Semilinear systems with a multi-valued nonlinear term

In-Sook Kim; Suk-Joon Hong

doi:10.1515/math-2017-0056

Article Open Access

Semilinear systems with a multi-valued nonlinear term

In-Sook Kim and Suk-Joon Hong

Published/Copyright: May 20, 2017

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$Open Mathematics$

From the journal Open Mathematics Volume 15 Issue 1

Abstract

Introducing a topological degree theory, we first establish some existence results for the inclusion h ∈ Lu − Nu 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

Lu−Nu=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

||Nu−λ12u||≤μ||u||+υfor allu∈H,

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

||Lu−ρ2u||≥ρ2||u||for allu∈D(L).

In this case, equation (1) admits a solution if there exist µ ∈ [0, ρ/2) and υ ∈ [0, ∞)such that

||Nu−ρ2u||≤μ||u||+υfor allu∈H.

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

L1u1−N1(u1,u2)=h1,L2u2−N2(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

L1u1−N1(u1,u2)∋h1,L2u2−N2(u1,u2)=h2, (2)

where L₁(u₁, u₂)= N_1,1(u₁)+ N_1,2(u₂), N_1,1 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

h∈Lu−Nu,

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

h∈Lu−Nu, (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

||a−ρ2u||≤μ||u||+O(||u||α)for all∈H,||u||→∞anda∈Nu. (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 B_r (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 → 2^H is said to be:

upper semicontinuous if the set F − 1 ( A ) = { u ∈ Ω F u ∩ A ≠ ∅ } is closed in Ω for every closed set A in H;
weakly upper semicontinuous if F^-1 (A) is closed in Ω for every weakly closed set A in H;
bounded if it maps bounded sets into bounded sets;
compact if it is upper semicontinuous and the image of any bounded set is relatively compact;
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 : H → E and Q : H → E^┴ be the orthogonal projections, respectively.

Definition 2.2

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

of class (S₊)p if for any sequence (u_n) in Ω and for any sequence (w_n) in H with w_n ∈ Fu_n such that u_n ⇀ u, Qu_n → Qu, and
limsupn→∞wn,P(un−u)≤0,
we have u_n → u;
P-pseudomonotone, written F ∈ (PM)_p, if for any sequence (u_n) in Ω and for any sequence (w_n) in H with w_n ∈ Fu_n such that u_n ⇀ u, Qu_n → Qu, and
limsupn→∞wn,P(un−u)≤0,
we have lim n → ∞ w n , P ( u n − u ) = 0 , and if u ∈ Ω and w_j ⇀ w for some subsequence (w_j) of (w_n) then w ∈ Fu;
P-quasimonotone, written F ∈ (QM)_p, if for any sequence (u_n) in Ω and for any sequence (w_n) in H with w_n ∈ Fu_n such that u_n ⇀ u and Qu_n → Qu, we have
liminfn→∞wn,P(un−u)≥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^ￌ < ∞. 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) ⊂ H → H is a closed densely defined linear operator with

ImL=(KerL)⊥,

and K : Im L → Im L ∩ D(L), the inverse of the restriction of L to Im L ∩ D(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 → 2^H is a multi-valued operator, where G is an open set in H. Then

h∈Lu−Nu,u∈G¯∩D(L)

if and only if

h~∈Qu−(KQ−P)Nu,u∈G¯,

where h̄ = (KQ − P)h.

Proof

Suppose that h̄ ∈ Qu − (KQ − P)Nu with u ∈ Ḡ. Then KQh − Ph = Qu − KQa + Pa for some a ∈ Nu. Since Qh = LQu − Qa = Lu − Qa and −Ph = Pa, we have

h=Ph+Qh=Lu−Qa−Pa∈Lu−Nu.

Conversely, suppose that h ∈ Lu − Nu with u ∈ Ḡ ∩ D(L). Then we get

h∼∈(KQ−P)(L−N)u=Qu−(KQ−P)Nu.

This completes the proof. □

Given an open bounded set G in H, we consider a class of semilinear operators

L−N,

where N : Ḡ → 2^H 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 − (KQ − P)N, we associate a family of Leray-Schauder type operators defined by

Fλ:=I−(KQ−λΨ2P)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 → 2^H 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

hλ∉Fλ(A)forallλ>λ0,

where h_λ := (KQ − λψ²P)h.

