Consider non-autonomous random Boissonade equations on a three-dimensional bounded domain with boundary of Lipschitz type, where homogeneous boundary conditions of Dirichlet type are equipped. This type of equations is a activator-inhibitor model that exhibits Turing structures for describing the relations between the genuine homogeneous 2D systems and the 3D monolayers. The underlying feature of the problem under consideration is a non-Lipschitz condition imposed on the nonlinear function controlling intensity of colored noise, leading to possible nonuniqueness of solutions with given initial data. A class of multivalued semiflows having non-autonomy and randomness is defined. The measurability and continuity properties of such semiflows are studied. Moreover, the long-time dynamics in the pullback sense for the semiflows is studied using the idea of energy equation. We also investigate the asymptotically autonomous robustness for the equations on a three-dimensional bounded domain with smooth boundary.
We consider a class of Hadamard type fractional differential equations with disturbance parameter and present the number $\varrho^*$ that sharply demarcates the existence and nonexistence of positive solutions for the corresponding boundary value problem across all non-negative parameter values by using fixed point theory. Specifically, we establish that no positive solutions exist if the parameter $\varrho> \varrho^*$, while at least one positive solution emerges when $\varrho$ lies in $[0,\varrho^*]$. Notably, within the subcritical regime $(0,\varrho^*)$, the problem admits at least two distinct positive solutions. Therefore, in this paper, the influence of parameter on solution is clearly demonstrated, with no ambiguities or indistinct regions. Some existing results failed to prove the existence of this type of point. The present study not only demonstrates the existence of one such point but also determines its imprecise range, thus complementing and enhancing previous findings. Finally, an example is presented and analyzed to illustrate the practical implications of our theoretical results.
We present a new result on realization of graphs as the {\it Reeb graphs} of Morse functions with given level sets. The {\it Reeb graph} of a Morse function on a closed manifold is the quotient space obtained by contracting connected components of its level sets to single points. Reeb graphs are graphs whose vertex sets consist of all points representing components with critical points. They are not only fundamental tools in geometry using differentiable maps, but also in applications of mathematics such as visualizations. Constructing explicit functions with prescribed {\it regular} level sets, was considered first by the author essentially and previously some cases where manifolds of the domains are at most $3$-dimensional have been studied. We generalize these cases by generalizing regular level sets to closed manifolds each component of which is the boundary of a nice compact manifold and construct Morse functions.
This paper is devoted to examine the controllability of a class of non-autonomous stochastic neutral functional differential equations with state-dependent delay and fractional Brownian motion. First, by employing operator theory and estimates on the nonlinear terms, we establish the existence of mild solutions for the considered system. Then, with the aid of Mönch's fixed point theorem and the properties of the measure of non-compactness, we derive sufficient conditions for the controllability of the system. Finally, an illustrative example is provided to demonstrate the applicability of the theoretical results. These findings offer a theoretical foundation for the analysis and control of complex systems in related fields.
In this paper, we investigate the existence of positive solutions to a class of $n$-th order nonlocal differential equations with convolution coefficients and derivative dependence. By constructing nonstandard cone and specific open set, we establish the existence result of positive solutions by means of fixed point index theory. Two examples are also given to illustrate the main result.
