Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps

We consider differential-operator equations with Wλ₀ -pseudomonotone operators. The problem of studying periodic solutions via the Faedo – Galerkin method has been considered. The important a priory estimates have been obtained. A topological description of resolvent operators is given.

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Дата:	2006
Автори:	Kasyanov, P.O., Mel'nik, V.S., Toscano, S.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2006
Назва видання:	Нелінійні коливання
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/178098
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps / P.O. Kasyanov, V.S. Mel'nik, S. Toscano // Нелінійні коливання. — 2006. — Т. 9, № 2. — С. 187-212. — Бібліогр.: 16 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1780982021-02-18T01:28:40Z Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps Kasyanov, P.O. Mel'nik, V.S. Toscano, S. We consider differential-operator equations with Wλ₀ -pseudomonotone operators. The problem of studying periodic solutions via the Faedo – Galerkin method has been considered. The important a priory estimates have been obtained. A topological description of resolvent operators is given. Розглянуто диференцiально-операторнi рiвняння з Wλ₀ -псевдомонотонними операторами. Розв’язано проблему вивчення перiодичних розв’язкiв методом Фаедо – Гальоркiна. Отримано важливi апрiорнi оцiнки. Наведено певний топологiчний опис резольвентних операторiв. 2006 Article Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps / P.O. Kasyanov, V.S. Mel'nik, S. Toscano // Нелінійні коливання. — 2006. — Т. 9, № 2. — С. 187-212. — Бібліогр.: 16 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178098 517.9 en Нелінійні коливання Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We consider differential-operator equations with Wλ₀ -pseudomonotone operators. The problem of studying periodic solutions via the Faedo – Galerkin method has been considered. The important a priory estimates have been obtained. A topological description of resolvent operators is given.
format	Article
author	Kasyanov, P.O. Mel'nik, V.S. Toscano, S.
spellingShingle	Kasyanov, P.O. Mel'nik, V.S. Toscano, S. Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps Нелінійні коливання
author_facet	Kasyanov, P.O. Mel'nik, V.S. Toscano, S.
author_sort	Kasyanov, P.O.
title	Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps
title_short	Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps
title_full	Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps
title_fullStr	Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps
title_full_unstemmed	Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps
title_sort	periodic solutions of nonlinear evolution equations with wλ₀-pseudomonotone maps
publisher	Інститут математики НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/178098
citation_txt	Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps / P.O. Kasyanov, V.S. Mel'nik, S. Toscano // Нелінійні коливання. — 2006. — Т. 9, № 2. — С. 187-212. — Бібліогр.: 16 назв. — англ.
series	Нелінійні коливання
work_keys_str_mv	AT kasyanovpo periodicsolutionsofnonlinearevolutionequationswithwl0pseudomonotonemaps AT melnikvs periodicsolutionsofnonlinearevolutionequationswithwl0pseudomonotonemaps AT toscanos periodicsolutionsofnonlinearevolutionequationswithwl0pseudomonotonemaps
first_indexed	2025-07-15T16:27:58Z
last_indexed	2025-07-15T16:27:58Z
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fulltext	UDC 517 . 9 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS ПЕРIОДИЧНI РОЗВ’ЯЗКИ НЕЛIНIЙНИХ ЕВОЛЮЦIЙНИХ РIВНЯНЬ З Wλ0 -ПСЕВДОМОНОТОННИМИ ВIДОБРАЖЕННЯМИ P. O. Kasyanov Kyiv Nat. Taras Shevchenko Univ. Volodymyrs’ka Str. 64, 01033, Kyiv, Ukraine e-mail: kasyanov@univ.kiev.ua V. S. Mel’nik Inst. Appl. and System Analysis Nat. Acad. Sci. Ukraine Peremogy Avenue 37, Build. 35, 03056, Kyiv, Ukraine e-mail: vsmelnyk@ukr.net S. Toscano Univ. Salerno Via Ponte Don Melillo 1, 84084 Fisciano (Salerno), Italy e-mail: speranza.toscano@virgilio.it We consider differential-operator equations with Wλ0 -pseudomonotone operators. The problem of stu- dying periodic solutions via the Faedo – Galerkin method has been considered. The important a priory estimates have been obtained. A topological description of resolvent operators is given. Розглянуто диференцiально-операторнi рiвняння зWλ0 -псевдомонотонними операторами. Роз- в’язано проблему вивчення перiодичних розв’язкiв методом Фаедо – Гальоркiна. Отримано важ- ливi апрiорнi оцiнки. Наведено певний топологiчний опис резольвентних операторiв. We obtain a condition for existence and uniqueness of a periodic solution of a system of nonlinear integro-differential equations with an impulsive effect. The solution is represented as a limit of periodic iterations. We give estimates for the convergence rate and the exact solution. Одержано умову iснування єдиного перiодичного розв’язку системи нелiнiйних iнтегро- диференцiальних рiвнянь з iмпульсною дiєю. Розв’язок подано у виглядi границi перiоди- чних iтерацiй. Наведено оцiнки швидкостi збiжностi i точного розв’язку системи. 1. Introduction. One of the most effective approaches to investigate nonlinear problems, defi- ned by partial differential equations with boundary values consists in their into equations in Banach spaces governed by nonlinear operators. In order to study these equations, modern methods of nonlinear analysis have been used [1 – 3]. In [4], by using a special basis, the Cauchy problem for a class of equations with operators of Volterra type has been studied. An important periodic problem for equations with monotone differential operators of Volterra type has been studied in [1]. Periodic solutions for pseudomonotone operators have been considered in [2]. In the present paper we introduce a new construction of bases to prove existence of peri- c© P. O. Kasyanov, V. S. Mel’nik, S. Toscano, 2006 ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 187 188 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO odic solutions of differential-operator equations by using the Faedo – Galerkin method for Wλ0-pseudomonotone operators. From the point of view of applications, we have essentially extended the class of operators considered by other authors (see [4 – 6]). 