Notes on Krasnoselskii-type fixed-point theorems and their application to fractional hybrid differential problems

Abstract: In this paper we prove a new version of Kransoselskii's fixed-point theorem under a (ψ, θ, ϕ)-weak contraction condition. The theoretical result is applied to prove the existence of a solution of the following fractional hybrid differential equation involving the Riemann-Liouville differential and integral operators orders of 0<α<1 and β>0:

where D^α is the Riemann-Liouville fractional derivative order of α, I^β is Riemann-Liouville fractional integral operator order of β>0, J=[t₀, t₀+a], for some fixed t₀∈ ℝ, a>0 and the functions f : J × ℝ → ℝ and g : J × ℝ × ℝ → ℝ satisfy certain conditions. An example is also furnished to illustrate the hypotheses and the abstract result of this paper.

Key Words and Phrases: Fixed-point theorem, Riemann-Liouville fractional derivative, hybrid initial value problem.

2010 Mathematics Subject Classification: 26A33, 34A08, 34A12, 47H07, 47H10.

Published on-line: July 1st, 2021.

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	Notes on Krasnoselskii-type fixed-point theorems and their application to fractional hybrid differential problems


	Fixed Point Theory, Volume 22, No. 2, 2021, 465-480, July 1st, 2021 DOI: 10.24193/fpt-ro.2021.2.31 Authors: H. Akhadkulov, T.Y. Ying, A.B. Saaban, M.S. Noorani and H. Ibrahim Abstract: In this paper we prove a new version of Kransoselskii's fixed-point theorem under a (ψ, θ, ϕ)-weak contraction condition. The theoretical result is applied to prove the existence of a solution of the following fractional hybrid differential equation involving the Riemann-Liouville differential and integral operators orders of 0<α<1 and β>0: where D^α is the Riemann-Liouville fractional derivative order of α, I^β is Riemann-Liouville fractional integral operator order of β>0, J=[t₀, t₀+a], for some fixed t₀∈ ℝ, a>0 and the functions f : J × ℝ → ℝ and g : J × ℝ × ℝ → ℝ satisfy certain conditions. An example is also furnished to illustrate the hypotheses and the abstract result of this paper. Key Words and Phrases: Fixed-point theorem, Riemann-Liouville fractional derivative, hybrid initial value problem. 2010 Mathematics Subject Classification: 26A33, 34A08, 34A12, 47H07, 47H10. Published on-line: July 1st, 2021. Abstract pdf Fulltext pdf Back to volume's table of contents