In this paper we find viscosity solutions to a system with two parabolic obstacle-type equations that involve two normalized \(p-\)Laplacian operators. We analyze a two-player zero-sum game played on two boards (with different rules in each board), in which at each board one of the two players has the choice of playing in that board or switching to the other board and then play. We prove that the game has a value and show that these value functions converge uniformly (when a parameter that controls the size of the steps made in the game goes to zero) to a viscosity solution of a system in which one component acts as an obstacle for the other component and vice versa. In this way, we find solutions to the parabolic two-membranes problem.

In this paper, the dynamical behaviors of a stochastic competition system with a saturation effect are analyzed. The existence and uniqueness of globally positive solution are proved in detail, and the sufficient conditions for stochastic permanence, strong persistence in the mean, weak persistence, and extinction are obtained respectively. Then the existence and uniqueness of stationary distribution are also obtained under some appropriate assumptions. Finally, several numerical simulations are provided to justify the analytical results.

This paper proposes an explicit numerical scheme based on Delannoy polynomials in conjunction with the tau method for solving the time-fractional diffusion equation involving the Caputo derivative. The proposed method constructs approximate solutions using shifted Delannoy polynomials as basis functions, allowing efficient and accurate treatment of the nonlocal nature of fractional derivatives. The method transforms the time-fractional diffusion problem into a system of algebraic equations, which can be solved explicitly. Several benchmark examples are provided to confirm the efficiency, accuracy, and applicability of the new scheme.

This paper develops a comprehensive theory for variable-exponent Bochner spaces\(L^{p(\cdot)}([0,T];X)\), establishing fundamental results on compact embeddings and maximal regularity with applications to nonlocal evolution equations. We extend the classical Aubin-Lions framework through innovative modular convergence techniques, proving sharp compactness criteria under log-Holder continuity conditions. For time-dependent fractional operators, including the fractional Laplacian\((-\Delta)^{s(t)}\) and Levy-type processes with variable order\(\alpha(t)\), we derive optimal maximal regularity estimates that reveal new connections between exponent functions\(p(t)\) and operator orders. A groundbreaking contribution is our systematic analysis of fractal dimension dynamics in variable-order fractional PDEs, characterizing how evolving regularity\(s(t)\) governs solution behavior. Furthermore, we develop novel functional-analytic tools for stochastic exponents\(p(t,\omega)\), yielding compact embedding results in\(L^{p(\cdot,\omega)}(X)\) spaces and boundedness properties for nonlinear operators. Combining techniques from modular function theory, refined interpolation methods, and stochastic analysis, our work provides powerful new approaches for problems in anomalous diffusion and heterogeneous media. These results significantly advance both the theoretical foundations and practical applications of variable-exponent spaces in modern PDE analysis.

<p>In this work, we seek conditions for the existence or nonexistence of solutions for nonlinear Riemann-Liouville fractional boundary value problems of order <span class="math inline">\(\alpha + 2n\)</span>, where <span class="math inline">\(\alpha \in (m-1, m]\)</span> with <span class="math inline">\(m \geq 3\)</span> and <span class="math inline">\(m, n \in \mathbb{N}\)</span>. The problem’s nonlinearity is continuous and also depends on a positive parameter upon which our constraints are established. Our approach involves constructing a Green’s function by combining the Green’s functions of a lower-order fractional boundary value problem and a right-focal boundary value problem <span class="math inline">\(n\)</span> times. Leveraging the properties of this Green’s function, we apply Krasnosel’skii’s Fixed Point Theorem to establish our results. Several examples are presented to illustrate the existence and nonexistence regions.</p>

We introduce Littlewood Paley functions defined in terms of a reparameterization of the Ornstein-Uhlenbeck semigroup obtaining that these operators are bounded in \(L^p\), \(1<p<\infty\), with respect to the unidimensional gaussian measure, by means of singular integrals theory. In addition, we study the Abel summability of the Fourier Hermite expansions considering their pointwise convergence and their convergence in the \(L^p\) sense, obtaining a version of Tauber's theorem.

This work introduces a unique family of bi-univalent functions utilising \(q\)-Gegenbauer polynomials. The estimates of the initial coefficients \(\left\vert a_{2}\right\vert\) and \(\left\vert a_{2}\right\vert\) for functions in this new class, together with the Fekete-Szegö functional, have been obtained. Subsequent to the specialisation of the parameters utilised in our principal findings, many novel outcomes are presented.

In this paper, we establish some new characterizations of a weight \(w\) such that discrete Hardy operator \(\mathcal{H}f(n):=\frac{1}{n}\sum\limits_{s=1}^{n}f(s)\) for quasi-nonincreasing sequence \(f(n)\) is bounded in the Banach space \(\ell _{w}^{p}(\mathbb{Z}_{+})\) when \(0<p<\infty .\) In particular, we will prove that \(\mathcal{H}f\) is bounded in \(\ell _{w}^{p}(\mathbb{Z}_{+})\) if and only if \(w\) belongs to the \(\beta\)-discrete Arino and Muckenhoupt class \(\mathcal{B}_{p,\beta }\). We prove that the self-improving property for the class \(\mathcal{B}_{p,\beta }\) holds, that is we prove that if \(w\in \mathcal{B}_{p,\beta }\) then there exists an \(\varepsilon >0\) such that \(w\in \mathcal{B}_{p-\varepsilon ,\beta }.\)

The current study focuses on the investigation and develop of a new approach called Hussein–Jassim method (HJM), suggested lately by Hassan et al.; specifically, we investigate its applicability to fractional ordinary delay differential equations in the Caputo fractional sense. Several examples are offered to demonstrate the method’s reliability. The results of this study demonstrate that the proposed method is highly effective and convenient for solving fractional delay differential equations.