2. Problem definition. Let (V1, ‖ · ‖V1) and (V2, ‖ · ‖V2) be some reflexive separable Banach spaces, continuously embedded in a Hilbert space (H, (·, ·)) such that V := V1 ∩ V2 is dense in the spaces V1, V2 and H. (2.1) After the identification H ∼= H∗ we get V1 ⊂ H ⊂ V ∗ 1 , V2 ⊂ H ⊂ V ∗ 2 , (2.2) with continuous and dense embeddings [1], where (V ∗ i , ‖·‖Vi ) is the space topologically conjugate to Vi with respect to the canonical bilinear form 〈·, ·〉Vi : V ∗ i × Vi → R, i = 1, 2, which coincides on H with the inner product (·, ·) on H . Let us consider the functional spaces Xi = Lri (S;H) ∩ Lpi (S;Vi), where S = [0, T ], 0 < T < +∞, 1 < pi ≤ ri < +∞, i = 1, 2. The spaces Xi are Banach spaces with the norms ‖y‖Xi = ‖y‖Lpi (S;Vi) + ‖y‖Lri (S;H). Moreover, Xi is a reflexive space. Let us also consider the Banach space X = X1 ∩X2 with the norm ‖y‖X = ‖y‖X1 + ‖y‖X2 . Since the spaces Lqi (S;V ∗ i )+Lr′i (S;H) and X∗ i are isometrically isomorphic, we identify them. Analogously, X∗ = X∗ 1 + X∗ 2 = Lq1(S;V ∗ 1 ) + Lq2(S;V ∗ 2 ) + Lr′1 (S;H) + Lr′2 (S;H), where ri −1 + r′i −1 = pi −1 + qi −1 = 1. Let us define a duality form on X∗ ×X , 〈f, y〉 = ∫ S (f11(τ), y(τ))H dτ + ∫ S (f12(τ), y(τ))H dτ + ∫ S 〈f21(τ), y(τ)〉V1 dτ+ + ∫ S 〈f22(τ), y(τ)〉V2 dτ = ∫ S (f(τ), y(τ)) dτ, where f = f11 + f12 + f21 + f22, f1i ∈ Lr′i (S;H), f2i ∈ Lqi (S;V ∗ i ). For each f ∈ X∗, ‖f‖X∗ = inf f = f11 + f12 + f21 + f22 : f1i ∈ Lr′ i (S;H), f2i ∈ Lqi (S;V ∗ i ) (i = 1, 2) max { ‖f11‖Lr′1 (S;H), ‖f12‖Lr′2 (S;H), ‖f21‖Lq1 (S;V ∗ 1 ), ‖f22‖Lq2 (S;V ∗ 2 ) } . Let A : X1 → X∗ 1 and B : X2 → X∗ 2 be single-valued nonlinear operators, L : D(L) ⊂ ⊂ X → X∗ be a linear closed densly defined operator. We consider the following problem: Ly +A(y) +B(y) = f, (2.3) y ∈ D(L), (2.4) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 189 where f ∈ X∗ is arbitrary and fixed. 3. Classes of maps. Let (Y, ‖ · ‖Y ) be a Banach space, W a normed space with a norm ‖ · ‖W . We consider W ⊂ Y with a continuous embedding. Further, yn ⇀ y in Y means that yn weakly converges to y in the space Y . If Y is not reflexive, then yn ⇀ y in Y ∗ means that yn *-weakly converges to y in the space Y ∗. Definition 3.1. Let D(A) be a subset of Y . A single-valued map A : D(A) ⊂ Y → Y ∗ is called: coercive if ‖y‖−1 Y 〈A(y), y〉Y → +∞ as ‖y‖Y → ∞, y ∈ D(A); weakly coercive if for each f ∈ Y ∗ there exists R > 0 such that 〈A(y)− f, y〉Y ≥ 0 as ‖y‖Y = R, y ∈ D(A); bounded if for any L > 0 there exists l > 0 such that ‖A(y)‖Y ∗ ≤ l ∀y ∈ D(A) : ‖y‖Y ≤ L; locally bounded if for any fixed y ∈ D(A) there exist constants m > 0 and M > 0 such that ‖A(ξ)‖Y ∗ ≤ M if ‖y − ξ‖Y ≤ m, ξ ∈ D(A); finite-dimensionally locally bounded if for each finite-dimensional subspace F ⊂ D(A), A ∣ ∣ F is locally bounded on (F, ‖ · ‖Y ). Definition 3.2. A single-valued map A : D(A) ⊂ Y → Y ∗ is called radially continuous if for any fixed y, ξ ∈ D(A), lim t→+0 〈A(y + tξ), ξ〉Y = 〈A(y), ξ〉Y ; an operator with semibounded variation on W (with (Y,W )-semibounded variation) if for all R > 0 and all y1, y2 ∈ D(A) with ‖y1‖Y ≤ R, ‖y2‖Y ≤ R, the inequality 〈A(y1)−A(y2), y1 − y2〉Y ≥ −C(R; ‖y1 − y2‖ ′ W ) is fulfilled; λ-pseudomonotone onW (Wλ-pseudomonotone), if for each sequence {yn}n≥1 ⊂ W∩D(A) such that yn ⇀ y0 in W with y0 ∈ D(A), the inequality lim n→∞ 〈A(yn), yn − y0〉Y ≤ 0, (3.1) implies existence of {ynk }k≥1 from {yn}n≥1 such that lim k→∞ 〈A(ynk ), ynk − w〉Y ≥ 〈A(y0), y0 − w〉Y ∀w ∈ D(A) ∩W ; (3.2) λ0-pseudomonotone on W (Wλ0-pseudomonotone), if for each sequence {yn}n≥1 ⊂ W ∩ ∩D(A) such that yn ⇀ y0 in W, A(yn) ⇀ d0 in Y ∗ with y0 ∈ D(A) and d0 ∈ Y ∗, ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 190 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO it follows from (3.1) that there exists {ynk }k≥1 ⊂ {yn}n≥1 such that (3.2) is true. The above single-valued map satisfies property (κ) if for each bounded set D in Y there exists c ∈ R such that 〈A(v), v〉Y ≥ −c‖v‖Y ∀v ∈ D; property (Π) if for each nonempty bounded subset B ⊂ D(A) and for each k > 0 such that 〈A(y), y〉Y ≤ k for each y ∈ B, it follows that there exists K > 0 such that ‖A(y)‖Y ∗ ≤ K for all y ∈ B. HereC(r1; · ) : R+ → R is a continuous function for each r1 ≥ 0 and such that τ−1C(r1; τr2) → → 0 as τ → +0 ∀ r1, r2 ≥ 0 and ‖· ‖′W is a (semi-)norm on Y , that is compact with respect to ‖· ‖W on W and continuous with respect to ‖ · ‖Y on Y ! Remark 3.1. The idea of passing to subsequences in the latter definition was adopted from Skrypnik’s work [7]. Let Y = Y1 ∩ Y2, where (Y1, ‖ · ‖Y1) and (Y2, ‖ · ‖Y2) are some Banach spaces. Definition 3.3. A pair of single-valued maps A : D(A) ⊂ Y1 → Y ∗ 1 and B : D(B) ⊂ Y2 → → Y ∗ 2 is called s-mutually bounded, if for each M > 0 and each bounded set D ⊂ Y there exists K > 0 such that from y ∈ D(A) ∩D(B) ∩D and 〈A(y), y〉Y1 + 〈B(y), y〉Y2 ≤ M, we have or ‖A(y)‖Y ∗ 1 ≤ K or ‖B(y)‖Y ∗ 2 ≤ K. Remark 3.2. A bounded map A : Y → Y ∗ satisfies property (κ) and property (Π); a λ- pseudomonotone on W map is λ0-pseudomonotone on W . The converse statement is true for bounded single-valued maps. If one of the operators of the pair (A;B) is bounded, then the pair (A;B) is s-mutually bounded. Moreover, if a pair (A;B) is s-mutually bounded, then the operator C = A + B : X → X∗ has property (Π). If a pair of operators is s-mutually bounded and each of them satisfies condition (Π), then the sum of the given operators also satisfies condition (Π). Now let W = W1 ∩W2, where (W1, ‖ · ‖W1) and (W2, ‖ · ‖W2) are Banach spaces such that Wi ⊂ Yi with a continuous embedding. Lemma 3.1. LetA : Y1 → Y ∗ 1 andB : Y2 → Y ∗ 2 be s-mutually bounded λ0-pseudomonotone operators onW1 andW2, respectively. Then the mapC := A+B : Y → Y ∗ is λ0-pseudomonotone on W . Remark 3.3. If a pair (A;B) is not s-mutually bounded, then the above proposition takes place only for maps that are λ-pseudomonotone on W1 and W2, respectively. ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 191 Proof. Let yn ⇀ y0 in W (it means that yn ⇀ y0 in W1 and yn ⇀ y0 in W2), C(yn) ⇀ d0 in Y ∗ and inequality (3.1) hold. Since the pair (A;B) is s-mutually bounded, from the estimate 〈C(yn), yn〉Y = 〈A(yn) +B(yn), yn〉Y = 〈A(yn), yn〉Y1 + 〈B(yn), yn〉Y2 ≤ k we have that either ‖A(yn)‖Y ∗ 1 ≤ C or ‖B(yn)‖Y ∗ 2 ≤ C. Then passing to a subsequence if necessary, we claim that A(yn) ⇀ d′0 in Y ∗ 1 and B(yn) ⇀ d′′0 in Y ∗ 2 . (3.3) From inequality (3.1) we have lim n→∞ 〈B(yn), yn − y0〉Y2 + lim n→∞ 〈A(yn), yn − y0〉Y1 ≤ lim n→∞ 〈C(yn), yn − y0〉Y ≤ 0, or, symmetrically, lim n→∞ 〈A(yn), yn − y0〉Y1 + lim n→∞ 〈B(yn), yn − y0〉Y2 ≤ lim n→∞ 〈C(yn), yn − y0〉Y ≤ 0. Let us consider the last inequality. It is obvious that there exists a subsequence {ym}m ⊂ ⊂ {yn}n≥1 such that 0 ≥ lim n→∞ 〈B(yn), yn − y0〉Y2 + lim n→∞ 〈A(yn), yn − y0〉Y1 ≥ ≥ lim m→∞ 〈B(ym), ym − y0〉Y2 + lim m→∞ 〈A(ym), ym − y0〉Y1 . (3.4) Hence, it follows that either lim m→∞ 〈A(ym), ym − y0〉Y1 ≤ 0 or lim m→∞ 〈B(ym), ym − y0〉Y2 ≤ 0. Without loss of generality we assume that lim m→∞ 〈A(ym), ym − y0〉Y1 ≤ 0. Then, from (3.3) and λ0-pseudomonotony of A on W1, it follows that there exists {ymk }k≥1 in {ym}m such that lim k→∞ 〈A(ymk ), ymk − v〉Y1 ≥ 〈A(y0), y0 − v〉Y1 ∀v ∈ Y1 ∩W1. (3.5) If we take in the last relation v = y0 we obtain that 〈A(ymk ), ymk − y0〉Y1 → 0 as k → +∞. Then, due to (3.4), lim k→∞ 〈B(ymk ), ymk − y0〉Y2 ≤ 0. In virtue of λ0-pseudomonotony of B on W2, passing to a