Denumerably many positive solutions for fractional order boundary value problems

Prasad, K. Rajendra, Khuddush, Mahammad and Rashmita, M.

Full PDF

creative_2020_29_2_191_203

DOI: https://doi.org/10.37193/CMI.2020.02.11

Issue no: Vol 29/2020 no. 2 Tags: positive solutions, Cone, Fractional derivative, homeomorphism, homomorphism, Krasnoselskii’s fixed point theorem

Description
Additional information

The sufficient conditions are derived for the existence of denumerably many positive solutions for fractional order boundary value problem

$\begin{aligned} &\mathfrak{D}^\varepsilon_{0^+}\Big[\upphi\big[\mathfrak{D}^\eta_{0^+}\upomega(t)\big]\Big]+f(\upomega(t))=0,~0<t<1,\\ &\hskip0.9cm\upomega(0)=\upomega^{\prime}(0)=\mathfrak{D}^\eta_{0^+} \upomega(0)=0,\\ &\hskip1.5cm\mathfrak{D}^\varepsilon_{0^+}\upomega(1)=I^{\delta}_{0^+}\upomega(1) \end{aligned}$

where $\mathfrak{D}^\varepsilon_{0^+},\,\mathfrak{D}^\eta_{0^+}$ denote fractional derivatives of Riemann-Liouville type with $0 < \varepsilon < 1,$ $2 < \eta \leq 3,$ $I^{\delta}_{0^+}$ $({\delta}\in \mathbb{R})$ denotes Riemann-Liouville fractional integral, $\upphi:\mathbb{R}\rightarrow\mathbb{R}$ is an increasing homeomorphism and positive homomorphism operator(IHPHO), by applying Hölder’s inequality and Krasnoselskii’s cone fixed point theorem in a Banach space.