SOLVABILITY OF NONLINEAR EQUATIONS

Transcription

1 SOLVABILITY OF NONLINEAR EQUATIONS PETRONELA CATANĂ Using some numerical characteristics for nonlinear operators F acting between two Banach spaces X and Y, we discuss the solvability of a linear ecuation λx Lx = y,y X. We extend the spectral sets defined by means of lower characteristics to discuss the solvability of the nonlinear equation λx F (x) =y, y X. AMS 2 Subject Classification: 47J1. Key words: measure of noncompactness, epi and k-epi operators, measure of solvability and stable solvability, nonlinear integral equation. 1. INTRODUCTION We use some numerical characteristics for nonlinear operators F between two Banach spaces X and Y over K (see [3], [2]), to describe mapping properties of F, such as compactness, Lipschitz continuity or quasiboundedness. We consider several subsets of K by means of the lower characteristics [F ] Lip, [F ] q, [F ] b and [F ] a, defined below, because these give us information on the solvability of the linear equation λx Lx = y, y X. Ourideaistouse these sets to provide information on the solvability of the nonlinear equation λx F (x) =y, y X. We will consider a more general problem of the form λj(x) F (x) =y, y X, where F and J are continuous nonlinear operators between the Banach spaces X and Y. Using the measure of solvability of F and the homotopy property of k-epi operators, we give a result for stably solvable operators, which can be proved as a direct consequence of the Rouché type estimate for stably solvable operators, involving the measure of stable solvability of F. These results are ilustrated by means of applications to nonlinear integral equation. We namely consider a Hammerstein integral equation and a Uryson integral equation of second kind. 2. PRELIMINARIES Let X and Y be two Banach spaces over K and F : X Y a continuous operator. We recall a useful topological characteristic in the theory and applications of both linear and nonlinear analysis. The measure of noncompactness MATH. REPORTS 9(59), 3 (27),

2 25 Petronela Catană 2 of a bounded subset M of X is defined by (2.1) α(m) =inf{ɛ : ɛ >, M has a finite ɛ-net in X}. Here, by a finite ɛ-net for M we understand a finite set {x 1,...,x n } X with the property that M [x 1 + B ɛ (θ)] [x n + B ɛ (θ)], for the closed ball with centre θ and radius ɛ>inx. Given the set F C(X, Y ) of all continuous operators from X into Y, we define (see [2]) F (x) F (y) (2.2) [F ] Lip =sup x y x y and F (x) F (y) [F ] Lip =inf x y x y and write F Lip(X, Y )if[f ] Lip < ; [F ] Lip =meansthatf is constant. We also consider another characteristics F (x) (2.3) [F ] Q = lim sup x x F (x) and [F ] q = lim inf x x of F C(X, Y ), and we write F Q(X, Y )if[f] Q < and call the operator F quasibounded; [F ] Q =meansthatf has strictly sublinear growth, more precisely, F (x) = o( x ) as x. We also consider F (x) (2.4) [F ] B =sup x x and F (x) [F ] b =inf x x and write F B(X, Y )if[f ] B < and call the operator F linearly bounded; [F ] B implies F =Θ. Let X and Y be two infinite dimensional Banach spaces. Recall that a continuous operator F : X Y is said to be α-lipschitz if there exists k> such that α(f (M)) kα(m) for any bounded subset M X. Set (2.5) [F ] A =inf{k : k>, α(f (M)) kα(m)}). We say that [F ] A is the measure of noncompactness of F or the α-norm of F ; if [F ] A 1 the operator F is called α-nonexpansive and α-contractive if the inequality is strict. We also introduce the lower characteristic (2.6) [F ] a =sup{k : k>, α(f (M)) kα(m)}. As in the linear case, equivalent representation, in infinite dimensional spaces are useful: α(f (M)) (2.7) [F ] A = sup α(m)> α(m) and [F ] a = inf α(m)> α(f (M)). α(m)