Proof

Let A be any closed subset of Ḡ such that h ∉ (L − N)(A ∩ D(L)). Assume that the assertion is not true. Then we find sequences (λ_n) in (0, ∞) and (u_n) in A with λ_n → ∞ such that h_{λ n} ∈ F_λn (u_n) for all n ∈ ℕ, that is,

un−KQ(an+h)+λnΨ2P(an+h)=0,

where a_n ∈ Nu_n. This equation is equivalent to

Qun−KQ(an+h)=0andPun+λnΨ2P(an+h)=0. (5)

Passing to subsequences if necessary, we may suppose that u_n ⇀ u and a_n ⇀ a for some u, a ∈ H. Then we have by (5) and the strong continuity of the operator K

Qun→KQ(a+h)=QuandΨ2Pan→−Ψ2Ph=Ψ2Pa,

which implies Pa = − Ph, by the injectivity of ψ. Since P and ψ are self-adjoint with P² = P and Pa_n ⇀ −Ph, it follows from (5) that

lim sup n → ∞ a n + h , P ( u n − u ) = lim sup n → ∞ P ( a n + h ) , P ( u n − u ) = lim sup n → ∞ P ( a n + h ) , − λ n Ψ 2 P ( a n + h ) = lim sup n → ∞ − λ n Ψ P ( a n + h 2 ≤ 0.

In view of N + h ∈ (S₊)P, this implies u_n → u ∈ A. Note that Nu is closed and convex and so weakly closed. Since N is weakly upper semicontinuous, we have a ∈ Nu. This yields to

KQh−Ph=Qu−KQa+Pa∈Qu−(KQ−P)Nu,

which contradicts the hypothesis that h ∉ (L − N)(A ∩ D(L)), in view of Lemma 2.3 with A in place of Ḡ. 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 h_λ ∉ F_λ(∂G) for all λ > λ₀ and d_LS (F_λ, G, h_λ) is constant for all λ > λ₀, where d_LS denotes the Leray-Schauder degree.

Proof

According to Lemma 2.4 with A = ∂G, we can choose a positive number λ₀ such that h_λ ∉ F_λ(∂G) for all λ > λ₀. For the second assertion, let λ₁, λ₂ be arbitrary numbers in (λ₀, ∞) such that λ₁ < λ₂. Then F_λ, λ ∈ [λ₁, λ₂], defines a Leray-Schauder type homotopy such that h_λ ∉ F_λ(∂G) for all λ ∈ [λ₁, λ₂]. It follows from the homotopy invariance property of the degree d_LS that

dLS(Fλ1,G,hλ1)=dLS(Fλ2,G,hλ2).

Since λ₁, λ₂ were arbitrarily chosen in (λ₀, ∞), we conclude that d_LS(F_λ, G, h_λ) 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 : Ḡ → 2^H is a weakly upper semicontinuous bounded multi-valued operator of class (S₊)_P, where G is any open bounded set in H. If h ∉ (L − N)(∂G ∩ D(L)), we define a degree function d by

d(L−N,G,h):=limλ→∞dLS(Fλ,G,hλ),

where F_λ = I − (KQ − λψ²P)N and h_λ = (KQ − λψ²P)h.

Definition 2.7

A multi-valued operator N : [0, 1] × Ḡ → 2^H is said to be a homotopy of class (S₊)_P if for any sequence (t_n, u_n) in [0, 1] × Ḡ and for any sequence (a_n) in H with a_n ∈ N(t_n, u_n) such that t_n → t, u_n ⇀ u, Qu_n ⇀ Qu, and

limsupn→∞an,P(un−u)≤0,

we have u_n → u.

Lemma 2.8

(Affine homotopy) If N₁, N₂ : Ḡ → 2^H are weakly upper semicontinuous bounded operators of class (S₊)_P, then N : [0, 1] × Ḡ → 2^H given by

N(t,u):=(1−t)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

L−N,

where N : [0,1] × Ḡ → 2^H 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 → 2^H 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 − N_t)(∂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−(KQ−λΨ2P)Nt and hλ(t)=(KQ−λΨ2P)h(t).