subsequence {ym′ k } ⊂ {ymk }k≥1, we find lim k→∞ 〈B(ym′ k ), ym′ k − w〉Y2 ≥ 〈B(y0), y0 − w〉Y2 ∀w ∈ Y2 ∩W2. (3.6) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 192 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO Then from relations (3.5) and (3.6) we finally obtain lim k→∞ 〈C(ym′ k ),ym′ k − x〉Y ≥ lim k→∞ 〈A(ym′ k ), ym′ k − x〉Y1 + lim k→∞ 〈B(ym′ k ), ym′ k − x〉Y2 ≥ ≥ 〈A(y0), y0 − x〉Y1 + 〈B(y0), y0 − x〉Y2 = 〈C(y0), y0 − x〉Y ∀x ∈ W ∩ Y. The lemma is proved. Lemma 3.2. Let A : Y1 → Y ∗ 1 and B : Y2 → Y ∗ 2 be coercive maps, that satisfy condition (κ). Then the operator C := A+B : Y → Y ∗ is coercive. Proof. We obtain this statement arguing by contradiction. Let {xn}n≥1 ⊂ Y with xm 6= 0̄ and ‖xn‖Y = ‖xn‖Y1 + ‖xn‖Y2 → +∞ as n → ∞, but sup n≥1 〈C(xn), xn〉Y ‖xn‖Y < +∞. (3.7) Let γA(r) := inf ‖v‖Y1 =r 〈A(v), v〉Y1 ‖v‖Y1 , γB(r) := inf ‖w‖Y2 =r 〈B(w), w〉Y2 ‖w‖Y2 , r > 0. We remark that γA(r) → +∞, γB(r) → +∞ as r → +∞. In the case ‖xm‖Y1 → +∞ as m → ∞ and ‖xm‖Y2 ≤ c ∀m ≥ 1, we get 〈A(xn), xn〉Y1 ‖xn‖Y ≥ γA(‖xn‖Y1) ‖xn‖Y1 ‖xn‖Y → +∞ as m → +∞ and, moreover, 〈B(xn), xn〉Y2 ‖xn‖Y ≥ −c1 ‖xn‖Y2 ‖xn‖Y → 0 as n → ∞, where c1 ∈ R is a constant as in condition (κ) with D = {y ∈ Y2 \| ‖y‖Y2 ≤ c}. Consequently, 〈C(xn), xn〉Y ‖xn‖Y = 〈A(xn), xn〉Y1 ‖xn‖Y + 〈B(xn), xn〉Y2 ‖xn‖Y → +∞ as n → ∞. This is in contradiction with (3.7). If ‖xn‖Y1 ≤ c ∀n ≥ 1 and ‖xn‖Y2 → ∞ as n → ∞ the reasoning is the same. When ‖xn‖Y1 → ∞ and ‖xn‖Y2 → ∞ as n → ∞, we get the contradiction +∞ > sup n≥1 〈C(xn), xn〉Y ‖xn‖Y ≥ γA(‖xn‖Y1) ‖xn‖Y1 ‖xn‖Y1 + ‖xn‖Y2 + + γB(‖xn‖Y2) ‖xn‖Y2 ‖xn‖Y1 + ‖xn‖Y2 ≥ min {γA(‖xn‖Y1), γB(‖xn‖Y2)} → +∞. The lemma is proved. Remark 3.4. Under the conditions of the latter lemma, it follows that the operator C = = A+B : Y → Y ∗ is weakly coercive. Definition 3.4. An operator L : D(L) ⊂ Y → Y ∗ is called monotone, if for each y1, y2 ∈ D(L) 〈Ly1 − Ly2, y1 − y2〉Y ≥ 0; ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 193 maximal monotone, if it is monotone and from 〈w − Lu, v − u〉Y ≥ 0 for all u ∈ D(L) it follows that v ∈ D(L) and Lv = w. Remark 3.5. If the reflexive Banach space Y is strictly convex with its conjugate then [2] (Lemma 3.1.1) the linear operator L : D(L) ⊂ Y → Y ∗ is maximal monotone and densly defined if and only if L is a closed unbounded operator such that 〈Ly, y〉Y ≥ 0 ∀y ∈ D(L) and 〈L∗y, y〉Y ≥ 0 ∀y ∈ D(L∗), where L∗ : D(L∗) ⊂ Y → Y ∗ is the conjugate operator of L in the sense of the theory of unbounded operators [8]. 4. Auxiliary statements. From (2.1) and (2.2), V = V1∩V2 ⊂ H with a continuous and dense embedding. As V is a separable Banach space, there exists a complete in V , and consequently in H , countable system of vectors, {hi}i≥1 ⊂ V . Let for each n ≥ 1, Hn = span{hi} n i=1. On Hn we consider the inner product induced from H that we again denote by (·, ·). Let Pn : H → Hn ⊂ H be the operator of orthogonal projection from H on Hn, i.e., ∀h ∈ H : Pnh = argmin hn∈Hn ‖h− hn‖H . Definition 4.1. We say that a triple ({hi}i≥1;V ;H) satisfies condition (γ) if sup n≥1 ‖Pn‖L(V,V ) < < +∞, i.e., there exists C ≥ 1 such that ∀v ∈ V ∀n ≥ 1 : ‖Pnv‖V ≤ C‖v‖V . Remark 4.1. When the system of vectors {hi}i≥1 ⊂ V is orthogonal in H , condition (γ) means that the given system is a Schauder basis in the space V (in particular in V1 and in V2) [9]. Remark 4.2. Since Pn ∈ L(V, V ), its conjugate operator P ∗ n ∈ L(V ∗, V ∗) and ‖Pn‖L(V,V ) = = ‖P ∗ n‖L(V ∗,V ∗). It is clear that for each h ∈ H Pnh = P ∗ nh. Hence, we identify Pn with P ∗ n . Then condition (γ) means that for each v ∈ V and n ≥ 1, ‖Pnv‖V ∗ ≤ C · ‖v‖V ∗ . Due to the equivalence between H∗ and H , it follows that H∗ n ∼= Hn. For each n ≥ 1 we consider the Banach space Xn = Lp0(S;Hn) ⊂ X , where p0 := max{r1, r2} with the norm ‖ · ‖Xn induced from the space X . This norm is equivalent to the natural norm in Lp0(S; Hn) [1]. The space Lq0(S;Hn) (q−1 0 + p−1 0 = 1) with the norm ‖f‖X∗ n := sup x∈Xn\{0} \|〈f, x〉\| ‖x‖X = sup x∈Xn\{0} \|〈f, x〉Xn \| ‖x‖Xn is isometrically isomorphic to the conjugate spaceX∗ n ofXn (further the given spaces are identi- fied); moreover, the map X∗ n ×Xn ∋ f, x → ∫ S (f(τ), x(τ))Hndτ = ∫ S (f(τ), x(τ))dτ = 〈f, x〉Xn ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 194 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO is a duality form on X∗ n×Xn. This is true due to X∗ n = Lq0(S;Hn) ⊂ Lq0(S;H) ⊂ Lr′1 (S;H)+ +Lr′2 (S;H)+Lq1(S;V ∗ 1 )+Lq2(S;V ∗ 2 ) = X∗ (see [1]). Let us remark that 〈·, ·〉 ∣ ∣ X∗ n×Xn = 〈·, ·〉Xn . Proposition 4.1. For each n ≥ 1, Xn = PnX , i.e., Xn = {Pny(·) \| y(·) ∈ X}, and we have 〈f, Pny〉 = 〈f, y〉 ∀y ∈ X and f ∈ X∗ n. Moreover, if triples ({hj}j≥1;Vi;H) , i = 1, 2, satisfy condition (γ) with C = Ci, then ‖Pny‖X ≤ max {C1, C2} · ‖y‖X ∀y ∈ X and n ≥ 1. Proof. For each y ∈ X let yn(·) := Pny(·), i.e., yn(t) = Pny(t) for almost all (a.a.) t ∈ S. Since Pn is linear and continuous on V1, V2 and H , we have that yn ∈ Xn ⊂ X . It follows from condition (γ) and the definitions of ‖ · ‖Lpi (S;Vi) and ‖ · ‖Lri (S;H) that ‖yn‖Lpi (S;Vi) ≤ ≤ Ci‖y‖Lpi (S;Vi) and ‖yn‖Lri (S;H) ≤ ‖y‖Lri (S;H). Thus ‖yn‖X ≤ max {C1, C2}‖y‖X . Now we prove that for all f ∈ X∗ n 〈f, yn〉 = 〈f, y〉. As f ∈ Lq0(S;Hn), we have 〈f, y〉 = ∫ S (f(τ), y(τ))dτ = ∫ S (f(τ), Pny(τ))dτ = ∫ S (f(τ), yn(τ))dτ = 〈f, yn〉, since for all n ≥ 1, h ∈ H and v ∈ Hn, (h− Pnh, v) = (h− Pnh, v)H = 0. The proposition is proved. For each n ≥ 1 we denote by In the canonical embedding of Xn in X (∀x ∈ Xn Inx = x), I∗n : X∗ → X∗ n its conjugate operator. We remark that ‖In‖L((Xn, ‖·‖X); (X, ‖·‖X)) = ‖I∗n‖L((X∗, ‖·‖X∗ ); (X∗ n, ‖·‖X∗ n )) = 1. Proposition 4.2. For each n ≥ 1 and f ∈ X∗, (I∗nf)(t) = Pnf(t) for a.a. t ∈ S. Moreover, if triples ({hj}j≥1;Vi;H) , i = 1, 2, satisfy condition (γ) with C = Ci, then for all f ∈ X∗ and n ≥ 1 ‖I∗nf‖X∗ ≤ max {C1, C2}‖f‖X∗ , i.e., sup n≥1 ‖I∗n‖L(X∗; X∗) ≤ max {C1, C2}. Proof. Let n ≥ 1 and f ∈ X∗ be fixed. Let us show that for a.a. t ∈ S, (I∗nf)(t) = Pnf(t). From Remark 4.2 it follows that for each x ∈ Xn 〈I∗nf, x〉 = 〈f, x〉 = ∫ S (f(τ), x(τ))dτ = = ∫ S (f(τ)− Pnf(τ), x(τ))dτ + ∫ S (Pnf(τ), x(τ))dτ = ∫ S (Pnf(τ), x(τ))dτ, (4.1) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 195 since for all u ∈ Hn and v ∈ V ∗, (v − Pnv, u) = 0. Let us prove the last equality. For each v ∈ V ∗ there exists a sequence {vk}k≥1 ⊂ H such that vk → v in V ∗ as k → ∞. It is clear that for each k ≥ 1, (vk−Pnvk, u) = (vk−Pnvk, u)H = = 0. From the continuity of (·, ·) on V ∗×V , it follows that (v−Pnv, u) = lim k→∞ (vk−Pnvk, u) = = 0. From (4.1) it follows that for a.a. t ∈ S (I∗nf)(t) = Pnf(t). Now we prove the second part of this proposition. Let condition (γ) be true, f ∈ X∗ and n ≥ 1 be arbitrary and fixed. Then from Remark 4.2 and condition (γ) it follows that for each f0 ∈ Lq0(S;H) and fi ∈ Lqi (S;V ∗ i ) such that f = f0 + f1 + f2, we have ‖I∗nf0‖Lq0 (S;H) + ‖I∗nf1‖Lq1 (S;V ∗) + ‖I∗nf2‖Lq2 (S;V ∗ 2 ) = =   ∫ S ‖Pnf0(τ)‖ q0 Hdτ   1 q0 +   ∫ S ‖Pnf1(τ)‖ q1 V ∗ 1 dτ   1 q1 +   ∫ S ‖Pnf2(τ)‖ q2 V ∗ 2 dτ   1 q2 ≤ ≤   ∫ S ‖f0(τ)‖ q0 Hdτ   1 q0 + C1   ∫ S ‖f1(τ)‖ q1 V ∗ 1 dτ   1 q1 + C2   ∫ S ‖f2(τ)‖ q2 V ∗ 2 dτ   1 q2 ≤ ≤ max {C1, C2} ( ‖f0‖Lq0 (S;H) + ‖f1‖Lq1 (S;V ∗ 1 ) + ‖f2‖Lq2 (S;V ∗ 2 ) ) , as C1, C2 ≥ 1. Hence, from the definition of ‖f‖X∗ it follows that ‖I∗nf‖X∗ ≤ max {C1, C2}‖f‖X∗ . The proposition is proved. From the last two propositions and properties of I∗n, we immediately obtain the following corollary. Corollary 4.1. For all n ≥ 1, X∗ n = PnX ∗ = I∗nX, i.e., X∗ n = {Pnf(·) \| f(·) ∈ X∗} = {I∗nf \| f ∈ X ∗} . Proposition 4.3. The set ⋃ n≥1 Xn is dense in (X, ‖ · ‖X). Proof. 