Fixed Point Theorems Over Extended ( ϕ , ψ )‐Metric Spaces and Applications in Differential Equations

The main aim of this research endeavor is to introduce a novel type of generalized metric space, termed an extended ( ϕ , ψ )‐metric space, to establish new fixed point results and employ them to prove the existence and uniqueness of solutions to first‐order differential equations. To achieve this goal, we introduce the concepts of ( α , θ )‐admissible Banach contraction, ( α , θ )‐admissible crooked Banach contraction, and ( α , θ )‐admissible ( φ , β )‐contraction in an extended ( ϕ , ψ )‐metric space. We augment our theoretical results by providing compelling and nontrivial illustrative examples.