3 3 Solvability of nonlinear equations 251 We now introduce several subsets of K by means of the lower characteristics [F ] Lip, [F ] q, [F ] b and [F ] a (see [2]): (2.8) σ Lip (F )={λ K :[λi F ] Lip =}, σ q (F )={λ K :[λi F ] q =}, σ b (F )={λ K :[λi F ] b =}, σ a (F )={λ K :[λi F ] a =}. For F L (X), these subspectra give information about the solvability of the linear equation (2.9) λx Lx = y, y X. Ourideaistousethespectralsetstoprovideinformationonthesolvability of the nonlinear equation (2.1) λx F (x) =y, y X. If λ σ Lip (F ) then the operator λi F is injective and equation (2.1) has at most one solution for a fixed y. The relation λ {σ Lip (F ),σ q (F ),σ b (F )} does not imply the surjectivity of the operator λi F, not even in the linear case. 3. THE MEASURE OF SOLVABILITY OF F We consider a general problem of the form (3.1) λj(x) F (x) =y, y Y, where F and J are continuous nonlinear operators between two Banach spaces X and Y. Definition 3.1 (see [2]). Let X and Y be Banach spaces over K. Denote by F (X) the family of all open, bounded, connected subsets Ω of X with θ Ω. A continuous operator F : Ω Y is called epi on ΩifF (x) θ on Ω and, for any compact operator G : Ω Y satisfying G(x) on Ω, the equation F (x) =G(x) has a solution x Ω. More generaly, we call F a k-epi operator on Ω, k if the property mentioned before holds for all operators with [G] A k (not only for compact operators). For F : Ω Y and Ω F (X) as before, we introduce (3.2) ν Ω (F )=inf{k : k>, F is not k-epi on Ω} (3.3) ν(f )= inf ν Ω(F ), Ω F (X) where ν(f ) stands for the measure of solvability of F. The homotopy property gives a continuation principle for epi and k-epi operators. It may be compared with its analogue property of the topological degree. We recall the homotopy property. Suppose that F : Ω Y is k -epi on Ωforsomek, that H : Ω [, 1] Y is continuous with H(x, ) θ

4 252 Petronela Catană 4 and α(h(m [, 1])) kα(m), M Ωforsomek k. Let S = {x Ω:F (x)+h(x, t) =θ for some t [, 1]}. If S Ω = then the operator F 1 = F + H(, 1) is k 1 -epi on Ωfork 1 k k. Theorem 3.1. Let F H(X, Y ) and J : X Y with ν(j) >. Fix λ K with λ ν(j) > [F ] A and let (3.4) S = {x X : λj(x) =tf (x) for some t (, 1]}. Then either S is bounded, or the operator λj F is k-epi on Ω for some Ω F (X) and every k λ ν(j) [F ] A. Proof. Applying the homotopy property of k-epi operators to the operator G = λj and the homotopy H(x, t) = tf (x), we get α(h(m [, 1])) α(co(f (M) {θ})) = α(f (M)) [F ] A α(m), M Ω. If S is bounded, we may find Ω F (X) such that S Ω =. Again, from homotopy, we conclude that the operator G + H(, 1) = λj F is k-epi on Ω, for k λ ν(j) [F ] A. Using a Rouché type estimate, one can show that λ sup J(x) < inf F (x), Ω F (x), x Ω x Ω without using the set S. Definition 3.2 (see [6]). We call stably solvable a continuous operator F : X Y if, given any compact operator G : X Y with [G] Q =, the equation F (x) =G(x) has a solution x X. Remark. Every stable solvable operator is surjective, take G(X) y, but the converse is not true. For F C(X, Y ) is called the number (3.5) µ(f )=inf{k : k, F is not k-stably solvable} the measure of stable solvability of F. On account of this definition we may use a Rouché type inequality, namely, (3.6) µ(f + G) µ(f ) max{[g] A, [G] Q } for F, G C(X, Y ) and the following result holds. Lemma 3.1. Let F, G C(X, Y ). If F is k-stably solvable with k [G] A and k [G] Q, then F + G is k -stably solvable for k k max{[f ] A, [F ] Q }. We next have Lemma 3.2. Suppose that F : Ω Y is strictly epi on Ω and G : Ω Y satisfies sup x Ω G(x ) < dist(θ, F( Ω)) and [G Ω ] <ν Ω (F ). Then F + G is strictly epi on Ω.

