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 H\ddot{o}lder’s inequality and Krasnoselskii’s cone fixed point theorem in a Banach space.

 

 

 

 

 

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Author(s)

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