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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


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Volume 15, Issue 1

Issues

Volume 13 (2015)

Semilinear systems with a multi-valued nonlinear term

In-Sook Kim
  • Corresponding author
  • Department of Mathematics, Sungkyunkwan University, Natural Science Campus, Seobu-ro 2066, Suwon 16419, Republic of Korea
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/ Suk-Joon Hong
  • Department of Mathematics, Sungkyunkwan University, Natural Science Campus, Seobu-ro 2066, Suwon 16419, Republic of Korea
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Published Online: 2017-05-20 | DOI: https://doi.org/10.1515/math-2017-0056

Abstract

Introducing a topological degree theory, we first establish some existence results for the inclusion hLuNu in the nonresonance and resonance cases, where L is a closed densely defined linear operator on a Hilbert space with a compact resolvent and N is a nonlinear multi-valued operator of monotone type. Using the nonresonance result, we next show that abstract semilinear system has a solution under certain conditions on N = (N1, N2), provided that L = (L1, L2) satisfies dim Ker L1 = ∞ and dim Ker L2 < ∞. As an application, periodic Dirichlet problems for the system involving the wave operator and a discontinuous nonlinear term are discussed.

Keywords: Semilinear system; Multi-valued operator; Operators of monotone type; Degree theory

MSC 2010: 47H04; 47H05; 47H11; 35A16; 35B10; 35L71

1 Introduction

Semilinear wave equation and abstract semilinear equation have been studied in many ways; see [1-4], for instance. To solve this problem, Mawhin and Willem [5, 6] employed the Leray-Schauder theory combined with monotone type operators in Galerkin arguments; see [1]. Berkovits and Tienari [7] introduced a topological degree theory for multi-valued operators of monotone type with so called elliptic super-regularization method to deal with hyperbolic problems with discontinuous nonlinearity.

Let H be a real separable Hilbert space. We first observe a semilinear equation LuNu=h,(1) where L is a closed densely defined linear operator on H with a compact resolvent and N is a nonlinear operator. In the self-adjoint case, it is known that equation (1) has a solution when ||Nuλ12u||μ||u||+υfor alluH, where λ1 is the first positive eigenvalue of L and µ ∈ [0, λ1/2), υ ∈ [0, ∞) are constants. More generally, if Ker L = Ker L*, L* being the adjoint operator of L, then there is a positive number ρ such that ||Luρ2u||ρ2||u||for alluD(L). In this case, equation (1) admits a solution if there exist µ ∈ [0, ρ/2) and υ ∈ [0, ∞)such that ||Nuρ2u||μ||u||+υfor alluH. The existence proof for the nonlinear equation was based on the Leray-Schauder theory. See [1, 8, 9].

Next, Berkovits and Fabry [10] considered a system of semilinear equations L1u1N1(u1,u2)=h1,L2u2N2(u1,u2)=h2, where L1, L2 are closed densely defined linear operators with dim Ker L1 = ∞ and dim Ker L2 < ∞ and N1, N2 are nonlinear operators.

In the present paper, our goal is to study the semilinear system including a multi-valued nonlinear term. Especially, we are interested in the following semilinear system L1u1N1(u1,u2)h1,L2u2N2(u1,u2)=h2,(2) where L1(u1, u2)= N1,1(u1)+ N1,2(u2), N1,1 is a weakly upper semicontinuous bounded multi-valued operator of monotone type, and N1, N2 are continuous bounded operators. The semilinear system (2) can be written as hLuNu, where L = (L1, L2) is as above and N = (N1, N2) is a weakly upper semicontinuous bounded multi-valued operator satisfying generalized (S+) condition with respect to the orthogonal projection to Ker L.

More generally, to find a solution of the semilinear inclusion hLuNu,(3) we introduce a topological degree theory for a wider class including the class (S+) with elliptic super-regularization method, following the basic lines of the Berkovits-Tienari degree for the class (S+) given in [7].

Using the fact that some linear injection has nonzero degree, we show that the inclusion (3) has a solution if there are µ ∈ [0, ρ/2) and α ∈ [0,1) such that ||aρ2u||μ||u||+O(||u||α)for allH,||u||andaNu.(4) Moreover, we are concerned with the solvability of the inclusion (3) under an additional h-dependent resonance type condition when µ = ρ/2 in (4) is allowed. For semilinear equations in a more general setting, we refer to [10].

Concerning abstract semilinear systems, it is emphasized that the nonlinear operator N = (N1, N2) is not necessarily of class (S+). Namely, instead of the Berkovits-Tienari degree for the class (S+), our degree theory plays an important role in the study of semilinear systems with mixed nonlinear terms like (2). In the nonresonance case, we prove that (2) is solvable under certain conditions on N = (N1, N2). Applying this result, we show the existence of weak solutions of periodic Dirichlet problem for the system involving the wave operator and a discontinuous nonlinear term. Actually, it was inspired by the works [7, 11].

2 Degree theory

Let H be a real Hilbert space. Given a nonempty subset Ω of H, let Ω and ∂Ω denote the closure and the boundary of Ω in H, respectively. Let Br (u) denote the open ball in H of radius r > 0 centered at u. The symbol → (⇀) stands for strong (weak) convergence.

Definition 2.1

A multi-valued operator F : Ω ⊂ H → 2H is said to be:

  1. upper semicontinuous if the set F1(A)={uΩFuA} is closed in Ω for every closed set A in H;

  2. weakly upper semicontinuous if F-1 (A) is closed in Ω for every weakly closed set A in H;

  3. bounded if it maps bounded sets into bounded sets;

  4. compact if it is upper semicontinuous and the image of any bounded set is relatively compact;

  5. of Leray-Schauder type if it is of the form I − C, where C is compact and I denotes the identity operator.