1. At first we prove that the set L∞(S;V ) is dense in the space (X, ‖ · ‖X). Let x ∈ X be arbitrary and fixed. Then for each n ≥ 1 we consider xn(t) := { x(t), ‖x(t)‖V ≤ n, 0, otherwise. (4.2) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 196 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO Obviously for all n ≥ 1, xn ∈ L∞(S;V ). From (2.2) it follows that there exists γ > 0 such that, according to (4.2), as i = 1, 2 and for a.a. t ∈ S, we get ‖xn(t)− x(t)‖H ≤ γ‖xn(t)− x(t)‖V → 0, ‖xn(t)− x(t)‖Vi ≤ ‖xn(t)− x(t)‖V → 0 as n → ∞, (4.3) ‖xn(t)‖H ≤ ‖x(t)‖H , ‖xn(t)‖Vi ≤ ‖x(t)‖Vi . (4.4) Further let φn H(t) = ‖xn(t)− x(t)‖p0 H , φn Vi (t) = ‖xn(t)− x(t)‖pi Vi . Hence from (4.3) and (4.4) we obtain φn H(t) → 0, φn Vi (t) → 0 as n → ∞, (4.5) \|φn H(t)\| ≤ 2p0‖x(t)‖p0 H =: φH(t), \|φn Vi (t)\| ≤ 2pi‖x(t)‖pi Vi =: φVi (t) (4.6) for a.a. t ∈ S. As x ∈ X , we have φH , φV1 , φV2 ∈ L1(S). So, because of (4.5) and (4.6), we can apply the Lebesgue theorem with integrable majorants φH , φV1 and φV2 , respectively (see [10]). Hence it follows that φn H → 0̄ and φn Vi → 0̄ in L1(S) as i = 1, 2. Consequently, ‖xn − x‖X → 0 as n → ∞. 2. Further, let for some linear variety L from V Υ(L) := { y ∈ (S → L) \| y is a simple function} (see [1, p. 152]). Let us prove that the set Υ(V ) is dense in the normed space (L∞(S, V ), ‖ · ‖X). Every arbitrary fixed element x ∈ L∞(S, V ) is measurable, according to Bochner, as a function from the class (S → V ). So, there exists a sequence {xn}n≥1 ⊂ Υ(V ) such that xn(t) → x(t) in V as n → ∞ for a.a. t ∈ S. (4.7) Since x ∈ L∞(S, V ) it follows that ess sup t∈S ‖x(t)‖V =: c < +∞. For each n ≥ 1 we introduce yn(t) :=      xn(t), ‖xn(t)‖V ≤ 2c, 0̄, otherwise. (4.8) From (4.7) and (4.8) it follows that yn ∈ Υ(V ) as n ≥ 1 and, moreover, yn(t) → x(t) in V as n → ∞ for a.a. t ∈ S. Hence, taking into account (2.2), we obtain that as i = 1, 2 and for a.a. t ∈ S yn(t) → x(t) in H, yn(t) → x(t) in V1, yn(t) → x(t) in V2 as n → ∞. ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 197 As in item 1, assuming φH ≡ φV1 ≡ φV2 ≡ max{(3c)p1 , (3c)p2 , (3cγ)p0} ∈ L1(S)), we obtain that yn → x in X as n → ∞. So, Υ(V ) is dense in (L∞(S, V ), ‖ · ‖X). 3. Since the set span{hn}n≥1 = ⋃ n≥1 Hn is dense in (V, ‖ · ‖V ) and (2.2) holds, it is clear that the set Υ( ⋃ n≥1 Hn) = ⋃ n≥1 Υ(Hn) is dense in (Υ(V ), ‖ · ‖X). In order to end the proof we remark that for each n ≥ 1, Υ(Hn) ⊂ Xn. The proposition is proved. Proposition 4.4. Let L : D(L) ⊂ X → X∗ be a linear maximal monotone operator. Then the normed space D(L) with the graph norm ‖y‖D(L) = ‖y‖X + ‖Ly‖X∗ is complete (hence, it is weakly complete). Proof. Let {yn}n≥1 ⊂ D(L) be a Cauchy sequence. Since X is a Banach space, there exists y ∈ X such that yn → y in X as n → ∞. (4.9) Analogously there exists χ ∈ X∗ such that Lyn → χ in X∗ as n → ∞. (4.10) Now we prove that 〈χ− Lu, y − u〉 ≥ 0 for each u ∈ D(L). Let u ∈ D(L) be arbitrary and fixed. In virtue of (4.9), (4.10) and since L is monotone onD(L), it follows that for each n ≥ 1 0 ≤ 〈Lyn − Lu, yn − u〉 → 〈χ− Lu, y − u〉 as n → ∞. Consequently, from the maximal monotony of L, the required statement follows. The proposition is proved. 5. Faedo – Galerkin method. For each n ≥ 1 let us set Ln := I∗nLIn : D(Ln) = D(L) ∩ ∩Xn ⊂ Xn → X∗ n, An := I∗nAIn : Xn → X∗ n, Bn := I∗nBIn : Xn → X∗ n, fn := I∗nf ∈ X ∗ n. Remark 5.1. We will also denote by I∗n the operators conjugate to the canonical embeddings of Xn in X1 and of Xn in X2, because these operators coincide with I∗n on X∗ 1 ∩ X ∗ 2 which is dense in X∗ 1 , X∗ 2 , X∗. Now we consider D(L) as a normed space with the graph norm ‖y‖D(L) = ‖y‖X + ‖Ly‖X∗ for each y ∈ D(L). We remark that if the linear operator L is closed and densely defined, then (D(L), ‖ · ‖D(L)) is a Banach space continuously embedded in X . In addition to the problem (2.3), (2.4) we consider the following class of problems: Lnyn +An(yn) +Bn(yn) = fn, (5.1)n yn ∈ D(Ln). (5.2)n ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 198 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO Remark 5.2. We consider on D(Ln) the graph norm ‖yn‖D(Ln) = ‖yn‖Xn + ‖Lnyn‖X∗ n for each yn ∈ D(Ln). Definition 5.1. We say that a solution to (2.3), (2.4) y ∈ D(L) is obtained via the Faedo – Galerkin method, if y is the weak limit of a subsequence {ynk }k≥1 from {yn}n≥1 in D(L), where for each n ≥ 1, yn is a solution to the problem (5.1)n, (5.2)n. 6. Choice of the basis. As it is well-known from [11] there exist separable Banach spaces that do not have a Schauder basis. Hence we need to introduce some constructions of a basis to satisfy condition (γ). Definition 6.1. We say that a system of vectors {hi}i≥1 of a separable Hilbert space (V ; (·, ·)V ), continuously and densely embedded in a Hilbert space (H; (·, ·)H), is a special basis for the pair of spaces (V ;H) if it satisfies the following conditions: {hi}i≥1 is orthonormal in (H, (·, ·)H); {hi}i≥1 is orthogonal in (V, (·, ·)V ); ∀i ≥ 1 (hi, v)V = λi(hi, v)H ∀v ∈ V , where 0 ≤ λ1 ≤ λ2, . . . , λj −→ ∞ as j −→ ∞. Lemma 6.1. If V is a Hilbert space, compactly and densely embedded in a Hilbert space H , then there exists a special basis {hi}i≥1 for (V ;H). Moreover, for an arbitrary such system, the triple ({hi}i≥1;V ;H) satisfies condition (γ) with the constant C = 1. Proof. From [12, p. 54 – 58], under these assumptions, it is well-known that there exists a special basis {hi}i≥1 for the pair (V ;H). So, in order to complete the proof it is enough to show that the triple ({hi}i≥1;V ;H) satisfies condition (γ) with the constant C = 1 for an arbitrary special basis {hi}i≥1 for (V ;H). Therefore, let Hn = span{hi} n i=1 and let us denote by Pn the operator of orthogonal projection from H to Hn. Obviously, Pm ∈ L(V ;V ). Further let us prove that for all n ≥ 1 ‖Pnh‖V ≤ ‖h‖V ∀h ∈ ⋃ m≥1 Hm. (6.1) Let n ≥ 1 be fixed. Then h ∈ ⋃ m≥1 Hm ⇒ ∃m0 ≥ n + 1 : h ∈ Hm0 . Whence, since {hi}i≥1 is orthonormal in H, we have h = m0 ∑ i=1 (h, hi)Hhi, Pnh = n ∑ i=1 (h, hi)Hhi. In order to obtain (6.1) it is necessary to show that Pnh is orthogonal to (h − Pnh) in V . In fact, (Pnh, h − Pnh)V = = ( n ∑ i=1 (h, hi)Hhi, m0 ∑ j=n+1 (h, hj)Hhj ) V = n ∑ i=1 m ∑ j=n+1 (h, hi)H(h, hj)H(hi, hj)V = 0, since {hi}i≥1 is orthogonal in V . So, from continuity of Pn on V we have that for all n ≥ 1 and v ∈ V ‖Pnv‖V ≤ ‖v‖V . The lemma is proved. Now let us make the same for Banach spaces. We consider that I is a subset of R. Let {Zα}α∈I be a family of Banach spaces such that for all α1, α2 ∈ I , α1 < α2, Zα2 ⊂ Zα1 with a continuous embedding; there exists a set Φ such that for all α ∈ I , Φ is dense in Zα; for all α0 ∈ I and x ∈ Φ, ‖x‖Zα → ‖x‖Zα0 as α → α0, α ∈ I . We also consider a Banach space H such that Zα ⊂ H with continuous embeddings for all α ∈ I and the set Φ is dense in H . ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 199 Let {hj}j≥1 ⊂ Φ be a system of vectors. Proposition 6.1. Let the above assumptions be true. If for some α0 ∈ I the triple ({hj}j≥1; Zα0 ;H) satisfies condition (γ) with the constant C ≥ 1, then the set of α ∈ I for which the triple ({hj}j≥1;Zα;H) satisfies the same condition with the same constant is closed in I . Proof. For an arbitrary α ∈ I , statement G(α) means that the triple ({hj}j≥1;Zα;H) sati- sfies condition (γ) with the constant C. We denote I+ = {α ∈ I \| G(α) is true} and I− = I \ I+. Let α ∈ I be an arbitrary cluster point of I+. Then there exists {αn}n≥1 ⊂ I+ such that αn → α. For each fixed element x ∈ Φ, by using the definition of I+, ∀m ≥ 1 ∀x ∈ Φ ∀n ≥ 1 : ‖Pnx‖Zαm ≤ C‖x‖Zαm and passing to the limit as m → +∞ in the last inequality, we obtain ‖Pnx‖Zα ≤ C‖x‖Zα ∀x ∈ Φ ∀n ≥ 1. Then from density of Φ in Zα and continuity, Pn on Zα, statementG(α) follows. So, α ∈ I+, i.e., the statement G(α) is true. The proposition is proved. Now we consider one application of the above proposition. But at first we need to give some definitions from the interpolation theory. For an interpolation pairA0, A1 (i.e., Banach spacesA0 andA1 that are continuously embed- ded in some linear topological space) let us consider the functional K(t, x) = inf x=x0+x1: x0∈A0, x1∈A1 (‖x0‖A0 + t‖x1‖A1) , t > 0, x ∈ A0 +A1. For fixed x ∈ A0 + A1, this map is a monotone increasing, continuous, concave function of the variable t > 0 (see [9], Lemma 1.3.1). For θ ∈ (0, 1) and 1 < p < +∞ let us consider the following space: (A0, A1)θ,p =    x ∈ A0 +A1 ∣ ∣ ∣ +∞ ∫ 0 [ t−θK(t, x) ]pdt t < +∞    ; (6.2) (A0, A1)θ,p with ‖x‖θ,p = ( ∫ +∞ 0 [t−θK(t, x)]p dt t ) 1 p it is a Banach space (for more details see [9]) and this results in (see [9], Theorem 1.3.3): A0 ∩A1 ⊂ (A0, A1)θ,p ⊂ A0 +A1 ∀θ ∈ (0, 1) ∀1 < p < +∞ (6.3) with dense and continuous embeddings. Definition 6.2. Let 1 ≤ r < 2. We say that a filter of Banach spaces {Zp}p≥r, a Hilbert space H, and a system of vectors, {hi}i≥1, complete in Zp ∀p ≥ r satisfy the main conditions, if ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 200 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO a) p2 > p1 > r Zp2 ⊂ Zp1 ⊂ H with continuous and dense embeddings; b) p2 > p > p1 > r (Zp1 , Zp2)θ,p = Zp, where θ = θ(p) ∈ (0, 1) : 1 p = 1− θ p1 + θ p2 ; c) Z2 is a Hilbert space; d) for some C ≥ 1 the triple ({hi}i≥1;Z2;H) satisfies condition (γ) with a constant C, the set I+(C) = {p ≥ 2 \| the triple ({hi}i≥1;Zp;H) does not satisfy condition (γ) with the constant C} (if it is not empty) contains its minimal element and the set I−(C) = {p ∈ [r; 2] \| the triple ({hi}i≥1;Zp;H) does not satisfy condition (γ) with the constant C} (if it is not empty) contains its maximal element. Lemma 6.2. Let 1 ≤ r < 2, {Zp}p≥r, be a filter of Banach spaces, H a Hilbert space H , and a system of vectors, {hi}i≥1, complete in Zp ∀p ≥ r satisfy the main conditions. Then, for all p > r the triple ({hi}i≥1;Zp;H) satisfies condition (γ). Remark 6.1. In the case Z2 ⊂ H with compact a embedding, due to Lemma 6.1, as the vector system {hi}i≥1 we can choose a special basis for the pair (Z2;H). In particular, the above result means that the special basis for (Z2;H) is a Schauder basis for an arbitrary space Zp as r < p ≤ 2. Proof. Let N > 2 and M ∈ (r, 2) be arbitrary fixed numbers. Now we apply Proposition 6.1 with I = (M,N), α0 = 2, Φ = ZN . In order to do this, it is sufficient to prove that ‖x‖Zq → ‖x‖Zp as q → p (q ∈ I) ∀p ∈ I ∀x ∈ ZN . (6.4) Let p be an arbitrary element of I (hence there exists δ such that [p− δ, p+ δ] ⊂ I), x be a fixed element of the space ZN . From (6.2) and the main condition b) for {Zp}p≥r and H , for all q ∈ [p− δ, p+ δ] it results in ‖x‖Zq = ‖x‖(ZM ,ZN )θ,q = ( +∞ ∫ 0 [ t−θK(t, x) ]q dt t ) 1 q , (6.5) where 1/q = (1− θ(q))/M + θ(q)/N , i.e., θ(q) = 1 M − 1 q 1 M − 1 N ∈ [θ(p− δ), θ(p+ δ)] = [ 1 M − 1 p−δ 1 M − 1 N , 1 M − 1 p+δ 1 M − 1 N ] ⊂ (0, 1). The following relations prove (6.4). Denote f(t, q) = [ t−θ(q)K(t, x) ]q 1 t ∀(t, q) ∈ (0,+∞)× [p− δ, p+ δ]. From (6.2) and (6.5) it follows that for each q ∈ [p− δ, p + δ], we have f(·, q) ∈ L1[0,+∞); moreover, for each t ∈ (0,+∞), f(t, ·) ∈ C([p− δ, p+ δ]). Furthermore, noticing that for each t > 0 and q ∈ [p− δ, p+ δ], [ t−θ(q)K(t, x) ]q 1 t ≤ max {[ t−θ(p−δ)K(t, x) ]p−δ , [ t−θ(p−δ)K(t, x) ]p+δ , ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 201 [ t−θ(p+δ)K(t, x) ]p−δ , [ t−θ(p+δ)K(t, x) ]p+δ}1 t =: g(t), and having in mind that (6.3) holds and x ∈ ZN = ZM ∩ ZN , we have +∞ ∫ 0 g(t)dt ≤ 4 max    +∞ ∫ 0 [ t−θ(p−δ)K(t, x) ]p−δ dt t , +∞ ∫ 0 [ t−θ(p−δ)K(t, x) ]p+δ dt t , +∞ ∫ 0 [ t−θ(p+δ)K(t, x) ]p−δ dt t , +∞ ∫ 0 [ t−θ(p+δ)K(t, x) ]p+δ dt t    = 4 max { ‖x‖p−δ (ZM ,ZN )θ(p−δ),p−δ , ‖x‖p+δ (ZM ,ZN )θ(p−δ),p+δ , ‖x‖p−δ (ZM ,ZN )θ(p+δ),p−δ , ‖x‖p+δ (ZM ,ZN )θ(p+δ),p+δ } . Thus, the theorem of continuous dependence of the Lebesgue integral on a parameter [13] (Theorem 8.1.1) assures the convergence (6.4). To finish the proof we remark that the set {p ≥ 2 \| the triple ({hi}i≥1;Zp;H) does not satisfy condition (γ) with the constant C} contains its minimal element (respectively, the set I−(C) = {p ∈ [r; 2] \| the triple ({hi}i≥1;Zp;H) does not satisfy condition (γ) with the constant C} contains its maximal element), which contradicts Proposition 6.1. Corollary 6.1. Let V1, V2 be Banach spaces, continuously embedded in the Hilbert space H . Let us assume that for some filter of Banach spaces {Zi p}p≥ri , ri ∈ [1; 2), i = 1, 2, there exists pi > ri such that Vi = Zi pi , within to equivalent norms. Moreover, there exists a Hilbert space Z ⊂ V1 ∩ V2, compactly and densely embedded in H, such that for a special basis {hj}j≥1 for the pair (Z;H) with {hj}j≥1 ⊂ ∩p>ri Zi p for some 0 ≤ µ1 ≤ µ2, ..., µj −→ ∞ as j −→ ∞ and si > 0, i = 1, 2, Zi 2 = { u ∈ H ∣ ∣ ∣ ∞ ∑ j=1 µsi j (u, hj) 2 < +∞ } is a Hilbert space with the inner product (u, v)Zi 2 = ∞ ∑ j=1 µsi j (u, hj)(v, hj); {Zi p}p≥ri together with H and the system of vectors {hi}i≥1 satisfies the main conditions. Then the triple ({hj}j≥1;Vi;H) satisfies condition (γ). Proof. Having in mind the proof of Lemmas 6.1 and 6.2, it is enough to show that {hj}j≥1 is a special basis for (Zi 2;H). In fact, since {hj}j≥1 is orthogonal in H we have ∀r, s ≥ 1 : (hr, hs)Zi 2 = ∞ ∑ j=1 µsi j (hr, hj)(hs, hj) = µsi r { 1, r = s, 0, r 6= s, ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 202 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO ∀r ≥ 1 : (hr, v)Zi 2 = ∞ ∑ j=1 µsi j (hr, hj)(v, hj) = µsi r (hr, v) ∀v ∈ Zi 2. The corollary is proved. Remark 6.2. In what follows we may assume that the triple of spaces V1, V2 and H satisfies the conditions of Corollary 6.1! 