A list of articles of the latest volume. The journal Opuscula Mathematica publishes original research articles that are of significant importance in all areas of Discrete Mathematics, Functional Analysis, Differential Equations, Mathematical Physics, Nonlinear Analysis, Numerical Analysis, Probability Theory and Statistics, Theory of Optimal Control and Optimization, Financial Mathematics and Mathematical Economic Theory, Operations Research, and other areas of Applied Mathematics. https://www.opuscula.agh.edu.pl AGH University of Science and Technology Press en Opuscula Mathematica 1232-9274 2300-6919 https://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg https://www.opuscula.agh.edu.pl This study focuses on a sequence of approximate minimizers for the functional \[J(u)=\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{p_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{p_{i}(x)}dx-\mu\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{q_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{q_{i}(x)}dx-\int\limits_{\Omega} F(u(x))dx,\] where \(\Omega\subset\mathbb{R}^N\) (\(N\geq 3\)) is a bounded domain, and \(p_i,q_i\in C(\overline{\Omega})\) with \(1\lt p_i,q_i\lt +\infty\) for all \(i \in \{1,\ldots,N\}\). We establish the convergence result to the infimum of \(J(u)\) when \(F:\mathbb{R}\to\mathbb{R}\) is a locally Lipschitz function of controlled growth, following the Galerkin method. As an application, we establish the existence of solutions to a class of Dirichlet inclusions associated to the functional. https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4605.pdf Galerkin-type minimizers to a competing problem for (\vec{p},\vec{q})-Laplacian with variable exponents Zhenfeng Zhang Mina Ghasemi Calogero Vetro anisotropic Sobolev space, Clarke generalized gradient, Dirichlet problem, Galerkin basis, \((\vec{p},\vec{q})\)-Laplacian with variable exponents doi:10.7494/OpMath.202511231 Opuscula Math. 46, no. 1 (2026), 101-119, https://doi.org/10.7494/OpMath.202511231 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202511231 https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4605.pdf 46 1 101 119
Title: Galerkin-type minimizers to a competing problem for (\vec{p},\vec{q})-Laplacian with variable exponents.

Author(s): Zhenfeng Zhang, Mina Ghasemi, Calogero Vetro.

Abstract: This study focuses on a sequence of approximate minimizers for the functional \[J(u)=\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{p_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{p_{i}(x)}dx-\mu\int\limits_{\Omega}\sum\limits_{i=1}^{N}\frac{1}{q_{i}(x)}\bigg|\frac{\partial u}{\partial x_{i}}\bigg|^{q_{i}(x)}dx-\int\limits_{\Omega} F(u(x))dx,\] where \(\Omega\subset\mathbb{R}^N\) (\(N\geq 3\)) is a bounded domain, and \(p_i,q_i\in C(\overline{\Omega})\) with \(1\lt p_i,q_i\lt +\infty\) for all \(i \in \{1,\ldots,N\}\). We establish the convergence result to the infimum of \(J(u)\) when \(F:\mathbb{R}\to\mathbb{R}\) is a locally Lipschitz function of controlled growth, following the Galerkin method. As an application, we establish the existence of solutions to a class of Dirichlet inclusions associated to the functional.
Keywords: anisotropic Sobolev space, Clarke generalized gradient, Dirichlet problem, Galerkin basis, \((\vec{p},\vec{q})\)-Laplacian with variable exponents.
Mathematics Subject Classification: 46E30, 47J22.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 101-119, https://doi.org/10.7494/OpMath.202511231.

 

]]> For \(0 \lt p \leq 1 \lt q \lt \infty\) and \(\gamma \gt 0\), we introduce the Calderón-Hardy spaces \(\mathcal{H}^{p}_{q,\gamma}(\mathbb{H}^{n})\) on the Heisenberg group \(\mathbb{H}^{n}\), and show for every \(f \in H^{p}(\mathbb{H}^{n})\) that the equation \[\mathcal{L}F=f\] has a unique solution \(F\) in \(\mathcal{H}^{p}_{q,2}(\mathbb{H}^{n})\), where \(\mathcal{L}\) is the sub-Laplacian on \(\mathbb{H}^{n}\), \[1 \lt q \lt \frac{n+1}{n} \quad \text{and} \quad (2n+2)\left(2+\frac{2n+2}{q}\right)^{-1} \lt p \leq 1.\] https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4604.pdf Calderón-Hardy type spaces and the Heisenberg sub-Laplacian Pablo Rocha Calderón-Hardy type spaces, Hardy type spaces, atomic decomposition, Heisenberg group, sub-Laplacian doi:10.7494/OpMath.202512221 Opuscula Math. 46, no. 1 (2026), 73-99, https://doi.org/10.7494/OpMath.202512221 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202512221 https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4604.pdf 46 1 73 99
Title: Calderón-Hardy type spaces and the Heisenberg sub-Laplacian.

Author(s): Pablo Rocha.