Let (H, 〈·,·〉) be a real separable Hilbert space and E a closed subspace of H. Let P : HE and Q : HE be the orthogonal projections, respectively.

Definition 2.2

A multi-valued operator F : Ω ⊂ H → 2H \ ∅ is said to be:

  1. of class (S+)p if for any sequence (un) in Ω and for any sequence (wn) in H with wnFun such that unu, QunQu, and limsupnwn,P(unu)0, we have unu;

  2. P-pseudomonotone, written F ∈ (PM)p, if for any sequence (un) in Ω and for any sequence (wn) in H with wnFun such that unu, QunQu, and limsupnwn,P(unu)0, we have limnwn,P(unu)=0, and if u ∈ Ω and wjw for some subsequence (wj) of (wn) then wFu;

  3. P-quasimonotone, written F ∈ (QM)p, if for any sequence (un) in Ω and for any sequence (wn) in H with wnFun such that unu and QunQu, we have liminfnwn,P(unu)0.

With P = I, we get the definitions for classes (S+), (PM), and (QM) in [7].

Throughout this paper, we will always assume that all multi-valued operators considered have nonempty closed convex values.

It is clear from the definitions that (S+) = (S+)P, (PM) = (PM)P, and (QM) = (QM)P if dim E < ∞. If all operators are assumed to be bounded and weakly upper semicontinuous, it is easy to see that (S+)P ⊂ (PM)P ⊂ (QM)P and the class (S+)P is invariant under (QM)P-perturbations.

Let H be a real separable Hilbert space. Suppose that L : D(L) ⊂ HH is a closed densely defined linear operator with ImL=(KerL), and K : Im L → Im LD(L), the inverse of the restriction of L to Im LD(L), is compact. Let P : H → Ker L and Q : H → Im L be the orthogonal projections, respectively.

To find solutions of a semilinear inclusion, we need the following equivalent formulation; see [7].

Lemma 2.3

Let L, K, P, Q be as above. Suppose that N : Ḡ → 2H is a multi-valued operator, where G is an open set in H. Then hLuNu,uG¯D(L) if and only if h~Qu(KQP)Nu,uG¯, where h̄ = (KQP)h.

Proof

Suppose that Qu − (KQP)Nu with u. Then KQhPh = QuKQa + Pa for some aNu. Since Qh = LQuQa = LuQa and −Ph = Pa, we have h=Ph+Qh=LuQaPaLuNu. Conversely, suppose that hLuNu with uD(L). Then we get h(KQP)(LN)u=Qu(KQP)Nu. This completes the proof.  □

Given an open bounded set G in H, we consider a class of semilinear operators LN, where N : 2H is a weakly upper semicontinuous bounded multi-valued operator of class (S+)P.

We introduce a topological degree theory for the above class, following the basic idea of the Berkovits-Tienari degree for the class (S+) given in [7]. To do this, we adopt elliptic super-regularization method as in [7, 12].

Let ψ : Ker L → Ker L be a compact self-adjoint linear injection. To each F = Q − (KQP)N, we associate a family of Leray-Schauder type operators defined by Fλ:=I(KQλΨ2P)Nforλ>0. For the Leray-Schauder degree theory for multi-valued operators, we refer to [13, 14].

We give a fundamental result which is useful for the construction of our degree and its properties.

Lemma 2.4

Suppose that G is an open bounded set in H and N : Ḡ → 2H is a weakly upper semicontinuous bounded operator of class (S+)P. For any closed set A ⊂ Ḡ such that h ∉ (L − N)(A ∩ D(L)), there exists a positive number λ0 such that hλFλ(A)forallλ>λ0, where hλ := (KQ − λψ2P)h.

Proof

Let A be any closed subset of such that h ∉ (LN)(AD(L)). Assume that the assertion is not true. Then we find sequences (λn) in (0, ∞) and (un) in A with λn → ∞ such that hλ nFλn (un) for all n ∈ ℕ, that is, unKQ(an+h)+λnΨ2P(an+h)=0, where anNun. This equation is equivalent to QunKQ(an+h)=0andPun+λnΨ2P(an+h)=0.(5) Passing to subsequences if necessary, we may suppose that unu and ana for some u, aH. Then we have by (5) and the strong continuity of the operator K QunKQ(a+h)=QuandΨ2PanΨ2Ph=Ψ2Pa, which implies Pa = − Ph, by the injectivity of ψ. Since P and ψ are self-adjoint with P2 = P and Pan ⇀ −Ph, it follows from (5) that limsupnan+h,P(unu)=limsupnP(an+h),P(unu)=limsupnP(an+h),λnΨ2P(an+h)=limsupnλnΨP(an+h20. In view of N + h ∈ (S+)P, this implies unuA. Note that Nu is closed and convex and so weakly closed. Since N is weakly upper semicontinuous, we have aNu. This yields to KQhPh=QuKQa+PaQu(KQP)Nu, which contradicts the hypothesis that h ∉ (LN)(AD(L)), in view of Lemma 2.3 with A in place of . Therefore, the assertion must be true. This completes the proof.  □

Corollary 2.5

Let G be an open bounded set in H. If h ∉ (L − N)(∂G ∩ D(L)), there is a positive number λ0 such that hλ ∉ Fλ(∂G) for all λ > λ0 and dLS (Fλ, G, hλ) is constant for all λ > λ0, where dLS denotes the Leray-Schauder degree.

