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Issue 2
Volume 6 (2018)

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Authors |
: | K. Selva Rani, M. Himaya Jaleela Begum and S. Syed Ali Fathima |

Title |
: | On Contra Weakly $\mu g$-Continuous Function in Generalized Topological Spaces |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 1−5 |

**Abstract: ** In this paper we introduce some new class of functions called contra w$\mu$g-continuous functions and contra w$\mu$g-irresolute functions using w$\mu$g-closed sets in generalized topological space. We investigate their relationships with other existing functions in generalized topological space. Also we prove that composition of two contra w$\mu$g-continuous need not be contra w$\mu$g-continuous.

**Keywords: ** w$\mu$g-closed sets, contra w$\mu$g-continuous functions, contra w$\mu$g-irresolute functions.

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Authors |
: | S. Rohini and S. Syed Ali Fathima |

Title |
: | On Soft $g^{**} \mu$-Closed Sets in Soft Generalized Topological Spaces |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 7−12 |

**Abstract: ** In this paper we introduce some new class of sets called soft $g\mu$-closed sets,soft $g^{*}\mu$-closed sets and soft $g^{**} \mu$-closed sets in soft generalized topological spaces and some of their properties are established.Further, by using soft $g^{**} \mu$-closed sets we introduce soft $T_{\mu_{g^{**}}}$-space.

**Keywords: ** Soft $g\mu$-closed sets, soft $g^{*}\mu$-closed sets, soft $g^{**} \mu$-closed sets and soft $T_{\mu_{g^{**}}}$-space.

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Authors |
: | M. Janaki, K. Kanagarajan and D. Vivek |

Title |
: | Existence Results for Katugampola-type Fractional Functional Differential Equations |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 13−20 |

**Abstract: ** In this paper, the initial value problem is discussed for a class of Katugampola-type fractional functional differential equations and the criteria on existence are obtained.

**Keywords: ** Katugampola fractional derivative, Fractional functional differential equations, Fractional neutral functional differential equations, Existence, Fixed point.

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Authors |
: | D. Vivek, K. Kanagarajan and S. Harikrishnan |

Title |
: | Analytic Study on Nonlocal Initial Value Problems for Pantograph Equations with Hilfer-Hadamard Fractional Derivative |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 21−32 |

**Abstract: ** This paper is concerned with the existence and stability of solutions for a new class of nonlocal initial value problems involving Hilfer-Hadamrad type fractional pantograph equations. The existence results are obtained with the aid of some classical fixed point theorems, while the uniqueness of solutions is established by means of classical contraction mapping principle. We also discuss Ulam-Hyers stability for proposed problem.

**Keywords: ** Fractional pantograph equations, Hilfer fractional derivative, Existence, Fixed point, Ulam stability.

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Authors |
: | Vadivel Sadhasivam, Nagamanickam Nagajothi and Muthusamy Deepa |

Title |
: | Oscillation of Fractional Nonlinear Partial Differential Equations with Continuous Distributed Deviating Arguments |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 33−42 |

**Abstract: ** In this article, we establish some oscillation criteria for the fractional order partial differential equation with continuous distributed deviating arguments of the form
\begin{align*}
\dfrac{\partial}{\partial t}\left[r(t)D_{+,t}^{\alpha}\left(u(x,t)-\int_{\gamma}^{\delta}{q_0}(t,\zeta)u(x,\rho(t,\zeta))d\eta(\zeta)\right)\right]&=a(t)\Delta u(x,t)+\int_{c}^{d}p(t,\xi)\Delta u[x,\tau(t,\xi)]d\omega(\xi)\\
-\int_{c}^{d}q(x,t,\xi)g\left(u[x,\sigma(t,\xi)]\right)d\omega(\xi)+f(x,t),~~(x,t)\in G&=\Omega\times \mathbb{R}_+,
\end{align*}
with subject to the boundary conditions
\begin{align*}
\dfrac{\partial u(x,t)}{\partial \nu}+\mu(x,t) u(x,t)=\psi(x,t), \hspace{0.15in} (x,t)\in\partial\Omega \times \mathbb{R}_{+}
\end{align*} and $u=\chi(x,t)$, $(x,t)\in\partial\Omega\times \mathbb{R}_{+}$. Using the generalized Riccati technique and integral averaging method, new oscillation criteria are obtained.

**Keywords: ** Fractional partial differential equation, continuous deviating arguments, oscillation.

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Authors |
: | D. Balraj, V. Piramanantham and A. H. Ansari |

Title |
: | Existence of Best Proximity Points for Proximal C-Contractions Satisfying Rational Expression on Ordered Metric Spaces |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 43−51 |

**Abstract: ** The aim of this paper is to present in the partial order metric space setting,the concept of C-class functions
has been introduced and existence of best proximity point has been proved.
Our rational type mapping generalisations in both contraction mapping and
also generalisations the results of Luong and Thuan \cite{e}. Moreover, some examples are given to support the main results.

