FIXED POINTS FOR PAIRS OF MAPPINGS IN d-COMPLETE TOPOLOGICAL SPACES

Several important lnetric space fixed point theorems are proved for a lnrge class of non-metric spaces. In some cases the metric space proofs need only minor changes. This is surprising since the distance function used need not be symmetric and need not sa.tisfy the

,/fl)r X such that < td and the metric topology td is non-discrete.Now (X,t) is d-conplete .iJc.. d(a',,x,,+l) < cxz implies that {x,} is a d-Cauchy sequence.Thus, x, x in d and lmrefore in the topology t.See [4] for the construction of td.It should also be noted that ,Jy COml,lete quasi-metric space (X,d) (d(x,y) d(y,x)) is a d-complete topological space.Tlmc are several competing definitions for a Cauchy sequence, but d(xn,xn+l) < will iq)ly that {x, is a Cauchy sequence for any reasonable definition.One reasonable definition is dcriw'd from requiring that the filter generated by {x,} be a Cauchy filter in the quasi- fiformity generated by d.This gives {x,} is a Cauchy sequence if for each e > 0 there exists positive integer no ,(e) and x x(e)in X such that {x,'n > x0} C_ {g X" d(z,y) < e}.Tlw metric space definition also holds if d(x,,x,+a) < c.Some fundamental theorems for d-c,,mplctc topological spaces are given in [4] and [5]. 2. RESULTS.The results through Corollary 2.2 axe generalizations of theorems due to Jungck [7].Even though only minor changes are needed in the proofs, they are given for completeness.
THEOREM 2.1.Let (X,t) be a Hausdorff d-complete topological space and f X X a w-continuous function.Then f has a fixed point in X if and only if there exists c (0,1) and a w-continuous function g:X X which commutes with f and satisifes (1) 9(X) C f(X) and d(gx,gy) <_ a d(fx,fy) for alt z,y X.Indeed, f and 9 have a unique common fixed point if (1) holds.
PROOF.Suppose f(a) a for some a X.Put g(x) a for all x X.Then g(fx) a and f(gx) f(a) a.Also, g(x) a f(a) for all x X so that g(X) C_ f(X).For any ,, (O,),d(gz,gy) d(a,a) 0 < , d(fz,fy).Since g is w-continuous, (1) holds.
COROLLARY 2.1.Let (X,t) be a Hausdorff d-complete topological space and f and g commuting mappings from X to X. Suppose f and g are w-continuous and g(S) C_ f(X).If there exists (0,1) and a positive integer k such that d(.qz,gy) < d(fx,fy) for tt ,u x, PROOF.Clearly, g commutes with f and g'(X) C_ g(X) C_ f(X).Applying the theorem t,,gk and f gives a uniquep X withp--g'(p) f(p).Since f and g comnmtc 9(p)   ,.l(.fp) f(g(p)) .,:l'(gp)or g(p) is a common fixed point of f and gk.Uniqueness gives 9(P) 1' f(P).
Letting f i, the identity, gives a generalization of Banach's theorem.If we also let k= 1, we get Banach's theorem in this new setting.COROLLARY 2.2.Let n be a positive integer and let k > 1.If g is a surjective w- contm.,.o.us self map of a Haudorff d-complete topological space X such that d(a"., anu) >_ d(z,U) Io art , U e X, then g has a unique fized point.
() If lim f " (5) If I in addition, f is the identity, then p for each x in X.
From Corollary 2.1, f and g have a unique common fixed point p.
He assumes (X,d) is a metric space and proves that for any xo G X, xn f'xo x f(x).Ve use his definition in the more generM setting to obtMn the following.
THEOREM 2.3.Let (X,t) be a d-complete Hausdorff topological space and f X X a w-continuous generalized contraction.Let Xo E X and put x, --w f'xo f(xn-1).Then limz, z and x f(x).
If one replaces topological by symmetrizable in the theorem, f is forced to be w-continuous.Even though Altman asserted uniqueness of the fixed point, examples exist [11] that show otherwise.
Our next result is the d-complete analogue of Theorem 2.1 of Park [9].THEOREM 2.4.Let g and h be selfmaps of a d-complete topological space X. if FIXED POINTS FOR PAIRS OF MAPPINGS IN d-COMPLETE TOPOLOGICAL SPACES 263 (i) there exists a. sequence {x,} C X such that u,+l := gu,,u2,+ := hu,+ and has a cluster point zn X, (ii) g and hg are w-continuous at , (i,) G(x) := d(x,gz),H(x) := d(z, hx) are orbitally continuous at , and (i'v) g and h satisfy d(gx,hy) < d(x,y) for each distinct z and y in {h-,} satisfying either x hy or y gx, then (1) g or h has a fixed potnt ,n {-7,} or ( 2) is a common fixed point of g and h and u, as oo.
The proof is the same as that of Theorem 2.1 of [9].
COROLLARY 2.3.Let g and h be w-continuous selfinaps of a d-complete space X satis- fyn.q d(hz,gy) <_ max{d(z,y), d(z,hz), d(y,gy), d(y,hz)} for all x, y in X, "where 0 < k < and d(x,gx) and d(x, hx) are orbitally continuous.Then g or h has a fixed point or g and h have a unique common fixed point.
The uniqueness of a common fixed point follows from the contractive definition.
We now demonstrate that several results in the literature follow as special cases of Theorem 2.4.While our list is not exhaustive, it indicates the generality of the theorem.COROLLARY 2.4.[3, Theorem 1].Let T, T be two selfmaps of a Hausdorff F-complete space X, F" X X ---, [0, oo) a continuou symmetric mapping such that F(x,y) 0 for x y and, for each pair of distinct x, y in X, F(Tx,Ty) < max{[F(x,y),F(x,Tx),F(y,Ty)] U min[F(y,Ty),F(y,Tx)]}, and for some Xo in X the sequence {x,} defined by x2n+ Tx2n,x2,,+2 Tx2n+, has subsequence converging to a point in X.If T and T2T or T and TT2 are continuous at then is a fixed point of T1 and T2 (or T or T2 has a fixed point).PROOF.
With g T, h T, the remaining conditions of Theorem 2.4 are satisfied.
PROOF.Now that the contra('tive definition implies that ,,max{d(x, Sz),d(y, Ty)}, and the result follows from Corollary 2.3.
COROLLARY 2.7.[10, Theorem 1].Let X be an F-complete Hausdorff space, Tl, T. continuous selfmaps of X.Let F" X x X IR+, F continuous and such that F(x,x) 0 for all x in X and F(Tx,Ty) < a,E(x,y) + a,F(x,Tx) + aaF(y,Ty) for each distinct x,y in X, where p and q are positive integers, a, >_ O,a + a2 -t" aa < 1. ff for some Xo X, the sequence {x,} C X defined by x2n+l Tz.,x2.+2Tz2.+, has a convergent subsequeuce, then (T or T2 has a fixed point) or T and T2 have a unique common .fixedpoini.
PROOF.Set y Tx to get F(Tx,TTx) < a,F(x,Tx) + a:F(x,Tx) + aaF(Tx,TTx), or F(T TIT) < + F(, T), 1 aa and the conditions of Theorem 2.4 are satisfied with f T', g T q.
The uniqueness follows as in [10].
REMARK.The words in parentheses in Corollaries 4-7 have been added by the present authors, in order for the theorems to be correctly stated.

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Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
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FIXED
POINTS FOR PAIRS OF NAPPINGS IN d-CONPLETE TOPOLOgiCAL SPACES 251 then f and 9 have a unique common fized point.