Existence and uniqueness of positive solution of a nonlinear differential equation with higher order Erdélyi-Kober operators

Kangqun Zhang; Kangqun Zhang

doi:10.3934/math.2024067

AIMS Mathematics

2024, Volume 9, Issue 1: 1358-1372. doi: 10.3934/math.2024067

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Research article

Existence and uniqueness of positive solution of a nonlinear differential equation with higher order Erdélyi-Kober operators

Kangqun Zhang ^,

School of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, China

Received: 30 October 2023 Revised: 27 November 2023 Accepted: 28 November 2023 Published: 08 December 2023
MSC : 26A33, 26D07, 34A08, 34A12

In this paper, the initial value problem of a nonlinear differential equation with higher order Caputo type modification of the Erdélyi-Kober fractional derivatives was studied. Based on the transmutation method, the well-posedness of initial value problem of the higher order linear model was proved and an explicit solution was presented. Then some new Gronwall type inequalities involving Erdélyi-Kober fractional integral were established. By applying these results and some fixed point theorems, the existence and uniqueness of the positive solution of the nonlinear differential equation were proved. The method is applicable to the fractional differential equation with any order $ \gamma\in (n-1, n] $.
- fractional differential equation,
- Caputo type fractional derivative,
- Erdélyi-Kober operators,
- Gronwall type integral inequality,
- positive solution
Citation: Kangqun Zhang. Existence and uniqueness of positive solution of a nonlinear differential equation with higher order Erdélyi-Kober operators[J]. AIMS Mathematics, 2024, 9(1): 1358-1372. doi: 10.3934/math.2024067

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Abstract

In this paper, the initial value problem of a nonlinear differential equation with higher order Caputo type modification of the Erdélyi-Kober fractional derivatives was studied. Based on the transmutation method, the well-posedness of initial value problem of the higher order linear model was proved and an explicit solution was presented. Then some new Gronwall type inequalities involving Erdélyi-Kober fractional integral were established. By applying these results and some fixed point theorems, the existence and uniqueness of the positive solution of the nonlinear differential equation were proved. The method is applicable to the fractional differential equation with any order $ \gamma\in (n-1, n] $.

References

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