**Keywords: ** Best proximity point, fixed point, $C$-class function.

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Authors |
: | M. Janaki, K. Kanagarajan and D. Vivek |

Title |
: | Analytic Study on Fractional Implicit Differential Equations with Impulses via Katugampola Fractional Derivative |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 53−62 |

**Abstract: ** In this paper, we establish certain conditions for the existence and uniqueness of solutions for such a class of fractional implicit differential equations using Katugampola fractional derivative in Caputo. The arguments are based on fixed point theorems. We provide an example, which shows the validity of our main results.

**Keywords: ** Generalized fractional derivative, Fractional implicit differential equations, Impulses, Fixed point.

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Authors |
: | Handan Akyar |

Title |
: | On the Cartesian Product of Generalized Fibonacci Graphs |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 63−70 |

**Abstract: ** In this article generalized Fibonacci graphs and their Cartesian product graphs are considered. These graph's certain fundamental properties, such as number of edges, planarity, diameter, radius, center, distance center, Wiener index, etc. are studied.

**Keywords: ** Graph Theory, Generalized Fibonacci Graphs, Cartesian Product.

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Authors |
: | Ritika |

Title |
: | Common Fixed Points of Generalized Quasi--Contractive Type Operators By a Three-Step Iterative Process in CAT(0) Spaces |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 70−77 |

**Abstract: ** The aim of this paper is to approximate common fixed points of a recent three step iterative process essentially due to Thakur et al \cite {16} for two generalized quasi--contractive type operators and establish strong convergence result for the same operators in $CAT(0)$ spaces. The results obtained in this paper extend and improve the recent ones announced by Saluja \cite {15} , Xu and Noor \cite {17}, Noor \cite {13}, Picard \cite {14}, Mann \cite {12}, Ishikawa \cite {10} and many others.

**Keywords: ** CAT(0) spaces, Generalized Quasi--Contractive type operators, Common Fixed Points, Strong convergence.

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Authors |
: | Md. Mamun Ar Rashid |

Title |
: | Generalized Semi-closed Sets in a Space |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 79−82 |

**Abstract: ** In this paper the concept of generalized semi-closed sets and generalized semi open sets are introduced and some of its properties are investigated in a space considered by Aiexandroff A.D. \cite{1} where arbitrary union of open sets may not be open.

**Keywords: ** Generalised semi-closed set (g*s closed set), generalized semi open set (g*s open set).

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Authors |
: | G. V. R. Babu, P. S. Kumar and G. Satyanarayana |

Title |
: | Fixed Points of Generalized $\alpha-\beta-\psi-$Contractive Mappings in Partially Ordered Complete Metric Spaces |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 83−22 |

**Abstract: ** In this paper, we introduce the notion of generalized $\alpha-\beta-\psi-$contractive mappings and provide sufficient conditions for the existence and uniqueness of a fixed points for such mappings in partially ordered metric spaces. We deduce some corollaries and provide examples in support of our main results. Our results generalize the results of Asgari and Badehian \cite{Asgari}.

**Keywords: ** Fixed point, generalized $\alpha-\beta-\psi-$ contractive map, $\alpha-\beta-$admissible map, partially ordered complete metric space.

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Authors |
: | N. P. Muktawar and D. S. Palimkar |

Title |
: | Asymptotic Attractivity and Stability for Random Functional Differential Equation |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 93−98 |

**Abstract: ** The random differential equations appears quite naturally in the study of changes or the rate of change of any processes like diffusion process or Brownian motion. The random differential equations have been the subject of rather extensive research area since long time. It have been utilized as models for a wide variety of random problems that have been encountered time and in the areas of all areas like Sciences and Environmental Sciences. In this paper, we have discussed an existence results for local asymptotic attractivity and asymptotic stability of random solution for nonlinear random functional differential equations in Banach algebra through hybrid random fixed point theorem. The results obtained, is generalize and extend the asymptotic attractivity and stability of random solutions for concerning random functional differential equations.

**Keywords: ** Asymptotic attractivity, Stability, random functional differential equation, Banach algebra.

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Authors |
: | Amit Gupta, Gajendra K. Saraswat and Ravendra Singh |

Title |
: | Slip Effects on Blood Flow in the Presence of a Bell-shaped Stenosed Artery: A Casson Fluid Model |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 99−106 |

**Abstract: ** In this a paper, one dimensional Casson fluid flow under the influence of bell shaped stenosed artery have been considered. Analytical and numerical techniques both are carried out to observe this situation. Analytical expressions for axial velocity, plug flow velocity, volumetric flow rate, pressure gradient and wall shear stress has been derived. Numerical techniques have been used to show the effects of yield stress, slip and no slip velocity on these flow variables. The results show that wall shear stress increases as the values of yield stress diminishes and increase in slip velocity reduces the wall shear stress. It is also observed that plug flow velocity increases as slip velocity increases, pressure gradient increases as the slip velocity increases whereas it reduces as yield stress diminishes and axial velocity increases as velocity slip increases.

