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This study focuses on investigating the numerical solution of second-kind nonlinear Volterra–Fredholm–Hammerstein integral equations (NVFHIEs) by discretization technique. The purpose of this paper is to develop an efficient and accurate method for solving NVFHIEs, which are crucial for modeling systems with memory and cumulative effects, integrating past and present influences with nonlinear interactions. They are widely applied in control theory, population dynamics and physics. These equations are essential for solving complex real-world problems.

Demonstrating the solution’s existence and uniqueness in the equation is accomplished by using the Picard iterative method as a key technique. Using the trapezoidal discretization method is the chosen approach for numerically approximating the solution, yielding a nonlinear system of algebraic equations. The trapezoidal method (TM) exhibits quadratic convergence to the solution, supported by the application of a discrete Grönwall inequality. A novel Grönwall inequality is introduced to demonstrate the convergence of the considered method. This approach enables a detailed analysis of the equation’s behavior and facilitates the development of a robust solution method.

The numerical results conclusively show that the proposed method is highly efficacious in solving NVFHIEs, significantly reducing computational effort. Numerical examples and comparisons underscore the method’s practicality, effectiveness and reliability, confirming its outstanding performance compared to the referenced method.

Unlike existing approaches that rely on a combination of methods to tackle different aspects of the complex problems, especially nonlinear integral equations, the current approach presents a significant single-method solution, providing a comprehensive approach to solving the entire problem. Furthermore, the present work introduces the first numerical approaches for the considered integral equation, which has not been previously explored in the existing literature. To the best of the authors’ knowledge, the work is the first to address this equation, providing a foundational contribution for future research and applications. This innovative strategy not only simplifies the computational process but also offers a more comprehensive understanding of the problem’s dynamics.

This study aims to discuss the numerical solutions of weakly singular Volterra and Fredholm integral equations, which are used to model the problems like heat conduction in engineering and the electrostatic potential theory, using the modified Lagrange polynomial interpolation technique combined with the biconjugate gradient stabilized method (BiCGSTAB). The framework for the existence of the unique solutions of the integral equations is provided in the context of the Banach contraction principle and Bielecki norm.

The authors have applied the modified Lagrange polynomial method to approximate the numerical solutions of the second kind of weakly singular Volterra and Fredholm integral equations.

Approaching the interpolation of the unknown function using the aforementioned method generates an algebraic system of equations that is solved by an appropriate classical technique. Furthermore, some theorems concerning the convergence of the method and error estimation are proved. Some numerical examples are provided which attest to the application, effectiveness and reliability of the method. Compared to the Fredholm integral equations of weakly singular type, the current technique works better for the Volterra integral equations of weakly singular type. Furthermore, illustrative examples and comparisons are provided to show the approach’s validity and practicality, which demonstrates that the present method works well in contrast to the referenced method. The computations were performed by MATLAB software.

The convergence of these methods is dependent on the smoothness of the solution, it is challenging to find the solution and approximate it computationally in various applications modelled by integral equations of non-smooth kernels. Traditional analytical techniques, such as projection methods, do not work well in these cases since the produced linear system is unconditioned and hard to address. Also, proving the convergence and estimating error might be difficult. They are frequently also expensive to implement.

There is a great need for fast, user-friendly numerical techniques for these types of equations. In addition, polynomials are the most frequently used mathematical tools because of their ease of expression, quick computation on modern computers and simple to define. As a result, they made substantial contributions for many years to the theories and analysis like approximation and numerical, respectively.

This work presents a useful method for handling weakly singular integral equations without involving any process of change of variables to eliminate the singularity of the solution.

To the best of the authors’ knowledge, the authors claim the originality and effectiveness of their work, highlighting its successful application in addressing weakly singular Volterra and Fredholm integral equations for the first time. Importantly, the approach acknowledges and preserves the possible singularity of the solution, a novel aspect yet to be explored by researchers in the field.

The purpose of this study is to provide a review of the existing research landscape on work-life balance and women’s career motivation. It examines the relationship between work-life balance and career motivation in the context of Indian women. Specifically, it explores how the work-life balance of women influences the motivational aspects of their careers.

The research uses a systematic literature review to identify and analyze relevant literature on work-life balance and women’s career motivation among Indian women from the Scopus database.

The study uncovers critical insights into the connection between work-life balance and women’s career decisions. It gives insight on how work-life balance significantly impacts women’s career choices. The SLR reveals a notable and consistent upward trend in the domains of work-life balance and career motivation among women.

The findings of this study can inform organizations in tailoring policies that foster women’s career growth while simultaneously supporting a healthy work-life balance. In addition, the research can empower women to make informed decisions about their careers and personal lives. Ultimately, it contributes to creating a more inclusive and gender-equitable work environment, promoting both women’s career aspirations and their overall well-being.

This research stands out in its examination of the relationship between work-life balance and women’s career motivation, particularly in the unique context of Indian women. While previous studies have explored these topics individually, this research bridges the gap by investigating their interplay. Moreover, the application of a systematic literature review approach to these variables in the context of Indian women represents a novel contribution.

Nowadays, the competition is not only emerging from within the banking sector, but nonbanking companies like nonbanking financial companies (NBFCs) and FinTech are also growing in size and numbers, offering innovative financial products and services, giving a stiff competition to Indian banks. Thus, this study aims to investigate whether competition from within and outside the banking sector enhances or reduces the financial stability of the banking industry.

The study uses Herfindahl–Hirschman index to measure market share and Z score to measure financial stability. The study further examines the role of NBFCs and FinTech companies in impacting the financial stability by introducing variables like innovation, cybercrimes, systemically important institutions, etc. Thereafter, panel regression has been applied.

Empirical results show a positive relation of market share with financial stability, implying that increased competition in the Indian banking industry erodes the market power, adversely affecting the profit margins which encourages banks to take more risk and which may impact financial stability. The study shows a positive impact of innovation on financial stability which implies that the competition is acting as an enabler for banks. The authors find a negative relation of systemic important NBFCs with financial stability. The authors observe a negative association of cybercrimes with financial stability, reflecting that competition emerging from FinTech sector has exposed banks to new risks.

The policymakers should make sure that the competition of banks with other financial institutions, such as FinTech sector, remains healthy; otherwise, it can jeopardize the entire financial system. It is for the policymakers to define a boundary for FinTech sector, as the development of this sector has exposed the banking industry to new kinds of risks potential to create financial instability. The banks should do a comprehensive check on the company to which it is granting loans, and the government should amend laws. Though big banks have huge potential, consolidations can pose challenges at a macroeconomic level.

FinTech firms are a new entrant in the financial world which are providing immense competition to the banking sector, and thus radically changing the entire financial system. Therefore, it is extremely vital to study and explore the role of NBFCs and the FinTech industry as the main variable to analyze bank competition, which to the best of the authors’ knowledge is completely missing in the previous studies.