A Hierarchical Approach to Nonlinear Rational and \(\varphi\)–Contractive Mappings in Fuzzy Metric Spaces with Applications
Makhan Satpute *
Department of Mathematics, Govt. Narmadapuram P.G. College, Narmadapuram, M.P., India.
Manoj Ughade
Department of Mathematics, Institute for Excellence in Higher Education, M.P., India.
Rashmi Tiwari
Department of Mathematics, Mahatma Gandhi Govt. P.G. College, Itarsi, M.P., India.
Kamal Vadhawa
Department of Mathematics, Govt. Narmadapuram P.G. College, Narmadapuram, M.P., India.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we develop a unified framework of nonlinear contractive mappings in fuzzy metric spaces (\(\mathfrak{X}\), ℳ, * ) by integrating classical contraction principles such as Banach-type, Kannan-type, and Reich-type contractions with newly introduced rational and \(\varphi\)–nonlinear contractive conditions. The proposed \(\varphi\)-nonlinear contraction, governed by a control function \(\varphi\) : [0, 1] → [0, 1] satisfying \(\varphi\)(s) > s for all s ∈ (0, 1), provides a significant generalization of standard linear contraction models. Under suitable conditions, we establish existence and uniqueness of fixed points for self-mappings in complete fuzzy metric spaces and develop a hierarchy of contractions that clarifies the structural relationships among the introduced classes. The theoretical results are supported by illustrative examples and applications to nonlinear integral equations and fractional differential equations, demonstrating the effectiveness of the proposed framework in addressing problems arising in nonlinear and applied analysis.
Keywords: Fuzzy metric space, nonlinear rational contraction, control function, fixed point, integral equations