5 5 Solvability of nonlinear equations 253 The proof of Lemma 3.2 (see[2]) shows that the Rouché type inequality (3.7) ν(f + G) ν(f ) [G] A holds for the characteristic ν(f ), thus paralleling (3.6). The next lemma connects the measure of solvability and the measure of stable solvability of F. Lemma 3.3. For any F C(X, Y ) we have µ(f ) ν(f ). The next theorem gives a similar result for stably solvable operators and can be proved using the Rouché type estimate (3.6) for stably solvable operators. Theorem 3.2. Suppose that F H(X, Y ) Q(X, Y ) and J : X Y satisfies µ(j) >. Fix λ K with λ µ(j) > max{[f ] A, [F ] Q }. Then the operator λj F is k-stably solvable for every k λ µ(j) max{[f ] A, [F ] Q }. In particular, equation (3.1) has a solution x X for every y Y. 4. SOME APPLICATIONS INVOLVING NONLINEAR INTEGRAL EQUATION (I) Let us first consider a Hammerstein integral equation of the form (4.1) λx(s) The nonlinear Hammerstein operator (4.2) H(x)(s) = k(s, t)f(t, x(t)) dt = y(s), s 1. k(s, t)f(t, x(t)) dt can be used as a composition H = KF of the nonlinear Nemytskij operator (4.3) F (x)(t) =f(t, x(t)) generated by the nonlinearity of f and the linear integral operator (4.4) ky(s) = k(s, t)y(t)dt generated by the kernel function k. Assume that k :[, 1] [, 1] R is continuous while f :[, 1] R R satisfies a Carathéodory condition and a growth condition of the form (4.5) f(t, u) a(t)+b(t) u, t 1, u R, with functions a, b L 1 [, 1]. We write x 1 for the L 1 -norm and define a scalar function h by (4.6) h(t) = max k(s, t), t 1. s 1

6 254 Petronela Catană 6 Proposition 4.1 (see [2]). Suppose that λ > hb 1. Then equation (4.1) has a solution x C[, 1] for y(s). Moreover, if a(t) in (4.5), then equation (4.1) has a solution x C[, 1] for every y C[, 1]. Proof. We apply Theorem 3.2 with X = Y = C[, 1], J = I. The nonlinear Hammerstein operator (4.2) being compact in X, weseethat[λi H] a = λ > and distinguish two cases for λ. First, suppose that λ/ σ b (H) i.e. [λi H] b >. Consider the set (4.7) S = {x X : λx = th(x) forsomet (, 1]}. For x S we have λ x H(x) ha 1 + hb 1 x, hence x ha 1. λ hb 1 So, the set S defined by (4.7) is bounded and Theorem 3.2 implies that the operator λi H is k-epi on X for k < λ, i.e., ν(λi H) >. The assumption [λi H] b > implies that λ ρ F (H), so the equation H(x) =λx has a solution. Second, suppose that λ σ b (H), i.e., [λi H] b =. Then we can find a sequence {x n } X such that λx n H(x n ) 1 n x n and λ x ha 1 hb 1 x 1 n x n. Hence ( λ hb 1 1 n ) x n ha 1, i.e., {x n } is bounded because λ > hb 1. Consequently, λx n H(x n ) asn. Let M := {x 1,x 2,...} and [λi H] a α(m) α((λi H)(M)) =. Then {x n } has a convergent subsequence and its limit is a solution of the equation H(x) =λx. Now, assume that a(t). Then Feng s spectral radius defined by r F (H) =sup{ λ : λ ρ F (H)}, where ρ F (H) ={λ K : λi H is F -regular} is the Feng resolvent set, satisfies H(x) r F (H) max{[h] A, [H] B } =sup hb 1, x θ x so λ ρ F (H) for λ > hb 1. (II) Another application refers to Uryson integral equation of the second kind (4.8) λx(s) k(s, t, x(t)) dt =, s 1.

