**6. Conclusions**

In the present paper, we introduced two new types of contractions: (*<sup>α</sup>*,*β*)-type *ϑ*-contraction and (*<sup>α</sup>*,*β*)-type rational *ϑ*-contraction. Based on their applications we proved new fixed points theorems. These results generalized some known ones from fixed point theory. To support our results, we provided two non trivial examples. The obtained results are noteworthy contributions to the current results of literature in the theory of fixed points. In this field, one can establish (*<sup>α</sup>*,*β*)-type *ϑ*-contraction and (*<sup>α</sup>*,*β*)-type rational *ϑ*-contraction for the multivalued mappings in the perspective of complete metric spaces and generalized metric spaces. To illustrate the application of the new fixed point theorems, we considered an integral boundary value problem for a Caputo fractional scalar equation of order from the interval (1,2) and proved the existence of the solution.

**Author Contributions:** Conceptualization, H.Z., H.A.F., and J.A.; Methodology, H.Z., H.A.F., S.H., and J.A.; Validation, H.Z., H.A.F., S.H., and J.A.; Formal Analysis, H.Z., H.A.F., S.H., and J.A.; Writing—Original Draft Preparation, H.Z., H.A.F., S.H., and J.A.; Writing—Review and Editing, H.Z., H.A.F., S.H., and J.A.; Funding Acquisition, J.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors received no direct funding for this work.

**Acknowledgments:** The authors are thankful to the editor and the referees for their useful comments and suggestions. **Conflicts of Interest:** The authors declare no conflict of interest.