Proof

According to Lemma 2.4 with A = ∂G, we can choose a positive number λ0 such that hλFλ(∂G) for all λ > λ0. For the second assertion, let λ1, λ2 be arbitrary numbers in (λ0, ∞) such that λ1 < λ2. Then Fλ, λ ∈ [λ1, λ2], defines a Leray-Schauder type homotopy such that hλFλ(∂G) for all λ ∈ [λ1, λ2]. It follows from the homotopy invariance property of the degree dLS that dLS(Fλ1,G,hλ1)=dLS(Fλ2,G,hλ2). Since λ1, λ2 were arbitrarily chosen in (λ0, ∞), we conclude that dLS(Fλ, G, hλ) is constant for all λ > λ0. This completes the proof.  □

We are now ready to define a topological degree for the semilinear class involving multi-valued operators of class (S+)P.

Definition 2.6

Let L, K, P, Q be as above. Suppose that N : → 2H is a weakly upper semicontinuous bounded multi-valued operator of class (S+)P, where G is any open bounded set in H. If h ∉ (LN)(∂GD(L)), we define a degree function d by d(LN,G,h):=limλdLS(Fλ,G,hλ), where Fλ = I − (KQλψ2P)N and hλ = (KQλψ2P)h.

Definition 2.7

A multi-valued operator N : [0, 1] × Ḡ → 2H is said to be a homotopy of class (S+)P if for any sequence (tn, un) in [0, 1] × Ḡ and for any sequence (an) in H with an ∈ N(tn, un) such that tn → t, un ⇀ u, Qun ⇀ Qu, and limsupnan,P(unu)0, we have un → u.

Lemma 2.8

(Affine homotopy) If N1, N2 : Ḡ → 2H are weakly upper semicontinuous bounded operators of class (S+)P, then N : [0, 1] × Ḡ → 2H given by N(t,u):=(1t)N1u+tN2ufor(t,u)[0,1]×G¯ is a weakly upper semicontinuous bounded homotopy of class (S+)P.

Given an open bounded set G in H, we consider a class of semilinear homotopies LN, where N : [0,1] × → 2H is a weakly upper semicontinuous bounded multi-valued homotopy of class (S+)P.

We make the following observation for establishing the homotopy invariance of the degree d.

Lemma 2.9

Suppose that N : [0, 1] × Ḡ → 2H is a weakly upper semicontinuous bounded homotopy of class (S+)P, where G is an open bounded set in H, and h : [0,1] → H is a continuous curve in H such that h(t) ∉ (L − Nt)(∂G ∩ D(L))for all t ∈ [0,1]. Then there is a positive number λ0 such that hλ(t)(Ft)λ(G)forallt[0,1]andallλ>λ0, where Nt=N(t,.),(Ft)λ=I(KQλΨ2P)Nt and hλ(t)=(KQλΨ2P)h(t).

Proof

Assume to the contrary that there exist sequences (λn) in (0,∞), (tn) in [0,1], and (un) in ∂G with λn → ∞ such that hλn (tn) ∈ Fλn (tn, un) for all n ∈ ℕ. This can be written as unKQ(an+h(tn))+λnΨ2P(an+h(tn))=0,(6) where anN(tn, un). Equation (6) is equivalent to QunKQ(an+h(tn))=0andPun+λnΨ2P(an+h(tn))=0.(7) Without loss of generality, we may suppose that tnt, unu, and ana for some t ∈ [0,1] and some u, aH. Then we have KQ(a + h(t)) = Qu and −ψ2Ph(t) = ψ2Pa and so Pa = −Ph(t). Since P and ψ are self-adjoint and Pan + Ph(tn) ⇀ 0, we obtain from (7) that limsupnan+h(tn),P(unu)=limsupnP(an+h(tn)),P(unu)=limsupnP(an+h(tn)),λnΨ2P(an+h(tn))=limsupnλnΨP(an+h(tn))20, which implies, in view of h(tn) → h(t), that limsupnan,P(unu)0. Hence it follows from N ∈ (S+)P that unu ∈ ∂G. Since N is weakly upper semicontinuous, we have aN(t, u). Therefore, we get KQh(t)Ph(t)=QuKQa+PaQu(KQP)N(t,u), which contradicts the hypothesis that h(t) ∉ (LNt)(∂GD(L)). This completes the proof.  □

The degree theory plays a decisive role in the study of semilinear inclusions. Especially, the homotopy invariance is a powerful property in the use of the degree, as we will see in Theorem 3.1 below.

Theorem 2.10

Let L and N be as in Definition 2.6. Suppose that G is any open bounded subset of H and h ∉ (L − N)(∂G ∩ D(L)). Then the degree d has the following properties:

  1. (Existence) If d (L − N, G, h) ≠ 0, then the inclusion h ∈ Lu — Nu has a solution in G ∩ D(L).

  2. (Additivity) If G1 and G2 are disjoint open subsets of G such that h(LN)[(G¯(G1G2))D(L)], then we have d(LN,G,h)=d(LN,G1,h)+d(LN,G2,h).

  3. (Homotopy invariance) Suppose that N : [0,1] × Ḡ → 2H is a weakly upper semicontinuous bounded homotopy of class (S+)P. If h : [0, 1] → H is a continuous curve in H such that h(t)LuN(t,u)forallt[0,1]andalluGD(L), then d (L − N(t, ·), G, h(t)) is constant for all t ∈ [0,1].

Proof.

  1. Argue by contraposition. If hLuNu for all uD(L), Lemma 2.4 implies that there exists a positive number λ0 such that hλFλ() for all λ > λ0. It follows from the corresponding property of the Leray-Schauder degree that dLS(Fλ, G, hλ) = 0 for all λ > λ0. By Definition 2.6, we have d (LN, G, h) = 0.

