Expansive-Type Fixed Point Theorems in Double Controlled Metric Type Spaces with an Integral Equation Application
Ranjana Maravi *
Department of Mathematics, Institute for Excellence in Higher Education (IEHE), Bhopal, Madhya Pradesh, India.
Manoj Ughade
Department of Mathematics, Institute for Excellence in Higher Education (IEHE), Bhopal, Madhya Pradesh, India.
S. S. Shrivastava
Department of Mathematics, Institute for Excellence in Higher Education (IEHE), Bhopal, Madhya Pradesh, India.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we develop an expansive fixed point theory in the framework of double controlled metric type spaces governed by two control functions. By combining surjectivity with a backward inverse–iteration technique, we establish existence and uniqueness results for several classes of expansive-type mappings, including Reich (a,b)–expansive, Dass–Gupta rational expansive, Θ–weighted expansive, orbitally localized expansive, and (α,β)–mixed expansive mappings.
The theoretical results are supported by illustrative examples and are applied to a nonlinear Fredholm integral equation by verifying an appropriate expansive condition for the associated integral operator. The results show that expansive methods can provide solvability beyond classical contraction-based approaches.
Keywords: Double controlled metric type space, expansive mapping, backward iteration, fixed point, Fredholm integral equation