Proof

Assume to the contrary that there exist sequences (λ_n) in (0,∞), (t_n) in [0,1], and (u_n) in ∂G with λ_n → ∞ such that h_λn (t_n) ∈ F_λn (t_n, u_n) for all n ∈ ℕ. This can be written as

un−KQ(an+h(tn))+λnΨ2P(an+h(tn))=0, (6)

where a_n ∈ N(t_n, u_n). Equation (6) is equivalent to

Qun−KQ(an+h(tn))=0andPun+λnΨ2P(an+h(tn))=0. (7)

Without loss of generality, we may suppose that t_n → t, u_n ⇀ u, and a_n ⇀ a for some t ∈ [0,1] and some u, a ∈ H. 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 Pa_n + Ph(t_n) ⇀ 0, we obtain from (7) that

limsupn→∞an+h(tn),P(un−u)=limsupn→∞P(an+h(tn)),P(un−u)=limsupn→∞P(an+h(tn)),−λnΨ2P(an+h(tn))=limsupn→∞−λnΨP(an+h(tn))2≤0,

which implies, in view of h(t_n) → h(t), that

limsupn→∞an,P(un−u)≤0.

Hence it follows from N ∈ (S₊)_P that u_n → u ∈ ∂G. Since N is weakly upper semicontinuous, we have a ∈ N(t, u). Therefore, we get

KQh(t)−Ph(t)=Qu−KQa+Pa∈Qu−(KQ−P)N(t,u),

which contradicts the hypothesis that h(t) ∉ (L − N_t)(∂G ∩ D(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:

(Existence) If d (L − N, G, h) ≠ 0, then the inclusion h ∈ Lu — Nu has a solution in G ∩ D(L).
(Additivity) If G₁ and G₂ are disjoint open subsets of G such that h∉(L−N)[(G¯∖(G1∪G2))∩D(L)], then we have
d(L−N,G,h)=d(L−N,G1,h)+d(L−N,G2,h).
(Homotopy invariance) Suppose that N : [0,1] × Ḡ → 2^H 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)∉Lu−N(t,u)forallt∈[0,1]andallu∈∂G∩D(L),
then d (L − N(t, ·), G, h(t)) is constant for all t ∈ [0,1].

Proof.

Argue by contraposition. If h ∉ Lu − Nu for all u ∈ Ḡ ∩ D(L), Lemma 2.4 implies that there exists a positive number λ₀ such that h_λ ∉ F_λ(Ḡ) for all λ > λ₀. It follows from the corresponding property of the Leray-Schauder degree that d_LS(F_λ, G, h_λ) = 0 for all λ > λ₀. By Definition 2.6, we have d (L − N, G, h) = 0.
Applying Lemma 2.4 with A = Ḡ\(G₁ ∪ G₂), we find a positive number λ₀ such that
hλ∉Fλ(G¯∖(G1∪G2))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.
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)=(KQ−λΨ2P)h(t) and (Ft)λ=I−(KQ−λΨ2P)Nt. For each fixed λ > λ₀, (F_t)_λ, t ∈ [0,1] defines a Leray-Schauder type homotopy such that h_λ(t) ∉ (F_t)_λ(u) for all (t, u) ∈ [0,1] × ∂G. Hence it follows from the homotopy invariance property of the degree d_LS that d_LS ((F_t)_λ, G, h_λ(t)) is constant for all t ∈ [0,1]. For any t₁, t₂ ∈ [0,1], we have by Definition 2.6

d(L−Nt1,G,h(t1))=limλ→∞dLS((Ft1)λ,G,hλ(t1))=limλ→∞dLS((Ft2)λ,G,hλ(t2))=d(L−Nt2,G,h(t2)).

We conclude that d (L − N(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(L−B,G,h)=±1.

Proof

Let G be any open bounded set in H and h ∈ (L − B)(G ∩ D(L)). By the injectivity of the operator L − B, we have (L − B)v = h for some v ∈ G ∩ D(L). In view of part (b) of Theorem 2.10, we can choose a positive number R with ||υ|| < R such that

d(L−B,G,h)=d(L−B,BR(0),h).

It is clear that (L − B)u ≠: th for all t ∈ [0,1] and u ∈ ∂B_R(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(L−B,BR(0),h)=d(L−B,BR(0),0).

By Definition 2.6, we have

d(L−B,BR(0),0)=limλ→∞dLS(Tλ,BR(0),0),

where T_λ = I − (KQ − λψ²P)B. Since T_λ is an injective Leray-Schauder type operator for large λ, it is known in [15] that d (L − B, 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⁡(L−αI,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) ⊂ H → H is a closed densely defined linear operator with Im L = (Ker L)^ￌ, and K : Im L → Im L ∩ D(L), the inverse of the restriction of L to Im L ∩ D(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:=ρ∈RLu2≥ρLu,ufor allu∈D(L). (8)

It is easily checked that

A=ρ∈RLu−ρ2u≥ρ2ufor allu∈D(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 L − N 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 → 2^H 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

a−ρ2u≤μu+O(uα)foru∈H,u→∞,anda∈Nu. (10)

If N is P-quasimonotone, then the range of the operator L − N is dense in H.
If N is P-pseudomonotone, then the inclusion
h∈Lu−Nu

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

Proof

Let h be an arbitrary element of H. We consider the homotopy equation

th∈Lu−(1−t)ρ2u−tNufort∈[0,1]andu∈D(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

degL−ρ2I,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 (u_n) in D(L) and (t_n) in [0,1] with ||u_n|| → ∞ such that

tnh=Lun−(1−tn)ρ2un−tnanfor alln∈N,

where a_n ∈ Nu_n. For all n ∈ ℕ, we have by (9) and (10)

ρ2un≤Lun−ρ2un=tnan−ρ2un+h≤μun+h+O(unα)

and hence

ρ2−μun≤h+O(unα),

which contradicts the unboundedness of the sequence (u_n) chosen. Thus, the solution set is bounded in H. Now we can choose a positive constant R such that

th∉Lu−(1−t)ρ2u−tNufor allt∈[0,1]and allu∈D(L)withu≥R. (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):=(1−t)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(L−N,BR(0),h)=degL−ρ2I,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−(1−t)ρ2I−tN,BR(0),th≠0.

By part (a) of Theorem 2.10, there exists a u_t ∈ B_R (0) ∩ D(L) such that

th=Lut−(1−t)ρ2ut−tat, (15)

where a_t ∈ Nu_t. According to the assertion (15), for a sequence (t_n) in (0, 1) with t_n → 1, there is a corresponding sequence (u_n) in B_R(0) ∩ D(L) such that

tnh=Lun−(1−tn)ρ2un−tnan,

where a_n ∈ Nu_n. Hence it follows that Lu_n − a_n → h and so h∈Im(L−N)¯. Since h ∈ H was arbitrary, we conclude that the range of L − N 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 (u_n) in B_R(0) ∩ D(L) such that Lu_n − a_n → h, where a_n ∈ Nu_n. Without loss of generality, we may suppose that u_n ⇀ u and a_n ⇀ a for some u, a ∈ H. Since Im L = (KerL)^ￌ and Pu_n ⇀ Pu, we have

limn→∞an,P(un−u)=limn→∞Lun,P(un−u)−limn→∞h,P(un−u)=0.

Since Qu_n = KQLu_n and K is compact, it is obvious that Qu_n → Qu. The P-pseudomonotonicity of the operator N implies that a ∈ Nu. Since the graph of L is weakly closed and Lu_n ⇀ a + h, we obtain that

u∈D(L)andh∈Lu−Nu.

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 → 2^H be a weakly upper semicontinuous bounded operator. Suppose that there are ρ ∈ (0, sup 𝓐) and α ∈ [0,1) such that

a−ρ2u≤ρ2u+O(uα)foru∈H,u→∞,anda∈Nu. (16)

Let h ∈ H be given and suppose that for any sequence (u_n) in D(L) such that ||u_n || → ∞ and ||Lu_n|| = o(||u_n||) for n → ∞, there exists an integer n₀ such that

an+h,Pun>0foralln≥n0andallan∈Nun. (17)

If N is P-quasimonotone, then h∈Im(L−N)¯.
If N is P-pseudomonotone, then the inclusion
h∈Lu−Nu
has a solution in D(L).

