Is a person who believes that it's possible to overcome their depression more likely to overcome it then a person who believes that there's no hope overcoming their depression?
The answer depends on many factors; and if therapy is received, as with all mental health conditions, it also depends on the form of therapy the depressed client receives and the rapport between client and the therapist.
CBT is often the go to form of therapy within IAPT services advocating for therapies which they can easily measure outcomes. However, as pointed out in my answer on the limitations of CBT,
There are a fair few factors which can prevent CBT from becoming effective, and a trained and certified CBT practitioner will be able to assess the suitability of CBT. If they operate ethically, they will not go ahead with providing CBT to someone who it would not benefit.
One of the biggest factors which doesn't allow CBT to work is that if the client is not willing or able to challenge their thoughts and behaviours, then CBT will not be effective.
There are articles covering the effect of positive expectations creating positive outcomes. Scheier & Carver (1993) pointed out that
People who see desired outcomes as attainable continue to strive for those outcomes, even when progress is slow or difficult. When outcomes seem significantly unattainable, people withdraw their effort and disengage themselves from their goals. Thus, people's expectations provide a basis for engaging in one or two very different classes of behavior; continued striving versus giving up.
Belief that your depression can be cured will help by a considerable margin.
There is evidence consistent with the suggestion of the cognitive model of depression that certain negative cognitions can produce and maintain the state of depression. There is also good evidence that depressed mood affects the relative accessibility of positive and negative cognitions. Thus, negative cognitions appear to produce depression, and, conversely, depression increases the probability of just those cognitions which will cause further depression. This reciprocal relationship between depression and cognition may form the basis of a vicious cycle which will perpetuate and intensify depression. (Teasdale, 1983).
Owen Lightsey (1994) also looked at positive automatic thoughts (PATs) as a "stress buffer".
Is it possible that positive automatic thoughts (PATs) may similarly protect against depression? Such a finding could have considerable clinical significance. Cognitive therapies tend to focus on elimination of negative cognitions and on replacing maladaptive beliefs with adaptive beliefs (e.g., Beck et al., 1979); there is little focus on PATs per se. If PATs are found to uniquely buffer the effects of stress, straightforward training in generation of PATs could help to confer resilience
Later in the article he pointed out that things can be a little complicated:
Results for the overall sample confirmed the hypothesis that PATs predict future happiness: Higher frequency of pre-existing PATs was associated with greater future happiness. This finding suggests that PATs have an impact not only on immediate well-being, as suggested by Goodhart (1985), but also on future well-being. Contrary to hypotheses, however, the overall PATs x Stressful Event interaction did not predict either depression or happiness. Unexpectedly, a subtype of PATs — PATs about social self-worth — behaved in a manner consistent with a stress buffer role: The interaction of PATs about social self-worth with stressful events predicted unique variance in depression, above the variance accounted for by other Factor x Stress interactions.
It won't necessarily mean you are doomed if you feel there is no end to your mental health problems; it just means it will take longer to help you. All depression can be dealt with effectively and successfully given enough time and proper support.
Beck, A. T., Rush, A. J., Shaw, B. F., & Emery, G. (1979). Cognitive therapy of depression. New York: Guilford Press.
Goodhart, D. E. (1985). Some psychological effects associated with positive and negative thinking about stressful event outcomes: Was Pollyanna right?. Journal of personality and social psychology, 48(1), 216. DOI: 10.1037/0022-35188.8.131.52
Lightsey, O. R. (1994). " Thinking positive" as a stress buffer: The role of positive automatic cognitions in depression and happiness. Journal of Counseling Psychology, 41(3), 325. DOI: 10.1037/0022-0184.108.40.2065
Scheier, M. F., & Carver, C. S. (1993). On the power of positive thinking: The benefits of being optimistic. Current directions in psychological science, 2(1), 26-30. DOI: 10.1111/1467-8721.ep10770572
Teasdale, J. D. (1983). Negative thinking in depression: Cause, effect, or reciprocal relationship?. Advances in Behaviour Research and Therapy, 5(1), 3-25. DOI: 10.1016/0146-6402(83)90013-9