SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS

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1 Fixed Point Theoy, Volume 5, No. 1, 2004, nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal Militay College of Canada P. O. Box 17000, STN Foces Kingston, Ontaio Canada, K7K 7B4 gisac@juno.com 2 Nonlinea Analysis Cente Univesity of Caiova 13 A. I. Cuza Steet Caiova Romania zace@cental.ucv.o Abstact. Let E be a locally convex space and f : E E a mapping. We say that the equation f(x) = 0 is almost solvable on A E if 0 f(a). In this pape some esults about the solvability and almost solvability ae given. Ou esults ae based on some classical fixed point theoems and on some geometical conditions. Key Wods and Phases: Nonlinea opeatos, fixed point theoems Mathematics Subject Classification: 47J05, 4FJ25, 46C Intoduction In the poof of Theoem 2 pesented in ou pape [5] we used the semiinne-poduct [, ] in the sense defined by G. Lume [8] and studied by J. R. Giles [4]. We indicated also anothe kind of semi-inne-poduct, this is the semi-inne-poduct defined in a Banach space by [x, y] = y lim x 0 + y + tx y t and pesented in the K. Deimling s book [3]. 71

2 72 G. ISAC AND C. AVRAMESCU We note that this semi-inne-poduct is not linea in the fist vaiable, as the Lume s semi-inne-poduct, but it is only sub-linea. Howeve, ou Theoem 2 poved in [5] is also valid when we eplace the Lume s semi-inne-poduct by the Deimling s semi-iine-poduct, because in the poof of this theoem we used only two popeties, which ae valid fo both semi-inne-poducts. Now, in the section 2 of this pape we give a new vaiant of Theoem 2 poved in [5]; some consequences of this esult ae also indicated. This esult is genealized in the section 3 fo othe kind of opeatos, which ae not compacts. 2. Solvability in Banach spaces 2.1. Peliminaies. Let (E, ) be a Banach space, > 0 a eal numbe and f : E E a completely continuous mapping, i.e. f is continuous and fo any bounded set D E we have that f(d) is a elatively compact set. We denote: B = {x E x }; S = {x E x = } We will conside in B the equation f(x) = 0 (2.1) Ou goal is to study the almost and the solvability of equation (2.1). We say that equation (2.1) is almost solvable if 0 f(b ). (2.2) The notion of almost solvability is justified by the fact that, the condition (2.2) is equivalent with the equality inf f(x) = 0, x B (2.3) 2.2. Function G. Suppose given a mapping G : E E R, satisfying the following popeties: (g 1 ) G(x, x) 0 fo all x S (g 2 ) G(λx, y) λg(x, y), fo all λ > 0 and all x, y S. We give some examples of functions G satisfying (g 1 ) and (g 2 ). Example 1. If (E,, ) is a Hilbet space, then G(, ) =,.

3 SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS 73 Example 2. If (E, ) is a Banach space, then G(, ) = [, ] l whee [, ] l is the semi-inne-poduct in the Lume s sense, o G(, ) = [, ] d whee [, ] d is the semi-inne-poduct in the Deimling s sense. Example 3. Let (H,, ) be a Hilbet space and E = C([0, 1], H) the nomed vecto space with the nom x = sup x(t) H, whee H is the t [0,1] nom of the space H. Then we may take o o G(x, y) = sup x(t), y(t) t [0,1] G(x, y) = G(x, y) = inf x(t), y(t) t [0,1] 1 0 x(t), y(t) dt. Notice that fo these examples, the condition (g 2 ) can be eplaced by (g 2 ) G(λx, y) = λg(x, y), λ > 0, x, y S. If the function G satisfies the following popety: thee exists k > 0 such that G(x, y) k x y, fo all x, y B, (2.4) then in this case, we say that the function G is subodinated to the nom. Remak that all paticula functions G defined above ae subodinated to the nom Main esult. The main esult of this section is contained in the following theoem. Theoem 2.1. Let (E, ) be a Banach space. Suppose that i) f : E E is a completely continuous mapping, ii) G : E E R is a mapping such that the popeties (g 1 ) and (g 2 ) ae satisfied fo a paticula > 0, iii) the following inequality is satisfied G(f(x), x) < 0, fo any x S. (2.5) Then the equation (2.1) is almost solvable in B. Poof. If thee is x B such that f(x) = 0, then the equation (2.1) is solvable and, consequently, almost solvable.

