviated Argument

   Parveen Kumar, Ankit Kumar, Ramesh K. Vats, Avadhesh Kumar

   Abstract

   In this paper, the fractional integro-differential control system of order \(\gamma \in (1,2]\) in a Banach space with deviated argument is considered. In order to study the trajectory controllability for the proposed control problem, the theory of fractional calculus, Gronwall’s inequality, and fractional order cosine family are used. Finally, we provide an example to illustrate our main results.

9. Shehu-Adomian Decomposition Method for Dispersive KdV-Type Equations

   Abey S. Kelil, Appanah R. Appadu

   Abstract

   In this paper, a new method known to be Shehu-Adomian decomposition method is proposed to solve homogeneous and non-homogeneous dispersive KdV-type equations. The Shehu-Adomian decomposition method is a combination of Shehu’s transform and Adomian Decomposition method. Some illustrative problems of dispersive KdV-type equations are solved to check the validity of the method. The approximate solutions are given in series form and the proposed method is a reliable and powerful technique to solve numerous physical problems in applications.

10. On Certain Properties of Perturbed Freud-Type Weight: A Revisit

    Abey S. Kelil, Appanah R. Appadu, Sama Arjika

    Abstract

    In this paper, monic polynomials orthogonal with deformation of the Freud-type weight function are considered. These polynomials fulfill linear differential equations with some polynomial coefficients in their holonomic form. The aim of this work is to explore certain characterizing properties of perturbed Freud-type polynomials such as nonlinear recursion relations, finite moments, differential-recurrence, and differential relations satisfied by the recurrence coefficients as well as the corresponding semiclassical orthogonal polynomials. We note that the obtained differential equation fulfilled by the considered semiclassical polynomials are used to study an electrostatic interpretation for the distribution of zeros based on the original ideas of Stieltjes.

11. Complex Chaotic Systems and Its Complexity

    Ajit K. Singh

    Abstract

    This article is deal with an attempt to study the complex chaotic system and its complexity. Chaos in the dynamical system is very complex pattern with the real variables and becomes more complex with the complex variables. But due to its real application in the physical systems, it is very useful to study its behaviour. This article starts with the Lorenz model of integer order and of real variables and in a very systematic way it explores to the fractional order to the complex variables and ends with the fractional order complex chaotic systems. Numerical algorithm and stability analysis are also presented through the simulation results.

12. On the Bertrand Pairs of Open Non-Uniform Rational B-Spline Curves

    Muhsin Incesu, Sara Yilmaz Evren, Osman Gursoy

    Abstract

    B-spline curves are used basically in Computer-Aided Design (CAD), Computer-Aided Geometric Design (CAGD), and Computer-Aided Modeling (CAM). In determining the invariants of curves and surfaces at any point, there are some difficulties in expressing it analytically and calculating its invariants at the desired point. For these curves and surfaces the way to overcome these difficulties is to design them with spline curves and surfaces. In this paper the second- and third-order derivatives of open Non-Uniform Rational B-Spline (NURBS) curves at the points \(t=t_{d}\), \(t=t_{m-d}\), and arbitrary point in domain of these curves are given. In addition, the Frenet vector fields and curvatures of these open NURBS curves were expressed by their control points. The relationships between control points were expressed when given two open NURBS curves occurred as Bertrand curve pairs at the points \(t=t_{d}\), \(t=t_{m-d}\), and arbitrary point in domain of these curves.

13. Convergence Analysis of a Sixth-Order Method Under Weak Continuity Condition with First-Order Fréchet Derivative

    Mona Verma, Pooja Sharma, Neha Gupta

    Abstract

    In this article, a semi-local convergence analysis of a well-established sixth-order method in Banach spaces is discussed. The analysis has been done under the H\(\ddot{o}\)lder continuity condition with the help of the recurrence relation technique. The relevance of our study lies in the fact that many examples which do not satisfy the Lipschitz continuity but satisfy the H\(\ddot{o}\)lder continuity. A convergence theorem has been established for the existence-uniqueness of the solution. A priori error bound expression is also derived. Finally, the convergence analysis is carried out on various examples. These examples include Fredholm, Hammerstein integral equation, and a boundary value problem that validated the theoretical development.

14. (m, n)-Paranormal Composition Operators

    Baljinder Kour, Sonu Ram

    Abstract

    In this paper, we prove some characterizations for the class of (m, n)-paranormal operators acting on the complex Hilbert space \(\mathcal {H}\). The class of (m, n)-paranormal operators is characterized in terms of the Radon–Nikodym derivative of the measure \(\lambda T^{-1}\) with respect to \(\lambda \). Moreover, we discuss the conditions under which the classes of composition operators, weighted composition operators, multiplication composition operators are (m, n)-paranormal.

15. On the Domain of q-Euler Matrix in and c

    Taja Yaying

    Abstract

    In this study, we present the Banach spaces \(e^{q}_0\) and \(e^{q}_{c}\) obtained by the domain of q-analog \(E^{q}\) of the Euler matrix of order 1 in the spaces \(c_0\) and c,  respectively. We exhibit certain topological properties and inclusion relations of these spaces. We obtain the bases and determine the Köthe duals of the spaces \(e^{q}_0\) and \(e^{q}_{c}.\) We characterize certain classes of matrix mappings from the spaces \(e^{q}_0\) and \(e^{q}_{c}\) to the space \(\mu \in \{\ell _{\infty },c,c_0,\ell _{1},bs,cs,cs_0\}\).

