The problem of relative exact controllability for a class of Riemann–Liouville fractional stochastic delay differential systems is considered. First of all, necessary and sufficient conditions for the relative controllability of linear systems are proved. Next, under the premise of several reasonable assumptions, we are able to prove the relative exact controllability of the corresponding linear stochastic system. Subsequently, we innovatively combine Rothe's fixed point theorem with Itô's isometry, taking them as core analytical tools to overcome the research difficulties in the relative exact controllability of nonlinear stochastic systems. Finally, the effectiveness of the proposed method is clearly demonstrated through a specific example.

This paper investigates a class of nonlinear multiscale stochastic differential equations (SDEs) with a focus on capturing time-evolving memory effects and logarithmic scaling via a variable-exponent Caputo-Hadamard fractional derivative. To overcome the analytical challenges posed by the logarithmic structure, we employ a logarithmic transformation to map the variable-exponent Caputo-Hadamard problem onto its Caputo analogue. Building on this framework, we establish the well-posedness of solutions using a perturbation method under non-Lipschitz conditions, providing the necessary theoretical foundation for studying the system's asymptotic behavior. Subsequently, we prove a homogenization principle that simplifies the complex multiscale dynamics under a specified homogenization condition. To bridge the gap between theoretical derivation and practical application, a novel numerical scheme is developed, combining a weighted quadrature rule with the Euler-Maruyama method. The rigor of our approach is further supported by a convergence analysis. Numerical experiments are presented to validate the theoretical findings and demonstrate the efficiency of the proposed model.

Within a unified framework, we establish the existence of general Poisson stable solutions for McKean–Vlasov stochastic differential equations. We show that if the coefficients are Poisson stable and satisfy appropriate conditions, then there exists a unique bounded solution that is globally asymptotically stable and preserves the same recurrence property as the coefficients. To illustrate our results, we provide an example of the stochastic heat equation.

This paper is devoted to the study of the global behavior of solutions to a class of strongly damped wave equations involving the p–Laplacian operator and logarithmic source terms. More precisely, we consider the problem u tt − Δ u t − Δ p u = λ | u | α − 2 uln ⁡ ( 1 + | u | ) + γ | ∇u | β ln ⁡ ( 1 + | ∇u | ) , ( x , t ) ∈ R N × ( 0 , + ∞ ) , where Δ p u = div ( | ∇u | p − 2 ∇u ) with p>2, and 0 $ ]]> λ , γ , α , β > 0 are given constants. The presence of logarithmic nonlinearities combined with nonlinear diffusion and strong damping introduces significant analytical difficulties, since the logarithmic terms exhibit a growth behavior that lies between polynomial and exponential nonlinearities. To overcome these challenges, we employ a rescaled test function method together with suitable logarithmic inequalities. Under appropriate conditions on the spatial dimension and the nonlinear exponents, we prove the nonexistence of nontrivial global weak solutions. Furthermore, we derive an explicit upper bound for the lifespan of local weak solutions in terms of the initial data and the parameters of the problem. The obtained results illustrate how strong damping, p–Laplacian diffusion, and logarithmic source terms interact with each other. In particular, this interaction leads to the appearance of a critical threshold phenomenon that differs from the one observed in the classical power-type nonlinear case.

A third-order operator with periodic coefficients is an L-operator in the Lax pair for the Boussinesq equation on a circle. The projection of the divisor of the Floquet solution poles for this operator coincides with the spectrum of the three-point Dirichlet problem. The sign of the norming constant of the three-point problem determines the sheet of the Riemann surface on which the pole lies. We solve the inverse problem for a third-order operator with three-point Dirichlet conditions when the spectrum and norming constant are known. We construct a mapping from the set of coefficients to the set of spectral data and prove that this mapping is an analytic bijection in the neighborhood of zero.

This paper addresses a novel class of Schrödinger-Kirchhoff-type problems involving a double-phase structure and subject to Neumann boundary conditions. The model is governed by ( η ( s ) , ζ ( s ) ) -Laplacian-like operators and includes a convection term that explicitly depends on both the solution and its gradient. By imposing appropriate growth conditions on this nonlinear term, we prove the existence of weak solutions. Our approach combines the Galerkin approximation method with the theory of Young measures, all within the analytical framework of Musielak–Orlicz Sobolev spaces with variable exponents. Our results offer a new contribution to the field, extending existing models by incorporating both non-standard growth and gradient-dependent convection effects.

This paper studies a nonlinear class of fractional difference problems depending on parameter-dependent summation constraints. An explicit formula for the Green's function and its positivity and bounds for a specific interval of parameters are found. By use of the Guo–Krasnoselskii fixed point theorem with an appropriate cone, we obtain the existence criteria for multiple distinct positive solutions. This is achieved through a ‘trapping mechanism’ based on a series of alternating growth conditions of the nonlinearity. In addition, the use of the obtained results is justified through an illustrative example.

This paper studies a new class of optimal control problems involving discrete and differential inclusions with multiple delays and state constraints. Under a suitable regularity condition, optimality conditions for problems with two delays are analysed. By applying a discretization approach in the form of Euler-Lagrange type inclusions, sufficient optimality conditions for problems with multiple delays are established. In such an adjoint inclusion, to each delay parameter corresponds one absolutely continuous function. The transition from a discrete problem to a discrete-approximate problem is achieved through equivalence relations and the associated locally adjoint mappings (LAMs). These relations enable the derivation of optimality conditions for discrete-approximate problems with multiple delays. Finally, by passing to the limit, sufficient optimality conditions for differential inclusions with multiple delays are obtained. In particular, applications of the developed results to first-order linear optimal control problems with single delays are presented.

The main concern of this work is to study the sufficient conditions for existence, uniqueness, and approximate controllability results for the nonlinear Caputo conformable fractional neutral-type delayed integro-differential system with nonlocal conditions in a Hilbert space. Since, the conformable derivative retains several fundamental properties of classical calculus including mean value theorem, Rolle's theorem, product, quotient, and linearity rules. This distinguishes it from traditional fractional derivatives such as Riemann-Liouville, Caputo, and Hilfer. Therefore, the conformable derivative is simpler and faster but ignores history while the Caputo conformable fractional derivative offers a balance capturing some memory with easier calculations. Firstly, the proposed system is reformulated into an equivalent fixed point problem implementing the Riemann-Liouville conformable fractional integral operator. The Schauder fixed point theorem is used to derive the existence of mild solution. The Banach contraction principle is then applied to show the uniqueness of mild solution. The main tools applied to derive the results are theory of fractional calculus, semigroup of bounded linear operators, and fixed point theorems. Further, the approximate controllability result for the proposed system is established under the consideration that the corresponding linear system is approximate controllable. An illustrative example is presented to demonstrate the applicability of theoretical results.

We derive a nonlinear Forchheimer-type boundary condition at the interface between a free fluid and a rigid porous wall using homogenization of the Navier–Stokes equations. The equations are considered on a rectangular, two-dimensional domain with a periodically perforated bottom boundary where Dirichlet conditions on the solid parts alternate with pressure conditions on the porous openings. The resulting law includes a quadratic correction in the normal velocity, accounting for the inertial effects. This generalizes classical linear models and provides a rigorous framework for nonlinear interface dynamics. The formal asymptotic expansion and the error estimates that justify the result are accompanied by numerical computations of the effective coefficients.