# Mathematica Moravica, Vol. 24, No. 2 (2020)

On coupled systems of fractional impulsive differential equations by using a new Caputo-Fabrizio fractional derivative
Mathematica Moravica, Vol. 24, No. 2 (2020), 1–19.

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Abstract. In this paper, we investigate the existence and uniqueness of solutions for coupled system of Caputo-Fabrizio fractional impulsive differential equations using the fixed point approach in generalized metric spaces. The compactness of solution sets of the system is also investigated. An example is provided to illustrate the developed theory.
Keywords. Fractional impulsive differential equations, Caputo-Fabrizio Fractional Derivative, Fixed point principle, Generalized metric space.

The influence of $\theta$-function to the class of MWP operators
Mathematica Moravica, Vol. 24, No. 2 (2020), 21–31.

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Abstract. In this work, taking into account the $\theta$-function, we present a general class of multivalued weakly Picard operators on complete metric space. We also provide an example showing that it includes some earlier classes as properly.
Keywords. Weakly Picard operator, multivalued mappings, $\theta$-contraction, fixed point.

Some fuzzy common fixed point theorems using common limit in the range property with an application
Mathematica Moravica, Vol. 24, No. 2 (2020), 33–49.

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Abstract. In the present paper, we prove some common fixed point theorems for mappings satisfying common limit in the range property in $M$-fuzzy metric space. Further, we prove fixed point theorem for $\phi$-contractive conditions in aforesaid spaces with the illustration of an example. As an application of our result, we study the existence and uniqueness of the solution of integral equation (Volterra integral equations of the second kind) with instances.
Keywords. $M$-fuzzy metric spaces, Common limit in the range property ((CLR) property), Weakly compatible mappings.

Initial Coefficient estimates for a classes of $m$-fold symmetric bi-univalent functions involving Mittag-Leffler function
Mathematica Moravica, Vol. 24, No. 2 (2020), 51–61.

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Abstract. The main object of the present paper is to use Mittag-Leffler function to introduce and study two new classes $\mathcal{R}_{\Sigma_{m}}(\gamma,\lambda,\eta,\delta,\tau;\alpha)$ and $\mathcal{R}_{\Sigma_{m}}^{*}(\gamma,\lambda,\eta,\delta,\tau;\beta)$ of $\Sigma_{m}$ consisting of analytic and $m$-fold symmetric bi-univalent functions defined in the open unit disk $U$. Also, we determine the estimates on the initial coefficients $\left| a_{m+1}\right|$ and $\left| a_{2m+1}\right|$ for functions in each of these new classes. Furthermore, we indicate certain special cases for our results.
Keywords. Analytic function, $m$-fold symmetric bi-univalent function, Coefficient bounds, Mittag-Leffler function.

Common fixed points under strict conditions
Mathematica Moravica, Vol. 24, No. 2 (2020), 63–70.

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Abstract. In this contribution, three new concepts called reciprocally continuous, strictly subweakly compatible and strictly subreciprocally continuous single and multivalued mappings are given for obtention some common fixed point theorems in a metric space. Our results improve and complement the results of Aliouche and Popa, Azam and Beg, Deshpande and Pathak, Kaneko and Sessa, Popa and others.
Keywords. Strictly subweakly compatible mappings, reciprocally continuous mappings, strictly subreciprocally continuous mappings.

Coefficient bounds for certain subclasses of bi-prestarlike functions associatedwith the Chebyshev polynomials
Mathematica Moravica, Vol. 24, No. 2 (2020), 71–82.

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Abstract. In this paper, we introduce and investigate a new subclass of bi-prestarlike functions defined in the open unit disk, associated with Chebyshev Polynomials. Furthermore, we find estimates of first two coefficients of functions in these classes, making use of the Chebyshev polynomials. Also, we obtain the Fekete-Szegö inequalities for function in these classes. Several consequences of the results are also pointed out as corollaries.
Keywords. Analytic functions, bi-univalent functions, prestarlike functions, Chebyshev polynomials, coefficient estimates, Fekete-Szegö inequalities.

New family of special numbers associated with finite operator
Mathematica Moravica, Vol. 24, No. 2 (2020), 83–98.

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Abstract. Using the notion of the generating function of a function, we define an operator with whom we manage to build a large family of numbers and polynomials. This technique permits to give the closed formulae and interesting combinatorial identities. Among others, these polynomials are a generalization of the Fubini numbers and polynomials.
Keywords. Generating function, operators, Fubini numbers and polynomials.

Some fixed point theorems for generalized $(\psi-\phi)$-weak contraction mappings in partial metric spaces
Mathematica Moravica, Vol. 24, No. 2 (2020), 99–115.

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Abstract. The aim of this paper is to introduce the concepts of generalized $(\psi-\phi)$-weak contraction mappings of type (A) and (B) and establish some fixed point theorems for said contraction mappings in complete partial metric spaces. Our results extend and generalize several results from the current existing literature.
Keywords. Fixed point, generalized $(\psi-\phi)$-weak contraction mapping, partial metric space.

New inequalities for $F$-convex functions pertaining generalized fractional integrals
Mathematica Moravica, Vol. 24, No. 2 (2020), 117–131.

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Abstract. In this paper, the authors, utilizing $F$-convex functions which are defined by B. Samet, establish some new Hermite-Hadamard type inequalities via generalized fractional integrals. Some special cases of our main results recaptured the well-known earlier works.
Keywords. Convex function, trigonometrically $\rho$-convex functions.

On relationships between q-products identities, $R_{\alpha}$, $R_{\beta}$ and $R_{m}$ functions related to Jacobi’s triple-product identity
Mathematica Moravica, Vol. 24, No. 2 (2020), 133–144.

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Abstract. The authors establish a set of two new relationships involving $q$-product identities, $R_{\alpha}, R_{\beta}$, and $R_{m}$ $(m = 1,2,3,\dots)$ functions; and answer a open question of Srivastava et al. [18]. The present work is motivated and based upon recent findings of Chaudhary et al. [8].
Keywords. Theta-function identities; $R_{\alpha}, R_{\beta}$, and $R_{m}$ functions; Jacobi's triple-product identity; $q$-Product identities; Rogers-Ramanujan continued fraction; Rogers-Ramanujan identities.