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 In this paper, we study the normalized solutions of the following critical growth Choquard equation with mixed local and nonlocal operators: \[\begin{split}-\Delta u +(-\Delta)^s u &= \lambda u +\mu |u|^{p-2}u +(I_{\alpha}*|u|^{2^*_{\alpha}})|u|^{2^*_{\alpha}-2}u \quad\text{in}\quad \mathbb{R}^N,\\ \| u\|_2 &= \tau,\end{split}\] where \(N\geq 3\), \(\tau\gt 0\), \(I_{\alpha}\) is the Riesz potential of order \(\alpha\in (0,N)\), \(2^*_{\alpha}=\frac{N+\alpha}{N-2}\) is the critical exponent corresponding to the Hardy-Littlewood-Sobolev inequality, \((-\Delta)^s\) is the nonlocal fractional Laplacian operator with \(s\in (0,1)\), \(\mu\gt 0\) is a parameter and \(\lambda\) appears as Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of the mass-subcritical perturbation \(\mu |u|^{p-2}u\) with \(2\gt p\gt 2+\frac{4s}{N}\) under some assumptions on \(\tau\). https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4612.pdf Solutions with prescribed mass for a critical Choquard equation driven by a local-nonlocal operator Nidhi Nidhi Konijeti Sreenadh normalized solution, Choquard equation, critical exponent, mixed local and nonlocal operator, \(L^2\)-subcritical perturbation, nonlinear Schrödinger equation driven by local-nonlocal operator doi:10.7494/OpMath.202603181 Opuscula Math. 46, no. 2 (2026), 235-266, https://doi.org/10.7494/OpMath.202603181 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202603181 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4612.pdf 46 2 235 266
Title: Solutions with prescribed mass for a critical Choquard equation driven by a local-nonlocal operator.

Author(s): Nidhi Nidhi, Konijeti Sreenadh.

Abstract: In this paper, we study the normalized solutions of the following critical growth Choquard equation with mixed local and nonlocal operators: \[\begin{split}-\Delta u +(-\Delta)^s u &= \lambda u +\mu |u|^{p-2}u +(I_{\alpha}*|u|^{2^*_{\alpha}})|u|^{2^*_{\alpha}-2}u \quad\text{in}\quad \mathbb{R}^N,\\ \| u\|_2 &= \tau,\end{split}\] where \(N\geq 3\), \(\tau\gt 0\), \(I_{\alpha}\) is the Riesz potential of order \(\alpha\in (0,N)\), \(2^*_{\alpha}=\frac{N+\alpha}{N-2}\) is the critical exponent corresponding to the Hardy-Littlewood-Sobolev inequality, \((-\Delta)^s\) is the nonlocal fractional Laplacian operator with \(s\in (0,1)\), \(\mu\gt 0\) is a parameter and \(\lambda\) appears as Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of the mass-subcritical perturbation \(\mu |u|^{p-2}u\) with \(2\gt p\gt 2+\frac{4s}{N}\) under some assumptions on \(\tau\).
Keywords: normalized solution, Choquard equation, critical exponent, mixed local and nonlocal operator, \(L^2\)-subcritical perturbation, nonlinear Schrödinger equation driven by local-nonlocal operator.
Mathematics Subject Classification: 35Q55, 35M10, 35J61, 35A01.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 235-266, https://doi.org/10.7494/OpMath.202603181.

 

]]> We provide an alternative approach, based on the Leray-Schauder fixed point index in cones, to a fixed point theorem for operator systems due to Precup. Our focus is on the case of operators whose components are entirely of compressive type. The abstract technique is applied to a system of second-order differential equations providing a coexistence positive solution by means of an eigenvalue type criterion. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4611.pdf On a fixed point theorem for operator systems and eigenvalue criteria for existence of positive solutions Laura M. Fernández-Pardo Jorge Rodríguez-López coexistence fixed point, fixed point index, positive solution, nonlinear systems doi:10.7494/OpMath.202602271 Opuscula Math. 46, no. 2 (2026), 219-234, https://doi.org/10.7494/OpMath.202602271 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202602271 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4611.pdf 46 2 219 234
Title: On a fixed point theorem for operator systems and eigenvalue criteria for existence of positive solutions.

Author(s): Laura M. Fernández-Pardo, Jorge Rodríguez-López.

Abstract: We provide an alternative approach, based on the Leray-Schauder fixed point index in cones, to a fixed point theorem for operator systems due to Precup. Our focus is on the case of operators whose components are entirely of compressive type. The abstract technique is applied to a system of second-order differential equations providing a coexistence positive solution by means of an eigenvalue type criterion.
Keywords: coexistence fixed point, fixed point index, positive solution, nonlinear systems.
Mathematics Subject Classification: 47H10, 47H11, 34B18, 34B16, 34C25.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 219-234, https://doi.org/10.7494/OpMath.202602271.