Proof

We consider the homotopy equation

th∈Lu−(1−t)ρ2u−tNufor(t,u)∈[0,1]×D(L). (18)

We have to show that the solution set

S=u∈D(L)|th∈Lu−(1−t)ρ2u−tNufor somet∈[0,1]

is bounded in H. Assume to the contrary that there exist sequences (u_n) in D(L) and (t_n) in (0,1] with ||u_n|| → ∞ such that

tnh=Lun−(1−tn)ρ2un−tnanfor alln∈N, (19)

where a_n ∈ Nu_n. Let ρ̄ ∈ A be any positive number with ρ̄ > p. For all u ∈ D(L), we have by (8)

Lu−ρ2u2=Lu2−ρLu,u+ρ2u2≥1−ρρ¯Lu2+ρ2u2.

Hence it follows from (19) and (16) that

1−ρρ¯Lun2+ρ2un2≤Lun−ρ2un2≤an−ρ2un+h2≤ρ2un+h+O(unα)2,

which implies

Lun=oun.

Set Z_n := u_n/||u_n|| and w_n := Lu_n. Then ||w_n|| = ||Lu_n||/||u_n|| → 0 implies Qz_n = Kw_n → 0. Since |〈Lu_n, Pu_n〉 = 0, we have by (19)

an+h,Pun=−(1−tn)tn−1ρ2un,Pun.

It follows from un,Pun=un2−un,Qun that

an+h,Pun=−(1−tn)tn−1ρ2un21−zn,Qzn.

Hence we obtain from Qz_n → 0 that

an+h,Pun≤0for 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 L_k : D(L_k) ⊂ H_k → H_k be a closed densely defined linear operator with Im L_k = (Ker L_k)^ￌ. Suppose that K_k : Im L_k → Im L_k, the inverse of the restriction of L_k to Im L_k ∩ D(L_k), is compact. For k = 1,2, let P_k : H_k → Ker L_k and Q_k : H_k → Im L_k be the orthogonal projections, respectively.

Define the diagonal operator L : D(L) ⊂ H → H 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 L ∩ D(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 → 2^H is a bounded multi-valued operator such that

N₁(υ,.) : H₂ → 2^H₁ is upper semicontinuous with compact values for each υ ∈ H₁
for each z ∈ H₂ there exist a positive number δ = δ(z) and a positive constant c = c(z) such that
b−d,υ′−υ≥cυ′−υ2
for all υ′,υ∈H1,z′∈Bδ(z),b∈N1(υ′,z′)andd∈N1(υ,z′).

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

Proof

Let (u_n) be any sequence in H and (a_n) any sequence in H with a_n ∈ Nu_n such that

un→u,Qun→Qu,andlimsupn→∞an,P(un−u)≤0. (20)

Let u_n = (υ_n, z_n) and u = (υ, z). Since dim Ker L₂ < ∞ and Qu_n → Qu, we have

zn→zinH2. (21)

Since Qu_n → Qu and N is bounded, we get by (21)

limsupn→∞an,P(un−u)=limsupn→∞an,un−u=limsupn→∞bn,υn−υ, (22)

where a_n = (b_n, c_n) ∈ Nu_n, that is, b_n ∈ N₁ (υ_n, z_n). By hypothesis (a), we see that a sequence (d_n) has a strongly convergent subsequence in H₁, denoted again by (d_n), where d_n ∈ N₁(υ, z_n), and hence

limn→∞dn,υn−υ=0,

which implies

limsupn→∞bn−dn,υn−υ=limsupn→∞bn,υn−υ. (23)

It follows from (20), (22), and (23) that

limsupn→∞bn−dn,υn−υ≤0.

Noting that b_n ∈ N₁ (υ_n, z_n) and d_n ∈ N₁ (υ, z_n), we get by hypothesis (b)

0≤limsupn→∞⁡cυn−υ2≤limsupn→∞bn−dn,υn−υ≤0.

Since υ_n → υ in H₁, we have by (21)

un=υn,zn→υ,z=u.

We conclude that N ∈ (S₊)_P. □

Proposition 4.2

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

N(υ,.) : H₂ → 2^H1 is upper semicontinuous with compact values for each υ ( H₁;
for each z H₂ there exists a positive number δ = δ(z) such that
b−d,υ′−υ≥0
for all υ′,υ∈H1,z′∈Bδ(z),b∈N1(υ′,z′),andd∈N1(υ,z′).

