**6. Conclusions**

In our study, Firstly, we established the equivalence between problem (5) and the Volterra integral equation. Secondly, Banach and Schaefer's fixed point theorems were used to establish the existence and uniqueness solutions for implicit fractional pantograph differential equation which involves *φ*-Hilfer fractional derivatives. Based on *φ*-Hilfer fractional derivatives, we found that the stability of Ulam-Hyers and generalized Ulam-Hyers allowed on the implicit fractional pantograph differential equation, supplemented with a nonlocal Riemann-Liouville condition. In addition, examples were given to illustrate our main results. Moreover, it worthy to mention the following remarks:


**Author Contributions:** The authors contributed equally in writing this article. All authors have read and agreed to the published version of the manuscript.

**Funding:** Petchra Pra Jom Klao Doctoral Scholarship for Ph.D. program of King Mongkut's University of Technology Thonburi (KMUTT). The Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. King Mongkut's University of Technology North Bangkok, Contract no. KMUTNB-63-KNOW-033.

**Acknowledgments:** The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. The first author was supported by the "Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi" (Grant No. 13/2561). Moreover, this research work was financially supported by King Mongkut's University of Technology Thonburi through the KMUTT 55th Anniversary Commemorative Fund.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
