if-statement with ((...)) string comparison not working in a bash script

(( .. )) is an arithmetic construct, and in arithmetic contexts, a string is taken as the name of a variable, and the value of that variable is used. This happens after $var expansions are expanded, so your script looks at variables called nfosys and Infosys. With both variables unset, both are taken to be zero, i.e. equal. But:

$ str1=foo str2=bar foo=1
$ (( $str1 == $str2 )) && echo true || echo false
false   

See e.g. Bash's manual on shell arithmetic:

Shell variables are allowed as operands; parameter expansion is performed before the expression is evaluated. Within an expression, shell variables may also be referenced by name without using the parameter expansion syntax. A shell variable that is null or unset evaluates to 0 when referenced by name without using the parameter expansion syntax.

For string comparison, use

[ "$STR_1" = "$STR_2" ]   # in any POSIX shell, or

[[ $STR_1 = "$STR_2" ]]   # in Bash/ksh/zsh

The former needs quotes due to word-splitting, the latter needs it only on the right-hand side (in bash / ksh) for $STR_2 not to be taken as a pattern.

Also see: What is the difference between the Bash operators [[ vs [ vs ( vs ((?

Using [] instead would give the wanted behavior:

STR_1="nfosys"
STR_2="Infosys"

if [ $STR_1 = $STR_2 ]
then
        echo "Strings are equal"
else
        echo "Strings are not equal"
fi

Output:

+ STR_1=nfosys
+ STR_2=Infosys
+ '[' nfosys = Infosys ']'
+ echo 'Strings are not equal'
Strings are not equal

A good rule of thumb is to also quote your variables, just in case they end up being wrongly interpreted or be expanded/split:

STR_1="nfosys"
STR_2="Infosys"

if [ "$STR_1" = "$STR_2" ]
then
        echo "Strings are equal"
else
        echo "Strings are not equal"
fi

Also, while [] is considered POSIX, using [[]] if your main target is bash, is better.

Editor's note: POSIX only knows single =.

Thanks All. i have restarted my session and change to single square bracket -

#!/usr/bin/bash

STR_1="nfosys"
STR_2="Infosys"

if [ $STR_1 == $STR_2 ]
then
        echo "Strings are equal"
else
        echo "Strings are not equal"
fi

If i execute with sh test.sh then getting error -

root@ip:~# sh test.sh 
test.sh: 6: [: nfosys: unexpected operator
Strings are not equal
root@ip:~# bash test.sh 
Strings are not equal

If i execute with bash then display correct output without any error

root@ip:~# bash test.sh 
Strings are not equal

Weird before posting question i had tried both ways - sh/bash and with ()/[]. But after restarting session seems working fine.

Editor's refreshed code adhering to POSIX, and some good codex (lower-case variable names, error print to stderr, fixed major == issue, etc.):

#!/bin/sh

str_1=nfosys
str_2=Infosys

if [ "$str_1" = "$str_2" ]; then
    echo "Strings are equal"
else
    echo "Strings are not equal" >&2
    exit 1
fi