  2. Applying Lemma 2.4 with A = \(G1G2), we find a positive number λ0 such that hλFλ(G¯(G1G2))for allλ>λ0. By the additivity of the Leray-Schauder degree, we have dLS(Fλ,G,hλ)=dLS(Fλ,G1,hλ)+dLS(Fλ,G2,hλ)for allλ>λ0. The conclusion follows directly from the definition of the degree d.

  3. In view of Lemma 2.9, we can choose a positive number λ0 such that hλ(t)(Ft)λ(G)for allt[0,1]and allλ>λ0,

where hλ(t)=(KQλΨ2P)h(t) and (Ft)λ=I(KQλΨ2P)Nt. For each fixed λ > λ0, (Ft)λ, t ∈ [0,1] defines a Leray-Schauder type homotopy such that hλ(t) ∉ (Ft)λ(u) for all (t, u) ∈ [0,1] × ∂G. Hence it follows from the homotopy invariance property of the degree dLS that dLS ((Ft)λ, G, hλ(t)) is constant for all t ∈ [0,1]. For any t1, t2 ∈ [0,1], we have by Definition 2.6 d(LNt1,G,h(t1))=limλdLS((Ft1)λ,G,hλ(t1))=limλdLS((Ft2)λ,G,hλ(t2))=d(LNt2,G,h(t2)). We conclude that d (LN(t, ·), G, h(t)) is constant for all t ∈ [0,1]. This completes the proof.  □

The following result says that some linear injection has nonzero degree. It will be a key tool for proving the existence of a solution for semilinear inclusions in the next section.

Lemma 2.11

Let B : H → H be a bounded linear operator of class (S+)P such that L − B is injective. Then for any bounded open set G ⊂ H and h ∈ (L − B)(G ∩ D(L)), we have d(LB,G,h)=±1.

Proof

Let G be any open bounded set in H and h ∈ (LB)(GD(L)). By the injectivity of the operator LB, we have (LB)v = h for some vGD(L). In view of part (b) of Theorem 2.10, we can choose a positive number R with ||υ|| < R such that d(LB,G,h)=d(LB,BR(0),h). It is clear that (LB)u ≠: th for all t ∈ [0,1] and u ∈ ∂BR(0). Letting h : [0,1] → H be defined by h(t) := th for t ∈ [0,1], we obtain from part (c) of Theorem 2.10 that d(LB,BR(0),h)=d(LB,BR(0),0). By Definition 2.6, we have d(LB,BR(0),0)=limλdLS(Tλ,BR(0),0), where Tλ = I − (KQλψ2P)B. Since Tλ is an injective Leray-Schauder type operator for large λ, it is known in [15] that d (LB, G, h) is +1 or −1. This completes the proof.  □

Corollary 2.12

If L − αI is injective for some positive constant α, then we have deg(LαI,Br(0),0)=±1foranypositivenumberr.

Proof

Note that αI ∈ (S+)P. Apply Lemma 2.11 with B = αI and h = 0.  □

3 Existence results

This section is devoted to the solvability of semilinear inclusions in the nonresonance and resonance cases, by using the degree theory in the previous section.

Let (H, 〈·, ·〉) be a real separable Hilbert space. Suppose that L : D(L) ⊂ HH is a closed densely defined linear operator with Im L = (Ker L), and K : Im L → Im LD(L), the inverse of the restriction of L to Im LD(L), is compact. Let P : H → Ker L and Q : H → Im L be the orthogonal projections, respectively. Here, Ker L may be infinite dimensional.

Set A:=ρRLu2ρLu,ufor alluD(L).(8) It is easily checked that A=ρRLuρ2uρ2ufor alluD(L).(9) It is known in [8, 9] that the set 𝓐 is a closed interval containing 0 as an interior point of 𝓐.

We present a nonresonance theorem on the surjectivity of LN when N is P-pseudomonotone. The basic idea of proof comes from Theorem 6.1 of [10], where, of course, semilinear equations in a more general setting were dealt with.

Theorem 3.1

Let L be as above. Let N : H → 2H be a weakly upper semicontinuous bounded operator. Suppose that there exist numbers ρ ∈ (0, sup 𝓐], µ ∈ [0, ρ/2), and α ∈ [0,1) such that aρ2uμu+O(uα)foruH,u,andaNu.(10)

  1. If N is P-quasimonotone, then the range of the operator L − N is dense in H.

  2. If N is P-pseudomonotone, then the inclusion hLuNu

    has a solution in D(L) for every h ∈ H.

Proof

Let h be an arbitrary element of H. We consider the homotopy equation thLu(1t)ρ2utNufort[0,1]anduD(L).(11) Since ρ ∈ 𝓐 implies, in view of (9), that the linear operator L − (p/2)I is injective, equation (11) with t = 0 has only the trivial solution. From Corollary 2.12, we know that degLρ2I,Br(0),0)0for any positive numberr.(12) We first claim that the set of solutions of (11) is bounded in H.In fact, we assume that there are sequences (un) in D(L) and (tn) in [0,1] with ||un|| → ∞ such that tnh=Lun(1tn)ρ2untnanfor allnN, where anNun. For all n ∈ ℕ, we have by (9) and (10) ρ2unLunρ2un=tnanρ2un+hμun+h+O(unα) and hence ρ2μunh+O(unα), which contradicts the unboundedness of the sequence (un) chosen. Thus, the solution set is bounded in H. Now we can choose a positive constant R such that thLu(1t)ρ2utNufor allt[0,1]and alluD(L)withuR.(13) There are three cases to consider. Firstly, we suppose that N ∈ (S+)P. Notice by Lemma 2.8 that N1:[0,1]×BR(0)¯2H defined by N1(t,u):=(1t)ρ2Iu+tNufor(t,u)[0,1]×BR(0)¯ is a weakly upper semicontinuous bounded homotopy of class (S+)P. Part (c) of Theorem 2.10 implies, in view of (13) and (12), that deg(LN,BR(0),h)=degLρ2I,BR(0),0)0.(14)

