Csdrhrt

Descartes' fundamental truth (cogito, ergo sum) would help me accept without any doubt that I do exist. So, I accept that I do exist without any doubt.

But there's another truth that --for me-- seem a consequence of Descartes' truth: the instinct of survival. I do exist, and I seem to try to keep such existence, to persist along time. It can be expressed as this: I exist in order to exist, or I live in order for my species to continue existing, or I live in order to persist.

The fact is that I seem to try to keep such existence. If I should decide, rationally or instinctively, there is no doubt that I want to exist. Things seem to follow the same rule: I cannot easily break life or rocks, or atoms. I drink a lot of water, eat fiber, follow moral, legal rules, etc. Clearly, I really do want to exist.

I'm not asking if my understanding is correct. I want to know if there's some philosophical approach that would explain (or rebate) the instinct of survival, or the tendency of things to keep existing. There should be an explanation to this yearning. Is there any?

Descartes 'I', which exists is not his body. The self-referential term refers (or so he supposes) to an immaterial - thinking, non-extended - substance. There is nothing he can do to preserve the existence of this substance. There is a lot he can do to preserve his bodily existence.

The philosopher who comes to most readily to mind as regards self-preservation is Spinoza. In Ethics, III, proposition 6, Spinoza states : 'Each thing, in so far as it is in itself, endeavours to persevere in its being'* (Unaquaeque res, quantumin se est, in suo persevare conatur).

There are always multiple strands of assumption behind Spinoza's statements and they can't be disentangled here. But perhaps the key idea is that no particular thing can have any determinateness or individuality unless it has some power of self-maintenance, some capacity to be active in relation to other things and not merely a passive recipient or subject of effects.

Its character and individuality depends on its necessarily limited power of self-maintenance; it can be distinguished as a unitary thing with a recognizable constancy of character in so far as, although a system of parts, it succeeds in maintaining its own characteristic coherence and balance of parts. (Stuart Hampshire, Spinoza, Harmondsworth : Penguin, 1970: 77.)

*tr. Spinoza, Ethics, ed. & tr. G.H.R. Parkinson, Oxford : OUP, 2000: 171.

This is clearly a compressed answer but there is unfortunately no space to explore the full background to Spinoza's position on self-perseverance. Hampshire's book, despite its age, retains a good deal of value and you could do worse than to start with this.

I want to know if there's some philosophical approach that would
explain (or rebate) the instinct of survival, or the tendency of
things to keep existing. There should be an explanation to this
yearning. Is there any?

The question was about instinct of survival, instinct wiki stipulates that

"Any behavior is instinctive if it is performed without being based
upon prior experience (that is, in the absence of learning), and is
therefore an expression of innate biological factors. "

So biology and more specifically evolution theory should hold the answers to that question. There are many misunderstandings around evolution theory. For instance the instinct of survival isnt the same as the tendency of things to keep existing.

Such phrasing:

I exist in order to exist, or I live in order for my species to
continue existing, or I live in order to persist

could easily lead you into thinking in terms of purpose. But nature and evolution have no purpose.

The best books on the matter of evolution are the ones from Stephen Jay Gould. They are a collection of essays detailing the harduous buildup of evolution theory and the many pitfalls scientists met along the way. They easily classify as epistemology and philosophy.
Once you can think about evolution straight, the issues of purpose & survival are easily dealt with.

In Derrida's The Postcard, To Speculate--on "Freud", he refers to the Life Drive. This is the primal instinct that forms in every life form, and is so basic as to be deeply unconscious.

Derrida arrives at the Life Drive through a development of Freud's Death Drive and ends up relating it to Nietzsche's Will to Power.

All three are aspects of the basic survival instinct in various levels of depth, from the instinct of a seed to grow, to the instinct for a mind to solve problems and gain mastery of its environment.

More details here:
https://philosophy.stackexchange.com/a/41100/5154

Interesting question and answers. I agree with Manu de Hanoi that survival is hard-wired into our biology via evolution. However, I don't think I'd characterize evolutionary theory as philosophy, therefore, I'd answer your question NO.

It's interesting to note that much philosophical discourse governs questions related to killing other people and suicide. Is it OK to kill other people and, if so, under what conditions?

But it's hard to understand why philosophers would ponder the "need" to survive. They may ask WHY we exist, HOW life began and maybe even explore the meaning of life.

Descartes said "I think, therefore I am." We could almost paraphrase that - "We exist, therefore we survive."

On the other hand, people who are in great pain may wish to end their lives. Philosophers may then ask why such people should continue to live.

Your question makes more sense (to me) in this context. This type of question appears to lie primarily in the domain of religion, though it's also a common question in health care (e.g. people being kept alive artificially).

Similarly, if we want to focus on the survival of our species, then we face a paradox: our survival is jeopardized by our sheer numbers. This is another special context that has generated some philosophical discourse, though I can't cite any specific references.

Schopenhauer takes this as a central fact -- that life craves life and cultivates life. After him, Nietzsche refines it to reflect the wish is not just to exist, but to have control over one's world to some degree, either individually or as part of a collective. (People do die of purposelessness. Some lives are worse than death. But the kind that end in suicide or surrender do not seem to be about suffering, but about pointlessness, fear of being destructive, or other unsurpassed shame.)

But philosophies like Buddhism and Taoism or their Western versions based in the influence of the "pre-Socratics" influenced by the Stoic, Epicurean, Eleatic, or Cynical schools do not see this as a necessary thing, or a philosophically valuable position. We have no proof that this is necessary, good, or natural, only that we constantly observe it.

Descartes 'cogito' is a response, in form at least, to Augustine's 'Dubito ergo sum.' One aspect of this latter notion is that if we had no doubts about the good will of the universe (the existence of God for Augustine) we might not have an instinct toward survival. We might achieve the Stoic or Buddhist ideal and accept the value of whatever happened with equanimity, in complete trust. (Quakers put this in a Biblical context reconciling two directives that seem to be at odds as a way of becoming like little children while still having put away childish things. They opposed both striving and frivolity.)

A more modern version of this same thought flows through some versions of Existential Psychotherapy in the form "The will to survive is the fear of change". We are willing to risk things in proportion to how sure we can be that the result will be worthwhile. If we have total trust, we can willingly risk everything we know, and we do not need to cling to existence. Without a fear of change, we could ideally see our existence as just another fact. Then we could set aside the fear of death and act based on a broader perspective.