7. The main resolvability theorem. Theorem 7.1. Let L : D(L) ⊂ X → X∗, A : X1 → X∗ 1 , and B : X2 → X∗ 2 be maps such that 1) L is linear maximal monotone and satisfies the following conditions: (L1) for each n ≥ 1 and xn ∈ D(Ln) = Xn ∩D(L), Lxn ∈ X ∗ n; (L2) for each n ≥ 1, the set D(Ln) is dense in Xn; (L3) for each n ≥ 1, Ln is a maximal monotone operator; 2) there exist Banach spacesW1 andW2 such thatW1 ⊂ X1,W2 ⊂ X2, andD(L) ⊂ W1∩W2 with a continuous embedding; 3) A is λ0-pseudomonotone on W1 and satisfies condition (Π); 4) B is λ0-pseudomonotone on W2 and satisfies condition (Π); 5) the pair (A;B) is s-mutually bounded and the sum C = A + B : X → X∗ is finite- dimensionally locally bounded and weakly coercive. Furthermore, let {hj}j≥1 ⊂ V be a complete system of vectors in V1, V2, H such that for i = 1, 2 the triple ({hj}j≥1;Vi;H) satisfies condition (γ). Then for each f ∈ X∗ the set KH(f) := { y ∈ D(L) ∣ ∣ y is a solution to (2.3), (2.4), obtained via the Faedo – Galerkin method } is nonempty and we have the representation KH(f) = ⋂ n≥1 [ ⋃ m≥n Km(fm) ] Xw , (7.1) where for each n ≥ 1, Kn(fn) = {yn ∈ D(Ln) \| yn is a solution of (5.1)n, (5.2)n} and [ · ]Xw is the closure of an operator in the space X with respect to the weak topology. Moreover, if the operator A+B : X → X∗ is coercive, then KH(f) is weakly compact. Remark 7.1. A sufficient condition for getting the weak coercivity of A+B is the following: A is coercive and satisfies condition (κ) on X1, B is coercive and satisfies condition (κ) on X2 (see Lemma 3.2). Remark 7.2. From condition L2 on the operator L and from Proposition 4.3, it follows that L is densly defined. ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 203 Proof. By Lemma 3.1 and Remark 3.2 we consider a λ0-pseudomonotone on W1 ∩W2 (and hence on D(L)), finite-dimensionally locally bounded, weakly coercive map, X ∋ y → C(y) := A(y) +B(y) ∈ X∗, which satisfies condition (Π). Let f ∈ X∗ be fixed. Now we use the weak coercivity condition for C. There exists R > 0 such that 〈C(y)− f, y〉 ≥ 0 ∀y ∈ X : ‖y‖X = R. (7.2) 7.1. Resolvability of the approximating problems. Lemma 7.1. For all n ≥ 1 there exists a solution of the problem (5.1)n, (5.2)n yn ∈ D(Ln) such that ‖yn‖X ≤ R. Proof. In order to obtain this result we need to prove that for each n ≥ 1 Cn := An +Bn = = I∗n(A+B) : Xn → X∗ n satisfies the following: i1) Cn satisfies condition (Π); i2) Cn is λ0-pseudomonotone on D(Ln), locally finite-dimensionally bounded; i3) 〈Cn(yn)− fn, yn〉 ≥ 0 ∀yn ∈ Xn : ‖yn‖Xn = R. Let us consider i1). Let B ⊂ Xn be some nonempty bounded subset and k > 0 be a constant such that 〈Cn(y), y〉 ≤ k for each y ∈ B. Since for each y ∈ Xn, 〈Cn(y), y〉 = 〈I∗nC(y), y〉 = 〈C(y), y〉, we have 〈C(y), y〉 ≤ k for each y ∈ B. Since C satisfies condition (Π), there exists K > 0 such that ‖C(y)‖X∗ ≤ K for all y ∈ B. Consequently, sup y∈B ‖Cn(y)‖X∗ ≤ K‖I∗n‖L(X∗;X∗ n) < +∞. Now we consider i2). Because of the boundedness of In ∈ L(Xn;X), I∗n ∈ L(X∗;X∗ n) and the locally finite-dimensional boundedness of C : X → X∗, it follows that Cn on Xn is locally finite-dimensional bounded. Now we prove the λ0-pseudomonotony of Cn on D(Ln). Let {ym}m≥0 ⊂ D(Ln) be an arbitrary sequence such that ym ⇀ y0 inD(Ln),Cn(ym) ⇀ d ∈ X∗ n asm → +∞ and inequality (3.1) hold. As D(Ln) ⊂ D(L) with continuous embedding, ym ⇀ y0 in D(L) as m → +∞. (7.3) Since for all m ≥ 1 〈I∗nC(ym), ym − y0〉 = 〈C(ym), ym − y0〉, ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 204 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO we have lim m→∞ 〈C(ym), ym − y0〉 = lim m→∞ 〈Cn(ym), ym − y0〉 ≤ 0. (7.4) Hence, lim m→∞ 〈C(ym), ym〉 ≤ lim m→∞ 〈Cn(ym), ym − y0〉+ lim m→∞ 〈Cn(ym), y0〉 ≤ 〈d, y0〉 < +∞. Since C satisfies condition (Π), we have that the sequence {C(ym)}m≥1 is bounded in X∗. Hence, for a subsequence, C(ym) ⇀ g in X∗ as m → ∞. Consequently from (7.3) and (7.4), we get the existence of a subsequence {ymk }k≥1 ⊂ {ym}m≥1 such that for all w ∈ X lim k→∞ 〈C(ymk ), ymk − w〉 ≥ 〈C(y0), y0 − w〉. This means that for each w ∈ Xn lim k→∞ 〈Cn(ymk ), ymk − w〉 ≥ 〈Cn(y0), y0 − w〉. So, Cn is λ0-pseudomonotone on D(Ln). Condition i3) holds thanks to (7.2). Now let us continue the proof of the lemma. From [14] (Theorem 2.1) with V = W = X = = Xn, A = Cn, B ≡ 0, L = Ln, D(L) = D(Ln), f = fn, r = R and the properties i1) – i3) for Cn, L2 − L3 for Ln, it follows that the problem (5.1)n, (5.2)n has at least one solution yn ∈ D(Ln) such that ‖yn‖X ≤ R. The lemma is proved. Let us remark that under condition (Π) imposed on Cn it is easy to find the next estimate (7.7) from which it is possible to use the λ0-pseudomonotony for C on D(Ln). 7.2. Passing to limit. Due to the Lemma 7.1 we have a sequence of Galerkin approximate solutions {yn}n≥1 that satisfies the next conditions: ∀n ≥ 1 : ‖yn‖X ≤ R, (7.5) ∀n ≥ 1 : yn ∈ D(Ln) ⊂ D(L), Lnyn + Cn(yn) = fn. (7.6) In order to prove the above theorem we need to obtain an important result. Lemma 7.2. Let, for some subsequence {nk}k≥1 from the natural scale, the sequence {ynk }k≥1 satisfy the next conditions: for all k ≥ 1, ynk ∈ D(Lnk ) = D(L) ∩Xnk ; for all k ≥ 1, Lnk ynk + Cnk (ynk ) = fnk ; ynk ⇀ y in X as k → ∞ for some y ∈ X . ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 205 Then, y ∈ KH(f). Proof. From the definitions of Lnk , Cnk and fnk for each k ≥ 1, it follows that 〈Cnk (ynk ), ynk 〉 = 〈fnk − Lnk ynk , ynk 〉 = 〈f − Lynk , ynk 〉 ≤ ≤ ‖f‖X∗ sup k≥1 ‖ynk ‖X =: K1 < +∞, where K1 is a constant which does not depend on k ≥ 1. Hence, due to property (Π) of the operator C, it follows that there exists K2 > 0 such that for each k ≥ 1 ‖C(ynk )‖X∗ ≤ K2 < +∞. (7.7) Using the condition L1 for L and Proposition 4.2, that for all k ≥ 1 ‖Lynk ‖X∗ = ‖Lnk ynk ‖X∗ = ‖I∗nk (f − C(ynk ))‖X∗ ≤ ≤ max {C1, C2} (‖f‖X∗ +K2) =: K3 < +∞, (7.8) where K3 is a constant which does not depend on k ≥ 1. Hence, for each k ≥ 1, ‖ynk ‖D(L) = ‖ynk ‖X + ‖Lynk ‖X∗ ≤ sup k≥1 ‖ynk ‖X +K3 =: K4 < +∞, where K4 is a constant that does not depend on k ≥ 1. Consequently, due to (7.7), Proposition 4.4 and Banach – Alaoglu theorem, there exists a subsequence {ym} of {ynk } such that for some y ∈ D(L) and d ∈ X∗ the next convergence takes place: ym ⇀ y in D(L), C(ym) ⇀ d in X∗. (7.9) 1. Let us prove that lim m→∞ 〈Lym + C(ym), ym − y〉 = 0. (7.10) Since the set ⋃ n≥1 Xn is dense in X , for each m there exists um ∈ Xm (for example um ∈ ∈ argmin vm∈Xm ‖y − vm‖X) such that um → y in X . So, due to (7.8), (7.7) we obtain that for each m \|〈Lym + C(ym), ym − y〉\| ≤ \|〈Lym + C(ym), ym − um〉\|+ \|〈Lym + C(ym), um − y〉\| ≤ ≤ \|〈f, ym − um〉\|+ (K3 +K2) · ‖y − um‖X → \|〈f, y − y〉\| = 0. 