## Issue 4 − EVolume 3 (2015)

ISSUE 4 − A ISSUE 4 − B ISSUE 4 − C ISSUE 4 − D ISSUE 4 − E ISSUE 4 − F
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 Article Type : Research Article Title : Homogenization of a Nonlinear Fibre-Reinforced Structure with Contact Conditions on the Interface Matrix-Fibres Country : Morocco Authors : H.Samadi || M.El Jarroudi

Abstract: We study the homogenization of a nonlinear problem posed in a fibre-reinforced composite with matrix-fibres interfacial condition depending on a parameter $\lambda=\lambda(\varepsilon)$, $\varepsilon$ being the size of the basic cell. Using $\Gamma$-convergence methods, three homogenized problems are determined according to the limit of the ratio$\gamma = \frac{\lambda(\varepsilon)}{\varepsilon}$. The main result is that the effective constitutive relations reveal non-local terms associated with the microscopic interactions between the matrix and the fibers.

Keywords: Composite material, periodic fibres, interfacial conditions, $\Gamma$-convergence, homogenized models.

National School of Applied Science, LABTIC, UAE, Tangier, Morocco.

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 Article Type : Research Article Title : A Note on the Continuability of Solutions of a Perturbed Second Order Nonlinear Differential Equation of Li\'{e}nard Type Country : Argentina Authors : Juan E. Napoles Valdes || Luciano M. Lugo Motta Bittencurt || Paulo M. Guzman

Abstract: In this note we study the continuability of the solutions of a Lienard type equation with forcing term under suitable assumptions.

Keywords: Continuability, second order, non-autonomous.

Juan E. Napoles Valdes
UNNE, FaCENA, Av. Libertad 5450, (3400) Corrientes, Argentina and UTN, FRRE, French 414, (3500) Resistencia, Argentina.
E-mail: jnapoles@exa.unne.edu.ar

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 Article Type : Research Article Title : On the Boundedness of Some Nonlinear Differential Equation of Second Order Country : Argentina Authors : Paulo M. Guzman || Juan E. Napoles Valdes || Luciano M.Lugo

Abstract: In this paper we study the boundedness of the solutions of some nonlinear differential equation using as a key tool the Second Lyapunov method, i.e. find sufficient conditions under which the solutions of this equation are bounded. Various particular cases and methodological remarks are included at the end of paper.

Keywords: Lyapunov's Second Method, boundedness, second order nonlinear equation.

Juan E. Napoles Valdes
UNNE, FaCENA, Av. Libertad 5450, (3400) Corrientes, Argentina and UTN, FRRE, French 414, (3500) Resistencia, Argentina.
E-mail: jnapoles@exa.unne.edu.ar

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 Article Type : Research Article Title : Approximating Fixed Point in CAT(0) Space by s-iteration Process for a Pair of Single Valued and Multivalued Mappings Country : India Authors : Manjula Tripathi || Anil Kumar Dubey

Abstract: Suppose $K$ is a closed convex subset of a complete CAT(0) space $X$. $T$ is mapping from $K$ to $X$. $F(T)$ is set of fixed point of $T$ which is nonempty. Sequence $\{x_n\}$ is defined by an element $x_1 \in k$ such that \begin{eqnarray*} x_{n+1} &=& P((1-\alpha_n) Tx_n \oplus \alpha_n y_n)\\ y_n &=& P((1-\beta_n) x_n \oplus \beta_n T x_n) \ \ \ \forall \geq 1 \end{eqnarray*} where $P$ is the nearest point projection from $X$ onto $k$. $\{\alpha_n\}, \{\beta_n\}$ are real sequences in (0,1) with the condition \begin{eqnarray*} \displaystyle \sum^{\infty}_{n=1}\alpha_n \beta_n (1-\beta_n) =\infty \end{eqnarray*} Then $\{x_n\}$ converges to some point $x^*$ in $F(T)$. This result is extension of the result of Abdul Rehman Razani and saeed Shabhani. [Approximating fixed points for nonself mappings in CAT(0) spaces Springer 2011:65]

Keywords: S-iteration, CAT(0) spaces, fixed point condition E, nonself mapping, condition C.

Anil Kumar Dubey
Assistant Professor, Bhilai Institute of Technology, Durg, India.
E-mail: anilkumardby70@gmail.com

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 Article Type : Research Article Title : On Certain Subclasses of Multivalent Functions Involving A Differintegral Operator Country : India Authors : C.Selvaraj || T.R.K.Kumar || G. Thirupathi

Abstract: In this paper, we introduce the class $S_{p}^{m,l,\lambda}\left(\alpha,\beta,\gamma,\sigma;A,B\right)$ of p-valent functions in the unit disc $\mathbb{U}=\left\{z :\left|z\right|<1\right\}$. We obtain coefficient estimate, distortion and closure thoerems and radii of close-to-convexity, starlikeness and convexity for this class.

