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In this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequences are introduced in the context of modular function spaces and proven to converge to the fixed point of a multivalued map T such that PρT, an associate multivalued map, is a ρ-contractive-like mapping.

The concepts of relative ρ-stability and weak ρ-stability are introduced, and conditions in which these multistep iterations are relatively ρ-stable, weakly ρ-stable and ρ-stable are established for the newly introduced strong ρ-quasi-contractive-like class of maps.

Noor type, Ishikawa type and Mann type iterative sequences are deduced as corollaries in this study.

The results obtained in this work are complementary to those proved in normed and metric spaces in the literature.

Healthcare providers are under pressure due to increasing and more complex demands for services. Increased pressure on budgets and human resources adds to an ever-growing problem set. Competent leaders are in demand to ensure effective and well-performing healthcare organisations that deliver balanced results and high-quality services. Researchers have made significant efforts to identify and define determining competencies for healthcare leadership. Broad terms such as competence are, however, inherently at risk of becoming too generic to add analytical value. The purpose of this study is to suggest a holistic framework for understanding healthcare leadership competence, that can be crucial for operationalising important healthcare leadership competencies for researchers, decision-makers as well as practitioners.

In the present study, a critical interpretive synthesis (CIS) was conducted to analyse competency descriptions for healthcare leaders. The descriptions were retrieved from peer reviewed empirical studies published between 2010 and 2022 that aimed to identify healthcare services leadership competencies. Grounded theory was utilised to code the data and inductively develop new categories of healthcare leadership competencies. The categorisation was then analysed to suggest a holistic framework for healthcare leadership competence.

Forty-one papers were included in the review. Coding and analysing the competence descriptions resulted in 12 healthcare leadership competence categories: (1) character, (2) interpersonal relations, (3) leadership, (4) professionalism, (5) soft HRM, (6) management, (7) organisational knowledge, (8) technology, (9) knowledge of the healthcare environment, (10) change and innovation, (11) knowledge transformation and (12) boundary spanning. Based on this result, a holistic framework for understanding and analysing healthcare services leadership competencies was suggested. This framework suggests that the 12 categories of healthcare leadership competencies include a range of knowledge, skills and abilities that can be understood across the dimension personal – and technical, and organisational internal and – external competencies.

This literature review was conducted with the results of searching only two electronic databases. Because of this, there is a chance that there exist empirical studies that could have added to the development of the competence categories or could have contradicted some of the descriptions used in this analysis that were assessed as quite harmonised. A CIS also opens for a broader search, including the grey literature, books, policy documents and so on, but this study was limited to peer-reviewed empirical studies. This limitation could also have affected the result, as complex phenomenon such as competence might have been disclosed in greater details in, for example, books.

The holistic framework for healthcare leadership competences offers a common understanding of a “fuzzy” concept such as competence and can be used to identify specific competency needs in healthcare organisations, to develop strategic competency plans and educational programmes for healthcare leaders.

This study reveals a lack of consensus regarding the use and understanding of the concept of competence, and that key competencies addressed in the included papers are described vastly different in terms of what knowledge, skills and abilities they entail. This challenges the operationalisation of healthcare services leadership competencies. The proposed framework for healthcare services leadership competencies offers a common understanding of work-related competencies and a possibility to analyse key leadership competencies based on a holistic framework.