Since your injury is minor, you won't likely find a lawyer who sees a fair opportunity to get paid for his or her work in this case. Consider suing in small claims court, but threaten it first and maybe they'll come across. The complex probably has a liability insurance policy, so send the claim in writing and see if you hear from an adjuster. They may pay you something, but probably only if they really think you'll file your suit in small claims court.
You will probably have to show some prior notice for liability to attach. Consult with an attorney in TX to know for sure.
Nothing in this communication should be construed as creating an attorney client relationship. This is for informational purposes only. Attorney will take no action on your behalf unless and until a written retainer agreement is signed. There are strict time deadlines on filing claims and, as such, you are advised to consult with and retain an attorney immediately to file such claims timely or you will lose any right to recovery.
You should take photographs of all physical evidence, including the fixture and the place that it was affixed immediately after the incident. You should save any physical evidence which may show why the ceiling lamp fell. The apartment complex is responsible for providing the apartment in a reasonably safe condition, so you would need that there was some negligence in maintance of the ceiling lamp.
You should document everything (photos and journal) and check if you have receipts for the damaged property. Speak with a local lawyer about your case. Your landlord owes you a duty to keep the property in a reasonably safe condition. Falling light fixtures does not seem like a reasonably safe condition.
This response is not intended to act as legal advice. I am not licensed to practice law in any state other than the State of Illinois. No attorney-client relationship is formed until you sign an attorney-client agreement with my office.
You will want to photograph the area. The question is whether the landlord had notice about the prior defect with the installation or if there was a problem with maintenance of the area. Cases are also evaluated based on the significance of the injuries. Businesses and carriers often fight a bit harder when the injuries are minor or temporary in nature.