GAZETA MATEMATICA˘ - SSMR

1 GAZETA MATEMATIC ASERIA AANUL XXXV (CXIV)Nr. 1 2/ 2017 ARTICOLESome results concerning fixed points of -contractionsLoredana Ioana1)Abstract. Existence and uniqueness of fixed points of -contractions, inthe general sense, are given. Three proofs of a variant of the Boyd-Wongtheorem are presented. Using the conclusion of the theorem, some otherresults concerning the attractor of a GIFS of -contractions are :fixed point; Boyd-Wong theorem; :Primary 54H25; Secondary and preliminariesIn the last few decades, fixed point theory has had a very flourishingdevelopment, mainly because of its large range of beginning of the theory was in a certain way the classical Banachtheorem, which states that ifXis a complete metric space andfis a mappingofXinto itself which satisfiesd(f(x), f(y)) d(x, y),for some [0,1) and allx, y X,thenfhas a fixed pointx, and the successive approximationsfn(x) convergetoxforx , the conditiond(f(x), f(y)) d(x, y) does not insure thatfhas a fixed 1969 appeared the theorem of Boyd-Wong, which replaces the con-dition of Banach withd(f(x), f(y)) (d(x, y)), x, y X,where : [0, ) [0, ) is a comparison nition mapping.]]]

2 [0, ) [0, )is called acomparison functionif1)Faculty of Mathematics and Computer Science, University of Bucharest, , 010014, Bucharest, is continuous;b) (r)< r, r > nition :X Xand : [0, ) [0, )be a called a -function, or a -contraction, if x, y X,we haved(f(x), f(y)) (d(x, y)).The class of -contractions enlarges the class of contractions in theBanach sense (every contraction being a -contraction).In [4], the following result is called Matkowski s Theorem (1975).Theorem (X, d)be a complete metric space andf:X Xbe a -contraction. LetFfdenote the set of fixed points forf. Then we have:i)Ff=Ffn={x },for eachn N ;ii)for eachx Xthe sequence of successive approximationsfn(x)offstarting fromxconverges tox ;iii)if, additionally, is a strict comparison function, thend(x, x ) (d(x, f(x))).]]]]

3 In [4] also appear other basic fixed point principles, like the ContractionPrinciple, Ciri c-Reich-Rus s Theorem (1971), Meir-Keeler s Theorem (1969),Krasnoselskii s Theorem (1972), Graphic Contraction Principle, Caristi-Browder s Theorem, Clarke s Theorem, Niemytzki-Edelstein s Theorem. Itis also proved the next general a nonempty set andf:X Xbe an the following statements are equivalent:(P1)There exists a metricdonXsuch thatf: (X, d) (X, d)is a Picardoperator;(P2)fis a Bessaga operator;(P3)There exist a comparison function :R+ R+and a completemetricdonXsuch thatf: (X, d) (X, d)is a -contraction;(P4)There exists a metricdonXsuch that the fixed point problem iswell-posed forfwith respect main result of [5] is the following (X, d)be a complete metric space andf:Xm Xbesuch that there exists :Rm Rwith the following properties:(a)(r s, r, s Rm+) ( (r) (s));(b)(r R+, r >0) ( (r.))

4 , r)< r);(c) is continuous;L. Ioana, Fixed points of -contractions3(d) k=0 k(r)<+ ;(e)for allr R+ (r,0, .. ,0) + (0, r,0, .. ,0) + + (0,0, .. ,0, r) (r, r, .. , r);(f )for allx0, x1, .. , xm X,d(f(x0, .. , xm 1), f(x1, .. , xm)) (d(x0, x1), .. , d(xm 1, xm)).Then:(i)Ff={x };(ii)for any x0 X, the sequence( xn)n N, xn=f( xn 1, .. , xn 1), con-verges tox ;(iii)for allx0, .. , xm 1 X, the sequence(xm+n)n Ndefined byxm+n=f(xn, .. , xn+m 1)converges tox andd(xn, x ) m k=0 (d0) nm +k,where d0= max(d(x0, x1), .. , d(xm 1, xm)).Conditions to obtain iterates to the fixed point of the equationx=f(x, .. , x) are presented in [5]. All the papers treat the fixed point theory,mainly because the theory is important in solving operatorial equations.

5 Thetopic of [6] (which is the resum e of the author s thesis) is a chapter ofthis domain, namely the metrical fixed point theory. The approach is mainlythe one of successive approximations (first initiated by E. Picard in the years1890 1894, and later developed by St. Banach (1922) and R. Caccioppoli(1930)).The metrical fixed point theory was treated by many authors, like , S. Reich, M. G. Maia, F. E. Browder, S. B. Nadler etc. After the year1968, the development of the metrical fixed point theory has been explosive,its subjects being mainly the treatment of the following types of operators:contraction operators, contractive operators (Edelstein), and main aim in [6] is the study of contraction operators that satisfy acondition of generalized contraction, a condition that assures the convergenceof the sequence of successive approximations to the unique fixed point of theoperator (called Picard type operator).

