# Mathematica Moravica, Vol. 21, No. 1 (2017)

Čebyšev's type inequalities for positive linear maps of selfadjoint operators in Hilbert spaces
Mathematica Moravica, Vol. 21, No. 1 (2017), 1–15.

Abstract. Some inequalities for positive linear maps of continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given. Applications for power function and logarithm are provided as well.
Keywords. Positive linear maps, selfadjoint operators, synchronous(asynchronous) functions, monotonic functions, Čebyšev inequality, functions of selfadjoint operators.

Some topological properties of the spaces $expX$, $\lambda X$ and $NX$
Mathematica Moravica, Vol. 21, No. 1 (2017), 17–25.

Abstract. In this paper we prove that the exponential functor $exp$ and the functor of superextension $\lambda$ preserve some topological properties with respect to the topology of any $T_{1}$-space, and the functor of complete linked systems $N$ preserves some topological properties with respect to the topology of any compact space.
Keywords. $\pi$-base, exponential functor $exp$, the functor of superextension $\lambda$, the functor of complete linked systems $N$.

Q-Convergence in graded ditopological texture spaces
Mathematica Moravica, Vol. 21, No. 1 (2017), 27–36.

Abstract. Convergence of graded difilters have been presented and investigated by the authors in [9]. In this paper, using graded Q-dinhd systems defined in [8] the authors define a different convergence type of graded difilters called Q-convergence which has some advantages and some disadvantages in comparison with the convergence defined in [9].

A fixed point theorem for $(\mu,\psi)$-generalized $f$-weakly contractive mappings in partially ordered 2-metric spaces
Mathematica Moravica, Vol. 21, No. 1 (2017), 37–50.

Abstract. The purpose of this paper is to introduce the notion of a $(\mu,\psi)$-generalized $f$-weakly contractive mapping in partially ordered 2-metric spaces and state a fixed point theorem for this mapping in complete, partially ordered 2-metric spaces. The main results of this paper are generalizations of the main results of [4, 10]. Also, some examples are given to illustrate the obtained results.
Keywords. Fixed point, 2-metric space, $(\mu,\psi)$-generalized $f$-weakly contractive mapping.

Harmonic starlikeness and convexity of integral operators generated by Poisson distribution series
Mathematica Moravica, Vol. 21, No. 1 (2017), 51–60.

Abstract. The purpose of the present paper is to establish connections between various subclasses of harmonic univalent functions by applying certain integral operator involving the Poisson distribution series. To be more precise, we investigate such connections with harmonic starlike and harmonic convex mappings in the plane.
Keywords. Harmonic, univalent functions, Poisson distribution series.

On $(\sigma,\delta) – (S,1)$ rings and their extensions
Mathematica Moravica, Vol. 21, No. 1 (2017), 61–68.

Abstract. Let $R$ be a ring, $\sigma$ an endomorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$. We recall that $R$ is called an $(S,1)$-ring if for $a,b\in R$, $ab = 0$ implies $aRb = 0$. We involve $\sigma$ and $\delta$ to generalize this notion and say that $R$ is a $(\sigma,\delta) - (S,1)$ ring if for $a,b\in R$, $ab = 0$ implies $aRb = 0$, $\sigma(a)Rb = 0$, $aR\sigma(b) = 0$ and $\delta(a)Rb = 0$. In case $\sigma$ is identity, $R$ is called a $δ - (S,1)$ ring.
In this paper we study the associated prime ideals of Ore extension $R[x;\sigma,\delta]$ and we prove the following in this direction:
Let $R$ be a semiprime right Noetherian ring, which is also an algebra over $\mathbb{Q}$ ($\mathbb{Q}$ is the field of rational numbers), $\sigma$ an automorphism of $R$ and $\delta$ a $\sigma$-derivation of $R$ such that $R$ is a $(\sigma,\delta) - (S,1)$ ring. Then $P$ is an associated prime ideal of $R[x;\sigma,\delta]$ (viewed as a right module over itself) if and only if there exists an associated prime ideal $U$ of $R$ (viewed as a right module over itself) such that $(P \cap R)[x;\sigma,\delta] = P$ and $P \cap R) = U$.
Keywords. Endomorphism, derivation, $(\sigma,\delta) - (S,1)$ ring, Ore extension, prime ideal.