**Keywords: ** Casson Fluid, Yield Stress, slip velocity, plug flow velocity.

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Authors |
: | Porkodi Chinniah and Arumuganathan Ramalingam |

Title |
: | An Integer Factorization Method Equivalent to Fermat Factorization |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 107−111 |

**Abstract: ** Integer factorization problem had a great impact on security of many public key cryptosystems. Advancement in factoring increases the need to develop innovative public key cryptosystems. In this paper, a special purpose factorization method to find the factors of a composite number, which is the product of two distinct primes, is proposed. Fermat's method is theoretically important and many modern factorization algorithms like the quadratic sieve, multiple polynomial quadratic sieve, and number field sieves are based upon this method. The proposed scheme is equivalent to Fermat's method, and hence it can be used in such popular modern factorization methods. The steps involved in the algorithm are proved theoretically. Algorithm is illustrated numerically using Mathematica.

**Keywords: ** Integer factorization problem, Division algorithm, Fermat's method, Greatest Common Divisor.

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Authors |
: | Swarnima Bahadur and Ravindra Kumar Katheriya |

Title |
: | On (0,1,2) Trigonometric Interpolation |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 113−118 |

**Abstract: ** The aim of this paper is to discuss the case of $(0,1,2)$ interpolation by trigonometric polynomial on the zeros of $\sin mx$ at the point $x_k$=$\frac{2\pi k}{n}$, where $k=0,1,2,\dots,n-1$, where n is even $(n=2m)$ and the convergence behavior of this trigonometric polynomial.
Let $R_n(x)$ be a trigonometric polynomial of order such that
\begin{align*}
R_n(x_k)&=a_k\\
R'_n(x_k)&=b_k\\
R''_n(x_k)&=c_k
\end{align*}
where $x_k$=$\frac{2\pi k}{n}$, $k=0,1,2,\dots,n-1$ and $n$ is even $(n=2m)$.

**Keywords: ** Trigonometric polynomial, Interpolation, Explicit representation, Estimates, Convergence.

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Authors |
: | K. Ramalakshmi and K. Palani |

Title |
: | Independent and Upper Steiner Domination Number of Graphs |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 119−124 |

**Abstract: ** In this paper, independent and upper steiner domination number of graphs are introduced. Also, these numbers were found for some standard graphs.

**Keywords: ** Steiner number, steiner domination number, independent steiner domination number, upper steiner domination number.

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Authors |
: | B. Prabhakar Reddy |

Title |
: | Viscous Dissipation Effects on Unsteady MHD Flow Past an Impulsively Started Infinite Isothermal Vertcal Plate in The Presence of Radiation |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 125−134 |

**Abstract: ** In this paper, the effects of viscous dissipation on an unsteady MHD viscous incompressible fluid flow past an impulsively started infinite isothermal vertical plate in the presence of radiation has been presented. The fluid is considered gray, absorbing-emitting radiation but a non-scattering medium. Thermal radiation effects are simulated via a radiation parameter $N$, based on the Rossseland diffusion approximation. The momentum and thermal boundary layer equations are non-dimensionalized using appropriate transformations and then solved numerically subject to physically realistic boundary conditions by using the Ritz finite element method. The influence of physical parameters on the velocity profiles $(u)$, temperature distribution $(\theta )$, skin-friction $(\tau )$, and the Nusselt number $(Nu)$ are presented with the help of graphs and tables. The obtained results show that a decrease in the velocity and temperature as the radiation parameter is increased. The velocity $u$ is decreased as the magnetic parameter increases and increases with increasing Grashof number and time parameter. The skin-friction increases with increase in the magnetic parameter and decreases with increasing Grashof number and Eckert number.

**Keywords: ** MHD, magnetic parameter, radiation parameter, isothermal vertical plate, Ritz finite element method.

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Authors |
: | Fernandes Gloria and Tabitha Agnes Mangam |

Title |
: | Isolate Domination in Total graphs |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 135−139 |

**Abstract: ** A dominating set $S$ of a graph $G$ is said to be an isolate dominating set of $G$ if induced subgraph on $S$ has at least one isolated vertex. An isolate dominating set $S$ is said to be a minimal isolate dominating set if no proper subset of $S$ is an isolate dominating set. The isolate domination number $\gamma_0$ is defined as the minimum cardinality of minimal isolate dominating set. In this paper, we investigate the isolate domination number of total graphs of certain classes of graphs.

**Keywords: ** Domination number, Isolate domination number, Total graph.

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Authors |
: | S. Lavanya and S. Ramya |

Title |
: | Edge Domination in Some Special Classes of Fuzzy Graphs |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 141−144 |

**Abstract: ** In this paper we find the exact edge domination number of some special classes of fuzzy graphs like complete fuzzy graph, star fuzzy graph complete bipartite fuzzy graph etc.