7 7 Solvability of nonlinear equations 255 We shall study the nonlinear Uryson operator (4.9) U(x)(s) = k(s, t, x(t)) dt generated by (4.8) in the space L 2 [, 1]. About the continuous nonlinear kernel function k :[, 1] [, 1] R we make the following assumptions: (4.1) sup k(s, t, u) β r (s, t) withm r = sup u r s 1 (4.11) sup k(s, t, u) k(σ, t, u) γ r (s, σ, t) with lim u r s σ β r (s, t)dt<, γ r (s, σ, t)dt =, (4.12) k(s, t, u) Ψ(s, t)(1 + u ) withm = Ψ(s, t) 2 dt ds <. Proposition 4.2 (see [2]). Suppose that λ > 4M. Then equation (4.8) has a solution x L 2 [, 1]. Proof. We apply Theorem 3.2 with X = Y = L 2 [, 1] and J = I. The nonlinear Uryson operator (4.9) is compact in X, under the assumptions (4.1) (4.12). For any x X we have ( 2 ( 2 U(x)(s) 2 = k(s, t, x(t)) dt) Ψ(s, t)(1 + x(t) )dt) ( )( ) Ψ(s, t) 2 dt (1 + x(t) ) 2 dt 4 ( ) Ψ(s, t) 2 dt (1 + x 2 ). (Wehaveusedthefactthat(a + b) p 2 p (a p + b p )fora, b andp 1.) So, we have ( ) U(x) 2 4 Ψ(s, t) 2 dt ds (1 + x 2 ) 4M(1 + x 2 ). Again, we can distinguish two cases: [λi U] b > and[λi U] b = In the first case, the set S = {x X : λx = tu(x) fort (, 1]} is bounded because for every x S we have λ 2 x 2 U(x) 2 4M(1 + x 2 ), hence x 2 4M λ 2 4M. By Theorem 3.2, again, the operator λi U is k-epi for k < λ, so that λ ρ F (U).

8 256 Petronela Catană 8 In the second case, we can find a sequence {x n } in X such that λx n U(x n ) x n /n. As before, the sequence {x n } is bounded and the estimate 1 n x n λ x n U(x n ) λ x n 2 M 1+ x n 2 implies that λ 2 ( ) 1 1 M x n n. Letting n, the unboundedness of {x n } would give λ 2 M, contradicting the choice of λ. So, we proved that {x n } is bounded and the proof can be completed as in Proposition 4.1. Acknowledgements. The author wants to express her gratitude to Professor Dan Pascali for his support in the preparation of this paper. REFERENCES [1] J. Applell, Some spectral theory for nonlinear operators. Nonlinear Anal. 3 (1997), [2] J. Appell, E.D. Pascale and A. Vignoli, Nonlinear Spectral Theory. De Gruyter Ser. Nonlinear Anal. Appl. 1. Walter de Gruyter & Co., Berlin, 24. [3] J. Appell and M. Dörfner, Some spectral theory for nonlinear operators. Nonlinear Anal. 28 (1997), [4] D.E. Edmunds and D.W.D. Evans, Spectral Theory and Differential Operators. Oxford Univ. Press, New York, [5] D.E. Edmunds and J.R.L. Webb, Remarks on nonlinear spectral theory. Boll. Un. Mat. Ital. B (6) 2 (1983), [6] W. Feng, A new spectral theory for nonlinear operators and its applications. Abstracts Appl. Anal. 2 (1997), [7] M. Furi, M. Martelli and A. Vignoli, Stably solvable operators in Banach spaces. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 6 (1976), [8] M. Furi, M. Martelli and A. Vignoli, Contributions to the spectral theory for nonlinear operators in Banach spaces. Ann. Mat. Pura Appl. 118 (1978), [9] M. Furi, M. Martelli and A. Vignoli, On the solvability of nonlinear operator equations in normed spaces. Ann. Mat. Pura Appl. 128 (198), [1] R.H. Martin, Jr., Nonlinear Operators and Differential Equations in Banach Spaces. Wiley-Interscience, New York, [11] E.U. Tarafdar, and H.B. Thompson, On the solvability of nonlinear noncompact operator equations. J. Austral. Math. Soc. Ser. A 43 (1987), Received 6 January 27 Carmen Sylva High-School Eforie Sud, Constanţa, Romania petronela catana@yahoo.com