FIXED POINTS FOR NON-SURJECTIVE EXPANSION MAPPINGS

The contractive conditions of Popa (Demonstr. Math. 1990, 23, 213-218) were further improved for four non-surjective expansion mappings, and some common fixed point theorems under semi-compatible pairs of mappings are proved. Our main findings bring improvements to a number of results in the non-metric setting. Some implications for mathematical physics are raised with respect to physical invariants.

(D2) the w-continuity of G at some point p in X implies: lim FGxn Gp, whenever {xn} is a sequence in X such that: lim Fxn lira Gxn p, for some p in X.
n----n-REMARK 2.1  The d-complete symmetrizable space forms an importanl class of examples of d-complete topological spaces [1].REMARK 2.2   Compatibility [12] is defined in metric while semi- compatibility [11] is defined in non metric setting.Therefore, axioms D1 and D2 are independent. 3. NOTATIONS AND AUXILIARY RESULT.Throughout this paper, we will adopt the following notations" M is an arbitrary set with values in a Hausdorff topological space (X, t); N is the set of all positive integers, FI + is the set of all non-negative real numbers, and I' is the family of all functions -(FI+)3 FI+, satisfying the following properties" ( 1) (qt) is continuous on (FI+)3.
Condition (-3) is that already used by Popa [5,7] for expansion mappings.Alternative forms will be discussed below.
We also need the following important statement.PROPOSITION 3.1 Let A, B, S and T be self-maps of M, such that each of th pairs A, S and B, T are semi-compatible, and that, for all x, y in M, and (3.1) d(Sx, Ty) > v(d(Ax, By), d(Ax, Sx), d(By, Ty)).
If there exists u, v, and z in M such that Au Su Bv Tv z, then Az Bz Sz z z.

PROOF
Since A and S are semi-compatible mappings, and Au Su z by property (D1), we have Az ASu SAu Sz.From 3.1 we also have: by property (g-3-C).From this contradiction, it follows that Sz z.By symmetry, Bz=Tz =z.

MAIN RESULTS.
We now state and prove our main two theorems and emphasize some of their corollaries and related theorems.
Therefore, we have Au Su Bv Tv z, and hence, by proposition 3.1, it follows that z is a common fixed point of A, B, S, and T.

