Exchange hybrid Unable to send mail between Office 365 and On Prem Exchange 2010 SP3

We are configuring exchange 2010 SP3   hybrid setup. when we try to send mail from office 365 user to on premises  users get following error

How to fix this issue..???

Delivery has failed to these recipients or groups:
Dixon (
The domain name in the email address is incorrect. Check the email address.

Diagnostic information for administrators:
Generating server:
Remote Server returned '554 5.4.4 SMTPSEND.DNS.NonExistentDomain; nonexistent domain'
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Vasil Michev (MVP)Commented:
That's most likely due to error in the outbound connector configuration. Check the connector settings, if you have made any changes recently make sure to rerun the HCW. You can also just run the connector wizard in EO, it will ask you to verify the settings at the end and should detect most issues.
harisaboobakerAuthor Commented:
The connector have default settings  ...we don't change anything in office 365 outbound connector

inbound and external mail flow are working fine .only office 365 to on premise mail flow are make above error. 365         to  other Domain -------------------> Working
2.other domain  to   office 365       -------------------->Working
3.On premise      to  office 365       --------------------->Working
4.Office 365         to    On  Premise  ---------------------> Not Working

What thing are check in office 365 outbound Connector
harisaboobakerAuthor Commented:
The current scenario diagram are below ..Current diagram
currently  MX record pointed to Symantec antispam cloud service
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Vasil Michev (MVP)Commented:
Might help if you post the connector settings.

To run the validation, go to the ECP -> Mail flow -> Connectors -> double-click the on-prem connector -> Next next next until you are presented with the Validate option, then run the validation and post any errors here.

Also, run a message trace in EO, see what it shows. Oh, and double-check if the domain intent is set to Internal Relay.
harisaboobakerAuthor Commented:
when we try to validate office 365 outbound connector (office 365 to on premise connector) validation fail. outbound connector validation test screen short are below...

harisaboobakerAuthor Commented:
The smart host name of office 365 outbound connector is FQDN of on premise Exchange server name (Local name)
 This is automatically take from when we specifying the FQDN of on premise Hub transport server in  Hybrid Configuration wizard

harisaboobakerAuthor Commented:
is the smart host name of office 365 outbound connector must be public name of on premise exchange server???
Vasil Michev (MVP)Commented:
Yes, use the external FQDN. EO cannot resolve the .loc name, thus the validation and mail flow will not work.

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