# Mathematica Moravica, Vol. 25, No. 1 (2021)

Existence, uniqueness, approximation of solutions and $\mathbb E_{\alpha}$–Ulam stability results for a class of nonlinear fractional differential equations involving $\psi$–Caputo derivative with initial conditions
Mathematica Moravica, Vol. 25, No. 1 (2021), 1–30.

Abstract. The main purpose of this paper is to study the existence, uniqueness, $\mathbb E_{\alpha}$–Ulam stability results, and other properties of solutions for certain classes of nonlinear fractional differential equations involving the $\psi$–Caputo derivative with initial conditions. Modern tools of functional analysis are applied to obtain the main results. More precisely using Weissinger's fixed point theorem and Schaefer's fixed point theorem the existence and uniqueness results of solutions are proven in the bounded domain. While the well known Banach fixed point theorem coupled with Bielecki type norm are used with the end goal to establish sufficient conditions for existence and uniqueness results on unbounded domains. Meanwhile, the monotone iterative technique combined with the method of upper and lower solutions is used to prove the existence and uniqueness of extremal solutions. Furthermore, by means of new generalizations of Gronwall's inequality, different kinds of $\mathbb E_{\alpha}$–Ulam stability of the proposed problem are studied. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility and correctness of the main results.
Keywords. $\psi$–Caputo fractional derivative, fixed point, existence, uniqueness, extremal solutions, $\mathbb E_{\alpha}$–Ulam stability, Bielecki norm, monotone iterative technique, upper and lower solutions.

Fixed point theorems of generalized multi-valued mappings in cone $b$-metric spaces
Mathematica Moravica, Vol. 25, No. 1 (2021), 31–45.

Abstract. The aim of this paper is to establish fixed points for multi-valued mappings, by adapting the ideas in [1] to the cone $b$-metric space setting.
Keywords. Cone $b$-metric space, Multi-valued mappings, Fixed point.

Fixed point theorems on a closed bal
Mathematica Moravica, Vol. 25, No. 1 (2021), 47–55.

Abstract. The aim of the paper is to obtain some fixed point theorems for extended $(\varphi, F)$-weak type contraction on a closed ball in metric spaces. Our results generalize some recently established results.
Keywords. $\alpha$-admissible, $\alpha$-$\eta$-continuous, $\alpha$-$\eta$-$GF$-contraction, $F$-contraction, fixed point.

A note on identities with derivations in prime rings
Mathematica Moravica, Vol. 25, No. 1 (2021), 57–62.

Abstract. The present paper's primary purpose is to study the connection between commutativity of rings and the behavior of its derivations in prime rings.
Keywords. Ideal, Martindale ring of quotients, prime ring, generalized polynomial identity (GPI).

Measures of noncompactness on $w$-distance spaces
Mathematica Moravica, Vol. 25, No. 1 (2021), 63–69.

Abstract. The aim of this paper is to provide a new framework for the study of measures of noncompactness in generalized metric spaces. Firstly, we introduce the notion of $w$-measure of noncompactness on metric spaces with a $w$-distance and extend the diameter and Kuratowski functionals to this setting. At the end we give a characterization of metric completeness via our main results, providing a new answer to the open question mentioned by Arandjelović in his PhD thesis [2].
Keywords. Measures of noncompactness, metric spaces, $w$-distance.

Coefficient estimates for families of bi-univalent functions defined by Ruscheweyh derivative operator
Mathematica Moravica, Vol. 25, No. 1 (2021), 71–80.

Abstract. The main purpose of this manuscript is to find upper bounds for the second and third Taylor-Maclaurin coefficients for two families of holomorphic and bi-univalent functions associated with Ruscheweyh derivative operator. Further, we point out certain special cases for our results.
Keywords. Holomorphic function, Bi-Univalent function, Coefficient estimates, Ruscheweyh derivative operator.

Qualitative study of a third order rational system of difference equations
Mathematica Moravica, Vol. 25, No. 1 (2021), 81–97.

Abstract. This paper is concerned with the dynamics of positive solutions for a system of rational difference equations of the following form $$u_{n+1}=\frac{\alpha u_{n-1}^{2}}{\beta +\gamma v_{n-2}},\text{ }v_{n+1}=% \frac{\alpha _{1}v_{n-1}^{2}}{\beta _{1}+\gamma _{1}u_{n-2}},\quad n=0,1,\dots,$$ where the parameters $\alpha ,\beta ,\gamma ,\alpha _{1},\beta _{1},\gamma_{1}$ and the initial values $u_{-i},v_{-i}\in (0,\infty )$, $i=0,1,2$. Moreover, the rate of convergence of a solution that converges to the zero equilibrium of the system is discussed. Finally, some numerical examples are given to demonstrate the effectiveness of the results obtained.
Keywords. System of difference equations, equilibrium, positive solutions, invariant subsets.

Some best proximity point results for multivalued mappings on partial metric spaces
Mathematica Moravica, Vol. 25, No. 1 (2021), 99–111.

Abstract. In this paper, we introduce two new concepts of Feng-Liu type multivalued contraction mapping and cyclic Feng-Liu type multivalued contraction mapping. Then, we obtain some new best proximity point results for such mappings on partial metric spaces by considering Feng-Liu's technique. Finally, we provide examples to show the effectiveness of our results.
Keywords. Best proximity point, multivalued mappings, partial metric space.

Fixed point for $F_{\perp}$-weak contraction
Mathematica Moravica, Vol. 25, No. 1 (2021), 113–122.

Abstract. In this paper, we establish some fixed point results for $F_{\perp}$-weak contraction in orthogonal metric space and we give an application for the solution of second order differential equation.
Keywords. Orthogonal metric space, $F_{\perp}$-contraction, $F_{\perp}$-weak contraction.