# Mathematica Moravica, Vol. 22, No. 1 (2018)

Strong convergence theorems for two finite families of generalized asymptotically quasi-nonexpansive mappings with applications
Mathematica Moravica, Vol. 22, No. 1 (2018), 1–14.

Abstract. In this paper, an implicit iteration process has been proposed for two finite families of generalized asymptotically quasi-nonexpansive mappings and establish some strong convergence theorems in the framework of convex metric spaces. Also, some applications of our result has been given. Our results extend and generalize several results from the current existing literature.
Keywords. Generalized asymptotically quasi-nonexpansive mapping, implicit iteration process, common fixed point, convex metric space, strong convergence.

Best proximity points of $\alpha-$$\beta-$$\psi-$proximal contractive mappings in partially ordered complete metric spaces
Mathematica Moravica, Vol. 22, No. 1 (2018), 15–29.

Abstract. In this paper, we define $\alpha-$$\beta-$$\psi-$proximal contractive mappings in partially ordered metric spaces and prove the existence of best proximity points of these maps in partially ordered complete metric spaces. These results extend/generalize the results of Asgari and Badehian, J. Nonl. Sci. and Appl., 2015. We provide illustrative examples in support of our theorems.
Keywords. Best proximity point, proximal contractive maps, admissible map, partially ordered complete metric space.

Well-posedness and asymptotic stability for the Lamé system with internal distributed delay
Mathematica Moravica, Vol. 22, No. 1 (2018), 31–41.

Abstract. In this work, we consider the Lamé system in 3-dimension bounded domain with distributed delay term. We prove, under some appropriate assumptions, that this system is well-posed and stable. Furthermore, the asymptotic stability is given by using an appropriate Lyapunov functional.
Keywords. Lamé system, delay terms, Lyapunov functions, decay rates.

Generalized $C^{\psi}_{\beta}-$ rational contraction and fixed point theorem with application to second order differential equation
Mathematica Moravica, Vol. 22, No. 1 (2018), 43–54.

Abstract. In this article, generalized $C^{\psi}_{\beta}$- rational contraction is defined and the existence and uniqueness of fixed points for self map in partially ordered metric spaces are discussed. As an application, we apply our result to find existence and uniqueness of solutions of second order differential equations with boundary conditions.
Keywords. Fixed point, $C^{\psi}_{\beta}-$ rational contraction, partially ordered metric spaces, differential equations.

Relation between $b$-metric and fuzzy metric spaces
Mathematica Moravica, Vol. 22, No. 1 (2018), 55–63.

Abstract. In this work we have considered several common fixed point results in $b$-metric spaces for weak compatible mappings. By applications of these results we establish some fixed point theorems in $b$-fuzzy metric spaces.
Keywords. Fuzzy contractive mapping, complete fuzzy metric space, common fixed point theorem, weakly compatible maps.

Hermite-Hadamard type inequalities for $\left(m,M\right)$-$\Psi$-convex functions when $\Psi=-\ln$
Mathematica Moravica, Vol. 22, No. 1 (2018), 65–79.

Abstract. In this paper we establish some Hermite-Hadamard type inequalities for $\left( m,M\right)$-$\Psi$-convex functions when $\Psi =-\ln .$ Applications for power functions and weighted arithmetic mean and geometric mean are also provided.
Keywords. Convex functions, special convexity, weighted arithmetic and geometric means, logarithmic function.

Fixed point theorems of generalised $S-\beta-\psi$ contractive type mappings
Mathematica Moravica, Vol. 22, No. 1 (2018), 81–92.

Abstract. In this paper, we introduce the concept of generalised $S-\beta-\psi$ contractive type mappings. For these mappings we prove some fixed point theorems in the setting of $S$-metric space.
Keywords. Generalised $S-\beta-\psi$ contractive mappings, $S$-metric space, fixed point, $\alpha$-admissible, $\beta$-admissible.

An analytical approach for systems of fractional differential equations by means of the innovative homotopy perturbation method
Mathematica Moravica, Vol. 22, No. 1 (2018), 93–105.

$(f,g)$-derivation of ordered $\Gamma$-semirings
Abstract. In this paper, we introduce the concept of $(f,g)$-derivation, which is a generalization of $f-$ derivation and derivation of ordered $\Gamma$-semiring and study some properties of $(f,g)-$derivation of ordered $\Gamma$-semirings. We prove that, if $d$ is a $(f,g)$-derivation of an ordered integral $\Gamma$-semiring $M$ then $\ker d$ is a $m-k-$ideal of $M$ and we characterize $m-k-$ideal using $(f,g)$-derivation of ordered $\Gamma$-semiring $M.$
Keywords. Ordered $\Gamma$-semiring, derivation, $\Gamma$-semiring, integral ordered $\Gamma$-semiring, $(f,g)$-derivation.