**Keywords: ** Edge domination, complete fuzzy graph, star fuzzy graph, regular fuzzy graph.

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Authors |
: | G. Ayyappan and S. Karpagam |

Title |
: | An $M^{[X]}/G(a,b)/1$ Queueing System with Breakdown and Second Optional Repair, Stand-by Server, Balking, Variant Arrival Rate and Multiple Vacation |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 145−156 |

**Abstract: ** In this paper, we discussed a Non-Markovian batch arrival general bulk service single server queueing system with server's breakdown and second optional repair, stand-by server, balking, variant arrival rate and multiple vacation. The main server's service time, vacation time, stand-by server's service time are all follows general distributions and breakdown and two types of repair time for main server follows exponential distributions. There is a stand-by server which is employed during the period for which the regular server remains under repair. The probability generating function of queue size at an arbitrary time and some performance measures of the system are derived. Extensive numerical results are also illustrated.

**Keywords: ** Non-Markovian queue, General bulk service, Multiple vacation, Breakdown and optional Repair, Stand-by server, Balking, Variant arrival rate.

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Authors |
: | M.M. Iddrisu |

Title |
: | q-Steffensen's Inequality for Convex Functions |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 157−162 |

**Abstract: ** In this paper, q- Steffensen's Inequality for convex functions is presented with illustrative examples. Review of research works on Steffensen's Inequality and q-calculus is extensively carried out. Methods of q- differentiability and monotonicity of functions are employed to establish the results.

**Keywords: ** Steffensen's Inequality, Convex functions, q-calculus.

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Authors |
: | K. Nantomah, M. M. Iddrisu and C. A. Okpoti |

Title |
: | On a $q$-analogue of the Nielsen's $\beta$-Function |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 163−171 |

**Abstract: ** Motivated by the Jackson's $q$-integral and the $q$-digamma function, we introduce a $q$-analogue of the Nielsen's $\beta$-function. We further study some identities, monotonicity and convexity properties, and some inequalities involving the new function.

**Keywords: ** Nielsen's $\beta$-function, Jackson's $q$-integral, $q$-digamma function, $q$-analogue, inequality.

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Authors |
: | Tresa Mary Chacko and D. Susha |

Title |
: | Binary Linear Topological Spaces |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 173−179 |

**Abstract: ** In this paper we define and study the concept of binary linear topological spaces (BLTS) and their properties. Here we prove that the binary product of two linear topological spaces is a BLTS. Also we have the main result that the binary product preserve metrizability and normability. Finally we construct a BLTS from a family of binary seminorms on a pair of vector spaces.

**Keywords: ** Linear Topological Spaces, Binary topology, locally convex, metrizable.

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Authors |
: | M. Tamilselvi and N. Revathi |

Title |
: | k-Harmonic Mean Labeling of Some Cycle Related Graphs |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 181−191 |

**Abstract: ** Somasundaram and Ponraj \cite{12} have introduced the notion of mean labeling of graphs. R. Ponraj and D. Ramya introduced Super mean labeling of graphs in \cite{5}. S. Somasundaram and S.S. Sandhya introduced the concept of Harmonic mean labeling in \cite{6} and studied their behavior in \cite{7,8,9}. In this paper, we investigate k-harmonic mean labeling of some graphs.

**Keywords: ** Harmonic mean labeling, harmonic mean graph, k-harmonic mean labeling, k-harmonic mean graph.

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Authors |
: | V. Rajendran, P. Sathishmohan and K. Lavanya |

Title |
: | On $\mathcal{NI}_{\widehat g}$-Closed Sets in Nano Ideal Topological Spaces |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 193−199 |

**Abstract: ** This paper focuses on $\mathcal {NI}_{\widehat {g}}$-Closed sets (nano $\mathcal {I} _{\widehat {g}}$-closed sets) and $\mathcal {NI}_{\widehat {g}}$-open sets (nano $\mathcal {I} _{\widehat {g}}$-closed sets) in nano ideal topological spaces and certain properties of these are investigated. We also investigate the concept of nano $\mathcal {I} _{\widehat {g}}$-closed sets and discussed their relationships with other forms of nano ideal sets. Further, we have given an appropriate examples to understand the abstract concepts clearly.

**Keywords: ** ${\mathcal {N}g}$-closed sets, ${\mathcal {N}}{\widehat{g}}$-closed sets, ${\mathcal {NI}}_{g}$-closed sets, ${\mathcal {NI}}_{\widehat {g}}$-closed sets and ${\mathcal {NI}}_{\widehat g}$-open sets.

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Authors |
: | P. Sathishmohan, V. Rajendran and P. Jeevitha |

Title |
: | On $\pi g^\ast$s-Irresolute Functions in Topological Spaces |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 201−207 |

**Abstract: ** The purpose of this paper is to defined and studied the notion of $\pi$ generalized star semi-closed sets in topological spaces and some of their basic properties are investigated. This new class of sets lies between the class of $\pi$g-closed sets and the class of $\pi$gs-closed sets. Further the notion of $\pi g^\ast$s-open sets, $\pi g^\ast$s-continuous functions and $\pi g^\ast$s-irresolute functions are defined and the composition of two $\pi g^\ast$s-irresolute functions are discussed. Several examples are provided to illustrate the behaviour of new sets and functions.

