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The objective of this work is to study the periodic solutions for a class of sixth-order autonomous ordinary differential equations x(6)+(1+p2+q2)x… .+(p2+q2+p2q2)x¨+p2q2x=εF(x,ẋ,x¨,x…,x… .,x(5)), where p and q are rational numbers different from 1, 0, −1 and p ≠ q, ε is a small enough parameter and F ∈ C² is a nonlinear autonomous function.

The authors shall use the averaging theory to study the periodic solutions for a class of perturbed sixth-order autonomous differential equations (DEs). The averaging theory is a classical tool for the study of the dynamics of nonlinear differential systems with periodic forcing. The averaging theory has a long history that begins with the classical work of Lagrange and Laplace. The averaging theory is used to the study of periodic solutions for second and higher order DEs.

All the main results for the periodic solutions for a class of perturbed sixth-order autonomous DEs are presenting in the Theorem 1. The authors present some applications to illustrate the main results.

The authors studied Equation 1 which depends explicitly on the independent variable t. Here, the authors studied the autonomous case using a different approach.

This study aimed to investigate the effect of sustainability disclosure (SD) as a mediator for the relationship between corporate governance (CG) and the performance of firms listed on the Amman Stock Exchange (ASE).

The study analysed 405 reports of firms listed on the ASE from 2014 to 2018. The direct and indirect impact of governance mechanisms on the firms' performance was examined using STATA 15. A four-step procedure for testing mediation was used to determine the mediating role of SD.

The results demonstrated that the board and audit committees' effectiveness positively and significantly influences the firm's performance. Additionally, the results demonstrated that SD partially mediates the relationship between CG and the firm's performance.

Research implications – This study supported the assumptions of agency, resource dependence and stakeholder theories as the basis to explain the relationship among board’s effectiveness, audit committee’s effectiveness, sustainability report and firm performance in developing economies. In addition, the results suggested that CG helps to enhance the firm's performance and sustainability reporting. Firms providing sustainable report are deemed more responsible and attract more returns to firms. Research limitations – The study only focused on reports from five years for non-financial firms listed on the ASE to test the assumed relationship between the variables.

This study contributed to the body of knowledge by examining the mediating role of SD between CG and firm performance. Investors, managers and regulators can obtain further insights, especially those seeking to improve a firm's performance in the emerging markets, through a sound CG system and extensive sustainability reporting.

This study focused on the direct and indirect impacts of CG and firm performance in an emerging and developing economy. The study used SD as the mediating variable in examining the indirect effect.

The role of female managers has been of increasing interest among scholars in recent years, especially regarding sustainability issues. The same could be said about the usefulness of social media in non-governmental organizations (NGOs) in demonstrating accountability to their stakeholders and attracting and keeping donors and volunteers. This paper aims to meet both research interests by analyzing to what extent women in top positions can foster stakeholders' engagement via social media.

Online engagement can be proxied using data from social media to develop a measure that summarizes the main actions social media users are able to use in order to show their reactions to social media publications. Facebook data were obtained using proprietary software (Facebook data model) developed by the research team to carry out data massive extraction, processing and exploration.

The results of the multivariate analysis show that female leadership in both top and environmental-specific positions enhance social media engagement, while a higher percentage of women on the board of directors exert the opposite effect.

This study is not without limitations. First, this research is focused on a specific type of non-governmental organization (environmental NGO). Second, this study does not include economic variables such as donation income or expense structure. Third, data come only from Facebook as the leading social network.

This paper advances in the scarce knowledge about the role of women and the levels of online engagement (interactive conversations) in NGOs focused on sustainability.