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| Fixed Point Theorems through Equivalent Distance with Application to Fractional IVP |
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| KeyWord:Fixed point, Banach contractions, fractional differential equation, equivalent distance |
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| Abstract: |
| We establish new fixed-point theorems for operators on equivalent metric spaces $\mathcal{E}_{A,B}$-distances. For a family of mappings $\{\mathcal{T}^t\}$ satisfying $\mathcal{E}(t,\mathcal{T}^t x,\mathcal{T}^t y) \leq k(t)\mathcal{E}(t,x,y)$, we prove existence and uniqueness of fixed points under the condition that $\sup k(t) < 1$ along some sequence. This framework significantly extends classical contraction principles while maintaining their convergence guarantees. As a principal application, we analyze fractional differential equations where our approach naturally handles non-local operators. The theory yields new existence and uniqueness results for Caputo-type fractional initial value problems, demonstrating how $\mathcal{E}_{A,B}$-distances can capture evolving metric structures in analysis. Several examples illustrate the computational verifiability of our conditions in concrete function spaces. |
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