Nonlinear functional-integral equations contain the unknown function nonlinearly, occurring extensively in theory developed to a certain extent toward different applied problems and solutions thereof. Nonlinear science serves to reveal the nonlinear descriptions of widely different systems, has had a fundamental impact on complex dynamics. Accordingly, this study provides the basic definitions and results regarding Banach spaces with the proof as well as applications. Besides, proving the existence and uniqueness of the solutions by the Banach theorem is carried out. The existence results of the equations are generally obtained based on fundamental methods by which the fixed-point theorems are frequently applied. Subsequently, the definition of the measure pertaining to noncompactness on Banach space along with properties is presented. Hence, the aim is to study nonlinear functional-integral equations in the Banach space $BC(R_{+})$ using the measure of noncompactness. In addition, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to $BC(R_{+})$ is looked into. As the last stage, the definitions of global attractivity and asymptotical stability for equations on Banach space $BC(R_{+})$ are given while proving the existence of solutions and global attractivity. Besides these, the existence and asymptotical stability are proven. Furthermore, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to $BC(R_{+})$ for an example nonlinear integral equation has been demonstrated by one example premediated and the existence of blow-up point has been examined in one and more than one points regarding the nonlinear integral equation. The example nonlinear integral equation as addressed in this study has been examined in terms of its blow-up point and asymptotic stability with computational complexity for the first time. By doing so, it has been demonstrated that the existence of blow-up point has been the case in one and more than one points and besides this, the examination of its asymptotic stability has shown that it is asymptotically stable at least in one point, yet has no blow-up point pertaining to $BC(R_{+})$. This has enabled thorough analysis of computational complexity of the related nonlinear functional-integral equations within the example both addressing dimension of input in Banach space and that of dataset with $N$ dimension. This applicable approach assures the accurate transformation of the complex problems towards optimized solutions through the generation of advanced mathematical models in nonlinear dynamic settings characterized by complexity.

非线性泛函积分方程非线性地包含未知函数，在理论上广泛出现，并在一定程度上针对不同的应用问题及其解法发展。非线性科学用于揭示广泛不同系统的非线性描述，对复杂动力学产生了根本性的影响。因此，本研究提供了有关 Banach 空间的基本定义和结果以及证明和应用。此外，还用 Banach 定理证明了解的存在性和唯一性。方程的存在性结果一般是根据经常应用不动点定理的基本方法得到的。随后，给出了有关 Banach 空间上非紧致性的测度的定义以及性质。因此，$乙C(R_{+})$使用非紧凑性度量。此外，存在全局吸引子和渐近稳定性的证明，至少有一个解决方案与$乙C(R_{+})$被调查。作为最后阶段，Banach 空间方程的全局吸引力和渐近稳定性的定义$乙C(R_{+})$在证明存在解决方案和全球吸引力的同时给出。除此之外，证明了存在性和渐近稳定性。此外，存在全局吸引子和渐近稳定性的证明，至少有一个解决方案与$乙C(R_{+})$例如，非线性积分方程已通过一个示例预先介导得到证明，并且在非线性积分方程的一个和多个点中已经检查了爆破点的存在。本研究中解决的示例非线性积分方程首次根据其爆发点和渐近稳定性以及计算复杂性进行了检查。通过这样做，已经证明在一个和多个点上都存在爆炸点，除此之外，对其渐近稳定性的检验表明，它至少在一个点上是渐近稳定的，但没有与$乙C(R_{+})$. 这使得能够对示例中相关非线性泛函积分方程的计算复杂性进行彻底分析，包括处理 Banach 空间中输入的维数和数据集的维数$ñ$方面。这种适用的方法通过在以复杂性为特征的非线性动态设置中生成高级数学模型，确保将复杂问题准确转换为优化解决方案。