TOPOLOGICAL DIVISOR OF ZERO PERTURBATION FUNCTIONS

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1 Jounal of Pue and Applied Mathematics: Advances and Applications Volume 4, Numbe, 200, Pages 97-4 TOPOLOGICAL DIVISOR OF ZERO PERTURBATION FUNCTIONS Dépatement de Mathématiques Faculté des Sciences de Monasti Avenue de L envionnement 509 Monasti Tunisia haiel.shii@gmail.com and 200 Mathematics Subject Classification: Pimay 46H05; Seconday 46H20, 47A30, 47A53. Keywods and phases: Banach algeba, left (esp., ight) topological diviso of zeo, Calin algeba, Fedholm opeatos, spectum, spectal adius, measues of noncompactness, * C - algeba. Received Septembe 4, 200 Abstact Let ( A, ) be a complex unital Banach algeba and let N Eq be the set of all algeba-noms on A equivalent to the given algeba-nom. In this pape, we intoduce the concept of ρd - petubation and ρd - petubation functions depending on a nom N Eq and elated to the notion of topological divisos of zeo. We pove that some usual measues of eithe non-compactness o nonstict-singulaity of opeatos, as well othe quantities ae ρd - petubation o ρd - petubation function. We pove seveal spectal adius fomulae fo ρd - petubation and ρd - petubation functions. In paticula, we pove that if P is a ρd - petubation o ρd - petubation function and x A, then = + ρ( x) = inf{ P ( x ) : N } lim P ( x ), 200 Scientific Advances Publishes

2 98 ( x ) = inf { P ( ) : N }, ρ x whee ρ ( x) denotes the spectal adius of x. Eq. Intoduction In the liteatue about the opeato algeba, many authos ([], [5], [0], [2], [4], [5], [8], [9], [23], [25], [26], [27], [28], [29], [30], [32], and [33],...) have intoduced and investigated seveal quantities elated to the opeatos, such as measues of non-compactness, measues of nonstict-singulaity, numeical ange,...etc. They have established fomulae fo the spectal adius and the essential spectal adius of a given opeato, in tems of the asymptotic behavio of its powes. The notion of petubation functions has been intoduced on the algeba of opeatos. Fo instance, in [0], [9], [3],..., the authos defined petubation functions, which ae invaiant by compact petubation. In this pape, we intoduce, on a given Banach algeba, new functions connected with the concept of topological divisos of zeo and depending on a nom of the algeba. In Section 3, ρd - petubation and ρd - petubation functions ae intoduced. These functions will be called ρz - petubation functions. Note that numbe of well-nown quantities is actually special examples of such functions (see the given examples). With these functions, we povide analogous fomulae to Gelfand s spectal adius ones (see Theoems 3.4 A, is a complex unital Banach algeba and and 3.5). Fo example, if ( ) is an algeba-nom on A equivalent to the given algeba-nom, we pove that if P is a ρ - petubation function and x A, then = + Z ρ( x) = inf { P ( x ) : N } lim P ( x ), and ρ ( x) = inf { ( x) : N Eq }, whee ρ ( x) denotes the spectal adius of x. P

3 TOPOLOGICAL DIVISOR OF ZERO PERTURBATION 99 In Section 4, A is supposed to be a C - algeba. Relying on Muphy- West s wo, [20], we establish fomula in tems of ρz - petubation function. As an example, we show that if x A and P is a ρ Z - petubation function, then ρ( x) = inf { P ( e xe ) : a = a A}. a a 2. Peliminaies Fist, we intoduce some notation and teminology. Let (, ) complex unital Banach algeba with unity e. We denote by ( A) A be a N the set of all algeba-noms on A equivalent to the given algeba-nom and satisfying e = fo all N Eq ( A). When no confusion can aise, we simply wite N Eq. Let x, y A such that xy = e, then x is called a left invese of y and y is said to be a ight invese of x. If an element x is both a left invese and a ight invese of y, then x is called a two-sided invese, o simply an invese of y. An element x with an invese in A is said to be invetible in A. Rema that this invese is unique and will be denoted by x. The set of all invetible elements of A is denoted by lnv ( A). An element with a left (esp., ight) invese is left (esp., ight) invetible. The set of all left (esp., ight) invetible elements of A is denoted by lnv ( A) (esp., lnv ( A) ). An element x A is called a left topological diviso of zeo, if inf { xa : a A, a = } = 0. Similaly, x is a ight topological diviso of zeo, if inf { ax : a A, a = } = 0. We will denote by lnv d ( A) = { x A : x is not left topological diviso of zeo}. The set lnv d ( A) is defined in the obvious way. Eq