Reduced and irreducible simple algebraic extensions of commutative rings
Mathematica Moravica, Vol. 21, No. 1 (2017), 69–86.

Abstract. Let $A$ be a commutative ring with identity and $\alpha$ be an algebraic element over $A$. We give necessary and sufficient conditions under which the simple algebraic extension $A[\alpha]$ is without nilpotent or without idempotent elements.
Keywords. Commutative rings, polynomials, discriminants, resultants, simple algebraic extensions, reduced rings, irreducible rings.

A generalization of modules with the property $(P^{\ast})$
Mathematica Moravica, Vol. 21, No. 1 (2017), 87–94.

Abstract. I.A- Khazzi and P.F. Smith called a module $M$ have the property $(P^{\ast})$ if every submodule $N$ of $M$ there exists a direct summand $K$ of $M$ such that $K\leq N$ and $\frac{N}{K}\subseteq Rad(\frac{M}{K})$. Motivated by this, it is natural to introduce another notion that we called modules that have the properties $(GP^{\ast})$ and $(N - GP^{\ast})$ as proper generalizations of modules that have the property $(P^{\ast})$. In this paper we obtain various properties of modules that have properties $(GP^{\ast})$ and $(N - GP^{\ast})$. We show that the class of modules for which every direct summand is a fully invariant submodule that have the property $(GP^{\ast})$ is closed under finite direct sums. We completely determine the structure of these modules over generalized $f$-semiperfect rings.
Keywords. Generalized $f$-semiperfect ring, the properties $(P^{\ast})$, $(GP^{\ast})$ and $(N-GP^{\ast})$.

Solving Benjamin-Bona-Mahony equation by using the sn-ns method and the tanh-coth method
Mathematica Moravica, Vol. 21, No. 1 (2017), 95–103.

Abstract. In this study, we consider the Benjamin Bona Mahony equation which is in the form of $u_{t} + u_{x} + uu_{x} - u_{xxt} = 0$. The sn-ns method and the tanh-coth method have been applied to this equation. And then, exact solutions have been obtained.
Keywords. Benjamin Bona Mahony equation (BBM), the sn-ns method, tanh-coth method, elliptic function solution, trigonometric solution.

On Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals
Mathematica Moravica, Vol. 21, No. 1 (2017), 105–123.

Abstract. In this paper, we have established some generalized integral inequalities of Hermite-Hadamard-Fejér type for generalized fractional integrals. The results presented here would provide generalizations of those given in earlier works.
Keywords. Integral inequalities, fractional integrals, Hermite-Hadamard-Fejér inequality.

Approximate fixed point theorems of cyclical contraction mapping
Mathematica Moravica, Vol. 21, No. 1 (2017), 125–137.

Abstract. Let $X_{i}$, $i=1,\dots,m$, are subsets of a metric space $X$ and also $T: \cup_{i=1}^{m}X \to \cup_{i=1}^{m}X_{i}$, and $T(X_{1})\subseteq X_{2},\ldots,T(X_{m-1})\subseteq X_{m},T(X_{m})\subseteq X_{1}$. We are going to consider element $X\in \cup_{i=1}^{m}X_{i}$ such that $d(x,Tx)\leq\epsilon$ for some $\epsilon>0$. The existence results regarding approximate fixed points proved for the several operators such as Chatterjea and Zamfirescu on metric space (not necessarily complete). These results can be exploited to establish new approximate fixed point theorems for cyclical contraction maps. In addition, there is a new class of cyclical operators and contraction mapping on metric space (not necessarily complete) which do not need to be continuous. Finally, some examples are presented to illustrate our results for new approximate fixed point theorems on cyclical contraction maps.