Secondly, we suppose that N is P-quasimonotone. For a moment, fix t ∈ (0,1). Note that N1 (t, ·) = (1 −t)(ρ/2)I + tN is of class (S+)P. Applying the assertion (14) in the first case with N1 (t, ·) in place of N, we see that degL(1t)ρ2ItN,BR(0),th0. By part (a) of Theorem 2.10, there exists a utBR (0) ∩ D(L) such that th=Lut(1t)ρ2uttat,(15) where atNut. According to the assertion (15), for a sequence (tn) in (0, 1) with tn → 1, there is a corresponding sequence (un) in BR(0) ∩ D(L) such that tnh=Lun(1tn)ρ2untnan, where anNun. Hence it follows that Lunanh and so hIm(LN)¯. Since hH was arbitrary, we conclude that the range of LN is dense in H. Thus, statement (a) holds.

Thirdly, we suppose that N is P-pseudomonotone. In virtue of the second case, we take a sequence (un) in BR(0) ∩ D(L) such that Lunanh, where anNun. Without loss of generality, we may suppose that unu and ana for some u, aH. Since Im L = (KerL) and PunPu, we have limnan,P(unu)=limnLun,P(unu)limnh,P(unu)=0. Since Qun = KQLun and K is compact, it is obvious that QunQu. The P-pseudomonotonicity of the operator N implies that aNu. Since the graph of L is weakly closed and Luna + h, we obtain that uD(L)andhLuNu. We have just proved that statement (b) is valid. This completes the proof.  □

Next, we show the existence of a solution of the semilinear inclusion under an additional h-dependent resonance type condition when µ = ρ/2 in condition (10) is allowed.

Theorem 3.2

Let L be as above. Let N : H → 2H be a weakly upper semicontinuous bounded operator. Suppose that there are ρ ∈ (0, sup 𝓐) and α ∈ [0,1) such that aρ2uρ2u+O(uα)foruH,u,andaNu.(16) Let h ∈ H be given and suppose that for any sequence (un) in D(L) such that ||un || → ∞ and ||Lun|| = o(||un||) for n → ∞, there exists an integer n0 such that an+h,Pun>0forallnn0andallanNun.(17)

  1. If N is P-quasimonotone, thenhIm(LN)¯.

  2. If N is P-pseudomonotone, then the inclusion hLuNu has a solution in D(L).

Proof

We consider the homotopy equation thLu(1t)ρ2utNufor(t,u)[0,1]×D(L).(18) We have to show that the solution set S=uD(L)|thLu(1t)ρ2utNufor somet[0,1] is bounded in H. Assume to the contrary that there exist sequences (un) in D(L) and (tn) in (0,1] with ||un|| → ∞ such that tnh=Lun(1tn)ρ2untnanfor allnN,(19) where anNun. Let ρ̄A be any positive number with ρ̄ > p. For all uD(L), we have by (8) Luρ2u2=Lu2ρLu,u+ρ2u21ρρ¯Lu2+ρ2u2. Hence it follows from (19) and (16) that 1ρρ¯Lun2+ρ2un2Lunρ2un2anρ2un+h2ρ2un+h+O(unα)2, which implies Lun=oun. Set Zn := un/||un|| and wn := Lun. Then ||wn|| = ||Lun||/||un|| → 0 implies Qzn = Kwn → 0. Since |〈Lun, Pun〉 = 0, we have by (19) an+h,Pun=(1tn)tn1ρ2un,Pun. It follows from un,Pun=un2un,Qun that an+h,Pun=(1tn)tn1ρ2un21zn,Qzn. Hence we obtain from Qzn → 0 that an+h,Pun0for some largen, which contradicts hypothesis (17). Thus, we have shown that the solution set S is bounded in H. The rest of proof proceeds in a similar way to that of Theorem 3.1.  □

4 Semilinear systems

In this section, we first examine under what conditions the operators are of class (S+)P or P-quasimonotone and then establish some existence results for semilinear systems in the nonresonance case.

Let H1, H2 be two real separable Hilbert spaces and let H = H1 × H2 be the Hilbert space with inner product defined by u,υ=u1,υ1+u2,υ2foru=(u1,u2),υ=(υ1,υ2)H1×H2. For k = 1,2, let Lk : D(Lk) ⊂ HkHk be a closed densely defined linear operator with Im Lk = (Ker Lk). Suppose that Kk : Im Lk → Im Lk, the inverse of the restriction of Lk to Im LkD(Lk), is compact. For k = 1,2, let Pk : Hk → Ker Lk and Qk : Hk → Im Lk be the orthogonal projections, respectively.

Define the diagonal operator L : D(L) ⊂ HH by setting Lu=L1u1,L2u2foru=(u1,u2)D(L)=D(L1)×D(L2). Then K : Im L → Im L, the inverse of L to Im LD(L), is compact, where Ku=(K1u1,K2u2)foru=(u1,u2)ImL. Let P : H → Ker L and Q : H → Im L be the orthogonal projections, respectively. We write Pu=P1u1,P2u2andQu=Q1u1,Q2u2foru=(u1,u2)H. In what follows, we suppose that dim Ker L1 = ∞ and dim Ker L2 < ∞.

We show that N = (N1, N2) is of class (S+)P under strong monotonicity on the first component N1. For the single-valued case, we refer to [12, Lemma 6.2].

Proposition 4.1

Suppose that N = (N1, N2) : H → 2H is a bounded multi-valued operator such that

  1. N1(υ,.) : H2 → 2H1 is upper semicontinuous with compact values for each υ ∈ H1

  2. for each z ∈ H2 there exist a positive number δ = δ(z) and a positive constant c = c(z) such that bd,υυcυυ2 for all υ,υH1,zBδ(z),bN1(υ,z)anddN1(υ,z).