4 00 The spectum of x A is denoted by σ( x ) = { λ C : x λe lnv ( A)} and its spectal adius ρ ( x) = sup { λ : λ σ( x)}. The left appoximate point spectum of x, denoted by σ ( x), is the set d σ ( x) = { λ C : x λe is a left topological diviso of zeo }; d and the ight appoximate point spectum of x, denoted by σ ( x), is the set σd ( x) = { λ C : x λe is a ight topological diviso of zeo}. It is well-nown that lnv ( A) lnv d ( A) and lnv ( A) lnv ( A), (2.) σ ( x) σ ( x) σ( x), x A, (2.2) d d σ( x) σ ( x) σ ( x), x A, (2.3) whee σ( x) denotes the bounday of σ ( x). d d d d 3. ρz - Petubation Functions In this section, we intoduce two petubation functions connected with the concept of left and ight topological divisos of zeo and depending on the nom of the algeba A. With these functions, we povide analogous fomulae to Gelfand s spectal adius ones. Recall that a function S : A C is absolutely homogeneous, if S ( λx) = λ S( x), λ C, x A. Definition 3.. Given, a ρ - petubation function on A N Eq is a non-negative absolutely homogeneous function following popeties: () ( x) x, x A; P (2) if x A such that P ( x) <, then x e lnv d ( A). d P with the

5 TOPOLOGICAL DIVISOR OF ZERO PERTURBATION 0 The ρd - petubation function is defined in the obvious way. A function is said to be a ρz - petubation, if it is a ρd - petubation o ρd - petubation function. P It is easy to see that if ( e) =. P is a ρ - petubation function, then Z Fom (2.3), it is not difficult to see the following. Poposition 3.2. Let N Eq and Γ : A R+ be an absolutely homogeneous function such that ρ( x) Γ ( x) x, x A, then Γ is a ρ - petubation and a ρ - petubation function. d d Example 3.. Fo evey x A, define: θ( x) = inf { ax : a lnv ( A), a = a = }, θ( x) = inf { xa : a lnv ( A), a = a = }. It is not difficult to show that the function θ (esp., d - θ ) is a ρ petubation and a ρ - petubation function. d Let us conside the following sets: S S S ( A) = { x A : x = }, ( A) { x lnv ( A) S ( A) : y S ( A) yx = e}, = ( A) { x lnv ( A) S ( A) : y S ( A) xy = e}. = Example 3.2. Fo evey x A, define: ω : x ω ( x) = inf { xa : a S ( A) \ lnv ( A)},

6 02 : x ω ( x) = inf { ax : a S ( A) \ lnv ( A)}. ω ω ω is a ρd - petubation function. Let x A such that ( x) <. We now that thee exists a S ( A) \ lnv ( A) such that xa. Let 0 b 0 A be a ight invese of a 0 such that b 0 =. We have x xa0b0 xa0 b0 < b0. So, xa x < b. Hence, by [22, Theoem.4.6], we get 0 0 Consequently, x e lnv d ( A). ( x e) a0 = xa0 a0 lnv ( A). 0 < In the same way, we pove that ω is a ρ - petubation function. d Example 3.3. Let us define τ : x τ ( x) = sup { ρ( ax) : a ( A)}, S τ : x τ ( x) = sup { ρ( xa) : a ( A)}, S τ : x τ ( x) = sup { ρ( axb) : a, b A, a b = }. It is not difficult to see that τ (esp., τ, τ ) is a ρd - petubation and a ρd - petubation function. Let A denote the dual space of A, i.e., the Banach space of all continuous linea functionals on A. Fo x A, the numeical ange of x, denoted V ( x), is defined by ( x ) = { f ( x) : f, f ( e) = = f }, V A and its numeical adius denoted by v ( x), is defined by v ( x) = sup { λ : λ V( x) }.

