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Raymond Smullyan has several works, including several books, that explore diverse generalizations of the kind of fixed-points that your question is about.

Run the following search on MathSciNet: ti:(self-reference) au:(smullyan)

You will find the works I am talking about (8 in all). Perhaps the main one is:

• Smullyan, Raymond M., Diagonalization and self-reference, Oxford Logic Guides. 27. Oxford: Clarendon Press. xv, 396 p. (1994). ZBL0810.03001.

I am not sure if your exact fixed-point result is in there, but he has many very general fixed points of a similar nature.

Let me also mention that I have a short note on self-reference results that I have used in some graduate seminars, A review of the Gödel fixed-point lemma with generalizations and applications, where I discuss the double fixed point and larger fixed-point systems, along with several applications, including the universal algorithm and other such results. Some of the fixed-point results are simpler than yours and you may know of them already, but the applications are interesting and other readers may find the presentation useful.

Raymond Smullyan has several works, including several books, that explore diverse generalizations of the kind of fixed-points that your question is about.

Run the following search on MathSciNet: ti:(self-reference) au:(smullyan)

You will find the works I am talking about (8 in all). Perhaps the main one is:

• Smullyan, Raymond M., Diagonalization and self-reference, Oxford Logic Guides. 27. Oxford: Clarendon Press. xv, 396 p. (1994). ZBL0810.03001.

I am not sure if your exact fixed-point result is in there, but he has many very general fixed points of a similar nature.

Let me also mention a short elementary note that I wrote for use in some graduate seminars, A review of the Gödel fixed-point lemma with generalizations and applications, where I discuss the double fixed point and larger fixed-point systems, along with several applications, including the universal algorithm and other such results. Some of the fixed-point results are simpler than yours and you may know of them already, but the applications are interesting and other readers may find the presentation useful.

Raymond Smullyan has several works, including several books, that explore diverse generalizations of the kind of fixed-points that your question is about.

Run the following search on MathSciNet: ti:(self-reference) au:(smullyan)

You will find the works I am talking about (8 in all). Perhaps the main one is:

• Smullyan, Raymond M., Diagonalization and self-reference, Oxford Logic Guides. 27. Oxford: Clarendon Press. xv, 396 p. (1994). ZBL0810.03001.

I am not sure if your exact fixed-point result is in there, but he has many very general fixed points of a similar nature.

On a much more elementary nature, I have a short note on self-reference results that I have used in some graduate seminars, A review of the Gödel fixed-point lemma with generalizations and applications, where I discuss the double fixed point and larger fixed-point systems. But these are probably below yours in complexity and you likely know all of it already, but I include it here in case other readers find it useful.

Raymond Smullyan has several works, including several books, that explore diverse generalizations of the kind of fixed-points that your question is about.

Run the following search on MathSciNet: ti:(self-reference) au:(smullyan)

You will find the works I am talking about (8 in all). Perhaps the main one is:

• Smullyan, Raymond M., Diagonalization and self-reference, Oxford Logic Guides. 27. Oxford: Clarendon Press. xv, 396 p. (1994). ZBL0810.03001.

I am not sure if your exact fixed-point result is in there, but he has many very general fixed points of a similar nature.

Let me also mention that I have a short note on self-reference results that I have used in some graduate seminars, A review of the Gödel fixed-point lemma with generalizations and applications, where I discuss the double fixed point and larger fixed-point systems, along with several applications, including the universal algorithm and other such results. Some of the fixed-point results are simpler than yours and you may know of them already, but the applications are interesting and other readers may find the presentation useful.

Raymond Smullyan has several works, including several books, that explore diverse generalizations of the kind of fixed-points that your question is about.

Run the following search on MathSciNet: ti:(self-reference) au:(smullyan)

You will find the works I am talking about (8 in all). Perhaps the main one is:

• Smullyan, Raymond M., Diagonalization and self-reference, Oxford Logic Guides. 27. Oxford: Clarendon Press. xv, 396 p. (1994). ZBL0810.03001.

Raymond Smullyan has several works, including several books, that explore diverse generalizations of the kind of fixed-points that your question is about.

Run the following search on MathSciNet: ti:(self-reference) au:(smullyan)

You will find the works I am talking about (8 in all). Perhaps the main one is:

• Smullyan, Raymond M., Diagonalization and self-reference, Oxford Logic Guides. 27. Oxford: Clarendon Press. xv, 396 p. (1994). ZBL0810.03001.

I am not sure if your exact fixed-point result is in there, but he has many very general fixed points of a similar nature.

On a much more elementary nature, I have a short note on self-reference results that I have used in some graduate seminars, A review of the Gödel fixed-point lemma with generalizations and applications, where I discuss the double fixed point and larger fixed-point systems. But these are probably below yours in complexity and you likely know all of it already, but I include it here in case other readers find it useful.