4.1-b
Unia_ueness of the common .fixedpoint.Let us suppose that there exists a second distinct common fixed point w of A, B, S, and T.Then, from relation 3.1, we have: which is a contradiction.
Hence, z is the unique common fixed point of A, B, S, and T. This completes the proof.REMARK 4.1 Theorem 4.1 improves and generalizes Theorem 1 of Popa [7] and theorem 3 of Khan, Khan and Sessa [6], to d-complete topological spaces, under semi-compatible conditions.
Two corollaries and an infering theorem are worth noting.COROLLARY 4.1 Let A, B, S, T be self-mappings of Hausdorff space M, suct that pairs A, S and B, T are semi-compatible and satisfy both conditions 4.2, 4. for all x, y in M with r a positive integer.
REMARK 4.2  It should be noted that corollary 4.1 improves and generalizes Theorem 1 of Popa [5] to this non-metric setting.
If we put b c 0 in Corollary 4.1, we obtain the following: (Ol:lOl.l.,l:lY 4.2 Let A, B, S, T be self-mappings of M, such that pairs A, S ant T are semi-compatible and satisfy both conditions 4.2, 4.3, and the following condition 4.7: (4.7):There exists a constant k (1:1+) with k > 1, such that d(Sx, Ty) >_ k.d(Ax, By), for all x, y in (M) then, A, B, S, and T have a unique common fixed point in M.
If we replace A=B S by S2, and T by TS in Theorem 4.1, then we obtain the following result in which it is important to note that the semi-compatibility coqdition is no longer necessary: THEOREM 4.2 Let S and T be self-mappings of M, such that S(M) TS(M) S(M) _ S2(M) and S(M) is d-complete.Suppose, in addition, that there exists such: (3F), Vx, y M, d(S2x, TSy)>_ t (d(Sx, Sy), d(Sx, S2x), d(Sy, TSy)).
Then S and T have a unique common fixed point in M. REMARK 4.3.aIf we define as in the proof of Corollary 4.1, then the resub obtained in this new setting improves and generalizes Theorem 2.4 of Pathak et al., (1996) [13].The original theorem of this type was proved by these authors in a complete metric space.Now let denote the family of all functions #:1:1 + -> I:1 + which are non- decreasing, upper semi-continuous from the right, with: (0) 0, (t) < and Z(n_-l--,**) n(t) < for each > 0, We finally formulate the following interesting theorem: THEOREM 4.3 Let A, B, S, T, self-mappings of M, such that pairs A, S, and B, 1 are semi-compatible and satisfying both of conditions 4.2, 4.3 and the following condition 4.8: (4.8) # (d(Sx, Ty)) _> max {d(Ax, By), d(Ax, Sx), d(By, Ty)} for all x, y in M where e ).
Then, A, B, S, T, have a unique common fixed point in M. PROOF -4-2a.Existence of a common fixed p0..int.
Let Su z for some u in X. Putting x u and y X2n+l in inequality 4.8 and then, letting limits as n -->oo, we obtain: (0) 0 _> d(Au, z) which implies that Au z.Since A(M) T(M), there is a point v in M such that Au Tv z.Again, replacing x by u and y by v in inequality 4.8, we obtain: (0) 0 d(Su, Tv) > d(Bv, z) which means that Bv z.Therefore, Au Su Bv Tv z.However, since A and S are semi-compatible mappings and Au Su z, then, by property D I, in definition 2.2, we have Az ASu SAu Sz.By property 4.8 we also have: which is a contradiction, since for each > 0: (t) < ** (for: d(Sz, z) > 0), (d(Sz, z)) < d(Sz, z) Therefore, Sz z, and by symmetry, Bz Tz z, which demonstrates the existence of z as a common fixed point of A, B, S, and T.
4-2b.Unioucncss of the common fixed t)oint.This property readily follows from property 4.8, which completes the proof.Now, the last two theorems infer from replacing the expression: max { d(Ax, By), d(Ax, Sx), d(By, Ty)} by d(Ax, By), and other terms.THEOREM 4.4  Let A, B, S, T, be mappings from M into itself, such that tM pairs A, S, and B, T are semi-compatible and fulfill on one hand the previous surjectivity is necessary to prove theorem 4.3 in complete metric space, it is not required for this purpose if one considers S(M) d-complete, i.e. a nonmetric setting.
REMARK 4.3.bIn Theorem 4.1 the results could not be extended to deviation instead of a symmetric, since a null distance between two points would not necessarily infer from the identity of these two points.The same remark holds for the demonstration of Theorem 4.3, with:  d(Au, z)<0 Au=z, while in contrast: 5(Au, z) _< 0 "-= Au z.
REMARK 4.4 If we take X M, and S and T surjective, in Theorem 4.4, with condition 4.10, then we obtain a result in this new setting.It is worthwhile mentioning that the original theorem of this type was proved in 1992 by Kang and Rhoades [8], under the condition of compatibility, in a complete metric space.
REMARK 4.5 If we put M X, we obtain the original theorem of this type.,namely Theorem 2.1 proved by Pathak, Kang and Ryu [13] for a complete metric space, in this new setting.
REMARK 4.6 If for0O, we define 0:1:1+ FI+ by0 (t) 1/k.t,where k > then from Theorem 4.4 we obtain Corollary 4.2, which improves Theorem 2.6 of Kang and Rhoades [8] in this non-distance metric setting.REMARK 4.7 If in Theorem 4.2 we replace max {-,-, .}by rain {-,.,-} and take X M, and if S and T are un-equal surjective, then the statement is false even if A B I, the identity mappings.Dafter and Kaneko (1992) [14], Kang (1993) [15], Rhoades [9], Taniguchi (1989)  [16], and Wang et al. [10].REMARK 5.1 -Theorem 2 transposed to topological aspects suggests that metric distances could surprisingly provide a finer filtering than the symmetric difference previously proposed by some of us (Bounias and ,Bonaly, 1996) [19] as a non-metric distance between sets.A metric distance between (A) and (B) can be defined by: d(A, B) {(x e A, y e B), inf d(x, y)} while the non-metric distance would be: Now, this raises an interesting problem.Let sets A, B, S, T, be such that A c S, B c T. What are the conditions for: A(A, B) c A(S, T), with respect to d(A, B) d(S, T)?
The latter result is nearly trivial in the metric space FI + However, the former involves the following necessary conditions: AcB _ (ScT) and" [B(A) c [s(T), CA(B) c CT(S)   to be related to conditions 3.1 and 4.8.
Interestingly, when AnB (SrT) the obtained set contains the fixed points of the mappings (F) of S x T into itself, and when it reduces to one point, it identifies with the fixed points.The role of topological dimensions of the involved sets will be further examined.Let Fix (S x T)ts T the set of fixed points of the mappings F: (S x T S x T).Then: Fix (S x T) s r (ScT) PROPOSITION 5.1 Let A, B, S, T topological spaces, such that A c S, B c T, and T are complete and having respective topological dimensions ns * nT, if A and B are closed, then the mappings (S x T)t s a0 have a common fixed point.PROOF.If A and B are closed, they contain a Brouwer's type fixed point, denoted a and b.Let //(a) and //(b) neighborhoods of these points.The topological continuity in S and T suffices to state that the reciprocal image of //(a) by any mapping f: A -> S is//(a) and the reciprocal image of//(b) by any mapping f:B T is //(b) Now, provided S and T have topological dimensions ns and nT, such that ns nT, space (ScT) is a closed and has a Brouwer's fixed point.Since AcB (ScT) this fixed point is u a as well as u b, that is a b u, and the proposition is proved in these strictly non-metric conditions.This brings us now to some last points more closely related to fundamental physics.