4 74 G. ISAC AND C. AVRAMESCU Suppose that 0 f(b ), we will show that 0 f(b ). Indeed, we suppose the contay, that is, we suppose that 0 f(b ). In this case we can conside the mapping F (x) = f(x) f(x), x B. We have that F : B E is well defined and continuous. We have also the inclusion F (B ) S B. As in the poof of Theoem 2 ([5]), we can show that F (B ) is elatively compact. Theefoe, applying the Schaude Fixed Point Theoem we obtain the existence of a point x B, such that F (x ) = x, x S. Then we have ( ) f(x ) 0 > G(f(x ), x ) = G x, x f(x ) G(x, x ) 0 which is impossible. Hence, we must have that 0 f(b ) and the poof is complete. We note that, if G satisfies the condition G(x, x) 0 if x 0 and only (g 2 ) then we can eplace (2.5) by Indeed, fom x = F (x ) it follows G(x, x)g(f(x), x) < 0 fo all x S f(x ) G(x, x ) = G(f(x ), x ), x S and consequently G(x, x )G(f(x ), x ) 0 which contadicts the pecedent inequality Some solvability esults. It is clea that if the hypothesis of the Theoem 2.1 ae satisfied and 0 (B ) implies 0 f(b ) (2.6) then (2.1) is solvable. Indeed, if (2.6) is valid then F has a fixed point and a contadiction is then obtained. Using this emak we can obtain the following coollay. Coollay 2.1. If the hypothesis of the Theoem 2.1 ae satisfies and if thee exists a > 0 such that f(x) f(y) a x y fo all x, y B (2.7)

5 SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS 75 then the equation (2.1) is solvable and has a unique solution. Poof. Suppose that the implication (2.6) is not valid, i.e. 0 f(b ) but 0 f(b ). If x m B such that f(x m ) 0, then we have x m x p 1 a f(x m) f(x p ), which implies that {x m } is a Cauchy sequence and consequently x m x B. Because f is continuous we have that f(x) = 0 which contadicts the hypothesis. Theefoe (2.6) holds. Because f is injective the solution of (2.1) is unique. Now, we conside the case whee (2.7) is eplaced by Set f(x) f(y) a x y fo all x, y B (2.8) M = sup x B f(x) Note that M is finite, because f is completely continuous. Theoem 2.2. Suppose that: i) f : B E is a completely continuous mapping, ii) inequality (2.8) is satisfied, iii) G is an inne-poduct subodinated to the nom, i.e. iv) the inequality is satisfied fo all x S, whee G(x, y) k x y, x, y E, k > 0. G(f(x), x) c (2.9) c = k[a + M]. (2.10) Then the equation (2.1) is solvable on B. Poof. Define the function g : B R by g(x) = G(f(x), x). By (2.9) we have On the othe hand g(x) c < 0 fo all x S. (2.11) g(x) g(y) = G(f(x) f(y), x) + G(f(y), x y) k[ f(x) f(y) x + f(y) x y ] k[a + M] x y

6 76 G. ISAC AND C. AVRAMESCU theefoe whee g(x) g(y) α x y, x, y B (2.12) By (2.10) and (2.13) it follows that Set It is easy to see that Because α = k[a + M] (2.13) c = α (2.14) δ k = c 2 k α. (2.15) δ k =. k=1 g(x) = g(x) g(y) + g(y) α x y + g(y), it follows that fo all y S and x B such that we have Set 1 = δ 1, x y δ 1 (2.16) g(x) αδ 1 c c 2. (2.17) C 1 = {x B 1 x }. By (2.15), it follows that 1 > 0. On the othe hand fo evey x C 1 thee exists y C 1 such that (2.16) hold. Consequently Continuing this pocess we get whee g(x) c 2 fo all x C 1. g(x) c 2 k fo all x C k C k = {x k x k 1 }, k 1, 0 =. Now we suppose that (2.6) is not tue, then 0 f(b ) but thee is a sequence (x m ) B such that f(x m ) 0. Theefoe g(x m ) 0, m. (2.18)

7 SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS 77 We want to show that x m 0, m. (2.19) If (2.19) is not valid, thee exists a subsequence, again denoted by (x m ), such that x m β > 0, β. Because k 0, thee exists k β. Theefoe (x m ) k C j and consequently g(x m ) c, fo all m 1 (2.20) 2k which contadicts (2.18). But, fom (2.19) we have that f(0) = 0 which contadicts the assumption 0 f(b ). j=1 3. A solvability esult in locally convex spaces 3.1. Peliminaies. Now we give an extension of Theoem 2.1 to locally convex spaces. To do this, we need to ecall the following fixed point theoem. Theoem (Aino-Gautie-Penot) Let (E, τ) be a metizable locally convex space and let C E be a weakly compact convex set. Then any weakly sequentially continuous mapping fom C into C has a fixed point. The eade can find a poof of this esult in [1] Functions B and G. Let E(τ) be a metizable locally convex space. Suppose given a continuous mapping B : E R satisfying the following popeties: (b 1 ) B(x) > 0 fo any x E, x 0, (b 2 ) B(λx) = λb(x) fo any x E and λ > 0, λ R. Given > 0, ( R) we denote by S B = {x E B(x) = }. Suppose also given a mapping G : E E R satisfying conditions (g 1 ), (g 2 ) whee the set S is eplaced by the set S B. We ecall that a mapping f : E E is called stongly continuous if fo any sequence {x n } n N in E, weakly convegent to an element x E, we have that {f(x n )} n N is τ-convegent to f(x ).

