#### Liu Yang, Hui Zhou, Chunfang Shen, Multiple positive solutions of a boundary value problem of the fractional thermostat model, Vol. 2019 (2019), Article ID 41, pp. 1-10

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DOI: 10.23952/jnfa.2019.41

Received November 19, 2018; Accepted September 9, 2019; Published September 27, 2019

Abstract. In this paper, by means of the Avery-Peterson fixed point theorem, we establish the existence result of at least triple positive solutions of a boundary value problem of the fractional thermostat model which the derivative of unknown function is involved in the nonlinear term explicitly. An example illustrating our main result is given. Our results complements previous work in the area of the fractional thermostat model.

Liu Yang, Hui Zhou, Chunfang Shen, Multiple positive solutions of a boundary value problem of the fractional thermostat model, Journal of Nonlinear Functional Analysis, Vol. 2019 (2019), Article ID 41, pp. 1-10.

Addendum posted by the editor on August 26, 2020

The main results presented in this paper are not correct. In the proof of Theorem 3.5 the statement “then $w(t) \in AC^2[0,1]$ (see Proposition 3.1 in [23])” is not correct, the result of [23] does not apply to this situation. The authors were sent a detailed report and were invited to make corrections but have not done so. The main difficulty is that solutions of the integral equation are $C^1$ functions but are not $C^2$ functions except in special cases, see Theorem 6.26 of K. Diethelm, The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type. Lecture Notes in Mathematics No.2004. Springer-Verlag, Berlin, 2010. The definition of the Caputo differential operator from Diethelm’s book should be used and not the Definition 2.2 of this paper.