6 There are presented comparison functions, and generalizations of them,like (c)-comparison functions,p-dimensional comparison functions, abstract -contractions. They all serve to obtain results that generalize the classi-cal Contraction Principle (even concerning the order of convergence of thesequence of successive approximations).4 ArticoleThe notion of -contraction plays an important part in [7], too. Variousmetrical fixed point theorems are established for (X, d) a metric space andf:X Xa mapping or an operator satisfying one of a number of conditionsof contraction type. Some of them are:(i)(Banach (1922)) There exists a numbera [0,1) such thatd(f(x), f(y)) ad(x, y), x, y X.(ii)(Nemytzki (1936), Edelstein (1962)) For allx, y X, x =y,d(f(x), f(y))< d(x, y).]

7 (iii)(Rakotch (1962), Boyd and Wong (1969), Browder (1968)) Thereexists :R+ R+such that (t)< tfort >0, is increasing,continuous andd(f(x), f(y)) (d(x, y)), x, y X.(iv)(Rus (1972)) There exists a numbera [0,1) such thatd(f2(x), f(x)) ad(x, f(x)), x X.(v)(Reich (1971), Rus (1971)) There exista, b R+, a+ 2b <1, suchthatd(f(x), f(y)) ad(x, y) +b[d(x, f(x))+d(y, f(y))], x, y also Guseman (1970), Yen (1972), Kannan (1968), Ciri c (1974), Zam-firescu (1972), Jachymski and Stein (1999).The study is centered around these generalized contractions in termsof Picard operators, weakly Picard operators, Bessaga operators and ResultsWe begin the presentation with a variant of the Boyd-Wong (X, d)be a complete metric space and : [0, ) [0, )a function with the properties:(a) is continuous;(b) (r)< r, r >0 ( (0) = 0);(c) is :X Xhas the property thatd(f(x), f(y)) (d(x, y)), x, y X,thenfhas a unique fixed pointx , for anyx0 X, the sequence(fn(x0))n 1converges Ioana, Fixed points of -contractions5 Remark 7.]]]

8 (1) A comparison function verifies the inequality (r) r, r [0, ).(2) Anyf:X Xwith the property thatd(f(x), f(y)) (d(x, y)), x, y Xis non-expansive, meaningd(f(x), f(y)) d(x, y), x, y particular,fis shall present three proofs for the theorem of the fixed that there arex1, x2 Xsuch thatf(x1) =x1andf(x2) =x2. Thenr:=d(x1, x2) =d(f(x1), f(x2)) (d(x1, x2)) = (r).Supposing thatr >0, we obtainr (r)< r, which represents acontradiction. Thereforer= 0, so thatx1= of a fixed X. We shall prove that the sequence(xn)ndefined byxn=fn(x0),n 1, is a Cauchy sequence in the completemetric space (X, d), so that (xn)nis a convergent sequence tox the relationxn+1=f(xn), n 1, andfcontinuous we thendeduce thatx=f(x) andxis the unique fixed point [0, ), then n(r) n 0 then n(0) = 0 for all positive >0 then 2(r)< (r)< retc.]]

9 , so that ( n(r))n 0is a decreas-ing sequence, minorated by 0. Put limn n(r) =l 0. The requirementsimposed on entail (l) =land thereforel= a positive integer and >0. We haved(fn(x0), fn+p(x0)) n(d(x0, fp(x0))).According to Lemma 8, the right-hand side converges to 0 asn . Thus, >0, n" Nsuch that for everyn n", p 1, d(fn(x0), fn+p(x0))< .This means (fn(x0))n 1is a Cauchy keep the notation introduced in the statement ofTheorem XandK={x0, f(x0), f2(x0), ..}.Ifd(f(x), f(y)) (d(x, y)), x, y X, thenKis a compact shall prove that the metric space (K, d) (X, d) is countablycompact. This implies that (K, d) is a compact metric {xn}n K. Each such element can be writtenxn= limp enpforsuitableenpin the orbitO(x0) ={x0, f(x0), f2(x0).}

10 } eachn Nwe denoteAn={xn, xn+1, ..}and we shall prove that(An)n 1fulfills the following conditions: Anis a closed set for anyn 1; the sequence (An)n 1is decreasing with respect to inclusion; diam(An) n (X, d) is a complete metric space, using the Cantor theorem, itfollows that n NAn = , which implies that is (K, d) countably compact, so(K, d) will be a compact metric the first two of the desired properties are obviously satisfied, it re-mains to prove that one has diam(An) n notice thatd(xn, xm) =d( limp enp,limp emp) = limp d(enp, emp).We consider the setsBn={xn, xn+1, ..}and suppose thatxisxp1,f(x) isxp2, and so on. We eliminate first max{p1, .. , pn 1}positions. Theelements left, meaningfn(x), fn+1(x), .., are in the sets fromBsonward(whenn s )d(fn(x), fn+k(x)) n(d(x, fk(x))) n 0 diam(Bs) s 0and Proposition 9 is the promised proof of the Boyd-Wong theorem, we need a well-known result, recalled a compact metric space(K, d)andf:K Ksatisfyd(f(x), f(y))< d(x, y), x =y K,thenfhas a unique fixed , for anyx K, the sequence(fn(x))n 1converges the conditions of the Boyd-Wong theorem, we can apply Proposi-tion 10 withKan arbitrary closed orbitO(x) ={x, f(x), f2(x).