2. Now we obtain that lim m→∞ 〈C(ym), ym − y〉 ≤ 0. (7.11) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 206 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO From (7.10), (7.9) and from monotonicity of L, we have lim m→∞ 〈C(ym), ym − y〉 = lim m→∞ 〈Lym + C(ym), ym − y〉 − lim m→∞ ( 〈Lym − Ly, ym − y〉+ + 〈Ly, ym − y〉 ) ≤ 0 + lim m→∞ ( − 〈Lym − Ly, ym − y〉 ) + lim m→∞ 〈Ly, y − ym〉 ≤ 0. It follows from (7.9) and (7.11) that we can use the λ0-pseudomonotonicity ofC onD(L). Hence, there exists a subsequence {yk}k from {ym}m such that ∀ω ∈ X : lim k→∞ 〈C(yk), yk − ω〉 ≥ 〈C(y), y − ω〉. (7.12) We remark that the last relation is true as implied by Proposition 4.3. In particular, from (7.11) and (7.12) it follows that lim k→∞ 〈C(yk), yk − y〉 = 0. 3. Let us prove that ∀u ∈ D(L) ⋂ ( ⋃ n≥1 Xn ) : 〈f − d− Ly + Lu, u〉 ≥ 0. (7.13) In order to prove (7.13) it is necessary to obtain that ∀u ∈ D(L) ⋂ ( ⋃ n≥1 Xn ) : lim k→∞ 〈Lyk − Ly + Lu, u〉 ≥ 0. (7.14) Since L is monotone and using (7.9), for each u ∈ D(L) ⋂ ( ⋃ n≥1 Xn ) we have lim k→∞ 〈Lyk − Ly + Lu, u〉 ≥ lim k→∞ 〈Lyk − Ly, u〉 = 0. Further let u ∈ D(L) ⋂ ( ⋃ n≥1 Xn ) be arbitrary and fixed. Then there exists n0 ≥ 1 such that u ∈ D(L) ∩Xn0 and, for each k : k ≥ n0, 〈Lyk, u〉 = 〈Lkyk, u〉 = 〈I∗k(f − C(yk)), u〉 = 〈f − C(yk), u〉 → 〈f − d, u〉. (7.15) So, (7.13) directly follows from (7.14) and (7.15). 4. Now we prove that Ly = f − d. Let us use (7.13). We obtain that for each t > 0 and u ∈ D(L) ⋂ ( ⋃ n≥1 Xn ) , 〈f − d− Ly, t · u〉 ≥ −〈t · Lu, t · u〉, which is equivalent to 〈f − d− Ly, u〉 ≥ −t · 〈Lu, u〉. ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 207 Hence, ∀u ∈ D(L) ⋂ ( ⋃ n Xn ) : 〈f − d− Ly, u〉 ≥ 0, and, by the Proposition 4.3, the last relation is equivalent to Ly = f − d. 5. In order to prove that y ∈ D(L) is a solution to (2.3), (2.4) it is enough to show that d = C(y). Because of (7.11), (7.12) and (7.9), it follows that for each ω ∈ X 〈C(y), y − ω〉 ≤ lim k→∞ 〈C(yk), yk − ω〉 ≤ ≤ lim k→∞ 〈C(yk), yk − y〉+ lim k→∞ 〈C(yk), y − ω〉 ≤ 〈d, y − ω〉, which is equivalent to the required statement. So, y ∈ KH(f). The lemma is proved. Using (7.5), (7.6), Lemma 7.2, the Banach – Alaoglu theorem and the topological property of the upper limit [15] (Property 2.29.IV.8) we see that ∅ 6= ⋂ n≥1 [ ⋃ m≥n Km(fm) ] Xw ⊂ KH(f). The converse inclusion is obvious; it follows from the same topological property of the upper limit and from D(L) ⊂ X with a continuous embedding. Now let us prove that KH(f) is weakly compact under the coercivity condition on the operator C = A + B : X → X∗. Because of (7.1) it is enough to show that the given set is bounded. We obtain this statement arguing by contradiction. If {yn}n≥1 ⊂ KH(f) is such that ‖yn‖X → +∞ as n → ∞, we obtain the contradiction +∞ ← 1 ‖yn‖X 〈C(yn), yn〉 ≤ 1 ‖yn‖X 〈Lyn + C(yn), yn〉 = = 1 ‖yn‖X 〈f, yn〉 ≤ ‖f‖X∗ < +∞. 8. An Application. 8.1. On searching the periodic solutions of differential-operator equati- ons via the Faedo – Galerkin method. Let A : X1 → X∗ 1 and B : X2 → X∗ 2 be single-valued maps. We consider the next problem: y′ +A(y) +B(y) = f, (8.1) y(0) = y(T ) (8.2) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 208 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO in order to find solutions via the Faedo – Galerkin method in the classW = {y ∈ X \| y′ ∈ X∗}, where the derivative y′ of an element y ∈ X is considered in the sense of the scalar distributions space D∗(S;V ∗) = L(D(S);V ∗ w), with V = V1 ∩ V2, V ∗ w equal to V ∗ and the topology σ(V ∗, V ) [16]. On W we consider the norm ‖y‖W = ‖y‖X + ‖y′‖X∗ for each y ∈ W . We also consider the spaces Wi = {y ∈ Xi \| y ′ ∈ X∗}, i = 1, 2. Remark 8.1. It is clear that the space W is continuously embedded in C(S;V ∗). Hence, the condition (8.2) makes sense. Together with the problem (8.1), (8.2) we consider the next class of problems in order to search for solutions in Wn = {y ∈ Xn \| y ′ ∈ X∗ n}: y′n +An(yn) +Bn(yn) = fn, (8.3)n yn(0) = yn(T ), (8.4)n where the maps An, Bn, fn were introduced in Section 5, the derivative y′n of an element yn ∈ ∈ Xn is considered in the sense of D∗(S;Hn). LetWper := { y ∈ W ∣ ∣ y(0) = y(T ) } , let us introduce a mapL : D(L) = Wper ⊂ X → X∗ in such way that Ly = y′ for each y ∈ Wper. The main solvability theorem gives the next corollary. Corollary 8.1. Let A : X1 → X∗ 1 and B : X2 → X∗ 2 be maps such that 1) A is λ0-pseudomonotone on W1 and satisfies condition (Π); 2) B is λ0-pseudomonotone on W2 and satisfies condition (Π); 3) the pair (A;B) is s-mutually bounded and the sum C = A + B : X → X∗ is finite- dimensionally locally bounded and weakly coercive. Furthermore, let {hj}j≥1 ⊂ V be a complete system of vectors in V1, V2, H such that, as i = 1, 2, the triple ({hj}j≥1;Vi;H) satisfies condition (γ). Then for each f ∈ X∗ the set Kper H (f) := { y ∈ W ∣ ∣ y is a solution to (8.1), (8.2), obtained via the Faedo – Galerkin method } is nonempty and we have the representation Kper H (f) = ⋂ n≥1 [ ⋃ m≥n Kper m (fm) ] Xw , where for each n ≥ 1 Kper n (fn) = {yn ∈ Wn \| yn is a solution to (8.3)n, (8.4)n} . Moreover, if the operator A+B : X → X∗ is coercive, then Kper H (f) is weakly compact. Proof. At first let us prove the maximal monotonicity of L on Wper. For v ∈ X , w ∈ X∗ such that, for each u ∈ Wper, 〈w−Lu, v−u〉 ≥ 0 is true, let us prove that v ∈ Wper and v′ = w. ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 209 If we take u = hϕx ∈ Wper with ϕ ∈ D(S), x ∈ V and h > 0, we get 0 ≤ 〈w − ϕ′hx, v − ϕhx〉 = 〈w, v〉− − (∫ S (ϕ′(s)v(s) + ϕ(s)w(s))ds, hx ) + 〈ϕ′hx, ϕhx〉 = = 〈w, v〉+ h〈v′(ϕ)− w(ϕ), x〉, where v′(ϕ) and w(ϕ) are values of the distributions v′ and w on ϕ ∈ D(S). So, for each ϕ ∈ D(S) and x ∈ V , 〈v′(ϕ) − w(ϕ), x〉 ≥ 0 is true. Thus we obtain v′(ϕ) = w(ϕ) for all ϕ ∈ D(S). It means that v′ = w ∈ X∗. Now we prove v(0) = v(T ). If we use [1] with u(t) ≡ v(T ) ∈ Wper, we obtain that 0 ≤ 〈v′ − Lu, v − u〉 = 〈v′ − u′, v − u〉 = = 1 2 ( ‖v(T )− v(T )‖2H − ‖v(0)− v(T )‖2H ) = − 1 2 ‖v(0)− v(T )‖2H ≤ 0 and then v(0) = v(T ). In order to prove this statement, it is enough to show that L satisfies the conditions L1 –L3. The condition L1 follows from the next proposition. Proposition 8.1. For each y ∈ X and n ≥ 1, Pny ′ = (Pny) ′, where the derivative of an element x ∈ X has to be considered in the sense of D∗(S;V ∗). Proof. It is sufficient to show that for any ϕ ∈ D(S), Pny ′(ϕ) = (Pny) ′(ϕ). In fact, from the definition of the derivative in the sense of D∗(S;V ∗) we have Pny ′(ϕ) = −Pny(ϕ ′) = −Pn ∫ S y(τ)ϕ′(τ)dτ = = − ∫ S Pny(τ)ϕ ′(τ)dτ = −(Pny)(ϕ ′) = (Pny) ′(ϕ) ∀ϕ ∈ D(S). The proposition is proved. Condition L2 follows from [1] (Lemma VI.1.5) and from the fact that the set C1(S;Hn) is dense in Lp0(S,Hn) = Xn. Condition L3 follows from [1] (Lemma VI.1.7) with V = H = Hn and X = Xn. The corollary is proved. 