**Keywords: ** $\pi g^\ast$s-closed sets, $\pi g^\ast$s-open sets, $\pi g^\ast$s-continuous functions, $\pi g^\ast$s-irresolute functions.

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Authors |
: | Parajal Rai |

Title |
: | A Study on Factoriangular Numbers |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 209−218 |

**Abstract: ** In this paper, we prove that there is no factoriangular number that is also factorial. An expression of factoriangular number explicitly in terms of triangular number is given. Bounds of the ratio of consecutive factoriangular number are given. It is shown that for n $\geq$ 5, there exists no $Ft_{n}$ which divides $Ft_{n+1}$. Patterns in factoriangular number modulo n is also observed. It is conjectuerd that there exists no factoriangular number that is a perfect square. It has also been conjectured that for n $\geq$ 6, 8n! + 1 is not a perfect square. A conjecture \cite{cas} that there is no factoriangular number that is an even perfect number is proved.

**Keywords: ** Factoriangular Numbers, Generalized Factoriangular Numbers, even perfect number.

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Authors |
: | A. Sahaya Sudha and K. R. Vijayalakshmi |

Title |
: | Arithmetic Operations of Hexagonal Intuitionistic Fuzzy Number Using Extension Principle |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 219−225 |

**Abstract: ** This paper presents an analysis of the basic definitions of the theory of intuitionistic fuzzy sets. Some arithmetic operations using membership and non-membership functions on Hexagonal intuitionistic fuzzy numbers are revealed and the ways to approve the properties using $(\alpha ,\beta )$ cut intervals of intuitionistic fuzzy arithmetic are proposed. The concept is illustrated with numerical example.

**Keywords: ** Extension principle, Hexagonal intuitionistic fuzzy number, Intuitionistic fuzzy sets, $(\alpha ,\beta )$ cut intervals.

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Authors |
: | Nistala V.E.S. Murthy and Chundru Maheswari |

Title |
: | Representation of Soft Subsets by Products |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 227−241 |

**Abstract: ** In this paper, we show that for any soft set over a universal set, 1. there is a crisp set such that the complete lattice of all soft subsets of the given soft set is complete epimorphic to a complete lattice of certain subsets of the crisp set, where the join in the later complete lattice is the meet induced join and 2. there is a crisp set such that the complete lattice of all regular soft subsets of the given soft set is complete isomorphic to a complete lattice of certain subsets of the crisp set, where the meet in the former complete lattice is the join induced meet and the join in the later complete lattice is the meet induced join.

**Keywords: ** Factorizable subset, (Extended) Soft (sub) set, Associated product set for a soft set.

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Authors |
: | Hamant Kumar |

Title |
: | Almost $\beta$-Regular Spaces and Almost $rg\beta$-Closed Functions |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 243−250 |

**Abstract: ** The aim of this paper is to introduce and study a new class of spaces, namely almost $\beta$-regular spaces which are generalizations of the concepts of almost regular spaces and $\beta$-regular spaces and to obtain some characterizations of almost $\beta$-regular spaces. Further, by using $rg\beta$-closed sets, we define almost $rg\beta$-closed functions and to obtain preservation theorems of almost $\beta$-regular spaces. The relationships among p-regular, $\beta$-regular, almost regular, almost p-regular, almost $\beta$-regular spaces are investigated. The main result of this paper is that almost $\beta$-regularity is preserved under M-$\beta$-open $\beta$-$rg\beta$-closed surjective R-maps.

**Keywords: ** $\beta$-open and regular open sets, almost regular and almost $\beta$-regular spaces, $\beta$-$g\beta$-closed, $\beta$-$rg\beta$-closed, almost $\beta$-closed, almost $rg\beta$-closed and almost continuous functions.

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Authors |
: | P. Srikanth Rao and Veena Kulkarni |

Title |
: | Fixed Point Theorem for Occasionally Weakly Compatible Mappings With Common Limit Range Property Satisfying Generalized $(\psi,\phi)$-Weak Contractive Conditions |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 251−261 |

**Abstract: ** In this paper, we prove some common fixed point theorems for occasionally weakly compatible mappings in metric spaces satisfying generalized $(\psi,\phi)$-contractive conditions under the common limit range property. We present fixed point theorem for four finite families of self-mappings which can be utilized to derive common fixed point theorems involving any number of finite mappings. Our results improve and extend the corresponding results of Mohammad Imdad, Sunny Chauhan and ZoranKadelburg(Mathematical Sciences 7(16): 1-8, 2013).

**Keywords: ** Metric space, Occasionally weakly compatible mappings, Property (E.A), Generalized weak contraction, Common limit range property, Common fixed point.

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Authors |
: | S. Parimala, Kalyanasundaram Madhu and Jayakumar Jayaraman |

Title |
: | Revisit of Ostrowski's Method and Two New Higher Order Methods for Solving Nonlinear Equation |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 263−270 |

**Abstract: ** In this paper, we have obtained the Ostrowski's method in a different way and proposed two new methods of order seven and thirteen. The efficiency index of Ostrowsky's method is 1.587 and that of the seventh order method and thirteenth order method are respectively 1.626 and 1.670, which are better than Newton's method (1.414) and Ostrowsky's method. Also it is observed from the numerical illustrations, the proposed methods take less number of iterations than Newton's method. Few other methods are compared with the proposed two methods, where the number of iterations for those methods are either same or more than the presented methods. Some examples are given to illustrate the performance of the new methods.

**Keywords: ** Non-linear equation, multi-point iteration, optimal order, Kung-Traub conjecture, higher-order method.

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Authors |
: | Manish Sharma |

Title |
: | A Fixed Point Result in Dislocated Quasi b-Metric Space |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 271−274 |

**Abstract: ** The purpose of this paper is to consider the notion of dislocated quasi b-metric space and establish a fixed point theorem using symmetric rational type contractive condition in the context of dislocated quasi b-metric space as a generalization of Banach contraction principle.

**Keywords: ** Dislocated quasi b-metric space, rational type contraction, fixed point.

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Authors |
: | Rafia Aziz and Rinku Verma |

Title |
: | Fixed Point Theorem for Mappings Satisfying Certain Rational Inequality |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 275−277 |

**Abstract: ** The authors of \cite{1,2} and \cite{3} proved some fixed point theorems in complete metric space. In this paper we prove some fixed point theorems in complete metric space for mappings satisfying certain rational inequality.

**Keywords: ** Fixed point, common fixed point, self mapping and complete metric spaces.

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Authors |
: | Ranu Sharma, Bhagyashri A. Deole and Smita Verma |

Title |
: | Some Properties of Tri-b Open Sets in Tri Topological Space |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 279−287 |

**Abstract: ** The main aim of this paper is to study tri b-open sets in tri topological spaces along with their several properties and characterization. We study tri b-continuous, tri-b separation and obtain some of their basic properties.

**Keywords: ** Tri topology, tri b-open sets,tri b-continuous function, tri-gb-closed, tri-b separation.

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Authors |
: | Deepak Jain |

Title |
: | Common Fixed Point Theorem for Compatible Mappings of Type (K) in Digital Metric Spaces |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 289−293 |

**Abstract: ** In this paper, first we introduce the notion of compatible mappings of type $(K)$ in digital metric spaces analogue to the notion of compatible mappings of type $(K)$ in metric spaces and then we prove a common fixed point theorem for pairs of compatible mappings of type $(K)$ in digital metric spaces.

**Keywords: ** Fixed point, digital image, compatible mappings of type $(K)$, digital metric space.

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Authors |
: | Manoj Kumar Sharma |

Title |
: | An Inventory Model for Deteriorating Products with Demand Appraise by Promotional Effort |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 295−301 |

**Abstract: ** In Modern era when trends in technology, fashion and consumer change rapidly; the major problem faced by the retailer is how they can manage the deteriorating stock at time of product's self-life. In general they use some promotions efforts like price discount to prop up demand. We have tried to deal with such problem in this paper. We study the inventory problem of non-instantaneous single period deteriorating items with two parameter weibull distribution. We are considering two types of demand in this paper. First, it is function of time and then contain price factor to grown promotional effort. The holding cost is considered as constant. An approach is proposed to maximize total profit and optimal order quantity. Numerical examples illustrate the theoretical results and sensitivity analysis done as well.

**Keywords: ** Deterministic inventory model, weibull distribution, promotional efforts, demand function.

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Authors |
: | Shradha Dwivedi |

Title |
: | An Alternative Failure Model of Three Parameter Exponential Distribution: Study of its Structural Properties and Parameters Estimation |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 303−312 |

**Abstract: ** Here we studied a three parameter exponential distribution which can work as an alternative model for various other available exponential distribution in Reliability and Life testing theory.It has its own structural properties which makes it a unique and flexible model for various real life data. It is a three parameter model with different shape parameter but having the same location and scale parameter.

**Keywords: ** Three parameter exponential distribution, Mode, Moment Estimator, Maximum Likelihood Estimator.

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Authors |
: | R. Mariappan and M. Murugalingam |

Title |
: | On $m$-$\delta$-$I$-Open Sets and $m$-$\delta$-$I$-Continuity |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 313−317 |

**Abstract: ** In this paper, we introduce and study the notions of $m$-$\delta$-$I$-open sets and $m$-$\delta$-$I$-continuity and their related notions in ideal minimal spaces. Also we investigate the decomposition of $m$-semi-$I$-continuous functions and $m$-$\alpha$-$I$-continuous functions using $m$-$\delta$-$I$-open sets.

