In this paper, we consider some boundary value problems composed by coupled systems of second-order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply the lower and upper solutions method, and defining an adequate auxiliary, homotopic, and truncated problem, it is possible to apply topological degree theory as the tool to prove the existence of solution. In short, it is proved that for the parameter values such that there are lower and upper solutions, then there is also, at least, a solution and this solution is localized in a strip bounded by lower and upper solutions. As far as we know, it is the first paper where Ambrosetti-Prodi differential equations are considered in couple systems with different parameters.

This work presents a discussion of Ambrosetti-Prodi-type second-order periodic problems. In short, the existence, non-existence, and multiplicity of solutions will be discussed on the parameter $\lambda$. The arguments rely on a Nagumo condition, to guarantee an apriori bound on the first derivative, lower and upper-solutions method, and the Leray-Schauder's topological degree theory. There are two types of new results based on the parameter's variation: An existence and non-existence theorem and a multiplicity theorem, proving the existence of a bifurcation point. An application to a damped and forced pendulum is studied, suggesting a method to estimate the critical values of the parameter.

Heroin addiction is a continuously progressing phenomenon that represents a major problem for world's the public health; this indicates that the development of new methodologies to address the issue at the international level is a crucial priority. To study its transmission dynamics, a new stochastic fractional delayed heroin model based on stochastic fractional delay differential equations (SFDDEs) was developed to focus on the positive aspects of randomness and memory effects. The positive, boundedness, existence, and uniqueness of the model were studied rigorously. The equilibria (i.e., heroin-free equilibrium and the present equilibrium, which gives a clue about both eradication and persistence cases), reproduction number, and sensitivity of parameters were analyzed. The local and global stability of the new model was studied around its steady states. Also, well-known theorems are presented to investigate the extinction and persistence of heroin. The Grunwald-Letnikove non-standard finite difference (GL-NSFD) method was used for the efficient computational analysis of the stochastic fractional delayed model. For the dynamical consistency of the model, the positivity and boundedness of an efficient method were studied rigorously. The given study focuses on delay strategies and fractional calculus that could be useful in formulating specific measures for regulating addiction. Moreover, the simulated results support the theoretical analysis of the model and validate it.

In 2022, Leukemia is the 13th most common diagnosis of cancer globally as per the source of the International Agency for Research on Cancer (IARC). Leukemia is still a threat and challenge for all regions because of 46.6% infection in Asia, and 22.1% and 14.7% infection rates in Europe and North America, respectively. To study the dynamics of Leukemia, the population of cells has been divided into three subpopulations of cells susceptible cells, infected cells, and immune cells. To investigate the memory effects and uncertainty in disease progression, leukemia modeling is developed using stochastic fractional delay differential equations (SFDDEs). The feasible properties of positivity, boundedness, and equilibria (i.e., Leukemia Free Equilibrium (LFE) and Leukemia Present Equilibrium (LPE)) of the model were studied rigorously. The local and global stabilities and sensitivity of the parameters around the equilibria under the assumption of reproduction numbers were investigated. To support the theoretical analysis of the model, the Grunwald Letnikov Nonstandard Finite Difference (GL-NSFD) method was used to simulate the results of each subpopulation with memory effect. Also, the positivity and boundedness of the proposed method were studied. Our results show how different methods can help control the cell population and give useful advice to decision-makers on ways to lower leukemia rates in communities.