Analysis of JS-contractions with applications to fractional boundary value problems. (English) Zbl 1539.47124
Summary: In this article, we modify JS-contractions by weakening the conditions on the function \(\theta\), where \(\theta : (0,\infty) \rightarrow (1,\infty)\) is a strictly increasing function. We prove fixed-point results for obtained contractions. Some examples are given to validate the results and modifications. We use our main theorem to establish the existence results for the solutions of the Atangana-Baleanu-Caputo fractional boundary value problem [A. Atangana and D. Baleanu, Therm. Sci. 20, No. 2, 763–769 (2016; doi:10.2298/TSCI160111018A)] with integral boundary conditions. We also present a new definition of \(\theta\)-Ulam stability and find the stability of our fractional boundary value problem.
MSC:
| 47N20 | Applications of operator theory to differential and integral equations |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H10 | Fixed-point theorems |
| 34A08 | Fractional ordinary differential equations |
| 39B82 | Stability, separation, extension, and related topics for functional equations |
Keywords:
\(\theta\)-contraction; modified JS-contraction; modified weak JS-contraction; existence results; \(\theta\)-Ulam stabilityReferences:
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