7 TOPOLOGICAL DIVISOR OF ZERO PERTURBATION 03 Fom ([24], [4, Poposition 6, p. 53] o [3, Theoem 6, p. 9]), we now that ρ( x) v ( x) x, x A. Example 3.4. By Poposition 3.2, we deduce that the function v : A R + defined by x v ( x) is a ρd - petubation and a ρ d - petubation. Example 3.5. Fo x A, let v v v ( x) sup { v ( ax) : a A, a = }, = ( x) sup { v ( xa) : a A, a = }, = ( x) sup { v ( axb) : a, b A, a b = }. = We can pove without any difficulty that the functions, v, v and v ae ρd - petubation and ρd - petubation functions. Example 3.6. Fo x A, we define the function v by v ( x) = inf { v ( axa ) : a lnv ( A), a a = }. It is clea that the function petubation functions. v is a ρ - petubation and a ρ - d d Poposition 3.3. Let homogeneous function such that N Eq and Γ : A R+ ρ( x) Γ ( x) x, x A, then the function Γ : A R +, defined by be an absolutely Γ ( x) = inf { Γ ( bxa) : a lnv ( A) \ lnv ( A), b A, ab = e, ab = }, is a ρ d- petubation and a ρd- petubation functions.

8 04 Poof. To veify popety (2), let x A such that Γ ( x) <. Fom the definition of Γ, we now that thee exist a lnv ( A) lnv ( A ) and 0 \ b0 A such that Γ ( bxa 0 0 ) <, ab 0 0 = e, and a0b 0 =. So, fom [2, Coollay, p. 2] (o [6, Poposition 3.2.8]), we deduce ρ ( x) = ρ ( xa b ) = ρ( b xa ) Γ ( b xa ) < This implies x e lnv ( A ). Example 3.7. Fo evey x A, define ( x) = inf { bxa: a lnv ( A) \ lnv ( A), b A, ab = e, ab = }, v ( x) = inf { v ( bxa) : a lnv ( A) \ lnv ( A), b A, ab = e, ab = }. It follows fom Poposition 3.3, that (esp., v ) is a ρ d- petubation and a ρd- petubation functions. Fo the convenience of the eade, some notions and definitions in opeatos theoy will be ecalled hee. Thoughout, ( X, ) denotes a Banach space and ( B ( X ), ) the algeba of bounded opeatos fom X into itself. Let T B ( X ) and a nom on X equivalent to, we will denote T as the algeba-nom fo T elative to the nom. The conjugate space of the Banach space X is denoted by X and the adjoint of a linea opeato T in B ( X ) by T. case whee X is a Hilbet space, we denote it by H. Fo an opeato T B ( X ), we denote, espectively, by N ( T) and R ( T), the enel and the ange of T. The identity opeato will be denoted by I. Let K ( X ) be the ideal of compact opeatos. We shall use π to denote the natual homomophism of B ( X ) onto the Calin algeba C ( X ) = B ( X) K( X ). In

9 TOPOLOGICAL DIVISOR OF ZERO PERTURBATION 05 An opeato T B ( X ) is called Fedholm (esp., semi-fedholm), if R ( T) is closed subspace of X and max { dim N( T), dim N ( T )} < (esp., min { dim N( T), dim N ( T )} < ). It is well nown (Atinson s theoem) that T is a Fedholm, if and only if π ( T) is invetible in C ( X ). Fo semi- Fedholm theoy and Calin algebas, see, fo instance, [6-8, 7]. If T is a semi-fedholm, the index of T is defined by ind ( T) = dim N( T) dim N ( T ). If T B ( X ), then σ e ( T ) = σ( π ( T )) denotes the essential spectum of T, ± while σ e ( T ) = { λ C : T λi is not semi-fedholm} denotes the semi- Fedholm spectum of T. Fo T B ( X ) and M a subspace of X, we denote by T M the estiction of T to M. An opeato T B ( X ) is stictly singula, if fo evey infinite dimensional closed subspace M of X, T M is not a homomophism. Fo moe details about stictly singula opeato, see [6, 3]. It is well nown that if K is a compact opeato, then K is stictly singula opeato. Vaious diffeent measues of non-compactness of bounded opeatos have appeaed in the liteatue. In the following, we show that the usual measues of non-compactness and othe elated quantities ae ρ Z - petubation functions. Example 3.8. Fo T B ( X ), we denote by σ ( T) = σ ( T) { µ : { µ } is a component of σ( T) }. e The function ρ : B ( X) R defined by e e + ρ ( T) = sup { λ: λ σ ( T)}, e e