 

]]> We study the problem of finding a minimum \(k\)-critical-bipartite graph of order \((n,m)\): a bipartite graph \(G=(U,V;E)\), with \(|U|=n\), \(|V|=m\), and \(n\gt m\gt 1\), which is \(k\)-critical-bipartite, and the tuple \((|E|, \Delta_U, \Delta_V)\), where \(\Delta_U\) and \(\Delta_V\) denote the maximum degree in \(U\) and \(V\), respectively, is lexicographically minimum over all such graphs. \(G\) is \(k\)-critical-bipartite if deleting any set of at most \(k=n-m\) vertices from \(U\) yields \(G'\) that has a complete matching, i.e., a matching of size \(m\). Cichacz and Suchan solved the problem for biregular bipartite graphs. Here, we extend their results to bipartite graphs that are not biregular. We prove tight lower bounds on the connectivity of \(k\)-critical-bipartite graphs, and we show that \(k\)-critical-bipartite graphs are expander graphs. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4610.pdf Minimum k-critical-bipartite graphs: the irregular case Sylwia Cichacz Agnieszka Görlich Karol Suchan fault-tolerance, interconnection network, bipartite graph, complete matching, algorithm, \(k\)-critical-bipartite graph doi:10.7494/OpMath.202602181 Opuscula Math. 46, no. 2 (2026), 201-218, https://doi.org/10.7494/OpMath.202602181 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202602181 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4610.pdf 46 2 201 218
Title: Minimum k-critical-bipartite graphs: the irregular case.

Author(s): Sylwia Cichacz, Agnieszka Görlich, Karol Suchan.

Abstract: We study the problem of finding a minimum \(k\)-critical-bipartite graph of order \((n,m)\): a bipartite graph \(G=(U,V;E)\), with \(|U|=n\), \(|V|=m\), and \(n\gt m\gt 1\), which is \(k\)-critical-bipartite, and the tuple \((|E|, \Delta_U, \Delta_V)\), where \(\Delta_U\) and \(\Delta_V\) denote the maximum degree in \(U\) and \(V\), respectively, is lexicographically minimum over all such graphs. \(G\) is \(k\)-critical-bipartite if deleting any set of at most \(k=n-m\) vertices from \(U\) yields \(G'\) that has a complete matching, i.e., a matching of size \(m\). Cichacz and Suchan solved the problem for biregular bipartite graphs. Here, we extend their results to bipartite graphs that are not biregular. We prove tight lower bounds on the connectivity of \(k\)-critical-bipartite graphs, and we show that \(k\)-critical-bipartite graphs are expander graphs.
Keywords: fault-tolerance, interconnection network, bipartite graph, complete matching, algorithm, \(k\)-critical-bipartite graph.
Mathematics Subject Classification: 05C35, 05C70, 05C85, 68M10, 68M15.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 201-218, https://doi.org/10.7494/OpMath.202602181.

 

]]> We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. As a continuation of our previous study on the existence and uniqueness of solutions, we prove here some strong stabilization properties. In particular, we show the finite time extinction of solutions induced by the nonlinear saturation mechanism, which, sometimes, can be understood as a bang-bang control. The analysis relies on refined energy methods. Our results provide a rigorous justification of nonlinear dissipation as an effective stabilization mechanism for this class of complex equations where the maximum principle fails. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4609.pdf Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization Pascal Bégout Jesús Ildefonso Díaz damped Ginzburg-Landau equation, saturated nonlinearity, finite time extinction doi:10.7494/OpMath.202603112 Opuscula Math. 46, no. 2 (2026), 185-199, https://doi.org/10.7494/OpMath.202603112 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202603112 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4609.pdf 46 2 185 199
Title: Damped nonlinear Ginzburg-Landau equation with saturation. Part II. Strong Stabilization.

Author(s): Pascal Bégout, Jesús Ildefonso Díaz.

Abstract: We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. As a continuation of our previous study on the existence and uniqueness of solutions, we prove here some strong stabilization properties. In particular, we show the finite time extinction of solutions induced by the nonlinear saturation mechanism, which, sometimes, can be understood as a bang-bang control. The analysis relies on refined energy methods. Our results provide a rigorous justification of nonlinear dissipation as an effective stabilization mechanism for this class of complex equations where the maximum principle fails.
Keywords: damped Ginzburg-Landau equation, saturated nonlinearity, finite time extinction.
Mathematics Subject Classification: 35Q56, 35B40, 93D40.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 185-199, https://doi.org/10.7494/OpMath.202603112.

 

]]> We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4608.pdf Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains Pascal Bégout Jesús Ildefonso Díaz damped Ginzburg-Landau equation, saturated nonlinearity, finite time extinction, maximal monotone operators, existence and regularity of weak solutions doi:10.7494/OpMath.202603111 Opuscula Math. 46, no. 2 (2026), 153-183, https://doi.org/10.7494/OpMath.202603111 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202603111 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4608.pdf 46 2 153 183
Title: Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains.