Then the operator N is of class (S+)P.

Proof

Let (un) be any sequence in H and (an) any sequence in H with anNun such that unu,QunQu,andlimsupnan,P(unu)0.(20) Let un = (υn, zn) and u = (υ, z). Since dim Ker L2 < ∞ and QunQu, we have znzinH2.(21) Since QunQu and N is bounded, we get by (21) limsupnan,P(unu)=limsupnan,unu=limsupnbn,υnυ,(22) where an = (bn, cn) ∈ Nun, that is, bnN1 (υn, zn). By hypothesis (a), we see that a sequence (dn) has a strongly convergent subsequence in H1, denoted again by (dn), where dnN1(υ, zn), and hence limndn,υnυ=0, which implies limsupnbndn,υnυ=limsupnbn,υnυ.(23) It follows from (20), (22), and (23) that limsupnbndn,υnυ0. Noting that bnN1 (υn, zn) and dnN1 (υ, zn), we get by hypothesis (b) 0limsupncυnυ2limsupnbndn,υnυ0. Since υnυ in H1, we have by (21) un=υn,znυ,z=u. We conclude that N ∈ (S+)P.  □

Proposition 4.2

Suppose that N = (N1, N2) : H → 2H is a bounded multi-valued operator such that

  1. N(υ,.) : H2 → 2H1 is upper semicontinuous with compact values for each υ ( H1;

  2. for each z   H2 there exists a positive number δ = δ(z) such that bd,υυ0 for all υ,υH1,zBδ(z),bN1(υ,z),anddN1(υ,z).

Then N is P-quasimonotone.

Proof

Let (un) be a sequence in H and (an) a sequence in H with anNun such that unuandQunQu. As in the proof of Proposition 4.1, an analogous argument shows that liminfnan,P(unu)=liminfnbndn,υnυ0, where an=(bn,cn)Nun,un=(υn,zn),u=(υ,z),bnN1(υn,zn),anddnN1(υ,zn). The last inequality follows from hypothesis (b), which proves that N is P-quasimonotone.  □

In Propositions 4.3 and 4.4 below, the main point is that monotone type hypothesis on the second component N2 is not required and so N = (N1, N2) is not necessarily of class (S+) or pseudomonotone.

Proposition 4.3

Suppose that N = (N1, N2) : H → 2H is bounded, where N1(υ, z) = N1,1(υ) + N1,2(z), such that

  1. N1,1 : H1 → 2H1 is weakly upper semicontinuous and of class (S+);

  2. N1,2 : H2H1 is continuous;

  3. N2 : HH2 is demicontinuous.

    Then N is of class (S+)P.

Proof

Let (un) be any sequence in H and (an) any sequence in H with anNun such that unu,QunQu,andlimsupnan,P(unu)0.(24) Let un = (υn, zn) and u = (υ, z). As in the proof of Proposition 4.1, we have znz in H2 and limsupnan,P(unu)=limsupnbn,υnυ,(25) where an = (bn, cn) ∈ Nun, that is, bnN1,1(υn) + N1,2 (zn). From hypothesis (b) we obtain that limsupnbnN1,2(zn),υnυ=limsupnbn,υnυ.(26) It follows from (24), (25), and (26) that limsupnbnN1,2(zn),υnυ=limsupnan,P(unu)0. Since N1,1 is of class (S+) and bnN1,2(zn) ∈ N1,1(υn), this implies υnυ in H1. Therefore, we have un = (υn, zn) → (υ, z) = u. This means that N is of class (S+)P.  □

Proposition 4.4

Suppose that N = (N1, N2) : H → 2H is bounded, where N1(υ, z) = N1,1(υ) + N1,2(z), such that

  1. N1,1 : H1 → 2H1 is weakly upper semicontinuous and pseudomonotone;

  2. N1,2 : H2H1 continuous;

  3. N2 : HH2 is weakly continuous.

    Then N is P-pseudomonotone.

Proof

Let (un) be any sequence in H and (an) any sequence in H with anNun such that unu,QunQu,andlimsupnan,P(unu)0. Let un = (υn, zn) and u = (υ, z). As in the proof of Proposition 4.3, we have znz in H2 and limsupnbnN1,2(zn),υnυ=limsupnan,P(unu)0, where an = (bn, cn) ∈ Nun, that is, bnN1,1(υn) + N1,2 (zn) and cn = N2(un). Since N1,1 is pseudomonotone, we have limnbnN1,2(zn),υnυ=0 and hence limnan,P(unu)=0. Suppose that aj = (bj, cj) ⇀ a = (b, c) for some subsequence (aj) of (an). By hypotheses (b) and (c), we get N1,2(zj)N1,2(z)andcj=N2(uj)N2(u)=c. The pseudomonotonicity of N1,1 implies that bN1,2(z) ∈ N1,1 (υ). Therefore, we have a=(b,c)(N1,1(υ)+N1,2(z),N2(u))=Nu. Consequently, N is P-pseudomonotone.  □

Proposition 4.5

Suppose that N = (N1, N2) : H → 2H is bounded, where N1 (υ, z) = N1,1 (υ) + N1,2(z), such that

  1. N1,1 : H1 → 2H1 is weakly upper semicontinuous and quasimonotone;

  2. N1,2 : H2H1 is continuous;

  3. N2 : HH2 is demicontinuous.

    Then N is P-quasimonotone.

