In this study, our focus is on stabilizing a competitive game involving an original equipment manufacturer (OEM) and a third-party remanufacturer (TPR). To assess the presence of chaos within the dynamics of this game, we employ various analytical tools, including spectral entropy (SE), bifurcation diagrams, and Lyapunov exponents. The unpredictable nature of chaotic dynamics significantly influences the market and has negative implications for the strategic decisions of both firms. Our approach to counteracting this chaotic behaviour and stabilizing the system revolves around the implementation of the Ott, Grebogi, and Yorke (OGY) method. Crucially, our analysis highlights that the marginal costs ( $c_n$ and $c_r$) incurred by the OEM and TPR emerge as pivotal factors in achieving stabilization within the game. To provide a tangible demonstration of the effectiveness of our proposed stabilization strategy in the context of this competitive environment, we conducted numerical simulations.

We consider a nonlinear class of hybrid pantograph equations governed by Caputo-type $(p,q)$-fractional derivatives and proportional delay arguments. By rewriting the problem in an equivalent integral form, we first established the existence of solutions through Dhage's hybrid fixed point theorem in a Banach algebra. Under additional Lipschitz conditions, the Banach contraction principle was employed to guarantee the existence and uniqueness of solutions. The results generalize and extend studies on fractional and hybrid pantograph equations within the $(p,q)$-fractional setting.

This article focuses on the problem of integral boundary value for Riemann-Liouville derivatives equipped with $\left(p,q\right)$-difference calculus in Banach space. To provide further clarification, our focus lies in establishing the existence of a solution to our problem using the measure of noncompactness (m.n.) and the Mönch's fixed point theorem. Our investigation in the Banach space encompasses two nonlinear terms with two distinct orders of derivatives. Our paper concludes with an illustrative example and conclusion.

In this manuscript, under the matrix measure and some sufficient conditions, we will overcame all difficulties and challenges related to the fundamental matrix for a generalized nonlinear neutral functional differential equations in matrix form with multiple delays. The periodicity of solutions, as well as the uniqueness under the considered conditions has been proved employing the fixed point theory. Our approach expanded and generalized certain previously published findings for example, we studied the uniqueness of a solution that was absent in some literature. Moreover, an example was given to confirm the main results.