Creating an statement to check multiple dates between a start and end date

It'll make it much easier for people to help you if you can post code / data which we can run directly. The easiest way to do this is to use a handy R function called dput, which generates instructions to exactly recreate any R object. So you might run dput(MY_DATA), or if your data is much larger than needed to demonstrate your question, dput(head(MY_DATA)) to get the first six rows, and paste the output of that into your question. </PSA>

Here's code to generate your example data:

my_data <- data.frame(
  stringsAsFactors = FALSE,
        Start.date = c("11/12/2018", "07/03/2018", "04/06/2018", "26/07/2018"),
          End.date = c("29/11/2019", "24/04/2019", "23/04/2019", "29/08/2019"),
            Date.1 = c("08/03/2021", "08/03/2021", "08/03/2021", "08/03/2021"),
            Date.2 = c(NA, "12/09/2016", "02/10/2017", "03/08/2015"),
            Date.3 = c(NA, NA, "05/10/2018", "02/10/2017"),
            Date.4 = c(NA, NA, NA, "23/01/2017")
)

Here's a tidyverse approach to first convert your day/month/year dates into data in R's Date type using lubridate::dmy, then to compare each of Date.1 thru Date.4 against your start dates, and then finally to show if there are any 1's (within range).

library(dplyr); library(lubridate)
my_data %>%
  mutate(across(.fns = ~dmy(.x))) %>%
  mutate(across(.cols = starts_with("Date"),
                .fns = ~coalesce(.x >= Start.date & .x <= End.date, FALSE)*1)) %>%
  mutate(Change = pmax(Date.1, Date.2, Date.3, Date.4))

coalesce(..., FALSE) used here to treat NA like FALSE.

(...)*1 to convert TRUE/FALSE to 1/0.

pmax(...) to grab the largest of the 1/0's, i.e. "are there any 1's?"

Edit: alternative to leave Date columns intact:

my_data %>%
  mutate(across(.fns = ~dmy(.x))) %>%
  mutate(across(.cols = starts_with("Date"), 
                .names = "Check_{.col}",
                .fns = ~coalesce(.x >= Start.date & .x <= End.date, FALSE)*1)) %>%
  rowwise() %>%
  mutate(Change = max(c_across(starts_with("Check")))) %>%
  select(-starts_with("Check"))

  Start.date End.date   Date.1     Date.2     Date.3     Date.4     Change
  <date>     <date>     <date>     <date>     <date>     <date>      <dbl>
1 2018-12-11 2019-11-29 2021-03-08 NA         NA         NA              0
2 2018-03-07 2019-04-24 2021-03-08 2016-09-12 NA         NA              0
3 2018-06-04 2019-04-23 2021-03-08 2017-10-02 2018-10-05 NA              1
4 2018-07-26 2019-08-29 2021-03-08 2015-08-03 2017-10-02 2017-01-23      0

library(tidyverse)
library(lubridate)

df <- read.table(textConnection("start_date;end_date;date_1;date_2;date_3;date_4
11/12/2018;29/11/2019;08/03/2021;NA;NA;NA
07/03/2018;24/04/2019;08/03/2021;12/09/2016;NA;NA
04/06/2018;23/04/2019;08/03/2021;02/10/2017;05/10/2018;NA
26/07/2018;29/08/2019;08/03/2021;03/08/2015;02/10/2017;23/01/2017"),
                 sep=";",
                 header = TRUE)
df %>%
  mutate(
    across(everything(), lubridate::dmy),
    change = ((date_1 > start_date & date_1 < end_date) |
                (date_2 > start_date & date_2 < end_date) |
                (date_3 > start_date & date_3 < end_date)
    ) %>%
      coalesce(FALSE) %>%
      as.integer()
  )
#>   start_date   end_date     date_1     date_2     date_3     date_4 change
#> 1 2018-12-11 2019-11-29 2021-03-08       <NA>       <NA>       <NA>      0
#> 2 2018-03-07 2019-04-24 2021-03-08 2016-09-12       <NA>       <NA>      0
#> 3 2018-06-04 2019-04-23 2021-03-08 2017-10-02 2018-10-05       <NA>      1
#> 4 2018-07-26 2019-08-29 2021-03-08 2015-08-03 2017-10-02 2017-01-23      0