Proof

Let (un) be a sequence in H and (an) a sequence in H with anNun such that unuandQunQu. If un = (υn, zn) and u = (υ, z), we have by the quasimonoto N1city of N1,1 liminfnan,P(unu)=liminfnbnN1,2(zn),υnυ0, where an = (bn, cn) ∈ Nun, that is, bnN1,1(υn) + N1,2(zn). Thus, N is P-quasimonotone.  □

We are now in position to prove the existence of a solution for semilinear systems, by using the nonresonance theorem in the previous section.

Theorem 4.6

Let L1, L2 be as above such that dim Ker L1 = ∞ and dim Ker L2 < ∞. Suppose that N = (N1, N2) : H → 2H is a weakly upper semicontinuous bounded operator such that N1 : H → 2H1, N2 : HH2 satisfy the conditions of Proposition 4.1. Further, suppose that there exist numbers ρ ∈ (0, sup A], ε [0, ρ/2), and α ∈ [0, 1) such that a1ρ2u1μu1+Ouα,N2(u1,u2)ρ2u2μu2+Ouα, for u = (u1, u2) ∈ H, ||u|| → ∞, and a1N1(u1, u2). Then for every (h1, h2) ∈ H1 × H2, the system {L1u1N1(u1,u2)h1L2u2N2(u1,u2)=h2 has a solution in D(L1D(L2).

Proof

Note that the norm induced by the inner product on the space H = H1 × H2 is equivalent to the norm given by (u1,u2)1:=u1+u2for(u1,u2)H1×H2. Apply Theorem 3.1, based on Proposition 4.1 and (S+)P ⊂ (PM)P.  □

We close this section with somewhat more concrete semilinear system in a viewpoint of applications.

Theorem 4.7

Let L1, L2 be as above such that dim Ker L1 = ∞ and dim Ker L2 < ∞. Suppose that N = (N1, N2) : H → 2H is a bounded operator such that N1 : H → 2H1, N2 : HH2 satisfy the conditions of Proposition 4.3 or Proposition 4.4. Moreover, suppose that there are ρ ∈ (0, sup A], ε ∈ [0, ρ/2), and α ∈ [0, 1) such that a1+N1,2(u2)ρ2u1μu1+Ouα,N2(u1,u2)ρ2u2μu2+Ouα, for u = (u1, u2) ∈ H, ||u|| → ∞, and a1N1,1(u1). Then for every (h1, h2) ∈ H1 × H2, the system L1u1N1,1(u1)N1,2(u2)h1L2u2N2(u1,u2)=h2 has a solution in D(L1) × D(L2).

Proof

Apply Theorem 3.1 with Proposition 4.3 or Proposition 4.4.  □

5 Application

In this section, we study the existence of periodic solutions for the system involving the wave operator and a discontinuous nonlinear term, based on a nonresonance theorem for semilinear systems in the previous section.

Motivated by the works [7, 11], we consider the following periodic problem υttυxxh1(x,t)g1(x,t,z)g_(x,t,υ),g¯(x,t,υ)in(0,π)×R,zttzxx4zg2(x,t,υ,z)=h2(x,t)in(0,π)×R,υ(0,)=υ(π,)=0=z(0,)=z(π,),υ,zare2πperiodic int,(27) where g : [0, ᅲ] × ℝ × ℝ → ℝ is a possibly discontinuous function in the third variable and g1, g2 satisfy the Carathéodory condition. Here, g_(x,t,s)=liminfηsg(x,t,η)andg¯(x,t,s)=limsupηsg(x,t,η). To seek a weak solution of the problem (27), we will consider the corresponding semilinear system; see Definition 5.1 and (30) below.

Let Ω = (0, ᅲ) × (0, 2ᅲ) and let H = L2(Ω) be the real Hilbert space with usual inner product 〈·, ·〉 and norm || ·||. Let Φnm(x, t)=ᅲ-1 sin(nx) exp(imt) for n ∈ ℕ and m ∈ ℤ. Each uH has a representation u=(n,m)N×Zunmφnm, where unm = 〈u, Φnm〉.

We define a linear operator L1 : D(L1) ⊂ HH by L1u:=(n,m)N×Z(n2m2)unmφnm, where D(L1)={uH|(n,m)N×Z|n2m2|2|unm|2<}. Then L1 is a self-adjoint densely defined operator and KerL1=span{φnn,φn,nnN}andImL1=(KerL1). Note that λ = 1 is the first positive eigenvalue of L1 which corresponds to (n, m) = (1,0) and L1u2L1u,ufor alluD(L1).(28) The partial inverse L-11 : Im L1 → Im L1D(L1) is given by L11u:=(n,m)Γ1(n2m2)1unmφnm, where Γ1={(n,m)N×Zn2m2}. Note that the spectrum of the operator L-11, denoted by σ(L-11), has no limit point except 0 and dim Ker (L-11λI) is finite for every nonzero λ ∈ σ(L-11). This implies that L-11 is compact. See e.g., [2, 4, 6].

Next, we define another linear operator L2 : D(L2) ⊂ HH by L2u:=(n,m)N×Z(n2m24)unmφnm, where D(L2)={uH|(n,m)N×Z|n2m24|2|unm|2<}. Then L2 is a closed densely defined operator and KerL2=span{φ20}andImL2=(KerL2). Note that λ = 1 is the first positive eigenvalue of L2 corresponding to (n, m) = (3, 2) and L2u2L2u,ufor alluD(L2).(29) The partial inverse L-12 : Im L2 → Im L2D(L2) given by L21u:=(n,m)Γ2(n2m24)1unmφnm, where Γ2={(n,m)N×Zn2m24}, is compact.

