Positive solutions for nonlinear fractional boundary value problems

Abstract

This article deals with the following fractional boundary value problems (FBVP) consisting fractional and right focal boundary conditions

$$\begin{aligned}{} & {} (D^{\mu }_{a} \varrho )(\zeta ) + f_1(\zeta , \varrho (\zeta )) = 0,~ \varrho (a) = 0,~ (D^{\nu }_{a}\varrho )(b) = 0,\quad a< \zeta < b, \end{aligned}$$

(1)

$$\begin{aligned}{} & {} (D^\mu _0 \varrho )(\zeta ) + f_2(\zeta , \varrho (\zeta )) = 0,~ \varrho (0) = 0,~ (D^{\mu -1}_0 \varrho )(1) = 0,\quad 0< \zeta < 1, \end{aligned}$$

(2)

where $D^{\mu }$ denotes the Riemann–Liouville (RL) fractional differential operator with $1 < \mu \le 2,~\text {and}~ 0\le \nu \le 1$. $f_1(\zeta , \varrho (\zeta )): [a, b]\times [a, \infty ) \rightarrow [a, \infty )$ and $f_2(\zeta , \varrho (\zeta )): [0, 1]\times [0, \infty ) \rightarrow [0, \infty )$ are continuous functions. Here, we establish positive solutions for FBVPs (1) and (2) by using Krasnosel’ skii theorem and Leggett–Williams theorem. The established results are validated by examples given in the application section.