5-2
Some physically relevant remarks, REMARK 5.2.1 Conditions (V-3-A) and (V-3-B) lead to the same scalar h, and condition (-3-C) defines a projection of (1:13)into (FI).It should be pointed that the case of a projection of (FI4)into (FI) will not be immediate, since major differences lie betwen respective topological properties of 3-spaces and 4-spaces.
The introduction of scalar h makes the case essentially relevant with linear physics.However, later, in corollary 4.1, exponent r addresses to Euclidean-like norms if it is an integer.In contrast, if it is not integer, th system could be related to fractal scaling.However, it does not match with the alternative non- distance coordinates defined through intersections of sets (Bounias and Bonaly, 1996) [19], since exponents should be a sequence of the following type: {r, r-l, r- 2 }, with coefficients (b, c) < 0 in relation 4.6.REMARK 5.2.2In remark 4.6, we have not called " (t) 1/k.t" a metric setting.In fact, it essentially deviates from so-called natural metrics, deriving from Euclidean ones, but it does represent a kind of metric.In contrast, the symmetric difference between sets and its newly defined norm [19], would allow topological generalizations escaping the critical problem of scale inconsistency, in physics.It would then be interesting to re-examine as follows the theory of fixed points with respect to distances defined this way.We thus raise the conjecture that our results on fixed points could further contribute to provide some foundations to the still needed basic justification of the invariance of some physical quantities (see Ashtekar and Magnon-Ashtekar, 1979)  [20].The question of antinomic parity conservation versus parity violation at extreme scales (see Magnon, 1996  [21] for review) could then find some clarification through basic topologies governing the embedded spacetime.
Lastly, we are currently working on purely mathematical aspects of biology [22, 23] in which semi-compatibility condition [11] could provide previously missing basis for the justification of some brain functions.

REMARK 4 . 8
Our results improve and generalize several previous results b