In this paper, the solvability of an operator-valued integral equation in Hölder spaces, i.e.,

$$w({\zeta }_1)=y({\zeta }_1)+w({\zeta }_1){\int }_{\mathbf{J}}\kappa ({\zeta }_1,\phi )(T_{1}w)(\phi )d\phi +z({\zeta }_1){\int }_{\mathbf{J}}h(\phi ,(T_{2}w)(\phi ))d\phi ,$$

for ${\zeta }_1\in \mathbf{J}=[0,1],$ is studied by using Darbo's fixed point theorem (FPT). The process of the measure of noncompactness of the operators which constitute an intermediary of contraction and compact mappings can be explained with the help of Darbo's FPT. The greater effectiveness of Darbo's FPT due to its non-involvement of the compactness property gives a better scope when dealing with the Schauder FPT, where compactness is an essential property. To obtain a unique solution, we apply the Banach fixed point theorem and discuss the Hyers-Ulam stability of the integral equation. We also give some important examples to illustrate the existence and uniqueness of the results.

This work investigates the dynamic complications of the Ricker type predator-prey model in the presence of the additive type Allee effect in the prey population. In the modeling of discrete-time models, Euler forward approximations and piecewise constant arguments are the most frequently used schemes. In Euler forward approximations, the model may undergo period-doubled orbits and invariant circle orbits, even while varying the step size. In this way, differential equations with piecewise constant arguments (Ricker-type models) are a better choice for the discretization of a continuous-time model because they do not involve any step size. First, the interaction between prey and predator in the form of the Holling-Ⅱ type is considered. The essential mathematical features are discussed in terms of local stability and the bifurcation phenomenon as well. Next, we apply the center manifold theorem and normal form theory to achieve the existence and directions of flip bifurcation and Neimark-Sacker bifurcation. Moreover, this paper demonstrates that the outbreak of chaos can stabilize in the considered model with a higher value of the Allee parameter. The existence of chaotic orbits is verified with the help of a one-parameter bifurcation diagram and the largest Lyapunov exponents, respectively. Furthermore, different control methods are applied to control the bifurcation and fluctuating phenomena, i.e., state feedback, the Ott-Grebogi-Yorke, and hybrid control methods. Finally, to ensure our analytical results, numerical simulations have been carried out using MATLAB software.

This study explores the asymptotic and oscillatory behavior of solutions to third-order functional differential equations. By employing Riccati transformation, we effectively eliminate the possibility of nonoscillatory solutions, allowing for the development of oscillation criteria that are applicable to a broad range of equation models. A key objective of this work is to relax traditional constraints commonly imposed on these criteria, thereby enhancing their general applicability. The results presented not only refine and extend existing theories but also contribute to a deeper understanding of the subject. Practical implications of the theoretical findings are demonstrated through several illustrative examples, highlighting their relevance and potential applications.