8 78 G. ISAC AND C. AVRAMESCU 3.3. Main esult. We have the following esult. Theoem 3.1. Let E(τ) be a metizable locally convex space and suppose given a mapping B : E R satisfying conditions (b 1 ) and (b 2 ) and a mapping G : E E R satisfying conditions (g 1 ), (g 2 ) with espect to S B, fo a paticula. Suppose that the set D B = conv(s B ), is weakly compact. If f : E E is a stongly continuous mapping such that G(f(x), x) < 0, fo all x S B, then eithe thee exists an element x D B such that f(x ) = 0 o 0 f(d B ). Poof. If thee exists an element x D B, such that f(x ) = 0, then in this case the poof is complete. Suppose that f(x) 0 fo any x D B. We show that in this case 0 f(d B ). Indeed, suppose that 0 f(d B ). In this case we conside the mapping F (x) = B(f(x)) f(x), fo any x DB. We have that F is well defined and F (D B ) S B D B. Because 0 f(d B ), we can show that F : D B D B is (w)-sequentially continuous. By Aino-Gautie-Penot Fixed Point Theoem thee exists an element x D B, such that F (x ) = x S B. We have that which implies that f(x ) = B(f(x )) x, 0 > G(f(x ), x ) B(f(x )) G(x, x ) 0 which is a contadiction. Theefoe, we must have 0 f(d B ), and the poof is complete. Remak 1. If the mapping B is also sequentially weakly continuous, then in this case, we can suppose in Theoem 3.1 that f is sequentially continuous fom the weak to the weak topology. Remak 2. The eade can find seveal continuity tests fo a nonlinea mapping f : E E with espect to the given topology and the weak topology in [7].

9 SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS 79 Remak 3. In Theoems 2.1 and 3.1 we can eplace the inequality G(f(x), x) < 0 by G(f(x), x) > 0, if G is homogeneous with espect to the fist vaiable, since the zeos of f and of f ae the same Second existence esult. In 1970, F. E. Bowde intoduced in [2] the notion of nonlinea opeato of class (S) + and fo this kind of nonlinea mapping he defined a topological degee. I. V. Skypnik in [10] pesented a genealization of this topological degee and its applications to patial diffeential equations. Let (E, ) be a Banach space and E the topological dual of E. We ecall that a mapping f : E E is demicontinuous on a set D E if fo any sequence {x n } n N D convegent in nom to x 0 D, we have the equality lim f(x n), u = f(x 0 ), u fo all u E, n whee, is the duality between E and E. Also, we say that f is of class (S) + on D, if fo any sequence {x n } n N D with {x n } weakly convegent to x 0 E and such that lim sup f(x n ), x n x 0 n 0, we have that {x n } n N is convegent in nom to x 0. Finally, we say that f : E E is ρ-stongly monotone if thee exists a continuous stictly inceasing function ρ : R + R + such that ρ(0) = 0 and f(x) f(y), x y ρ( x y ) fo any x, y E. We have the following esult. Theoem 3.2. Let (E, ) be a sepaable eflexive Banach space. Let f : E E be a demicontinuous ρ-stongly monotone mapping. If thee exists > 0 such that f(x), x > 0 fo any x S, then the equation f(x) = 0 has a solution in B. Poof. This esult is a consequence of Theoem 4.4 poved in [10] (p. 47), because we take D = B. We have 0 B \ S and the assumption f(x), x > 0 implies also that f(x) 0 fo any x S. Moeove, in [6] was poved that any ρ stongly monotone mapping is of class (S) +. Theefoe, all the assumption of Theoem 4.4 ([10], p. 47) ae satisfied and the theoem follows. We note that, the poof of this Theoem 4.4 is based on the Skypnik-topological degee.

10 80 G. ISAC AND C. AVRAMESCU Refeences [1] O. Aino, Gautie, J. P. Penot, A fixed point theoem fo sequentially continuous mappings with applications to odinay diffeential equations, Funkcialaj Ekvacioj, 27(1984), [2] F. E. Bowde, Nonlinea elliptic bounday value poblems and the genealized topological degee, Bull. Ame. Math. Soc., 76(1970), [3] K. Deimling, Nonlinea Functional Analysis, Spinge-Velag, [4] J. R. Giles, Classes of semi-inne-poduct spaces, Tans. Ame. Math. Soc., 129(1967), [5] G. Isac, C. Avamescu, On some geneal solvability theoems, Foth-comping: Appl. Math. Lettes. [6] G. Isac, M. S. Gowda, Opeatos of class (S) 1 +, Altman s condition and the complementaity poblem, J. Faculty Sci. Univ. Tokyo, Sec. IA, 40(1993), [7] L. J. Lipkin, Weak continuity and compactness of nonlinea mappings, Nonlinea Anal. Theoy, Methods, Appl., 5, 11(1981), [8] G. Lume, Semi-inne-poduct spaces, Tans. Ame. Math. Soc., 100(1961), [9] J. Schaude, Defixpunktsatz functional aumen, Studia Math., 2(1930), [10] I. V. Skypnik, Methods fo Analysis of Nonlinea Elliptic Bounday Value Poblems, AMS, Tanslations of Mathematical Monogaphs, 139(1994).