**Keywords: ** Ideal minimal space, $m$-$\delta$-$I$-open sets, $m$-$\delta$-$I$-closed sets, $m$-$\delta$-$I$-continuity.

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Authors |
: | T. Nagalakshmi and G. Uthra |

Title |
: | A New Approach to Find an Optimal Solution of a Fuzzy Project Management Problem by Fuzzy Dynamic Programming |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 319−325 |

**Abstract: ** This paper mainly focuses on a new approach to find an optimal solution of a fuzzy Project Management problem with the help of Fuzzy Dynamic Programming. Project Management deals with the allocation of available foremen to the existing projects to complete the projects within the stipulated time. Moreover, it is known that fuzziness prevails in all fields. Hence, a general Project Management problem with fuzzy parameters is considered where the estimated time to complete each project is taken as Generalized Trapezoidal Fuzzy Numbers. The solution is obtained by the method of FDP by framing fuzzy forward and fuzzy backward recursive equations. It is observed that the solutions obtained by both the equations are the same. This approach is illustrated with a numerical example. This feature of the proposed approach eliminates the imprecision and fuzziness in Project Management models. The application of Fuzzy set theory in the field of dynamic Programming is called Fuzzy Dynamic Programming.

**Keywords: ** Fuzzy dynamic Programming, Generalized Trapezoidal Fuzzy numbers, Fuzzy Project Management Problem.

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Authors |
: | Fawziah M. Al-saar, Kirtiwant P. Ghadle and Priyanka A. Pathade |

Title |
: | The Approximate Solutions of Fredholm Integral Equations by Adomian Decomposition Method and its Modification |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 327−336 |

**Abstract: ** In this paper, a modification of Adomian decomposition method (MADM) for solving linear and nonlinear Fredholm integral equations of the second kind has been introduced. This new method is resulted from Adomian decomposition method (ADM) by a simple modification. This modification is based on the existence of Taylor expansion. To illustrate the modification on ADM, some examples are presented. Comparison of the result of applying the MADM and ADM revealing the new technique is very effective and convenient.

**Keywords: ** Adomian decomposition method, modified Adomian decomposition method, linear and nonlinear Fredholm integral equations of the second kind, Taylor expansion.

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Authors |
: | M. Perumalsamy, P. Arul Paul Sudhahar and R. Vasanthi |

Title |
: | The Upper Edge Fixed Steiner Number of a Graph |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 337−343 |

**Abstract: ** For a non-empty set W of vertices in a connected graph G, the Steiner distance $d(W)$ of W is the minimum size of a connected subgraph of G containing W. Necessarily, each such subgraph is a tree and is called a Steiner tree with respect to W or a Steiner W-tree. $S(W)$ denotes the set of vertices that lies in Steiner W-trees. Let G be a connected graph with at least 2 vertices. A set $W \subseteq V(G)$ is called a Steiner set of G if $S(W) = V(G)$. The Steiner number $s(G)$ is the minimum cardinality of a Steiner set. Let G be a connected graph with at least 3 vertices. For an edge $e = xy$ in G, a set $W \subseteq V(G) -\{x, y\}$ is called an edge fixed Steiner set of G if $W' = W \cup \{x, y\}$ is a Steiner set of G. The minimum cardinality of an edge fixed Steiner set is called the edge fixed Steiner number of G and is denoted by $s_e (G)$. Also the Steiner W-tree necessarily contains the edge e and is called edge fixed Steiner W-tree. In this paper, the concept of upper edge fixed Steiner number of a graph G denoted by $s_e^+(G)$ is studied. Also the graphs in which the upper edge fixed Steiner number is equal to n or $n-1$ are characterized. It is shown that for every pair a, b of integers with $a \geq 3$ and $b \geq 3$, there exists a connected graph G with $s_e(G) = a$ and $s_e^{+}(G)= b$.

**Keywords: ** Steiner set, edge fixed Steiner set, Steiner number, edge fixed Steiner number, minimal edge fixed Steiner set, upper edge fixed Steiner number.

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Authors |
: | S. Padmasekaran, R. Asokan and E. Nakkeeran |

Title |
: | Lie Classical Method for (2+1)-dimensional Rosenau Equation |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 345−350 |

**Abstract: ** In this paper, a (2+1)-dimensional Rosenau equation is subjected to Lie's classical method. Classification of its symmetry algebra into one- and two-dimensional subalgebras is carried out in order to facilitate its reduction systematically to (1+1)-dimensional PDE and then to first order ODE.