Then N is P-quasimonotone.

Proof

Let (u_n) be a sequence in H and (a_n) a sequence in H with a_n ∈ Nu_n such that

u n ⇀ u and Q u n → Q u .

As in the proof of Proposition 4.1, an analogous argument shows that

liminfn→∞an,P(un−u)=liminfn→∞bn−dn,υn−υ≥0,

where an=(bn,cn)∈Nun,un=(υn,zn),u=(υ,z),bn∈N1(υn,zn),anddn∈N1(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 → 2^H is bounded, where N₁(υ, z) = N_1,1(υ) + N_1,2(z), such that

N_1,1 : H₁ → 2^H₁ is weakly upper semicontinuous and of class (S₊);
N_1,2 : H₂ → H₁ is continuous;
N₂ : H → H₂ is demicontinuous.

Then N is of class (S₊)_P.

Proof

Let (u_n) be any sequence in H and (a_n) any sequence in H with a_n ∈ Nu_n such that

un→u,Qun→Qu,andlimsupn→∞an,P(un−u)≤0. (24)

Let u_n = (υ_n, z_n) and u = (υ, z). As in the proof of Proposition 4.1, we have z_n → z in H₂ and

limsupn→∞an,P(un−u)=limsupn→∞bn,υn−υ, (25)

where a_n = (b_n, c_n) ∈ Nu_n, that is, b_n ∈ N_1,1(υ_n) + N_1,2 (z_n). From hypothesis (b) we obtain that

limsupn→∞bn−N1,2(zn),υn−υ=limsupn→∞bn,υn−υ. (26)

It follows from (24), (25), and (26) that

limsupn→∞bn−N1,2(zn),υn−υ=limsupn→∞an,P(un−u)≤0.

Since N_1,1 is of class (S₊) and b_n − N_1,2(z_n) ∈ N_1,1(υ_n), this implies υ_n → υ in H₁. Therefore, we have u_n = (υ_n, z_n) → (υ, z) = u. This means that N is of class (S₊)_P. □

Proposition 4.4

Suppose that N = (N₁, N₂) : H → 2^H is bounded, where N₁(υ, z) = N_1,1(υ) + N_1,2(z), such that

N_1,1 : H₁ → 2^H₁ is weakly upper semicontinuous and pseudomonotone;
N_1,2 : H₂ → H₁ continuous;
N₂ : H → H₂ is weakly continuous.

Then N is P-pseudomonotone.

Proof

Let (u_n) be any sequence in H and (a_n) any sequence in H with a_n ∈ Nu_n such that

un→u,Qun→Qu,andlimsupn→∞an,P(un−u)≤0.

Let u_n = (υ_n, z_n) and u = (υ, z). As in the proof of Proposition 4.3, we have z_n → z in H₂ and

limsupn→∞bn−N1,2(zn),υn−υ=limsupn→∞an,P(un−u)≤0,

where a_n = (b_n, c_n) ∈ Nu_n, that is, b_n ∈ N_1,1(υ_n) + N_1,2 (z_n) and c_n = N₂(u_n). Since N_1,1 is pseudomonotone, we have

limn→∞bn−N1,2(zn),υn−υ=0

and hence

limn→∞an,P(un−u)=0.

Suppose that a_j = (b_j, c_j) ⇀ a = (b, c) for some subsequence (a_j) of (a_n). By hypotheses (b) and (c), we get

N1,2(zj)→N1,2(z)andcj=N2(uj)→N2(u)=c.

The pseudomonotonicity of N_1,1 implies that b − N_1,2(z) ∈ N_1,1 (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 → 2^H is bounded, where N₁ (υ, z) = N_1,1 (υ) + N_1,2(z), such that

N_1,1 : H₁ → 2^H₁ is weakly upper semicontinuous and quasimonotone;
N_1,2 : H₂ → H₁ is continuous;
N₂ : H → H₂ is demicontinuous.

Then N is P-quasimonotone.

Proof

Let (u_n) be a sequence in H and (a_n) a sequence in H with a_n ∈ Nu_n such that

un→uandQun→Qu.

