Mathematica Moravica, Vol. 29, No. 1 (2025)

M.P. Chaudhary , Salem Guiben , Kamel Mazhouda
Some identities associated with theta functions and tenth order mock theta functions
Mathematica Moravica, Vol. 29, No. 1 (2025), 1–14.
doi: https://doi.org/10.5937/MatMor2501001C
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Abstract. The main objective of this paper is to present some identities associated with theta functions and tenth order mock theta functions. Several closely-related identities such as (for example) $q$-product identities and Jacobi's triple-product identity are also considered.
Keywords. Theta functions, mock theta functions, $q$-product identities.

Shishir Jain , Yogita Sharma , Shobha Jain , Sumit Sharma
Fixed point results of $\alpha$-$\beta_Y$-$F$-Geraghty type contractive mapping on modular $b$-metric spaces
Mathematica Moravica, Vol. 29, No. 1 (2025), 15–29.
doi: https://doi.org/10.5937/MatMor2501015J
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Abstract. In this paper, we generalize $\alpha$-$\beta_Y$-$F$-Geraghty type contraction in modular $b$-metric spaces and prove some fixed point results that are justified by suitable examples. The obtained results improve and extend some well known fixed point results in the literature.
Keywords. Fixed point, Geraghty type contraction, $F$-contraction, modular $b$-metric space.

Gamaliel Morales
On the Lichtenberg hybrid quaternions
Mathematica Moravica, Vol. 29, No. 1 (2025), 31–41.
doi: https://doi.org/10.5937/MatMor2501031M
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Abstract. In this study, we define Lichtenberg hybrid quaternions. We give the Binet's formula, the generating functions, exponential generating functions and sum formulas of these quaternions. We find some relations between Jacobsthal hybrid quaternions, Mersenne hybrid quaternions and Lichtenberg hybrid quaternions. Also, Vajda's identity and, as consequences, Catalan's identity, d'Ocagne's identity and Cassini's identity are presented.
Keywords. Hybrid numbers, Jacobsthal numbers, Jacobsthal quaternions, Lichtenberg numbers, Mersenne numbers, Mersenne quaternions.

John R. Graef , Abdelghani Ouahab
Existence and uniqueness of solutions to a coupled system of implicit fractional differential equations
Mathematica Moravica, Vol. 29, No. 1 (2025), 43–69.
doi: https://doi.org/10.5937/MatMor2501043G
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Abstract. Using Perov's fixed point theorem, the authors establish the existence and uniqueness of solutions to the coupled system of implicit fractional differential equations \[ \begin{cases} {}^{c}\!D^{\alpha}x(t) = f_{1}(t,x(t),y(t), {}^{c}\!D^{\alpha}x(t)), & t\in J, \\ {}^{c}\!D^{\beta}y(t) = f_{2}(t,x(t),y(t), {}^{c}D\!^{\beta}y(t)), & t\in J, \\ x(0) = L_1[x], \quad x'(0) = L_2[x], \\ y(0) = L_3[y], \quad y'(0) = L_4[y], \end{cases} \] where $\alpha,\beta \in [1,2)$, $J=[0,1]$, ${}^{c}\!D^{\alpha}$ and ${}^{c}\!D^{\beta}$ are Caputo fractional derivatives, $f_{i}: [0,1]\times\mathbb{R}^3 \to \mathbb{R}$ are continuous functions for $i=1, 2$, and the functionals $L_{j}$, $ j=1,2,3,4$, are Stieltjes integrals. A second existence result is obtained by using a vector version of a fixed point theorem for a sum of two operators due to Krasnosel'skii. There is also a study of the structure of the set of solutions to the problem. Examples illustrate the results.
Keywords. Fractional differential equation, implicit differential equation, nonlocal conditions, Perov's fixed point theorem.

Deepjyoti Borgohain
Fractional differentiation composition operators from $S_p$ spaces to $H_q$ spaces
Mathematica Moravica, Vol. 29, No. 1 (2025), 71–82.
doi: https://doi.org/10.5937/MatMor2501071B
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Abstract. Let $S_p$ be the space of functions analytic on the unit disk and whose derivatives belong to the Hardy space. In this article, we investigate the boundedness and compactness of the fractional differentiation composition operators from $S_p$ spaces into Hardy spaces. Furthermore, we derive a sufficient condition for the boundedness of the fractional differentiation composition operators on $S_p$ spaces. These results extends some well-known results in literature.
Keywords. Gaussian hypergeometric function, boundedness, Hardy spaces.

Houari Bouzid , Abdelkader Benali , Abdelkrim Salim
On pantograph multi-point boundary value problem with Caputo $q$-fractional derivative
Mathematica Moravica, Vol. 29, No. 1 (2025), 83–98.
doi: https://doi.org/10.5937/MatMor2501083B
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Abstract. The purpose of this research is to investigate the existence and uniqueness of several solutions to the multi-point boundary value problem of nonlinear fractional differential equations involving two fractional derivatives. We demonstrate the existence of solutions by applying a number of fixed point theorems, including Banach's fixed point theorem, nonlinear alternative of Leray-Schauder type, and Leray-Schauder degree. Finally, two examples are presented to demonstrate our results.
Keywords. Fractional differential equations (FDE), $q$-Riemann-Liouville integral, Fixed point theorem, Existence, Leray-Schauder alternative.

Neha Pauriyal , Mahesh C. Joshi
$K$-$d$-frames and their duals
Mathematica Moravica, Vol. 29, No. 1 (2025), 99–111.
doi: https://doi.org/10.5937/MatMor2501099P
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Abstract. In this paper, we define a linear bounded operator for double sequences and give a new generalization of frame called $K$-$d$-frame. We establish that $K$-$d$-frame is square summable in norm for finite dimensional separable Hilbert spaces and prove some results on properties of frame operators and $K$-$d$-duals.
Keywords. Frame, $K$-frame, $K$-$d$-frame, $K$-$d$-frame operator, $d$-Bessel sequence.