ISSN 1064–5624, Doklady Mathematics, 2009, Vol 80, No 1, pp 482–486 © Pleiades Publishing, Ltd., 2009 Published in Russian in Doklady Akademii Nauk, 2009, Vol 427, No 2, pp 155–159 MATHEMATICS On a Semilinear Boundary Value Problem for Degenerate Parabolic Pseudodifferential Equations1 Yu V Egorova, Nguyen Minh Chuongb, and Dang Anh Tuanc Presented by Academician V.S Vladimirov February 18, 2009 Received March 6, 2009 DOI: 10.1134/S1064562409040085 In this article a semilinear boundary value problem is studied for a degenerate parabolic pseudodifferential equation The main result generalizes the famous theorem of Agranovich and Vishik (see [1]) We prove the existence of a solution using the Rothe theorem on a fixed point (see [2]) +∞ ᏸ u ( x, t ) = 1/2 ( 2π ) The space H(l, p, δ)(ޒ ⎛ ( ξ', q ) δ, d = ⎜ ξi ⎝i = ∑ p/ ( + δ i ) xn pδ i + q p/d⎞ tion of the space ∞ C0 ( u ( x, t ) dt ⎛ = ⎜ ⎝ , ) is defined as the comple- ޒ n – 1) v 1/ p ⎟ ⎠ – qt n–1 Let p ∈ (1, ∞), l ∈ ޒ+, q = µ + iτ ∈ ރ, µ ≥ 0, d ≥ 1, d ∈ ޚ, δ = (δ1, δ2, …, δn – 1, 0), δi ≥ 0, i = 1, 2, …, n – For each ξ' = (ξ1, ξ2, …, ξn – 1) ∈ ޒn – 1, xn ∈ ޒ+, set n–1 ∫e l, p, d, δ, ޒ n–1 ⎞ ( + ( ξ', q ) δ, d ) Ᏺ n – v ( ξ', q ) dξ'⎟ ⎠ n–1 ∫ ޒ with the norm ⎛ p ( ξ', x n, q ) δ, d = ⎜ ξi xn ⎝i = ∑ pδ i + q p/d⎞ n ∞ completion of the space P( ޒ+ ) = {u ∈ C (ޒn): suppu ∈ 1/ p ⎟ ⎠ ޒ+ } using the norm n , u ⎛ = ⎜ ⎝j = ∫ ޒ e – i 〈 x', ξ'〉 ∑ ∫ ∫ ( + ( ξ', q ) u ( x', x n, q ) d x', n l, p, d, δ, ޒ+ +∞ l Ᏺ n – u ( ξ', x n, q ) = ( n – )/2 ( 2π ) The space H(l, p, δ)( ޒ+ ) (l ∈ ޚ+) is defined as the n n–1 1/ p p lp ޒ n–1 δ, d + ( ξ', x n, q ) δ, d ) (l – j) p - n–1 ∫ Ᏺ n – v ( ξ', q ) ⎞p p j × D n Ᏺ n – u ( ξ', x n, q ) d x n dξ' ⎟ ⎠ The space Pl, p(d, à, , + ì (0, +)) is defined n = ( n – )/2 ( 2π ) ∫ ޒ e – i 〈 x', ξ'〉 v ( x', q ) d x' n–1 ∞ as the completion of the space P(ޒn) = {u ∈ C (ޒn × (–∞, +∞)): suppu ⊂ ޒ+ × (0, +∞)} with the norm n Define the Laplace transform u n P l, p ( d, µ, δ, ޒ+ × ( 0, +∞ ) ) +∞ ⎛ l p(l – j) ⎜ = ( + ( ξ', q ) δ, d + ( ξ', x n, q ) δ, d ) ⎜ ⎝ j = ޒnξ', τ ∑∫ ∫ a Université Paul Sabatier, Toulouse, France Institute of Mathematics, Hanoi, Vietnam c Hanoi National University, Hanoi, Vietnam b - The ∫ article was translated by the authors 482 ⎞p p j àt ì D n n u ( e u ( x, t ) ) ( ξ', x n, τ ) d x n dξ' dτ ⎟ ⎠ ON A SEMILINEAR BOUNDARY VALUE PROBLEM ∂Ω, are defined in the standard way using a unity partition Consider a pseudodifferential operator L(x, D, q, y) with a symbol The space Pl, p(d, à, , n ì (0, +∞)) is defined as ∞ C0 ( {v ∈ × ޒ the completion of the space P∞(ޒ n – × (0, +∞)} with the norm (–∞, +∞)): suppv ⊂ ޒ n – 1) = v P l, p ( d, µ, δ, ޒ n–1 n–1 ∑ L ( x, ξ, q, d ) = × ( 0, +∞ ) ) ∫ where m ∈ ޚ+, γi = + δi, i = 1, 2, …, n, lα ≥ 0, (0) is defined as the comThe space El, p(d, µ, δ, ޒ pletion of the space ᏸP(ޒn – 1) with the norm E l, p ( d, µ, δ, ޒ n–1 ) (0) aα ( · ) ∈ C ( ޒ ޒτ (1) E l, p ( d, µ, δ, ޒ ⎛ = ⎜ ⎝ v p l, p, q, d, δ, ޒ ޒτ n + ∑ n (1) b j ( ·, ξ' ) ∈ ∞ C0 ( (1) (1) b j ( x', · ) ∈ C ( ޒ n–1 ∫ γn – ∀ξ' ∈ ޒ α' β α b j ( x', λ ξ 1, λ ξ 2, …, λ γn – e – i 〈 x', ξ'〉 ∀λ > 0, (1) ξ n – ) = a α ( x, ξ' ), n–1 \ { }, ∀λ > l a α( x, ξ')x nα (ξ' ) q D n n Ᏺ n – u(ξ', x n, q) dξ' γ γ ∀λ > 0, ∀ξ' ∈ ޒ n–1 \ { } ), n–1 ξn – ) = λ k – dj ( ) b j ( x, \ { }, ∀ x' ∈ ޒ ) ޒ, ∀ξ' ∈ ޒ γn – n–1 ξ' ), n–1 ∫ ޒ e i 〈 x', ξ'〉 n–1 × b ( x', ξ', q, d ) Ᏺ n – v ( ξ', q )dξ' n–1 ξn – ) = ∀λ > 0, ∀ξ' ∈ ޒ DOKLADY MATHEMATICS b ( x', D', q, d )v ( x', q ) = ( n – )/2 ( 2π ) , ∀ j, n λ ξ 2, …, λ (0) ξ n – ) = a α ( ξ' ), A pseudodifferential operator b(x', D', q, d) with a symbol b(x', ξ', q, d) is denned as usually: \ { }, ∀ j, (1) γ2 \ { }, Let us introduce the operator j supp b j (·, ξ') ⊂ Kj, Kj are compact sets independent of ξ', γ (0) b j ( λ ξ 1, γn – \ { }, γ ∀x ∈ ޒ, + b j ( x', ξ' ) )q , where ∀ x' ∈ ޒ n–1 n–1 α + dβ ≤ 2m ޒ 〈 γ , α〉 + dβ – l α = 2m j=0 \ { } ), γ (1) n–1 a α ( x, λ ξ 1, λ ξ 2, …, λ k n–1 γ ∀ξ' ∈ ޒ (1) (0) j ( ξ' ) γ (0) dτ⎟ ⎠ Let us now set (0) ∀ξ' ∈ ޒ n a α ( λ ξ 1, λ ξ 2, …, λ ⎞p L ( x, D, q, d )u ( x, q ) = ( n – )/2 ( 2π ) b j ∈ C(ޒ \ { } ), (1) ∫ ∑ (b n–1 supp a α (·, ξ') ⊂ Kα, where Kα are compact sets independent of ξ' The spaces Pl, p(d, µ, δ, ∂Ω × (0, +∞)), Pl, p(d, µ, δ, ì (0, +)), El, p(d, à, , ) and El, p(d, µ, δ, Ω), where Ω is a bounded set in ޒn with smooth boundary b ( x', ξ', q, d ) = ∞ a α ( ·, ξ' ) ∈ C ( ) ޒ, The space El, p(d, µ, δ, ޒis defined as the completion of the space ᏸP∞(ޒn) with the norm n +) (1) \ { } ), a α ( x, · ) ∈ C ( ޒ ∀ x, ∀α , n +) v n–1 - ⎞p p v l, p, q, d, δ, ޒn – dτ⎟ ⎠ ∫ (1) a α ( x, ξ' ) = a α ( ξ' ) + a α ( x, ξ' ), n – 1) v α β l a α ( x, ξ' )x nα ξ n q , α + dβ ≤ 2m 〈 γ , α〉 + dβ – l α = 2m ⎛ ⎞ pp pl – µt = ⎜ + ( ξ', q) δ, d ) Ᏺn [ e v ( x', t) ](ξ', τ) dξ' dτ⎟ ⎜ ⎟ ⎝ ޒnξ', τ ⎠ ⎛ = ⎜ ⎝ 483 L ( x n, D, q, d )U ( x, q ) = F ( x, q ), B j ( D, q, d )U ( x, q ) k – dj ( ) λ b j ( ξ' ), \ { }, Vol 80 Consider the problem 2009 = G j ( x', q ), j = 1, 2, …, m, where No x=0 x ∈ ޒ+ , n (1) (2) 484 EGOROV et al ∑ L ( x n, D, q, d ) = l xn, q), ≤ k ≤ m, ≤ j ≤ 2m, are continuous in (ξ', xn, q) Moreover, α β a α ( ξ' )x nα ξ q , α + dβ ≤ 2m 〈 γ , α〉 + dβ – l α = 2m aα ∈ C ( ޒ n–1 γ1 2m γ2 \ { } ), a α ( λ ξ 1, λ ξ 2, …, λ = a α ( ξ' ), ∀λ > 0, ∀ξ' ∈ ޒ ∑ B j ( ξ', q, d ) = n–1 γn – γ γ = E l – 2m, p (d, à, , ì E j=0 à, , n–1 ) to El, p(d, µ, δ, ޒ n +, ޒ α>0 n – 1) l, p (d, µ, δ, α, ޒ+ , ޒn – 1) n such that (i) for any α > 0: R0 is bounded from El, p(d, µ, δ, α) to El, p(d, µ, δ); (ii) for any α > 0, (F, G) ∈ El, p(d, µ, δ, α), Ꮽ0R0(F, G) = (F, G) + Q(F, G), where Q is bounded from El, p(d, µ, δ, α) to E l, p (d, µ, δ, α); ∩ l>0 (iii) U ∈ El, p(d, µ, δ, ޒ+ and U(x', xn) = for xn ≥ α with some α > 0, then R0Ꮽ0U = U Proof Remark that (F, G) ∈ El, p(d, µ, δ, α) for almost all q ∈ C, Req = µ, (F, G) ∈ H(l, p, δ) and F(x', xn) = for xn ≥ α It is evident that there exists a unique function v(ξ', xn), such that L(xn, ξ', Dn, q, d)v(ξ', xn) = Ᏺn – 1F(ξ', xn, q), v(ξ', xn) = 0, for sufficiently large xn n bounded from the space El, p(d, µ, δ, ޒ+ ) to the space n El, p(d, µ, δ, ޒ+ , ޒn – 1) n The operator Ꮽ(xn, D, q, d) and problem (1), (2) are elliptic, if l a α (ξ') x nα ξαqβ ≠ 0, α + dβ ≤ 2m 〈 δ, α〉 + dβ – l α = 2m ∀ξ ∈ ޒn, q ∈ ރ, |ξ| + |q| ≠ 0, ∀xn > 0; (ii) the equation L0(xn, ξ', ξn, q, d) = has m solutions ξn with positive imaginary parts for xn > 0, |ξ'| + |q| ≠ 0; (iii) the Cauchy problem L ( x n, ξ', D n, q, d )v ( x n, q ) = 0, x n > 0; B j ( ξ', D n, q, d )v ( 0, q ) = h j , ∪E bounded operator R0 from ) ( d, l – m j – p ∑ ≤ k ≤ m n Proposition The operator Ꮽ(xn, D, q, d) is (i) L0(xn, ξ, q, d) = ≠0 ޒ+ , ޒn – 1)|F(x', xn) = for xn > α}, then there exists a m n +) ≤ j ≤ m – 1, xn = n \ { } n–1 ≤ C ( ( ξ', q ) δ, d ) El, p(d, µ, δ, α, ޒ+ , ޒn – 1) = {(F, G) ∈ El, p(d, µ, δ, α, For l ≥ l0, we set n p Lp Theorem Let Ꮽ be an ellipic operator If Set l0 = max{2m, mj + 1}, Ꮽ(xn, D, q, d)u = (L(xn, D, q, d)u, Bj(D, q, d) u xn = ) E l, p ( d, µ, δ, ޒ+, ޒ j for ξ' + q ≠ 0, ξ n – ) = b jkm ( ξ' ), n–1 D n Φ k ( ξ', x n, q ) detB j ( ξ', D n, q, d )Φ k ( ξ', x n, q ) \ { } ), ∀ξ' ∈ ޒ ∀λ > 0, c xn Suppose that k l γn – e j = 0k = \ { }, b jkl ( ξ' )ξ n q , n–1 b jkm ( λ ξ 1, λ ξ 2, …, λ ∑∑ ξn – ) k + dl = m j b jkl ∈ C ( ޒ m j = 1, 2, …, m has a unique solution, tending to as xn → +∞ for any hj ∈ ޒ Proposition Suppose that L0(xn, ξ, q, d) ≠ 0, ∀q ∈ ރ, |ξ| + |q| ≠ 0, ∀xn > 0, and the equation L0(x, ξ', ξn, q, d) = has m solutions ξn with positive Imξn for |ξ'| + |q| ≠ 0, xn > Then the space Nξ', q of solutions to the equation L(xn, ξ', Dn, q, d)v(xn, q) = has dimension m, with the base Φ1(ξ', xn, q), …, Φm(ξ', xn, q) and D n Φk(ξ', j Denote G ξ', q the map: (F, G) ۋv Then G ξ', q is a linear bounded map from Lp(0, α) to H2m, p(ޒ+) continuous in (ξ', q) Set R ( F, G ) = Ᏺ n – [ θG ξ', q ( Ᏺ n – F ( ξ', x n, q ) ) –1 + ( – θ) Ꮽ ξ', q ( Ᏺ n – F ( ξ', x n, q ) Ᏺ n – G ( ξ', q ) ) ], –1 ⎧ if ( ξ', q ) δ, d < θ = θ ( ( ξ', q ) δ, d ) = ⎨ ⎩ if ( ξ', q ) δ, d ≥ Consider the problem ∂ L ⎛ x n, D, -, d⎞ u ( x, t ) = f ( x, t ), ⎝ ∂t ⎠ ∂ B j ⎛ D, -, d⎞ u ( x, t ) ⎝ ∂t ⎠ xn = x ∈ ޒ+ , n = g ( x', t ), t > 0, t > 0, j = 1, 2, …, m DOKLADY MATHEMATICS Vol 80 No 2009 ON A SEMILINEAR BOUNDARY VALUE PROBLEM Theorem Suppose that the operator Ꮽ(xn, D, q, d) ∞ C0 ( ޒthere exists an ε > 0, is elliptic Then for any ψ ∈ such that for < ε ≤ ε0, there exists a linear operator R with the following properties: (i) for any a ≥ 0: R is bounded from the space n P l, p (d, µ, δ, a, ޒ+ × (0, +∞), ޒn – × (0, +∞)) to n) ∪ The operator Ꮽ(x, D, q, d) = (L(x, D, q, d), Bj(x, D, q, d)|∂Ω is elliptic at a point y ∈ Ω, if for y ∉ ∂Ω there is a neighborhood U ∩ ∂Ω = , y ∈ U, in which ϕiLψi has the principal part Pl, p(d, à, , ( ì 0, +)); (ii) for any a > 0, ( f, g) ∈ Pl, p(d, à, , a, ( ì 0, +), ( × ޒ0, +∞)) n + α + dβ = 2m –γ x 1, …, ε –γ n L i ( x, ξ, q ) ∑ = l α + dβ = 2m 〈 γ , α〉 + dβ – l α = 2m for x ∈ U i \∂Ω, B ij ( x', ξ, q, d ) = n (0, +∞)) → ∩P µ, δ, ޒ n–1 l, p (d, × (0, +∞)); is bounded; ∑ b jkl ( x', ξ' )ξ n q k l Then the trivial solution of the Cauchy problem L i ( 0, x n, ξ', D n, q, d )v ( x n ) = 0, (iii) if u Pl, p(d, à, , ( ì 0, +)) and u(x', xn, t) = 0, for xn ≥ a, then n + ∂ ∂ R ⎛ Ꮽ ⎛ x n, D, -, d⎞ + ψ ε ( x ) ⎛ Ꮽ ⎛ x, D, -, d⎞ ⎝ ⎝ ⎝ ⎝ ∂t ⎠ ∂t ⎠ ∂ – Ꮽ ⎛ x n, D, -, d⎞ ⎞ ⎞ u = u ⎝ ∂t ⎠ ⎠ ⎠ Let Ω be a bounded set in ޒn with smooth boundary ∂Ω Consider the problem ∂ L ⎛ x, D, -, d⎞ u ( x, t ) = f ( x, t ), ⎝ ∂t ⎠ x ∈ Ω, ∂ B j ⎛ x, D, -, d⎞ u ( x, t ) = g j ( x, t ), ⎝ ∂t ⎠ t > 0, where ξ + q ≠ 0, k + dl = m j l=0 N {Ui, ϕi } i = α β ϕ i ( x )a iα ( x, ξ' )x nα ξ q ≠ and ϕiBjψi has the principal part x n ), the operator T: Pl, p(d, à, , + ì (0, +), ޒn – × ∞ ξ + q ≠ 0; for y ∈ ∂Ω there is a neighborhood Ui ∩ ∂Ω ≠ , y ∈ Ui, in which ϕiLψi has the principal part ∂ – Ꮽ ⎛ x n, D, -, d⎞ ⎞ ⎞ R ( f , g ) = ( f , g ) + T ( f , g ), ⎝ ∂t ⎠ ⎠ ⎠ where ψ ε ( x ) = ψ ( ε β ϕ ( x )a iα ( x, ξ )q ≠ for x ∈ U i , n–1 ∂ ⎞ ∂ ⎛ Ꮽ ⎛ x , D, , d + ψ ε ⎛ Ꮽ ⎛ x, D, -, d⎞ ⎝ 0⎝ n ⎝ ⎝ ∂t ⎠ ∂t ⎠ ∑ L i ( x, ζ, q, d ) = a>0 n + 485 t > 0, (3) x ∈ ∂Ω, (4) B ij ( 0, ξ', D n, q, d )v ( ) = 0, j = 1, 2, …, m, for ξ' + q ≠ is the only its bounded solution Theorem Let the operator Ꮽ(x, D, q, d) be elliptic Then for all sufficiently large µ = Req and for any ( f, g) ∈ Pl + 1, p(d, µ, δ, Ω × (0, +∞), ∂Ω × (0, +∞)), problem (3), (4) has a unique solution u ∈ Pl, p(d, à, , ì (0, +)) Consider the problem ∂ L ⎛ x, D, -, d⎞ u ( x, t ) ⎝ ∂t ⎠ = f ( x , t , u ( x , t ), …, D x 2m – j = 1, 2, …, m, ∂ B j ⎛ x, D, -, d⎞ u ( x, t ) ⎝ ∂t ⎠ = is a partition of unity in Ω and ∞ the functions ψi ∈ C (Ω) are such that suppψi ⊂ Ui and ψi(x) = in a neighborhood of suppϕi After the Laplace transform problem (3), (4) goes to the problem: L ( x, D, q, d ) ᏸ u ( x, q ) = ᏸ f ( x, q ), x ∈ Ω, B j ( x, D, q, d ) ᏸ u ( x, q ) = ᏸ g j ( x, q ), x ∈ ∂Ω, Vol 80 No m –1 = g j ( x', t, u ( x, t ), …, D x j ∂Ω u ( x, t ) ), t > 0, (6) j = 1, 2, …, m; ∂ u ( x, t ) -k ∂t k = 0, l k = 0, 1, …, -0 , d (7) t=0 where f, qj satisfy the following conditions: (i) the maps (x, t, u) ۋf (x, t, u) and (x, t, uj) ۋgj(x, t, uj) satisfy the Caratheodory conditions, i.e they are j = 1, 2, …, m DOKLADY MATHEMATICS u ( u, t ) ), x ∈ Ω, t > 0; (5) 2009 486 EGOROV et al continuous in u = (u1, u2, …, uN), uj = (u1, u2, …, u N j ) for almost all x, and measurable in x, t for all u, uj, (ii) the maps u(x, t) ( ۋf (x, t, u(x, t)), gj(x, t, j u (x, t))), where u ( x, t ) = ( u ( x, t ), …, D x 2m – j (iii) there exists a r > 0, such that ∂ –1 ≥ lim sup Ꮽ ⎛ x, D, -, d⎞ × ( f , g j ) ⎝ ⎠ ∂t µ → +∞ u ( x, t ); m –1 u ( x , t ) = ( u ( x , t ), … , D x j from Pl, p(d, à, , ì (0, +)) to Pl, p(d, à, , ì (0, +), ∂Ω × (0, +∞)) are compact; u ( x, t ) ) where ( f , g j) ⎧1 j = sup ⎨ - ( f ( x, t, u ( x, t ) ), g j ( x, t, u ( x, t ) ) ) r ⎩ t > 0, u λ, µ P l, p ( d, à, , ì ( 0, + ), × ( 0, +∞ ) ) Theorem If Ꮽ(x, D, q, d) is elliptic, f, gj satisfy conditions (i)–(iii), then for all sufficiently large µ = Req, problem (5)–(7) has a solution u ∈ (Pl, p(d, µ, δ, Ω × (0, +∞)) Proof For any w ∈ Pl, p(d, µ, δ, Ω × (0, +∞)) the problem ∂ L ⎛ x, D, -, d⎞ u ( x, t ) = f ( x, t, w ( x, t ) ), ⎝ ∂t ⎠ x ∈ Ω, P l, p ( d, à, , ì ( 0, +∞ ) ) j By condition (iii) there are r > and µ0 > 0, such that for w Pl, p(d, à, , ì (0, +)) with P l, p ( d, à, , ì ( 0, +∞ ) ) –1 ⎛ -x, D, = r, ∀µ ≥ µ0, we have -Ꮽ ⎝ ∂ - , d - ì ||(f, gj)||, Thus u Pl, p ( d, à, , ì ( 0, +∞ ) ) ≤ r ∂t By (ii) the map w ۋu is compact from the closed sphere S r = { w ∈ P l, p ( d, à, , ì ( 0, + ) ) w to the ball P l, p ( d, µ, δ, Ω × ( 0, +∞ ) ) = r} u P l, p ( d, µ, δ, Ω × ( 0, +∞ ) ) ∂ B j ⎛ x, D, -, d⎞ u ( x, t ) ⎝ ∂t ⎠ ∂Ω x ∈ ∂Ω, ⎫ = r ⎬ ⎭ = g j ( x, t, w ( x, t ) ), t > 0, j = 1, 2, …, m, where µ = Req, has a unique solution u Pl, p(d, à, , ì (0, +∞)), such that ∂ –1 ≤ Ꮽ ⎛ x, D, -, d⎞ ⎝ ∂t ⎠ × ( f ( x , t , w ( x , t ) ), g j ( x , t , w ( x , t ) ) ) w λ, µ , P l, p ( d, µ, δ, Ω × ( 0, +∞ ), ∂Ω × ( 0, +∞ ) ) B r = { w ∈ P l, p ( d, à, , ì ( 0, +∞ ) ) w P l, p ( d, µ, δ, Ω × ( 0, +∞ ) ) ≤ r } Therefore, by the Rothe theorem, this map posses a fixed point in the ball Br of the space Pl, p(d, à, , ì (0, +)), which is a solution to the problem (5)–(7) REFERENCES M S Agranovich and M I Vishik, Usp Mat Nauk 19 (3), 53–161 (1964) E Rothe, Compositio Math 5, 177–197 (1937) Yu V Egorov, Nguyen Minh Chuong, and Dang Anh Tuan, C R Acad Sci Paris, Ser 337, 451–456 (2003) Yu V Egorov, Nguyen Minh Chuong, and Dang Anh Tuan, Dokl Math 74, 874–877 [Dokl Akad Nauk 411, 732–735 (2006)] DOKLADY MATHEMATICS Vol 80 No 2009 .. .ON A SEMILINEAR BOUNDARY VALUE PROBLEM ∂Ω, are defined in the standard way using a unity partition Consider a pseudodifferential operator L(x, D, q, y) with a symbol The space Pl, p(d,... 2009 ON A SEMILINEAR BOUNDARY VALUE PROBLEM Theorem Suppose that the operator Ꮽ(xn, D, q, d) ∞ C0 ( ޒthere exists an ε > 0, is elliptic Then for any ψ ∈ such that for < ε ≤ ε0, there exists a. .. qj satisfy the following conditions: (i) the maps (x, t, u) ۋf (x, t, u) and (x, t, uj) ۋgj(x, t, uj) satisfy the Caratheodory conditions, i.e they are j = 1, 2, …, m DOKLADY MATHEMATICS