8.2. Example. Let us consider a bounded domain Ω ⊂ R n with a sufficiently smooth boundary ∂Ω, S = [0, T ], Q = Ω × (0;T ), ΓT = ∂Ω × (0;T ). Let, as i = 1, 2, mi ∈ N, N i 1(respectively N i 2) by the number of derivatives respect to the variable x of order ≤ mi − ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 210 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO −1 (respectively mi) and { Ai α(x, t, η, ξ) } \|α\|≤mi be a family of real functions defined on Q × ×RN i 1 ×RN i 2 . Let Dku = {Dβu, \|β\| = k} be the differentiations with respect to x, δiu = {u, Du, ..., Dmi−1u}, Ai α(x, t, δiu,D miv) : x, t → Ai α(x, t, δiu(x, t),D miv(x, t)). Moreover, let ψ : R → R be a convex coercive function belonging to C1(R) with bounded derivative ψ′. Let us consider the next problem with Dirichlet boundary conditions: ∂y(x, t) ∂t + ∑ \|α\|≤m1 (−1)\|α\|Dα(A1 α(x, t, δ1y,D m1y)) + ∑ \|α\|≤m2 (−1)\|α\|Dα(A2 α(x, t, δ2y,D m2y))+ +ψ′(y(x, t)) = f(x, t) in Q, (8.5) y(x, 0) = y(x, T ) in Ω, (8.6) Dαy(x, t) = 0 on ΓT as \|α\| ≤ mi and i = 1, 2. (8.7) Let us assume that H = L2(Ω) and Vi = Wmi,pi 0 (Ω) with pi ∈ (1, 2] such that Vi ⊂ H with a continuous embedding. Under suitable conditions on the coefficients Ai α, the given problem can be written as y′ +A1(y) +A2(y) + ϕ′ G(y) = f, y(0) = y(T ), (8.8) where f ∈ X∗ = L2(S;L2(Ω))+Lq1(S;W−m1,q1(Ω))+Lq2(S;W−m2,q2(Ω)) (p−1 i +q−1 i = 1), ϕ′ G is the Gateaux derivative of the functional ϕ(y) = ∫ Q ψ(y(x, t))dxdt in the space L2(S;L2(Ω)). Each element y ∈ W that satisfies (8.8) is called a generalized solution to the problem (8.5) – (8.7). Choice of basis. Due to Corollary 6.1 and [9] (Theorem 4.3.1.2), under the main condition d), for the complete system of vectors in the spaces Wmi,pi 0 (Ω), we can take the special basis for the pair (H max{m1;m2}+ε 0 (Ω);L2(Ω)) with a suitable ε ≥ 0. Definition of the operatorsAi. LetAi α(x, t, η, ξ), defined onQ×RN i 1×RN i 2 , satisfy the next conditions: for almost all x, t ∈ Q, the map η, ξ → Ai α(x, t, η, ξ) is continuous on RN i 1 ×RN i 2 ; for all η, ξ, the map x, t → Ai α(x, t, η, ξ) is measurable on Q; (8.9) for all u, v ∈ Lpi(0, T ;Vi) =: Vi, Ai α(x, t, δiu,D miu) ∈ Lqi(Q). (8.10) Then for each u ∈ Vi the map w → ai(u,w) = ∑ \|α\|≤mi ∫ Q Ai α(x, t, δiu,D miu)Dαwdxdt ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 PERIODIC SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS WITH Wλ0 -PSEUDOMONOTONE MAPS 211 is continuous on Vi and also there exists Ai(u) ∈ V ∗ i such that ai(u,w) = 〈Ai(u), w〉. (8.11) Conditions on Ai. Similarly to [2] (Sections 2.2.5, 2.2.6, 3.2.1) we have Ai(u) = Ai(u, u), Ai(u, v) = Ai1(u, v) +Ai2(u), where 〈Ai1(u, v), w〉 = ∑ \|α\|=mi ∫ Q Ai α(x, t, δiu,D miv)Dαwdxdt, 〈Ai2(u), w〉 = ∑ \|α\|≤mi−1 ∫ Q Ai α(x, t, δiu,D miu)Dαwdxdt. We add the next conditions: 〈Ai1(u, u), u− v〉 − 〈Ai1(u, v), u− v〉 ≥ 0 ∀u, v ∈ Vi; (8.12) if uj ⇀ u in Vi, u ′ j ⇀ u′ in V∗i and if 〈Ai1(uj , uj)−Ai1(uj , u), uj − u〉 → 0, then Ai α(x, t, δuj ,D miuj) ⇀ Ai α(x, t, δu,Dmiu) in Lqi(Q); (8.13) coercivity. (8.14) Remark 8.2. Similarly to [2] (Theorem 2.2.8), sufficient conditions for getting (8.12), (8.13) are ∑ \|α\|=mi Ai α(x, t, η, ξ)ξα 1 \|ξ\|+ \|ξ\|pi−1 → +∞ as \|ξ\| → ∞ for almost all x, t ∈ Q and \|η\| bounded; ∑ \|α\|=mi (Ai α(x, t, η, ξ)−Ai α(x, t, η, ξ∗))(ξα − ξ ∗ α) > 0 as ξ 6= ξ∗ for almost all x, t ∈ Q and η. The next condition allows for coercivity: ∑ \|α\|=mi Ai α(x, t, η, ξ)ξα ≥ c\|ξ\|pi for sufficiently large \|ξ\|. A sufficient condition to get (8.10) (see [2, p. 332]) is \|Ai α(x, t, η, ξ)\| ≤ c[\|η\|pi−1 + \|ξ\|pi−1 + k(x, t)], k ∈ Lqi (Q). (8.15) ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2 212 P. O. KASYANOV, V. S. MEL’NIK, S. TOSCANO By analogy with the proof of [2] (Theorem 3.2.1 and Statement 2.2.6) we get the next proposition. Proposition 8.2. Let the operator Ai : Vi → V∗i , i = 1, 2, defined in (8.11), satisfy (8.9), (8.10), (8.12), (8.13) and (8.14). Then Ai is pseudomonotone on Wi. Moreover it is bounded if (8.15) holds. Due to the last statement and to Corollary 8.1, it follows that under the listed above conditi- ons, for all f ∈ X∗ there exists R > 0 such that KH(f) := { y ∈ W ∣ ∣ y is a generalized solution to the problem (8.5) – (8.7), obtained via the Faedo – Galerkin method } is nonempty, weakly compact in the closed ball from the space X with the center in the origin and radius R, and also representation (7.1) holds. 1. Gaevsky H., Greger K., and Zaharias K. Nonlinear the operator equations and the operator-differential equations (Russian transl.) — M.: Mir, 1978. — 337 p. 2. Lions J. L. Quelques méthodes de résolution des problemès aux limites non lineaires. — Paris: Dunod Gauthier-Villars, 1969. — 587 p. 3. Duvaut G., Lions J. L. Inequalities in mechanics and in phisycs (Russian transl.) — M.: Nauka, 1980. — 383 p. 4. Mel’nik V. S., Toscano L. About nonlinear differential-operator equations in Banach spaces with maps of pseudo-monotone type // System Research and Inform. Technol. — 2004. — 3. — P. 63 – 81. 5. Zgurovskiy M. Z., Mel’nik V. S., and Novikov A. N. Applied methods of analysis and control of nonlinear processes anf fields (in Russian). — Kiev: Naukova Dumka, 2004.– 588 p. 6. Ivanenko V. I., Mel’nik V. S. Variational methods in control problems for systems with distributed parameters (in Russian). — Kiev: Naukova Dumka, 1988. — 286 p. 7. Skrypnik V. I. Methods of research nonlinear boundary value problems (in Russian). — M.: Nauka, 1990. — 442 p. 8. Iosida K. Functional analysis (Russian transl.). — M.: Mir, 1967. — 624 p. 9. Triebel H. Interpolation theory, function spaces, differential operators (Russian transl.). —M.: Mir, 1980. — 664 p. 10. Halmosh P. R. The measure theory (Russian transl.). — M.: Izd-vo inostr. lit., 1953. — 292 p. 11. Enflo P. A counterexample to the approximation problem in Banach spaces // Acta Math. — 1973. — 130. — P. 309 – 317. 12. Temam R. Infinite-dimentional dynamical systems in mechanics and physics. — New York, 1988. 13. Dorogovtsev A. Ya. The elements of teory of measure and integral (in Russian). — Kiev: Vyshcha Shcola, 1989. — 150 p. 14. Mel’nik V. S. About operational inclusions in Banach spaces with densely defined operators // System Research and Inform. Technol. — 2003.– 3.— P. 120 – 126. 15. Kuratowski K. Topology (Russian transl.). — M.: Mir, 1966. — Pt 1. — 595 p. 16. Reed M., Simon B. Methods of modern mathematical physics (Russian transl.). — M.: Mir, 1977. — P. 1. — 359 p. Received 31.03.2006 ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 2

Periodic solutions of nonlinear evolution equations with Wλ₀-pseudomonotone maps

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