16. Study on Some Particular Class of Nonlinear Integral Equation with a Hybridized Approach

    Nimai Sarkar, Mausumi Sen

    Abstract

    This article deals with a particular class of integral equations involving pure delay term. The existence of a solution is described using fixed point theory. Moreover, a hybridized scheme is proposed to investigate the approximate solution. In this context, boundary element method is used with piecewise linear interpolation. Also, an algorithm is there for error estimation and in support of the considered numerical method stability analysis is done. This testimony completely demonstrates the comprehensive study of the considered class of integral equation and understanding the behaviour of the approximate solution in the presence of delay.

17. Investigation of the Existence Criteria for the Solution of the Functional Integral Equation in the Space

    Dipankar Saha, Mausumi Sen, Santanu Roy

    Abstract

    This work manifests the credibility of Darbo’s fixed point theory towards the solvability of nonlinear functional convolution integral equation with deviating argument. The solution space is taken to be the space of Lebesgue integrable functions defined on \(\mathbb {R_+}\). The concept of measure of noncompactness in correlation with the compactness criterion, i.e., Kolmogorov–Riesz compactness theorem in \(L^{p}(\mathbb {R_+})\) space has been taken. Then under certain suitable hypotheses and by the assistance of Darbo’s fixed point theory, sufficient conditions for the existence of the solution have been introduced. Finally, some examples have been taken to justify the result.

18. Functional Inequalities for the Generalized Wright Functions

    Sourav Das, Khaled Mehrez

    Abstract

    In this work, our aim is to obtain some mean value inequalities for the generalized Wright function. Mainly, we establish Turán, Redheffer, Wilker and Lazarević-type inequalities for the generalized Wright function. Furthermore, the monotonicity properties of ratios for partial sums of the series of these functions are discussed. Finally, some other related inequalities are also derived as a consequence.

19. An Information-Theoretic Entropy Related to Ihara Function and Billiard Dynamics

    Supriyo Dutta, Partha Guha

    Abstract

    This article aims to establish a connection between the dynamical billiards and information theory. We propose two generalized information-theoretic entropies based on the Ihara zeta functions associated with a combinatorial graph representing a billiard dynamical system, rigorously discussed in [5, 6].

20. On a New Subclass of Sakaguchi Type Functions Using -Derivative Operator

    S. Baskaran, G. Saravanan, K. Muthunagai

    Abstract

    The authors have introduced a new subclass of bi-univalent functions consisting of Sakaguchi type functions involving \((\mathfrak {p},\mathfrak {q})\)-derivative operator. Further, the estimation of bounds for \(|a_2|\) and \(|a_3|\) has been obtained. The authors have stated a few examples in this paper.

21. Some Double Integral Formulae Associated with Q Function and Galue-Type Struve Function

    Nirmal Kumar Jangid, Sunil Joshi, Sunil Dutt Purohit

    Abstract

    In this study, with the aid of Edward’s double integral formula, we establish some double integral formula; our results are associated with Q function and Galue-type Struve function. We often examine their special cases in the form of recognized functions such as the generalized Mittag–Leffler function and the generalized Struve function. The findings of our present paper would be both useful and helpful in the study of applied science and engineering problems.

22. Time-Dependent Analytical and Computational Study of an M/M/1 Queue with Disaster Failure and Multiple Working Vacations

    Madhu Jain, Mayank Singh, Rakesh Kumar Meena

    Abstract

    An M/M/1 working vacation (WV) queueing model with disaster failure is considered to examine time-dependent behavior. When the system is in busy mode, it can fail such that all the customers in the system are flushed out and never returns; such type of failure is known as disaster failure. The server is allowed to go for a WV after each busy period for a random duration of time. In the duration of WV, the server reduces the service rate rather than halting the service. After completing the vacation period, the server can take any number of vacation until he found some customers waiting in the queue; this vacation policy is known as multiple vacation policy. The transient analytical formulae for the queue size distributions are formulated by solving Chapman–Kolmogorov equations using continued fractions, modified Bessel function and probability generating function methods. Moreover, various queueing performance measures are given, and real-time performance is evaluated by computing the performance measures numerically.

23. Usual Stochastic Ordering Results for Series and Parallel Systems with Components Having Exponentiated Chen Distribution

    Madhurima Datta, Nitin Gupta

    Abstract

    In this paper, we have considered two n-component series systems and two n-component parallel systems. The random variables corresponding to each of these components are assumed to be independent and non-identically distributed. When the random variables followed Exponentiated Chen distribution (denoted as \(ECD(\alpha ,\beta ,\lambda )\) where \(\alpha ,\beta , \lambda \) are the 3 parameters), the systems can be compared based on the usual stochastic ordering. Some counterexamples were constructed to show that the hazard rate and reversed hazard rate orderings cannot be obtained under certain conditions.