10 06 is a ρ d- petubation and a ρd- petubation functions. We pove fist that σ ( T) σ ( T). Let λ σ ( T), assume that λ σ ( ). So T λ I is e e T Fedholm. Now, by [3, Coollay V..7] and [3, Theoem V..6], we now that thee exists ε > 0 such that T µ I is Fedholm and dim N ( T µ I) and dim N (( T µ I) ) ae constant fo all µ D ( λ, ε) \{ λ}, whee D ( λ, ε) denotes the open dis centeed at λ with adius ε. Since λ σ ( T), we deduce that dim N( T µ I) = dim N( ( T µ I) ) = 0, µ D ( λ, ε) \ { λ}. This implies that the singleton set { λ } is a component of σ ( T). Now, if ρ e ( T ) <, we deduce that ρ ( T ) <. So, T I is an invetible opeato. Example 3.9. Similaly as in Example 3.8, we pove that ± ± e e σ ( T) = σ ( T) { µ :{ µ } isacomponent of σ( T) }, is a ρ d- petubation and a ρd- petubation functions. Example 3.0. If Ω is a non-empty bounded subset of X, then Kuatowsi measue of non-compactness of Ω, denoted / ( Ω), is given by v/ ( Ω ) = inf { > 0 : Ω can be coveed by finitely many sets of diamete }. Fo T B ( X ) the Kuatowsi measue of non-compactness of T is denoted by Ψ ( T), and defined by v/ ( T ( Ω) ) Ψ ( T) = sup : v/ ( Ω ) > 0. v/ ( Ω) v

11 TOPOLOGICAL DIVISOR OF ZERO PERTURBATION 07 Fom [9, Lemma I.2.8], we now that Ψ ( T) = Ψ ( T + K) fo all K K ( X ), thus Ψ induces a map Ψ : C ( X) R + such that Ψ π = Ψ. Fom [9, Theoem I.4.4], it is not difficult to veify that Ψ is a ρ - petubation and a ρ - petubation functions. d d Example 3.. The measue of non-compactness of Housdoff. We define the function β : B ( X) R + by β ( T) = inf > 0 : T ( B ( 0, ) ) B ( xi, ), i= whee B ( xi, ) = { x X : x xi< }. Fom [9, Lemma I.2.8], we now that β ( ) T =β ( T + K), K K ( X ), thus β induces a map Φ : C ( X) R + such that Φ π=β. Fom [9, Theoem I.4.4], it is not difficult to see that Φ is a - petubation n ρ d and a ρd- petubation functions. Example 3.2. Measue of non-compactness (see [8] and [9, p. 24]). Fo T B ( X ), let : T T = inf { T :codim < }. m m M M By [9, Coollay I.2.22] (see also [8, p. 9]), we now that Tm= T + Km, K K( X ). Thus induces a map Θ : C ( X) R such that Θ π ( T) = T fo m m + all T B ( X ). The function Θ m is a ρ d- petubation and a ρ d- petubation functions. Clealy, Θ m has popety (). The popety (2) follows fom [8, Theoem 6.]. Example 3.3. Measues of non-stict singulaity (see [4, 23, 26, 29]). Fo T B ( H ), we mae the following definitions: m m

12 08 M G ( T) = inf { TN: N subspace of M, dim N = + }, ( T) = sup { G ( T) : M subspace of H, dim M = + }. M In [23, p. 062], we now that ( K) = 0 K is stictly singula ( T + K) = ( T), T B ( H ). Thus induces a map : ( ) C H R + such that π=. The function is a - petubation since popety (2) is a consequence of ρ d [23, Theoem 2.2]. Example 3.4. Measues of non-stict singulaity (see [, 23, 30]). Fo T B ( X ), let τ ( T) = sup { m( T M ): M subspace of X, dim M = + }, whee m( T ) = inf { T ( x) : x M, x = }. In [23, p. 062], we now that M τ ( K) = 0 K is stictly singula τ ( T + K) = τ ( T), T B ( X). Thus τ induces a map τ : C ( X) R such that τ π= τ. + τ is a ρ - petubation and a ρ - petubation functions. Indeed, d popety () in Definition 3. is obvious. The popety (2) is a consequence of [23, Coollay 2.5]. We pove fo a ρ d- petubation function an analogue of Gelfand s spectal adius fomula. Theoem 3.4. Let x A, N Eq, and let P be a ρ d - petubation function. Then d