**Keywords: ** Nonlinear PDE, Lie's Classical Method, Lies Algebra, Symmetry Reductions.

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Authors |
: | K. Alaguraja, R. Asokan and S. Padmasekaran |

Title |
: | Symmetry Reductions of (2+1)-dimensional Modified Equal Width Wave Equation with Damping Term by Lie Classical Method |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 351−356 |

**Abstract: ** In this paper, we are consider a (2+1)-dimensional Modified Equal Width Wave equation with damping term is $u_t + u + u^{3}u_x -\mu(u_{xxt} + u_{yyt})=0$, subjected to Lie’s classical method. Classification of its symmetry algebra into one- and two-dimensional subalgebras is carried out in order to facilitate its reduction systematically to (1+1)-dimensional PDE and then to first order ODE.

**Keywords: ** Nonlinear PDE, Lie's Classical Method, Lie's Algebra, Symmetry group.

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Authors |
: | G. Kumar and R. Jeyabalan |

Title |
: | Completely Generalized Semi-precontinuous Mapping in $\it IVF$-topological Spaces |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 357−364 |

**Abstract: ** In this paper we introduce $ivf$-completely generalized semi-pre continuous mappings. We investigate some of its properties. Also we provide some characterization of $ivf$-completely generalized semi-pre continuous mappings.

**Keywords: ** $ivf$-set, $ivf$-topological space, $ivf$-point, $ivf$-generalized semi-preclosed set, $ivf$-generalized semi-precontinuous mappings, $ivf$-semi-pre$T_{1/2}$ space.

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Authors |
: | S. M. Vaghasiya and G. V. Ghodasara |

Title |
: | Edge Difference Cordial Labeling of Graphs |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 365−371 |

**Abstract: ** We put up a dissimilar of difference cordial labeling namely as edge difference cordial labeling. As interchange the roles of
vertices and edges in difference cordial labeling. Let $G$ be a $(p,q)$ graph. Let $k$ be an integer with $1\leq k\leq q$ and
$f:E(G)\rightarrow \{1,2,\ldots, k\}$ be a map. For each vertex $v$, assign the label $\min|(f(e_i)-f(e_j))|$. The function $f$ is called
an edge difference cordial labeling of $G$ if $f$ is one-to-one map and $|v_{f}(1)-v_{f}(0)|\leq1$ where $v_{f}(x)$ denotes the number of
vertices labeled with $x \ (x\in\{1,2,\ldots,k\})$, $e_{f}(1)$ and $e_{f}(0)$ respectively denote the number of edges labeled with 1 and
not labeled with 1. A graph with an edge difference cordial labeling is called an edge difference cordial graph. In this paper we investigate
some results on newly defined idea.

**Keywords: ** Cordial labeling, edges in difference, cordial graph.

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Authors |
: | Rehana Khan and Q. H. Khan |

Title |
: | Multi-variable Hermite-Tricomi Functions: Generating Relations and Applications |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 373−382 |

**Abstract: ** In this paper, the authors uses some operational methods combining with the principle of monomiality to introduce the Hermite Tricomi function of four variables and three parameters. Series definition, differential equation and certain other properties of are derived. Generating relations of these functions are also derived by using Lie-algebraic method. Further,as applications of these functions, certain new and known generating relations involving other forms of Hermite Tricomi functions are established.

**Keywords: ** Hermite Tricomi functions, Lie-algebra representation, generating relations.

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Authors |
: | R. Eswari and V. Piramanantham |

Title |
: | Existence of Positive Periodic Solutions in a Non-selective Harvesting Impulsive Predator-prey System with Multiple Delays |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 383−393 |

**Abstract: ** In this paper, we propose a non-selective harvesting impulsive predator-prey system with multiple delays. By applying the continuation theorem of coincidence degree theory, we establish a set of sufficient conditions on the existence of at least one positive periodic solution. The result not only improves but also generalizes that for the case with impulses. Some examples are provided to illustrate the result.

**Keywords: ** Impulsive predator-prey system, Continuation theorem, Periodic solution.

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Authors |
: | Pushpendra Semwal and Komal |

Title |
: | Fixed Point Results for F-contractive Mappings |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 395−402 |

**Abstract: ** In this paper, we present a common fixed point theorem for a pair of self mappings, using F-contraction condition, in the setting of metric space.Further a fixed point theorem for a self map and application in dynamic programming are given. Our result extend the theorems of Wardoski, Cosentino and Vetro, Choudhury etc.

**Keywords: ** Fixed point, F-Contraction, complete metric space.

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Authors |
: | Mohammed S. Abdo and Satish K. Panchal |

Title |
: | Uniqueness and Stability Results of Fractional Neutral Differential Equations with Infinite Delay |

Volume, Issue, Year |
: | 6(2−A)(2018) |

Pages |
: | 403−410 |

**Abstract: ** In this paper, some uniqueness and Ulam-Hyers stability
results for neutral functional differential equations with fractional order
and infinite delay are proved. The problem that is discussed includes the
Caputo fractional derivative operator. The technique used is a variety of
tools of fractional calculus properties and Banach's fixed point theorem. An
example of the obtained results is given.

**Keywords: ** Fractional differential equations, Functional-differential equations with fractional derivatives, Stability, Fixed point theorem.