# Mathematica Moravica, Vol. 26, No. 1 (2022)

Generalized orthopair fuzzy matrices based on Hamacher operations
Mathematica Moravica, Vol. 26, No. 1 (2022), 1–26.

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Abstract. The objective of this paper is to apply the concept of intuitionistic fuzzy matrices and pythagorean fuzzy matrices to q-rung orthopair fuzzy matrices. In this paper, we introduce the Hamacher operations of q-rung orthopair fuzzy matrices and prove some desirable properties of these operations, such as commutativity, idempotency and monotonicity. Further, we prove De Morgan's laws over complement for these operations. Then we constructed the scalar multiplication $({n._{h}}A)$ and exponentiation $(A^{\wedge_{h}n})$ operations of q-rung orthopair fuzzy matrices and investigate their algebraic properties. Finally, we prove some properties of necessity and possibility operators of q-rung orthopair fuzzy matrices.
Keywords. Hamacher sum, Hamacher product, Scalar multiplication, Exponentiation, Necessity and possibility.

Certain integral representations involving hypergeometric functions in two variables
Mathematica Moravica, Vol. 26, No. 1 (2022), 27–36.

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Abstract. Various integral representations of hypergeometric functions have been introduced and investigated due to their important applications in divers fields. In this article, we define some new Euler-type integral representations for the Horn's functions of two variables $G_{1},G_{2},G_{3}$ and $H_{1}$.
Keywords. Beta function, Horn double functions, Appell functions, Eulerian integrals.

A generalization of the array type polynomials
Mathematica Moravica, Vol. 26, No. 1 (2022), 37–46.

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Abstract. We introduce a generalization of the array type polynomials by using two specific generating functions and investigate some of its basic properties in the sequel. A recurrence relation and two summation formulas involving these polynomials are also given.
Keywords. Appell polynomials, array polynomials, Stirling numbers of the second kind, Apostol--Bernoulli polynomials, generating function, recurrence relation.

Oscillation of even order nonlinear dynamic equations on time-scales
Mathematica Moravica, Vol. 26, No. 1 (2022), 47–55.

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Abstract. In this paper, the authors discuss the oscillatory behavior of solutions to a class of even order nonlinear dynamic equations on time scales. The results are established by a comparison with n-th order delay dynamic inequalities or first-order delay dynamic equations whose oscillatory characters are known. Several corollaries are obtained for special cases.
Keywords. Oscillation, super-linear neutral term, dynamic equations.

Fixed point theorems for cyclic contractions in $S$-metric spaces involving $C$-class function
Mathematica Moravica, Vol. 26, No. 1 (2022), 57–76.

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Abstract. In this paper, we study the class of cyclic contractions in the setting of $S$-metric spaces involving $C$-class function and establish some fixed point theorems in the setting of complete $S$-metric spaces. We support our results with some examples. Our results extend and generalize several results from the existing literature (see, e.g., [3, 8, 9, 14, 15, 20] and many others) to the case of more general ambient space and contraction condition.
Keywords. Fixed point, cyclic contraction, $S$-metric space, $C$-class function.

The complex-type Padovan$-p$ sequences
Mathematica Moravica, Vol. 26, No. 1 (2022), 77–88.

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Abstract. In this paper, we define the complex-type Padovan-$p$ sequence and then give the relationships between the Padovan-$p$ numbers and the complex-type Padovan-$p$ numbers. Also, we provide a new Binet formula and a new combinatorial representation of the complex-type Padovan-$p$ numbers by the aid of the $n$th power of the generating matrix of the complex-type Padovan-$% p$ sequence. In addition, we derive various properties of the complex-type Padovan-$p$ numbers such as the permanental, determinantal and exponential representations and the finite sums by matrix methods.
Keywords. The complex-type Padovan-$p$ sequence, matrix, representation.

Global and local existence of solution for fractional heat equation in $\mathbb{R}^N$ by Balakrishnan definition
Mathematica Moravica, Vol. 26, No. 1 (2022), 89–101.

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Abstract. Our aim here is to collect and to compare two definitions of the fractional powers of non-negative operators that can be found in the literature; we will present the proof of an equivalence and compare properties of that notions in different approaches. Then we will apply next this equivalence in the study of global and local existence of solution for the semilinear Cauchy problem in $\mathbb{R}^N$ with fractional Laplacian $\left\{\begin{array}{c} u_t = -(-\Delta)^\alpha u + f(x,u), \\ u(0,x) = u_0(x), \quad x \in \mathbb{R}^N. \end{array}\right.$
Keywords. Fractional powers of operator, Balakrishinan, global solvability, Heat Equation.

On a family of bi-univalent functions related to the Fibonacci numbers
Mathematica Moravica, Vol. 26, No. 1 (2022), 103–112.

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Abstract. In this study, we construct a new family of holomorphic biunivalent functions in the open unit disc by the help of q-analogue of Noor integral operator, principle of subordination and Fibonacci polynomials. Also we obtain coefficient bounds and Fekete Szegö inequality for functions belonging this family. We have illustrated relevant families and consequences.
Keywords. q-analogue of Noor integral operator, Fibonacci polynomials, Bi-univalent functions.

Fixed points for occasionally weakly biased mappings of type $(A)$
Mathematica Moravica, Vol. 26, No. 1 (2022), 113–122.

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Abstract. In this paper, in the first step, we will introduce the concept of occasionally weakly biased mappings of type $(A)$ which is a convenient generalization of the concept of weakly biased mappings of type $(A)$. In the second step, we will show that this new definition coincides with our concept of occasionally weakly biased mappings given in . In the third and last step we will give an example which verifies the validity of our result.
Keywords. Weakly $f$-biased (respectively $g$-biased) of type $(A)$ mappings, occasionally weakly $f$-biased (respectively $g$-biased) of type $(A)$ mappings, implicit relation, unique common fixed point theorems, metric space.

A geometric approach to the Proinov type contractions
Mathematica Moravica, Vol. 26, No. 1 (2022), 123–132.

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Abstract. In this paper, we prove some fixed-circle, fixed-disc and fixed-ellipse results on metric spaces. To do this, we define the notions of Proinov type $a_{0}$-contraction and generalized Proinov type $a_{0}$-contraction. Also, we give some illustrative examples to show the validity of our obtained results. Finally, we present a nice application to exponential linear unit activation functions.
Keywords. Fixed circle, fixed disc, fixed ellipse, Proinov type $a_{0}$-contraction, generalized Proinov type $a_{0}$-contraction.