13 TOPOLOGICAL DIVISOR OF ZERO PERTURBATION 09 Poof. We fist pove that ρ( x) = inf { P ( x ) : N } = lim + P ( x ). ρ( x) inf { P ( x ) : N }. Let N and λ C, such that λ > ( x ). Fom Definition 3., we have P P ( x ) P ( x ) <. λ λ So, x λ e lnv d ( A). Since σ d ( x ) = σd ( x), we deduce that By (2.3), we get This implies sup { λ C : λ σ ( x)} ( x ). d ρ ( x) ( x ). P P ρ( x) inf { P ( x ) : N }. () On the othe hand, by Definition 3., fo any, we have So, P Since,, we deduce that N Eq ( x ) x. lim P ( ) lim x x. (2) + + lim x = lim x = ρ( x). (3) + +

14 0 Now, (2) and (3) imply that lim P ( x ) ρ( x). + Now, the esult is an immediate consequence of () and (3). This completes the poof. Rema. Let us ema that Theoem 6.3 in [8] follows fom Theoem 3.4 (see, Example 3.2). Similaly, one can pove the following: Theoem 3.5. Let x A, N Eq, and let P be a ρ d - petubation function. Then ρ( x) = inf { P ( x ) : N } = lim + P ( x ). Fo the convenience of the eade, the following esult fom [3, Lemma 8, p. 2] is stated. Lemma 3.6. Let x A and ε > 0. Then, thee exists N Eq such that x ρ( x) + ε. Theoem 3.7. Let P be a ρ - petubation function. Then d ρ( x) = inf { ( x) : N }, x A. P Poof. Let N Eq and λ C, such that λ > P ( x ). Fom Definition 3., we have P Eq ( x ) P ( x ) <. λ λ This implies x λe lnv d ( A), and so sup{ : λ σ ( x)} ( x). λ d P

15 TOPOLOGICAL DIVISOR OF ZERO PERTURBATION Now by (2.3), we obtain ρ( x ) inf { ( x) : N }. () P Fo the convese, let ε > 0. By Lemma 3.6, we now that thee exists ε N Eq such that x ε ρ( x) + ε. This poves that Theefoe, P ε ( x) x ρ( x) + ε. ε Eq inf { ( x) : N } ρ( x) + ε. P Letting ε 0, we obtain Now, by () and (2), we get Eq inf { P ( x) : N Eq } ρ( x). (2) inf { ( x) : N } = ρ( x). P Eq This completes the poof. We can obtain the following theoem in the same way as Theoem 3.7. Theoem 3.8. Let P be a ρ - petubation function. Then d ρ( x) = inf { ( x) : N }, x A. P Eq 4. Case of C - Algebas We conclude this pape with othe esults on ρz - petubation functions in the case of C - algebas. Thoughout this section, we will assume that A is a unital C - algeba.

16 2 Then Theoem 4.. Let x A and let P be a ρd - petubation function. a ρ( x) = inf { P ( e xe ) : a = a A} a = inf { P ( bxb ) : b lnv ( A)}. Poof. Fom the poof of Theoem 3.7, we see that ρ( x) P ( x). Since ρ( bxb ) = ρ( x) fo all b lnv ( A), we deduce that ρ( x ) inf { P ( bxb ) : b lnv ( A)} a a inf { P ( e xe ) : a = a A}. Since P ( x) x, we conclude fom [20, Poposition 4], that a a ρ( x) = inf { e xe : a = a A} a inf{ P ( e xe ) : a = a A}. a This completes the poof of the theoem. We can obtain the following theoem in the same way as Theoem 4.. Then Theoem 4.2. Let x A and let P be a ρd - petubation function. a ρ( x) = inf { P ( e xe ) : a = a A} a = inf { P ( bxb ) : b lnv ( A)}. Refeences [] K. Astala and H. O. Tylli, On the bounded compact appoximation popety and measues of non-compactness, J. Funct. Anal. 70 (987), [2] B. Aupetit, Popiétés Spectales des Algèbes de Banach, Lectue Notes in Mathematics 735 (979). [3] F. F. Bonsall and J. Duncan, Numeical anges of opeatos on nomed spaces and of elements of nomed algebas, London Math. Soc. Lectue Note Seies, 97.