Author(s): Pascal Bégout, Jesús Ildefonso Díaz.

Abstract: We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schrödinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains.
Keywords: damped Ginzburg-Landau equation, saturated nonlinearity, finite time extinction, maximal monotone operators, existence and regularity of weak solutions.
Mathematics Subject Classification: 35Q56, 35A01, 35A02, 35D30, 35D35.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 153-183, https://doi.org/10.7494/OpMath.202603111.

 

]]> A subset \(J\) of vertices is said to be a \((1,k)\)-dominating set if every vertex \(v\) not belonging to the set \(J\) has a neighbour in \(J\) and there exists also another vertex in \(J\) within the distance at most \(k\) from \(v\). In this paper, we study the problem of the existence of independent \((1,k)\)-dominating sets for \(k\in\{1,2\}\) in the tensor product and in the strong product of two graphs. We give complete characterisations of these graph products, which have independent \((1,1)\)-dominating sets or independent \((1,2)\)-dominating sets, with respect to the properties of their factors. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4607.pdf On the existence of independent (1,k)-dominating sets for k\in\{1,2\} in two products of graphs Paweł Bednarz Adrian Michalski Natalia Paja dominating set, independent set, multiple domination, secondary domination, tensor product, strong product doi:10.7494/OpMath.202601201 Opuscula Math. 46, no. 2 (2026), 139-152, https://doi.org/10.7494/OpMath.202601201 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202601201 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4607.pdf 46 2 139 152
Title: On the existence of independent (1,k)-dominating sets for k\in\{1,2\} in two products of graphs.

Author(s): Paweł Bednarz, Adrian Michalski, Natalia Paja.

Abstract: A subset \(J\) of vertices is said to be a \((1,k)\)-dominating set if every vertex \(v\) not belonging to the set \(J\) has a neighbour in \(J\) and there exists also another vertex in \(J\) within the distance at most \(k\) from \(v\). In this paper, we study the problem of the existence of independent \((1,k)\)-dominating sets for \(k\in\{1,2\}\) in the tensor product and in the strong product of two graphs. We give complete characterisations of these graph products, which have independent \((1,1)\)-dominating sets or independent \((1,2)\)-dominating sets, with respect to the properties of their factors.
Keywords: dominating set, independent set, multiple domination, secondary domination, tensor product, strong product.
Mathematics Subject Classification: 05C69, 05C76.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 139-152, https://doi.org/10.7494/OpMath.202601201.

 

]]> In this note, we prove that every graph can be edge-labelled with red labels \(1,2\) and blue labels \(1,2\) so that vertices with any sum of incident red labels induce a \(1\)-degenerate graph, while vertices with any sum of incident blue labels induce a \(0\)-degenerate graph. This result stands as a closer step towards the so-called Standard \((2,2)\)-Conjecture (stating that \(0\)-degeneracy can be achieved in both colours), and provides some insight on the surrounding field, which covers the 1-2-3 Conjecture, the 1-2 Conjecture, and other close problems. Along the way, we also describe how many related problems are interconnected, and raise new problems and questions for further work on the topic. https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4606.pdf Some remarks and results on the Standard (2,2)-Conjecture Olivier Baudon Julien Bensmail Lyn Vayssieres 1-2-3 Conjecture, 1-2 Conjecture, proper labelling, labelling doi:10.7494/OpMath.202602101 Opuscula Math. 46, no. 2 (2026), 127-137, https://doi.org/10.7494/OpMath.202602101 Opuscula Mathematica 2026 2026 https://doi.org/10.7494/OpMath.202602101 https://www.opuscula.agh.edu.pl/vol46/2/art/opuscula_math_4606.pdf 46 2 127 137
Title: Some remarks and results on the Standard (2,2)-Conjecture.

Author(s): Olivier Baudon, Julien Bensmail, Lyn Vayssieres.

Abstract: In this note, we prove that every graph can be edge-labelled with red labels \(1,2\) and blue labels \(1,2\) so that vertices with any sum of incident red labels induce a \(1\)-degenerate graph, while vertices with any sum of incident blue labels induce a \(0\)-degenerate graph. This result stands as a closer step towards the so-called Standard \((2,2)\)-Conjecture (stating that \(0\)-degeneracy can be achieved in both colours), and provides some insight on the surrounding field, which covers the 1-2-3 Conjecture, the 1-2 Conjecture, and other close problems. Along the way, we also describe how many related problems are interconnected, and raise new problems and questions for further work on the topic.
Keywords: 1-2-3 Conjecture, 1-2 Conjecture, proper labelling, labelling.
Mathematics Subject Classification: 05C78, 05C15, 68R10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 46, no. 2 (2026), 127-137, https://doi.org/10.7494/OpMath.202602101.

 

]]> 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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