Abstract: For \(0 \lt p \leq 1 \lt q \lt \infty\) and \(\gamma \gt 0\), we introduce the Calderón-Hardy spaces \(\mathcal{H}^{p}_{q,\gamma}(\mathbb{H}^{n})\) on the Heisenberg group \(\mathbb{H}^{n}\), and show for every \(f \in H^{p}(\mathbb{H}^{n})\) that the equation \[\mathcal{L}F=f\] has a unique solution \(F\) in \(\mathcal{H}^{p}_{q,2}(\mathbb{H}^{n})\), where \(\mathcal{L}\) is the sub-Laplacian on \(\mathbb{H}^{n}\), \[1 \lt q \lt \frac{n+1}{n} \quad \text{and} \quad (2n+2)\left(2+\frac{2n+2}{q}\right)^{-1} \lt p \leq 1.\]
Keywords: Calderón-Hardy type spaces, Hardy type spaces, atomic decomposition, Heisenberg group, sub-Laplacian.
Mathematics Subject Classification: 42B25, 42B30, 42B35, 43A80.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 73-99, https://doi.org/10.7494/OpMath.202512221.

 

]]> In this short note we study a Harnack inequality for \(k\)-Hessian equations that involve nonlinear lower-order terms which depend on the solution and its gradient. https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4603.pdf A short note on Harnack inequality for k-Hessian equations with nonlinear gradient terms Ahmed Mohammed Giovanni Porru \(k\)-Hessian equation, \(k\)-convex solutions, Harnack inequality, Liouville property doi:10.7494/OpMath.202512261 Opuscula Math. 46, no. 1 (2026), 55-72, https://doi.org/10.7494/OpMath.202512261 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202512261 https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4603.pdf 46 1 55 72
Title: A short note on Harnack inequality for k-Hessian equations with nonlinear gradient terms.

Author(s): Ahmed Mohammed, Giovanni Porru.

Abstract: In this short note we study a Harnack inequality for \(k\)-Hessian equations that involve nonlinear lower-order terms which depend on the solution and its gradient.
Keywords: \(k\)-Hessian equation, \(k\)-convex solutions, Harnack inequality, Liouville property.
Mathematics Subject Classification: 35J60, 35J70, 35B45, 35B53.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 55-72, https://doi.org/10.7494/OpMath.202512261.

 

]]> This study investigated the asymptotic behavior of linear differential systems incorporating a proportional derivative-type (PD) differential operator. Building on the classical asymptotic convergence property of Wintner, a generalized Wintner-type asymptotic result was established for such systems. The proposed framework encompasses a wide class of time-varying coefficient matrices and extends classical asymptotic theory to equations governed by PD operators. An illustrative example is presented to demonstrate the applicability of the proposed theorem. https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4602.pdf Wintner-type asymptotic behavior of linear differential systems with a proportional derivative controller Kazuki Ishibashi Wintner-type, asymptotic behavior, linear differential systems, proportional-derivative controller doi:10.7494/OpMath.202512101 Opuscula Math. 46, no. 1 (2026), 41-54, https://doi.org/10.7494/OpMath.202512101 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202512101 https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4602.pdf 46 1 41 54
Title: Wintner-type asymptotic behavior of linear differential systems with a proportional derivative controller.

Author(s): Kazuki Ishibashi.

Abstract: This study investigated the asymptotic behavior of linear differential systems incorporating a proportional derivative-type (PD) differential operator. Building on the classical asymptotic convergence property of Wintner, a generalized Wintner-type asymptotic result was established for such systems. The proposed framework encompasses a wide class of time-varying coefficient matrices and extends classical asymptotic theory to equations governed by PD operators. An illustrative example is presented to demonstrate the applicability of the proposed theorem.
Keywords: Wintner-type, asymptotic behavior, linear differential systems, proportional-derivative controller.
Mathematics Subject Classification: 34D05, 26A24.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 41-54, https://doi.org/10.7494/OpMath.202512101.

 

]]> The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an auxiliary problem in the Borel plane overcomes the absence of adequate domains which would guarantee summability of the formal solution. Moreover, several exponential decay rates of the difference of analytic solutions with respect to the perturbation parameter at the origin are observed, leading to several asymptotic levels relating the analytic and the formal solution. https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4601.pdf Parametric formal Gevrey asymptotic expansions in two complex time variable problems Guoting Chen Alberto Lastra Stéphane Malek singularly perturbed, formal solution, several complex variables, Cauchy problem doi:10.7494/OpMath.202512271 Opuscula Math. 46, no. 1 (2026), 5-40, https://doi.org/10.7494/OpMath.202512271 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202512271 https://www.opuscula.agh.edu.pl/vol46/1/art/opuscula_math_4601.pdf 46 1 5 40
Title: Parametric formal Gevrey asymptotic expansions in two complex time variable problems.

Author(s): Guoting Chen, Alberto Lastra, Stéphane Malek.

Abstract: The analytic and formal solutions to a family of singularly perturbed partial differential equations in the complex domain involving two complex time variables are considered. The analytic continuation properties of the solution of an auxiliary problem in the Borel plane overcomes the absence of adequate domains which would guarantee summability of the formal solution. Moreover, several exponential decay rates of the difference of analytic solutions with respect to the perturbation parameter at the origin are observed, leading to several asymptotic levels relating the analytic and the formal solution.
Keywords: singularly perturbed, formal solution, several complex variables, Cauchy problem.
Mathematics Subject Classification: 35C10, 35R10, 35C15, 35C20.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 1 (2026), 5-40, https://doi.org/10.7494/OpMath.202512271.

 

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