17 TOPOLOGICAL DIVISOR OF ZERO PERTURBATION 3 [4] F. F. Bonsall and J. Duncan, Complete Nomed Algebas, Spinge-Velag, 973. [5] R. Bouldin, The essential minimum modulus, Indiana Univ. Math. J. 30 (98), [6] S. R. Caadus, W. E. Plaffenbege and B. Yood, Calin Algebas and Algebas of Opeatos on Banach Spaces, M. Dee, New Yo, 974. [7] J. B. Conway, A Couse in Functional Analysis, Second Edition, Spinge-Velag, New Yo, 990. [8] R. G. Douglas, Banach Algeba Techniques in Opeato Theoy, Academic Pess, 972. [9] D. E. Edmunds and W. D. Evans, Spectal Theoy and Diffeential Opeatos, Claendon Pess, Oxfod, 987. [0] F. Galaz-Fontes, Measues of non-compactness and uppe semi-fedholm petubation theoems, Poc. Ame. Math. Soc. 8 (993), [] F. Galaz-Fontes, Appoximation by semi-fedholm opeatos, Poc. Ame. Math. Soc. 20 (994), [2] H. A. Gindle and A. E. Taylo, The minimum modulus of a linea opeato and its use in spectal theoy, Studia Math. 22 (962/63), 5-4. [3] S. Goldbeg, Unbounded Linea Opeatos, McGaw-Hill, New Yo, 966. [4] M. Gonzalez and A. Matinon, Opeational quantities chaacteizing semi-fedholm opeatos, Studia Math. 4 (995), [5] P. Gopalaj and A. Stöh, On the essential lowe bound of elements in von Neumann algebas, Integal Equations Opeato Theoy 49 (2004), [6] R. Kadison and J. Ringose, Fundamentals of the Theoy of Opeato Algebas, Academic Pess, Olando, Vol. I, 983. [7] T. Kato, Petubation theoy fo nullity, deficiency and othe quantities of linea opeatos, J. Analyse Math. 6 (958), [8] A. Lebow and M. Schechte, Semigoups of opeatos and measues of noncompactness, J. Funct. Anal. 7 (97), -26. [9] M. Mehta, Fonctions petubation et fomules du ayon spectal essentiel et de distance au spectale essentiel, J. Opeato Theoy 5 (2004), 3-8. [20] G. J. Muphy and T. T. West, Spectal adius fomulae, Poc. Edinbugh Math. Soc. (2) 22(3) (979), [2] A. Pietsch, Opeatos Ideals, VEB Deutsch Velag de Wissenschaften, Belin, 978. [22] C. E. Ricat, Geneal Theoy of Banach Algebas, Van Nostand, Pinceton, 960. [23] M. Schechte, Quantities elated to stictly singula opeatos, Indiana Univ. Math. J. 2 (972), [24] J. G. Stampfli and J. P. Williams, Gowth conditions and the numeical ange in a Banach algeba, Tôhou Math. J. 20 (968),

18 4 [25] H. O. Tylli, On the asymptotic behaviou of some quantities elated to semi- Fedholm opeatos, J. London Math. Soc. 3(2) (985), [26] H. O. Tylli, The essential nom of an opeato is not self-dual, Isael J. Math. 9 (995), [27] J. Zemáne, Geometic intepetations of the essential minimum modulus, Invaiant subspaces and othe topics, Opeato Theoy: Adv. Appl. Bihäuse, Basel-Boston, Mass. 6 (982), [28] J. Zemáne, The sujectivity adius, pacing numbes and boundedness below of linea opeatos, Int. Equa. Ope. Theoy 6 (983), [29] J. Zemáne, The semi-fedholm adius of a linea opeato, Bull. Polish Acad. Sci. Math. 32 (984), [30] J. Zemáne, Geometic chaacteistics of semi-fedholm opeatos and thei asymptotic behaviou, Studia Math. 80 (984), [3] S. Živović, Semi-Fedholm opeatos and petubation functions, Filomat (997), [32] S. Živović, Semi-Fedholm opeatos and petubations, Publ. Inst. Math. (Beogad) (N.S.) 6(75) (997), [33] S. Živović, Measues of non-stict-singulaity and non-stict-cosingulaity, Mat. Vesni 54 (2002), -7. g

SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS

Fixed Point Theoy, Volume 5, No. 1, 2004, 71-80 http://www.math.ubbcluj.o/ nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal