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CONTINUATION THEORY FOR GENERAL CONTRACTIONS IN GAUGE SPACES ADELA CHIS¸ AND RADU PRECUP Received 9 March 2004 and in revised form 30 April 2004 A continuation principle of Leray-Schauder type is presented for contractions with re- spect to a gauge structure depending on the homotopy parameter. The result involves the most general notion of a contractive map on a gauge space and in particular yields homotopy invariance results for several types of generalized contractions. 1. Introduction One of the most useful results in nonlinear functional analysis, the Banach contraction principle, states that every contraction on a complete metric space into itself has a unique fixed point which can be obtained by successive approximations starting from any ele- ment of the space. Further extensions have tr ied to relax the metrical structure of the space, its complete- ness, or the contraction condition itself. Thus, there are known versions of the Banach fixed point theorem for contractions defined on subsets of locally convex spaces: Mari- nescu [18, page 181], in gauge spaces (spaces endowed w ith a family of pseudometrics): Colojoar ˘ a[5] and Gheorghiu [11], in uniform spaces: Knill [16], and in syntopogenous spaces: Precup [21]. As concerns the completeness of the space, there are known results for a space endowed with two metrics (or, more generally, with two families of pseudometrics). The space is assumed to be complete with respect to one of them, while the contraction condition is expressedintermsofthesecondone.ThefirstresultinthisdirectionisduetoMaia[17]. The extensions of Maia’s result to gauge spaces with two families of pseudometrics and to spaces with two syntopogenous structures were given by Gheorghiu [12]andPrecup [22], respectively. As regards the contra ction condition, several results have been established for vari- ous types of generalized contractions on metric spaces. We only refer to the earlier pa- pers of Kannan [15], Reich [27], Rus [29], and ´ Ciri ´ c[4], and to the survey article of Rhoades [28]. Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:3 (2004) 173–185 2000 Mathematics Subject Classification: 47H10, 54H25 URL: http://dx.doi.org/10.1155/S1687182004403027 174 Continuation theory We may say that almost every fixed point theorem for self-maps can be accompanied by a continuation result of Leray-Schauder type (or a homotopy invariance result). An elementary proof of the continuation principle for contractions on closed subsets of a Banach space (another proof is based on the degree theory) is due to Gatica and Kirk [10]. The homotopy invariance principle for contractions on complete metric spaces was established by Granas [14] (see also Frigon and Granas [8] and Andres and G ´ orniewicz [2]), extended to spaces endowed with two metrics or two vector-valued metrics, and completed by an iterative procedure of discrete continuation along the fixed points curve by Precup [23, 24] (see also O’Regan and Precup [19]andPrecup[26]). Continuation results for contractions on complete gauge spaces were given by Frigon [7] and for gener- alized contractions in the sense of Kannan-Reich-Rus and ´ Ciri ´ c, by Agarwal and O’Regan [1] and the first author [3]. However, until now, a unitary continuation theory for the most general notion of a contraction in gauge spaces has not been developed. The goal of this paper is to fill this gap solving this way a problem stated in Precup [25]. We are also motivated by a num- ber of papers which have been published in the last decade, such as those of Frigon and Granas [9] and O’Regan and Precup [20], and also by the applications to integral and differential equations in locally convex spaces, see Gheorghiu and Turinici [13]. 2. Preliminaries 2.1. Gauge spaces. Let X be any set. A map p : X × X → R + is called a pseudometric (or a gauge)onX if p(x,x) = 0, p(x, y) = p(y,x), and p(x, y) ≤ p(x,z)+p(z, y)forevery x, y,z ∈ X. A family ᏼ ={p α } α∈A of pseudometrics on X (or a gauge structure on X)is said to be separating if for each pair of points x, y ∈ X with x = y, there is a p α ∈ ᏼ such that p α (x, y) = 0. A pair (X,ᏼ) of a nonempty set X and a separating gauge structure ᏼ on X is called a gauge space. It is well known (see Dugundji [6, pages 198–204]) that any family ᏼ of pseudometrics on a set X induces on X astructureᐁ of uniform space and conversely, any uniform structure on X is induced by a family of pseudometrics on X. In addition, ᐁ is separating (or Hausdorff)ifandonlyifᏼ is separating. Hence we may identify the gauge spaces and the Hausdorff uniform spaces. For the rest of this section we consider a gauge space (X,ᏼ) with the gauge structure ᏼ ={p α } α∈A .Asequence(x n ) of elements in X is said to be Cauchy if for every ε>0and α ∈ A, there is an N with p α (x n ,x n+k ) ≤ ε for all n ≥ N and k ∈ N. The sequence (x n )is called convergent if there exists an x 0 ∈ X such that for every ε>0andα ∈ A, there is an N with p α (x 0 ,x n ) ≤ ε for all n ≥ N. A gauge space is called sequentially complete if any Cauchy sequence is convergent. A subset of X is said to be sequentially closed if it contains the limit of any convergent sequence of its elements. 2.2. General contractions on gauge spaces. We now recall the notion of contraction on a gauge s pace introduced by Gheorghiu [11]. Let (X,ᏼ) be a gauge space with ᏼ = {p α } α∈A .AmapF : D ⊂ X → X is a contraction if there exists a function ϕ : A → A and a ∈ R A + , a ={a α } α∈A such that A. Chis¸ and R. Precup 175 p α  F(x),F(y)  ≤ a α p ϕ(α) (x, y) ∀α ∈ A, x, y ∈ D, (2.1) ∞  n=1 a α a ϕ(α) a ϕ 2 (α) ···a ϕ n−1 (α) p ϕ n (α) (x, y) < ∞ (2.2) for every α ∈ A and x, y ∈ D.Here,ϕ n is the nth iteration of ϕ. Notice that a sufficient condition for (2.2)isthat ∞  n=1 a α a ϕ(α) a ϕ 2 (α) ···a ϕ n−1 (α) < ∞, (2.3) sup  p ϕ n (α) (x, y):n = 0,1,  < ∞∀α ∈ A, x, y ∈ D. (2.4) The above definition contains as particular cases the notion of contraction on a sub- set of a locally convex space introduced by Marinescu [18], for which ϕ 2 = ϕ, and the most worked notion of contraction on a gauge space as defined in Tarafdar [30], which corresponds to ϕ(α) = α and a α < 1forallα ∈ A. Given a space X endowed with two gauge structures ᏼ ={p α } α∈A and ᏽ ={q β } β∈B , in order to precise the gauge structure with respect to which a topological-type notion is considered, we will indicate the corresponding gauge structure in light of that notion. So, we will speak a bout ᏼ-Cauchy, ᏽ-Cauchy, ᏼ-convergent, and ᏽ-convergent sequences; ᏼ-sequentially closed and ᏽ-sequentially closed sets; ᏼ-contractions and ᏽ-contractions, andsoforth.AlsowesaythatamapF : X → X is (ᏼ,ᏽ)-sequentially continuous if for every ᏼ-convergent sequence (x n )withthelimitx, the sequence (F(x n )) is ᏽ-convergent to F(x). We now state Gheorghiu’s fixed point theorem of Maia type for self-maps of gauge spaces [12]. Theorem 2.1 (Gheorghiu). Let X be a nonempty set endowed with two separating gauge structures ᏼ ={p α } α∈A and ᏽ ={q β } β∈B and let F : X → X be a map. Assume that the following conditions are satisfied: (i) there is a function ψ : A → B and c ∈ (0,∞) A , c ={c α } α∈A such that p α (x, y) ≤ c α q ψ(α) (x, y) ∀α ∈ A, x, y ∈ X; (2.5) (ii) (X,ᏼ) is a s equentially complete gauge space; (iii) F is (ᏼ,ᏽ)-sequentially cont inuous; (iv) F is a ᏽ-contraction. Then F has a unique fixed point which can be obtained by successive approximations starting from any element of X. The following slight extension of Gheorghiu’s theorem will be used in the sequel. Theorem 2.2. Let X be a set endowed with two separating gauge structures ᏼ ={p α } α∈A and ᏽ ={q β } β∈B ,letD 0 and D be two nonempty subsets of X with D 0 ⊂ D,andletF : D → X be a map. Assume that F(D 0 ) ⊂ D 0 and D is ᏼ-closed. In addition, assume that the following conditions are satisfied: 176 Continuation theory (i) there is a function ψ : A → B and c ∈ (0,∞) A , c ={c α } α∈A such that p α (x, y) ≤ c α q ψ(α) (x, y) ∀α ∈ A, x, y ∈ X; (2.6) (ii) (X,ᏼ) is a s equentially complete gauge space; (iii) if x 0 ∈ D 0 , x n = F(x n−1 ) for n = 1,2, , and ᏼ-lim n→∞ x n = x for some x ∈ D, then F(x) = x; (iv) F is a ᏽ-contraction on D. Then F has a unique fixed point which can be obtained by successive approximations starting from any element of D 0 . Proof. Take any x 0 ∈ D 0 and consider the sequence (x n ) of successive approximations, x n = F(x n−1 ), n = 1,2, Since F(D 0 ) ⊂ D 0 ,onehasx n ∈ D 0 for all n ∈ N.By(iv),(x n ) is ᏽ-Cauchy. Next, (i) implies that (x n )isalsoᏼ-Cauchy, hence it is ᏼ-convergent to some x ∈ D, in virtue of (ii). Now, (iii) guarantees that F(x) = x. The uniqueness is a consequence of (iv).  2.3. Generalized contractions on metric spaces. It is worth noting that a number of fixed point results for generalized contractions on complete metric spaces appear as direct consequences of Theorem 2.2. Here are two examples. Let (X, p) be a complete metric space and F : X → X amap. (1) Assume that F satisfies p  F(x),F(y)  ≤ a  p  x, F(x)  + p  y,F(y)  + bp(x, y) (2.7) for all x, y ∈ X,wherea, b ∈ R + , a>0, and 2a +b<1. We associate to F a family of pseudometrics q k , k ∈ N,givenby q k (x, y) =        a r k − b k r k (r − b)  p  x, F(x)  + p  y,F(y)  +  b r  k p(x, y)forx = y, 0forx = y. (2.8) Here, r = (a + b)/(1 − a)andb<r<1. By induction, we can see that q k  F(x),F(y)  ≤ rq k+1 (x, y) ∀k ∈ N, x, y ∈ X. (2.9) It is clear that ᏽ ={q k } k∈N is a separating gauge structure on X and from (2.9)wehave that F is a ᏽ-contraction on X. In this case, ϕ : N → N is given by ϕ(k) = k +1anda k = r for all k ∈ N.Also,foranyk ∈ N,(2.2) means  ∞ n=1 r n q k+n (x, y) < ∞, which according to (2.8)istruesince0≤ b<1andb<r<1. Corollary 2.3 (Reich-Rus). If (X, p) is a complete metric space and F : X → X satisfies (2.7), then F has a unique fixed point. Proof. Let ᏼ ={p} and ᏽ ={q k } k∈N .Here,A ={1} and B = N.InTheorem 2.2,con- dition (i) holds because q 0 = p, (ii) reduces to the completeness of (X, p), and (iv) was explained above. Now we check (iii). Assume x 0 ∈ X, x n = F(x n−1 )forn = 1,2, ,and A. Chis¸ and R. Precup 177 ᏼ-lim n→∞ x n = x, that is, p(x,x n ) → 0asn →∞.From(2.7), we have p  x n ,F(x)  = p  F  x n−1  ,F(x)  ≤ a  p  x n−1 ,x n  + p  x, F(x)  + bp  x n−1 ,x  . (2.10) Passing to the limit, we obtain p(x,F(x)) ≤ ap(x,F(x)), whence p(x, F(x)) = 0, that is, F(x) = x. Now the conclusion follows from Theorem 2.2.  (2) Assume that F satisfies p  F(x),F(y)  ≤ amax  p(x, y), p  x, F(x)  , p  y,F(y)  , p  x, F(y)  , p  y,F(x)  (2.11) for all x, y ∈ X and some a ∈ [0,1). Then F is a ᏽ-contraction, where ᏽ ={q k } k∈N and q k (x, y) =          max  p  F i (x), F j (x)  , p  F i (y),F j (y)  , p  F i (x), F j (y)  : i, j = 0,1, ,k  for x = y, 0forx = y. (2.12) We have q 0 = p and from (2.11)weobtain p  F i (x), F j (x)  ≤ aq k (x, y), p  F i (y),F j (y)  ≤ aq k (x, y), p  F i (x), F j (y)  ≤ aq k (x, y), (2.13) for all i, j ∈{0,1, ,k} and x, y ∈ X. It follows that q k  F(x),F(y)  ≤ aq k+1 (x, y) (2.14) and also q k (x, y) = max  p  x, F i (x)  , p  y,F i (y)  , p  x, F i (y)  , p  y,F i (x)  , (2.15) where the maximum is taken over i ∈{0,1, ,k}. If, for example, q k (x, y) = p(x,F i (x)) for some i ∈{1,2, ,k},then q k (x, y) ≤ p  x, F(x)  + p  F(x),F i (x)  ≤ p  x, F(x)  + aq k (x, y). (2.16) Hence q k (x, y) ≤ 1 1 − a p  x, F(x)  ≤ 1 1 − a q 1 (x, y). (2.17) Generally, we can prove similarly that q k (x, y) ≤ 1 1 − a q 1 (x, y) (2.18) for all k ∈ N and x, y ∈ X. This shows that (2.3) holds for the gauge structure ᏽ ={q k } k∈N and a k = a for every k ∈ N. 178 Continuation theory Corollary 2.4 ( ´ Ciri ´ c). If (X, p) is a complete metric space and F : X → X satisfies (2.11), then F has a unique fixed point. Proof. Here again ᏼ ={p}, ᏽ ={q k } k∈N ,andq 0 = p. To check (iii), assume x 0 ∈ X, x n = F(x n−1 )forn = 1,2, ,andᏼ-lim n→∞ x n = x, that is, p(x,x n ) → 0asn →∞.From(2.11), we obtain p  x n ,F(x)  = p  F  x n−1  ,F(x)  ≤ amax  p  x n−1 ,x  , p  x n−1 ,F  x n−1  , p  x, F(x)  , p  x n−1 ,F(x)  , p  x, F  x n−1  . (2.19) Passing to the limit, we obtain p(x,F(x)) ≤ ap(x,F(x)), whence p(x, F(x)) = 0, that is, F(x) = x.ThuswemayapplyTheorem 2.2.  3. Continuation theorems in gauge spaces For a map H : D × [0,1] → X,whereD ⊂ X, we will use the following notations: Σ =  (x, λ) ∈ D × [0,1] : H(x,λ) = x  , ᏿ =  x ∈ D : H(x,λ) = x for some λ ∈ [0,1]  , Λ =  λ ∈ [0,1] : H(x,λ) = x for some x ∈ D  . (3.1) Now we state and prove the main result of this paper: a continuation principle for con- tractions on spaces with a gauge structure depending on the homotopy parameter. Theorem 3.1. Let X be a set endowed with the separating gauge structures ᏼ ={p α } α∈A and ᏽ λ ={q λ β } β∈B for λ ∈ [0,1].LetD ⊂ X be ᏼ-sequentially closed, H : D × [0,1] → X a map, and assume that the following conditions are satisfied: (i) for each λ ∈ [0,1], there exists a function ϕ λ : B → B and a λ ∈ [0,1) B , a λ ={a λ β } β∈B such that q λ β  H(x,λ),H(y,λ)  ≤ a λ β q λ ϕ λ (β) (x, y), ∞  n=1 a λ β a λ ϕ λ (β) a λ ϕ 2 λ (β) ···a λ ϕ n−1 λ (β) q λ ϕ n λ (β) (x, y) < ∞ (3.2) for every β ∈ B and x, y ∈ D; (ii) there exists ρ>0 such that for each (x,λ) ∈ Σ, there is a β ∈ B with inf  q λ β (x, y):y ∈ X \ D  >ρ; (3.3) (iii) for each λ ∈ [0,1], there is a function ψ : A → B and c ∈ (0,∞) A , c ={c α } α∈A such that p α (x, y) ≤ c α q λ ψ(α) (x, y) ∀α ∈ A, x, y ∈ X; (3.4) A. Chis¸ and R. Precup 179 (iv) (X, ᏼ) is a sequentially complete gauge space; (v) if λ ∈ [0, 1], x 0 ∈ D, x n = H(x n−1 ,λ) for n = 1,2, , and ᏼ-lim n→∞ x n = x, then H(x,λ) = x; (vi) for every ε>0,thereexistsδ = δ(ε) > 0 with q λ ϕ n λ (β)  x, H(x,λ)  ≤  1 − a λ ϕ n λ (β)  ε (3.5) for (x,µ) ∈ Σ, |λ − µ|≤δ,allβ ∈ B,andn ∈ N. In addition, assume that H 0 := H(·,0) has a fixed point. Then, for each λ ∈ [0,1],the map H λ := H(·,λ) has a unique fixed point. Proof. We prove that there exists a number h>0suchthatifµ ∈ Λ,thenλ ∈ Λ for every λ satisfying |λ − µ|≤h. This, together with 0 ∈ Λ, clearly implies Λ = [0,1]. First we note that from (ii) it follows that for each (x,λ) ∈ Σ, there exists β ∈ B such that the set B(x,λ,β) =:  y ∈ X : q λ ϕ n λ (β) (x, y) ≤ ρ ∀n ∈ N  (3.6) is included in D. Let µ ∈ Λ and let H(x,µ) = x. From (vi), there is h = h(ρ) > 0, independent of µ and x, such that q λ ϕ n λ (β)  x, H(x,λ)  = q λ ϕ n λ (β)  H(x,µ),H(x, λ)  ≤  1 − a λ ϕ n λ (β)  ρ (3.7) for |λ − µ|≤h and all n ∈ N. Consequently, if |λ − µ|≤h and y ∈ B(x,λ,β), then q λ ϕ n λ (β)  x, H(y, λ)  ≤ q λ ϕ n λ (β)  x, H(x,λ)  + q λ ϕ n λ (β)  H(x,λ),H(y,λ)  ≤  1 − a λ ϕ n λ (β)  ρ + a λ ϕ n λ (β) q λ ϕ n+1 λ (β) (x, y) ≤  1 − a λ ϕ n λ (β)  ρ + a λ ϕ n λ (β) ρ = ρ. (3.8) Hence, for |λ − µ|≤h, H λ is a self-map of D 0 := B(x,λ,β). Now Theorem 2.2 guarantees that λ ∈ Λ for |λ − µ|≤h.  Assuming a c ontinuity property of H,wederivefromTheorem 3.1 the following result. Theorem 3.2. Let X be a set endowed with the separating gauge structures ᏼ ={p α } α∈A and ᏽ λ ={q λ β } β∈B for λ ∈ [0,1].LetD ⊂ X be ᏼ-sequentially closed, H : D × [0,1] → X a map, and assume that the following conditions are satisfied: 180 Continuation theory (a) for each λ ∈ [0,1], there exists a function ϕ λ : B → B and a λ ∈ [0,1) B , a λ ={a λ β } β∈B such that q λ β  H(x,λ),H(y,λ)  ≤ a λ β q λ ϕ λ (β) (x, y), (3.9) sup  q λ ϕ n λ (β) (x, y):n ∈ N  < ∞, (3.10) sup  ∞  n=1 a λ β a λ ϕ λ (β) a λ ϕ 2 λ (β) ···a λ ϕ n−1 λ (β) : λ ∈ [0,1]  < ∞, (3.11) for all β ∈ B and x, y ∈ D; (b) there exists a set U ⊂ D such that H(x,λ) = x for all x ∈ D \ U and λ ∈ [0,1];and for each (x,µ) ∈ Σ,thereisβ ∈ B, δ>0,andγ>0 such that for ever y λ ∈ [0,1] with |λ − µ|≤γ,  y ∈ X : q λ β (x, y) <δ  ⊂ U; (3.12) (c) for each λ ∈ [0,1], there is a function ψ : A → B and c ∈ (0, ∞) A , c ={c α } α∈A such that p α (x, y) ≤ c α q λ ψ(α) (x, y) ∀α ∈ A, x, y ∈ X; (3.13) (d) (X,ᏼ) is a s equentially complete gauge space; (e) H is (ᏼ,ᏼ)-sequentially continuous; (f) for every ε>0,thereexistsδ = δ(ε) > 0 with q λ ϕ n λ (β)  x, H(x,λ)  ≤  1 − a λ ϕ n λ (β)  ε (3.14) for (x,µ) ∈ Σ, |λ − µ|≤δ,andallβ ∈ B and n ∈ N. In addition, assume that H 0 := H(·,0) has a fixed point. Then, for each λ ∈ [0,1],the map H λ := H(·,λ) has a unique fixed point. Proof. Conditions (i), (iii), (iv), and (vi) in Theorem 3.1 are obviously satisfied. Assume (ii) is false. Then, for each n ∈ N \{0}, there is (x n ,λ n ) ∈ Σ and y nβ ∈ X \ D with q λ n β  x n , y nβ  ≤ 1 n for every β ∈ B. (3.15) Clearly we may assume that λ n → λ. Fix an arbitrary β ∈ B. From (f) we see that for a given ε>0, there is a number N = N(ε) > 0suchthat q λ ϕ i λ (β)  x n ,H  x n ,λ  ≤ ε 4C (3.16) for all n ≥ N and i ∈ N,whereC is any positive number with 1+ ∞  i=1 a λ β a λ ϕ λ (β) ···a λ ϕ i−1 λ (β) ≤ C<∞, λ ∈ [0,1]. (3.17) A. Chis¸ and R. Precup 181 Now, for n,m ≥ N, using (a), we obtain q λ β  x n ,x m  = q λ β  H  x n ,λ n  ,H  x m ,λ m  ≤ q λ β  H  x n ,λ n  ,H  x n ,λ  + q λ β  H  x m ,λ m  ,H  x m ,λ  + q λ β  H  x n ,λ  ,H  x m ,λ  ≤ q λ β  H  x n ,λ n  ,H  x n ,λ  + q λ β  H  x m ,λ m  ,H  x m ,λ  + a λ β q λ ϕ λ (β)  x n ,x m  ≤ ε 2C + a λ β q λ ϕ λ (β)  x n ,x m  . (3.18) Similarly, q λ ϕ λ (β)  x n ,x m  ≤ ε 2C + a λ ϕ λ (β) q λ ϕ 2 λ (β)  x n ,x m  (3.19) and, in general, q λ ϕ i λ (β)  x n ,x m  ≤ ε 2C + a λ ϕ i λ (β) q λ ϕ i+1 λ (β)  x n ,x m  (3.20) for all i ∈ N. It follows that for all n,m ≥ N and every l ∈ N,wehave q λ β  x n ,x m  ≤ ε 2C  1+ l  i=1 a λ β a λ ϕ λ (β) ···a λ ϕ i−1 λ (β)  + a λ β a λ ϕ λ (β) ···a λ ϕ l λ (β) q λ ϕ l+1 π (β)  x n ,x m  ≤ ε 2 + a λ β a λ ϕ λ (β) ···a λ ϕ l λ (β) M  λ,β,x n ,x m  . (3.21) Here, M(λ,β,x, y):= sup{q λ ϕ n λ (β) (x, y):n ∈ N}.Accordingto(3.11), for each couple [n,m]withn,m ≥ N,wemayfindanl such that a λ β a λ ϕ λ (β) ···a λ ϕ l λ (β) M  λ,β,x n ,x m  ≤ ε 2 . (3.22) Hence q λ β  x n ,x m  ≤ ε ∀n,m ≥ N. (3.23) Thus the sequence (x n )isᏽ λ -Cauchy. Now (c) guarantees that (x n )isᏼ-Cauchy. Further- more, (d) implies that (x n )isᏼ-convergent. Let x = ᏼ-lim n→∞ x n .Clearlyx ∈ D.Then, from (e), ᏼ-lim n→∞ H(x n ,λ n ) = H(x,λ). Hence H(x,λ) = x. Now we claim that q λ n β  x, x n  −→ 0asn −→ ∞ . (3.24) Indeed, since (x,λ) ∈ Σ and λ n → λ, from (f) it follows that for a given ε>0, there is a number N 0 = N 0 (ε) > 0suchthat q λ n ϕ i λ n (β)  x, H  x, λ n  ≤ ε 2C (3.25) 182 Continuation theory for all n ≥ N 0 and i ∈ N.Then,forn ≥ N 0 ,wehave q λ n β  x, x n  = q λ n β  x, H  x n ,λ n  ≤ q λ n β  x, H  x, λ n  + q λ n β  H  x, λ n  ,H  x n ,λ n  ≤ ε 2C + a λ n β q λ n ϕ λn (β)  x, x n  . (3.26) Furthermore, as above, we deduce that q λ n β  x, x n  ≤ ε ∀n ≥ N 0 . (3.27) This proves our claim. Also (b) guarantees q λ n β  x, y nβ  ≥ δ (3.28) for a sufficiently large n and some β ∈ B.Now,from 0 <δ≤ q λ n β  x, y nβ  ≤ q λ n β  x, x n  + q λ n β  x n , y nβ  , (3.29) we derive a contradiction. This contradiction shows that (ii) holds. Also (v) immediately follows from (e). Thus Theorem 3.1 applies.  Remark 3.3. In particular, if the gauge structures reduce to metric structures, that is, ᏼ ={p} and ᏽ λ = ᏽ ={q}, p and q being two metrics on X, Theorem 3.2 becomes the first part of Theorem 2.2 of Precup [23] (with the additional assumption that there is a constant c>0withp(x, y) ≤ cq(x, y)forallx, y ∈ X). 4. Homotopy results for generalized contractions on metric s paces In this section, we test Theorem 3.1 on generalized contractions on complete metric spaces. We begin with a continuation result for generalized contractions of Reich-Rus type. Theorem 4.1. Let (X, p) beacompletemetricspace,D a closed subset of X,andH : D × [0,1] → X a map. Assume that the following conditions are satisfied: (A) there exist a,b ∈ R + with a>0 and 2a + b<1 such that p  H λ (x), H λ (y)  ≤ a  p  x, H λ (x)  + pd  y,H λ (y)  + bp(x, y) (4.1) for all x, y ∈ D and λ ∈ [0,1]; (B) inf{p(x, y):x ∈ ᏿, y ∈ X \ D} > 0; (C) for each ε>0,thereexistsδ = δ(ε) > 0 such that p  H(x,λ),H(x, µ)  ≤ ε for |λ − µ|≤δ, all x ∈ D. (4.2) In addition, assume that H 0 := H(·,0) has a fixed point. Then, for each λ ∈ [0,1], the map H λ := H(·,λ) has a unique fixed point. 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Equations int´grales dans les espaces localement convexes, Rev e Roumaine Math Pures Appl 23 (1978), no 1, 33–40 (French) A Granas, Continuation method for contractive maps, Topol Methods Nonlinear Anal 3 (1994), no 2, 375–379 R Kannan, Some results on fixed points, Bull Calcutta Math Soc 60 (1968), 71–76 R J Knill, Fixed points of uniform contractions, J Math Anal Appl 12 (1965), 449–455 M G Maia, Un’osservazione... exists δ > 0 such that λ qm x,H(x,λ) ≤ (1 − r)ε (4.3) for every (x,µ) ∈ Σ, |λ − µ| ≤ δ, and m ∈ N This can be proved by using (C) if we observe λ that qm (x,H(x,λ)) depends only on p(x,Hλ (x)) = p(H(x,µ),H(x,λ)), and a r m − bm a ≤ , r m (r − b) r − b b r m ≤ 1 (4.4) Similarly, Theorem 3.1 yields a continuation result for generalized contractions in the ´ c sense of Ciri´ Theorem 4.2 Let (X, p) be... des contractions multivoques e [Leray-Schauder-type results for multivalued contractions] , Topol Methods Nonlinear Anal 4 (1994), no 1, 197–208 (French) , R´sultats de type Leray-Schauder pour des contractions sur des espaces de Fr´chet [Leraye e Schauder-type results for contractions on Frechet spaces], Ann Sci Math Qu´ bec 22 (1998), e no 2, 161–168 (French) J A Gatica and W A Kirk, Fixed point theorems . CONTINUATION THEORY FOR GENERAL CONTRACTIONS IN GAUGE SPACES ADELA CHIS¸ AND RADU PRECUP Received 9 March 2004 and in revised form 30 April 2004 A continuation principle of Leray-Schauder. 267–281. [24] , The continuation principle for generalized contractions, Bull. Appl. Comput. Math. (Bu- dapest) 1927 (2001), 367–373. [25] , Continuation results for mappings of contractive type, Semin. Fixed. suggestions, espe- cially for pointing out the necessity of taking supremum over λ in (3.11). References [1] R. P. Agarwal and D. O’Regan, Fixed point theory for generalized contractions on spaces with