Firstly, we suppose that g : [0, ᅲ] × ℝ × ℝ → ℝ is 2ᅲ-periodic in the second variable such that (g1) and g are superpositionally measurable, that is, (·, ·, u(·, ·)) and ̱g(·, ·, u(·, ·)) are measurable on Ω for any measurable function u : Ω → ℝ;

(g2) g satisfies the growth condition: g(x,t,s)k0(x,t)+c0sfor almost all(x,t)Ωand allsR, where k0H is nonnegative and c0 is a positive constant;

(g3) there is a positive constant α such that (g(x,t,s)g(x,t,η))(sη)α|sη|2for almost all(x,t)Ωand alls,ηR.

We define a multi-valued operator N1,1 : H → 2H by setting N1,1(u):={wH|g_(x,t,u(x,t))w(x,t)g¯(x,t,u(x,t))for almost all(x,t)Ω}. Under conditions (g1) and (g2), the multi-valued operator N1,1 is bounded, upper semicontinuous, and Nu is nonempty, closed, and convex for every uH; see [16, Theorem 1.1]. Under additional condition (g3), the operator N1,1 is of class (S+).

Secondly, we suppose that g1 : [0, ᅲ] × ℝ × ℝ → ℝ is 2ᅲ-periodic in the second variable such that (g4)g1 satisfies the Carathéodory condition, that is, g1 (·, ·, s) is measurable on Ω for all s ∈ ℝ and g1(x, t, ·) is continuous on ℝ for almost all (x, t) ∈ Ω;

(g5)there are a nonnegative measurable function k1H and a positive constant c1 such that g1(x,t,s)k1(x,t)+c1|s|for almost all(x,t)Ωand allsR. We define the Nemytskii operator N1,2 : HH by N1,2(u)(x,t):=g1(x,t,u(x,t))foruHand(x,t)Ω. Under conditions (g4) and (g5), it is obvious that the operator N1,2 is bounded and continuous; see e.g., [17].

Thirdly, we suppose that g2 : [0, ᅲ] × ℝ × ℝ2 → ℝ is 2ᅲ-periodic in the second variable such that (g6) g2 satisfies the Caratheodory condition;

(g7)there exist a nonnegative function k2H and a positive constant c2 such that g2(x,t,s,p)k2(x,t)+c2(|s|+|p|)for almost all(x,t)Ωand all(s,p)R2. The Nemytskii operator N2 : H × H → H given by N2(u,υ)(x,t):=g2(x,t,u(x,t),υ(x,t))for(u,υ)H×Hand(x,t)Ω is clearly bounded and continuous under conditions (g6) and (g7).

Definition 5.1

A point (υ, z) ∈ H × H is said to be a weak solution of the problem (27) if there exists a point wN1,1(υ) such that υ,yttyxxw,yN1,2(z),y=h1,yandz,yttyxx4yN2(υ,z),y=h2,y for all yC2, where C2 denotes the space of twice continuously differentiable functions y : Ω̄ → ℝ such that y(0,)=y(π,)=0andy(,0)y(,2π)=yt(,0)yt(,2π)=0. In view of the above definitions, (27) has a weak solution (υ, z) in H × H if and only if (υ, z) ∈ D(L1) × D(L2) is a solution of the semilinear system L1υN1,1(υ)N1,2(z)h1L2zN2(υ,z)=h2.(30)

Theorem 5.2

Let g, g1 : [0, ᅲ] × ℝ × ℝ → ℝ and g2 : [0, ᅲ] × ℝ × ℝ2 → ℝ satisfy the conditions (g1)-(g7). Suppose that there are ∈ [0, 1/2) and β1, β2 [0,∞) such that w(x,t)+g1(x,t,p)12sμ|s|+β1,g2(x,t,s,p)12pμ|p|+β2, for almost all (x, t) ∈ Ω and all (s, p) ∈ ℝ2 with |(s, p)| → ∞, where ̱g (x, t, s) ≤ w(x, i) ≤ ḡ (x, t, s). Then for every (h1, h2) ∈ H × H, the given problem (27) has a weak solution.

Proof

Let L = (L1, L2) and N = (N1, N2) be defined as above. Notice that L1, L2 are closed densely defined linear operators with dim Ker L1 = ∞ and dim Ker L2 < ∞. Moreover, N1,1 is a bounded upper semicontinuous multi-valued operator of class (S+), and the Nemytskii operators N1,2 and N2 are bounded and continuous. It follows from (28) and (29) that Lu2Lu,ufor alluD(L), which means that 1 ∈ A, in the sense of (8). By hypotheses, we have w+N1,2(z)12υμυ+ξ1,N2(υ,z)12zμz+ξ2, for all υ, zH, ||(υ, z)|| → ∞, and all wN1,1 (υ), where ξ1, ξ2 are some constants. Applying Theorem 4.7 with ρ= 1 and α = 0, the system L1υN1,1(υ)N1,2(z)h1L2zN2(υ,z)=h2(31) has a solution for every (h1, h2) ∈ H × H. Therefore, (27) has a weak solution. This completes the proof.  □

Remark 5.3

Berkovits-Tienari [7] studied the periodic Dirichlet problem of the form uttuxx[g_(x,t,u),g¯(x,t,u)]in(0,π)×R, where g(x, t, ·) is nondecreasing on ℝ, based on the degree for the class (S+) introduced in [7]. In this note, the main point is that when treating semilinear system (27) with mixed nonlinear terms, certain monotonicity assumptions on g1 and g2 are not required. As was seen in Section 4, the class (S+)P is more general than the class (S+) and our degree theory for the class (S+)P plays a decisive role in the study of semilinear systems.

Acknowledgement

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03931517).

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About the article

Received: 2016-12-19

Accepted: 2017-03-21

Published Online: 2017-05-20


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 628–644, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0056.

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© 2017 Kim and Hong. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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