The goal of the current study is to introduce a new class of proximal-point mappings that are associated with a new class of ${\mathcal{B}}^l$-co-monotone mappings that are being defined. The ${\mathcal{B}}^l$-co-monotone mapping is the sum of co-coercive and symmetric monotone mappings and an extension of the $C_n$-monotone mapping. The investigation is further discussed, along with its application, which involves a variational inclusion problem (VIP) in Banach spaces. Moreover, the study proposes an iterative algorithm and systematically investigates the convergence characteristics of its generated sequences. For the purpose of illustrating our findings, a simplified numerical example is created to show the convergence graph by using the MATLAB 2015a.

This study explores certain discoveries regarding fixed points under a functional contraction incorporating two test functions within the setup of a metric space equipped with a locally transitive relation. The outcomes investigated herewith refined, expanded, and strengthened several previously established fixed‐point findings. We provide numerous illustrative instances that demonstrated the credibility of our findings. We investigate the existence and uniqueness of solutions to specific nonlinear integral equations in order to ensure the practical use of our findings.

This manuscript seeks to study couple fixed-point findings over a relational extended contraction through a locally, finitely transitive binary relation and a pair of generalized altering distance functions. Various prior discoveries are expanded, refined, modified, compiled, and enhanced by our findings. We craft some instances for demonstrating the practical relevance of our findings. We embark on our research in an attempt to recognize a unique solution to a boundary value problem connected with a second-order ordinary differential equation.

Mathematical modeling techniques have been used extensively during the human immunodeficiency virus (HIV) epidemic. Drug injection causes increased HIV spread in most countries globally. The media is crucial in spreading health awareness by changing mixing behavior. The published studies show some of the ways that differential equation models can be employed to explain how media awareness programs influence the spread and containment of disease (Greenhalgh et al. Appl Math Comput. 2015;251:539–563). Here we build a differential equation model which shows how disease awareness programs can alter the HIV prevalence in a group of people who inject drugs (PWIDs). This builds on previous work by Greenhalgh and Hay (1997) and Liang et al. (2016). We have constructed a mathematical model to describe the improved model that reduces the spread of the diseases through the effect of awareness of disease on sharing needles and syringes among the PWID population. The model supposes that PWIDs clean their needles before use rather than after. We carry out a steady state analysis and examine local stability. Our discussion has been focused on two ways of studying the influence of awareness of infection levels in epidemic modeling. The key biological parameter of our model is the basic reproductive number . is a crucial number which determines the behavior of the infection. We find that if is less than one then the disease‐free steady state is the unique steady state and moreover whatever the initial fraction of infected individuals then the disease will die out as time becomes large. If exceeds one there is the disease‐free steady state and a unique steady state with disease present. We also showed that the disease‐free steady state is locally asymptotically stable if is less than one, neutrally stable if is equal to one and unstable if exceeds one. In the last case, when is greater than one the endemic steady state was locally asymptotically stable. Our analytical results are confirmed by using simulation with realistic parameter values. In nontechnical terms, the number is a critical value describing how the disease will spread. If is less than or equal to one then the disease will always die out but if exceeds one and disease is present the disease will sustain itself and moreover the numbers of PWIDs with disease will tend to a unique nonzero value.

This paper investigates the resilient leader–follower consensus problem for a class of memory-dependent fractional-order multi-agent systems (FOMASs) with mixed communication delays, uncertain nonlinear dynamics, norm-bounded parametric uncertainties, and intermittent Denial-of-Service (DoS) attacks. The goal is to ensure that the follower agents follow the leader’s path even in the event of weak links, network changes, and lost communications. An adaptive radial basis function neural network (RBFNN) is used as an online approximator to cover the unmodeled nonlinearity. The two complementary delay configurations studied are a discrete-delay protocol, which features a fixed delay on the state, communication channel, and control input; and a distributed-delay protocol, which carries a memory of past states on the control input but a cumulative memory of past states on the DoS-induced input disturbance. In both scenarios, adequate linear matrix inequality (LMI) conditions are obtained in a Lyapunov–Razumikhin stability setting ensuring leader-following consensus, plus a less demanding version of the LMI conditions expressed in terms of eigenvalues, tailored for Kronecker-structured, large-scale networks. A condition of an average dwell-time is used to establish an average condition on the frequency of DoS attack to certify the robustness against intermittent communication disruption. The effectiveness of the proposed RBFNN-based resilient consensus framework is demonstrated by numerical simulation on a network of six followers, with the discrete-delay and distributed-delay configurations each validated separately; the scalability of the eigenvalue-based feasibility-verification test relative to the full LMI synthesis is examined with growing network size, and an ablation study is carried out to isolate the effect of the RBFNN compensation and the dwell-time-consistent gain design. The fully combined discrete-and-distributed-delay case is formulated but not numerically simulated in this work.