• Volume 125, Issue 3

August 2015,   pages  271-447

• A note on neighborhood total domination in graphs

Let 𝐺 = (𝑉 ,𝐸) be a graph without isolated vertices. A dominating set 𝑆 of 𝐺 is called a neighborhood total dominating set (or just NTDS) if the induced subgraph 𝐺[𝑁(𝑆)] has no isolated vertex. The minimum cardinality of a NTDS of 𝐺 is called the neighborhood total domination number of 𝐺 and is denoted by 𝛾nt(𝐺). In this paper, we obtain sharp bounds for the neighborhood total domination number of a tree. We also prove that the neighborhood total domination number is equal to the domination number in several classes of graphs including grid graphs.

• On the coefficient conjecture of Clunie and Sheil-Small on univalent harmonic mappings

In this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for a class of univalent harmonic functions which includes functions convex in some direction. Next, we prove growth and covering theorems and some related results. Finally, we propose two conjectures, an affirmative answer to one of which would then imply, for example, a solution to the conjecture of Clunie and Sheil-Small.

• Certain fractional integral operators and the generalized multi-index Mittag-Leffler functions

In this paper, we obtain formulas of fractional integration (of Marichev–Saigo–Maeda type) of the generalized multi-index Mittag-Leffler functions 𝐸𝛾,𝜅[(𝛼𝑗, 𝛽𝑗)𝑚; 𝑧] generalizing 2𝑚-parametric Mittag-Leffler functions studied by Saxena and Nishimoto (J. Fract. Calc. 37 (2010) 43–52). Some interesting special cases of our main results are considered too.

• Soliton solutions for a quasilinear Schrödinger equation via Morse theory

In this paper, Morse theory is used to show the existence of nontrivial weak solutions to a class of quasilinear Schrödinger equation of the form

$$-𝛥 u - \frac{p}{2^{p-1}} u 𝛥_p (u^2) = f(x, u)$$

in a bounded smooth domain $𝛺 \subset \mathbb{R}^N$ with Dirichlet boundary condition.

• Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations

In this article, we study the concept of Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations. We establish the results with Lipschitz condition and without Lipschitz condition on the forcing term. An interesting example is presented to illustrate the main findings. The results proven are new and complement the existing ones.

• Existence of positive solutions for systems of second order multi-point boundary value problems on time scales

In this paper, we establish the existence of positive solutions for systems of second order multi-point boundary value problems on time scales by applying Guo– Krasnosel’skii fixed point theorem.

• Some spherical analysis related to the pairs ($U(n),H_n$) and ($U(p, q),H_n$), 𝑝 + 𝑞 = 𝑛

In this paper, we define the normalized spherical transform associated with the generalized Gelfand pair $(U(p, q),H_n)$, where $H_n$ is the Heisenberg group 2𝑛 + 1-dimensional and 𝑝 + 𝑞 = 𝑛. We show that the normalized spherical transform $\mathfrak{F}(f)$ of a Schwartz function 𝑓 on $H_n$ restricted to the spectrum of the Gelfand pair ($U(n),H_n$) is the Gelfand transform $\hat{g}$ of a radial Schwartz function 𝑔 on $H_n$. Moreover, by the Godement–Plancherel inversion formula the function 𝑔 can be recovered explicitly from $\mathfrak{F}(f)$.

• Functional equations in matrix normed spaces

Using the fixed point method, we prove the Hyers–Ulam stability of the Cauchy additive functional equation and the quadratic functional equation in matrix normed spaces.

• ℎ-𝑝 Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-II: Proof of stability theorem

This is the second of a series of papers devoted to the study of ℎ-𝑝 spectral element methods for three dimensional elliptic problems on non-smooth domains. The present paper addresses the proof of the main stability theorem.We assume that the differential operator is a strongly elliptic operator which satisfies Lax–Milgram conditions. The spectral element functions are non-conforming. The stability estimate theorem of this paper will be used to design a numerical scheme which give exponentially accurate solutions to three dimensional elliptic problems on non-smooth domains and can be easily implemented on parallel computers.

• # Proceedings – Mathematical Sciences

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