Keywords: p-Valent analytic functions; Linear operator; Generalized neighborhoods; Fractional calculus operators; Hypergeometric function; Differential subordination.

T.R.K.Kumar
Department of Mathematics, R.M.K.Engineering College, R.S.M.Nagar, Kavaraipettai, Tamilnadu, India.
E-mail: kumartrk@yahoo.com

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 Article Type : Research Article Title : Fuzzy Quasi-maximal Spaces Country : India Authors : S.Anjalmose || G.Thangaraj

Abstract: In this paper the concept of Fuzzy Quasi-maximal spaces are introduced. Several properties and examples of fuzzy Quasi-maximal spaces are also studied. Relations between fuzzy Quasi-maximal space and fuzzy Baire space, fuzzy D-Baire space are studied.

Keywords: Fuzzy dense, fuzzy nowhere dense, fuzzy first category, fuzzy Quasi-maximal space, fuzzy Baire space and fuzzy D-Baire space.

S.Anjalmose
Department of Mathematics, St. Joseph's College of Arts $\&$ Science (Autonomous), Manjakuppam, Cuddalore, Tamil Nadu, India.
E-mail: ansalmose@gmail.com

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 Article Type : Research Article Title : Continuous and Contra Continuous Functions in Bi-topological Spacese Country : India Authors : Parvinder Singh

Abstract: The concept of bi-topological spaces was first introduced by J.C. Kelly \cite{2} in 1963. Many authors such as Levine \cite{3} contributed as he defined the semi-open sets and semi-continuity in bi-topological spaces. Maheshwari and Prasad \cite{5} contributed semi-open sets and semi-continuity to bi-topological spaces. The notion of $\beta$-open sets contributed by Mashhour et. al. \cite{6} and Andrijevic \cite{1} define Semi pre-open sets. In this paper we discuss pre-continuity and semi pre-continuity in bi-topological spaces. LellisThivager et.al. \cite{4} introduces $g^{\ast}$-closed sets topological spaces and initiated the concepts of ultra space by using $(1,2)\alpha$-open sets in bi-topological spaces and proved that each $(1,2)\alpha$-open sets is (1, 2) semi-open and (1, 2) pre-open but the converse of each is not true. R-Devi and S.Sampath Kumar and M. Caldas \cite{7} introduced and studied a class of sets and maps between bi-topological spaces Called supra $\alpha-$open sets and supra $\alpha$-continuous maps respectively.

Keywords: Bi-topological Spaces, Continuous Functions, Contra Continuous Functions.

Parvinder Singh
P.G. Department of Mathematics, S.G.G.S. Khalsa College, Mahilpur (Hoshiarpur), India.
E-mail: parvinder070@gmail.com

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 Article Type : Research Article Title : The New Proof of Euler's Inequality Using Spieker Center Country : India Authors : Dasari Naga Vijay Krishna

Abstract: If R is the Circumradius and r is the Inradius of a non-degenerate triangle then due to EULER we have an inequality referred as \lq\lq Euler's Inequality\rq\rq which states that $R\geq 2r$, and the equality holds when the triangle is Equilateral. In this article let us prove this famous inequality using the idea of \lq Spieker Center \rq.

Keywords: Euler's Inequality, Circumcenter, Incenter, Circumradius, Inradius, Cleaver, Spieker Center, Medial Triangle, Stewart's Theorem.

Dasari Naga Vijay Krishna
Department of Mathematics, Keshava Reddy Educational Instutions, Machiliaptnam, Kurnool, India.
E-mail: vijay9290009015@gmail.com

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 Article Type : Research Article Title : The Global Maximal Domination Number of a Graph Country : India Authors : V.R.Kulli

Abstract: A maximal dominating set D of a graph G is a global maximal dominating set if D is also a maximal dominating set of $\bar{G}$. The global maximal domination number $\gamma_{gm}(G)$ of G is the minimum cardinality of a global maximal dominating set. In this paper, bounds for $\gamma_{gm}(G)$ and exact values of $\gamma_{gm}(G)$ for some standard graphs are obtained. We characterize maximal dominating sets of G which are global maximal dominating sets. Also Nordhaus-Gaddum type results are obtained.

Keywords: Dominating set, maximal dominating set, global maximal dominating set, global maximal domination number.

V.R.Kulli
Department of Mathematics, Gulbarga University, Gulbarga, India.
E-mail: vrkulli@gmail.com

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