If u_n = (υ_n, z_n) and u = (υ, z), we have by the quasimonoto N₁city of N_1,1

liminfn→∞an,P(un−u)=liminfn→∞bn−N1,2(zn),υn−υ≥0,

where a_n = (b_n, c_n) ∈ Nu_n, that is, b_n ∈ N_1,1(υ_n) + N_1,2(z_n). 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 → 2^H is a weakly upper semicontinuous bounded operator such that N₁ : H → 2^H1, N₂ : H → H₂ 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

a1−ρ2u1≤μu1+Ouα,N2(u1,u2)−ρ2u2≤μu2+Ouα,

for u = (u₁, u₂) ∈ H, ||u|| → ∞, and a₁ ∈ N₁(u₁, u₂). Then for every (h₁, h₂) ∈ H₁ × H₂, the system

{ L1u1−N1(u1,u2)∋h1L2u2−N2(u1,u2)=h2

has a solution in D(L₁)×D(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 → 2^H is a bounded operator such that N₁ : H → 2^H1, N₂ : H → H₂ 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)−ρ2u1≤μu1+Ouα,N2(u1,u2)−ρ2u2≤μu2+Ouα,

for u = (u₁, u₂) ∈ H, ||u|| → ∞, and a₁ ∈ N_1,1(u₁). Then for every (h₁, h₂) ∈ H₁ × H₂, the system

L1u1−N1,1(u1)−N1,2(u2)∋h1L2u2−N2(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

υ t t − υ x x − h 1 ( x , t ) − g 1 ( x , t , z ) ∈ g _ ( x , t , υ ) , g ¯ ( x , t , υ ) in ( 0 , π ) × R , z t t − z x x − 4 z − g 2 ( x , t , υ , z ) = h 2 ( x , t ) in ( 0 , π ) × R , υ ( 0 , ⋅ ) = υ ( π , ⋅ ) = 0 = z ( 0 , ⋅ ) = z ( π , ⋅ ) , υ , z are 2 π − periodic in t , (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)=liminfη→s⁡g(x,t,η)andg¯(x,t,s)=limsupη→s⁡g(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 Φ_nm(x, t)=ￗ^-1 sin(nx) exp(imt) for n ∈ ℕ and m ∈ ℤ. Each u ∈ H has a representation

u=∑(n,m)∈N×Zunmφnm,

where u_nm = 〈u, Φ_nm〉.

We define a linear operator L₁ : D(L₁) ⊂ H → H by

L1u:=∑(n,m)∈N×Z(n2−m2)unmφnm,

where

D(L1)={u∈H|∑(n,m)∈N×Z|n2−m2|2|unm|2<∞}.

Then L₁ is a self-adjoint densely defined operator and

KerL1=span{φnn,φn,−nn∈N}andImL1=(KerL1)⊥.

Note that λ = 1 is the first positive eigenvalue of L₁ which corresponds to (n, m) = (1,0) and

L1u2≥L1u,ufor allu∈D(L1). (28)

The partial inverse L^-1₁ : Im L₁ → Im L₁ ∩ D(L₁) is given by

L1−1u:=∑(n,m)∈Γ1(n2−m2)−1unmφnm,

where Γ1={(n,m)∈ N×Zn2≠m2}. Note that the spectrum of the operator L^-1₁, denoted by σ(L^-1₁), has no limit point except 0 and dim Ker (L^-1₁ − λI) 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) ⊂ H → H by

L2u:=∑(n,m)∈N×Z(n2−m2−4)unmφnm,

where

D(L2)={u∈H|∑(n,m)∈N×Z|n2−m2−4|2|unm|2<∞}.

Then L₂ is a closed densely defined operator and

Ker L 2 = span { φ 20 } and Im L 2 = ( Ker L 2 ) ⊥ .

Note that λ = 1 is the first positive eigenvalue of L₂ corresponding to (n, m) = (3, 2) and

L2u2≥L2u,ufor allu∈D(L2). (29)

The partial inverse L^-1₂ : Im L₂ → Im L₂ ∩ D(L₂) given by

L2−1u:=∑(n,m)∈Γ2(n2−m2−4)−1unmφnm,

where Γ2={(n,m)∈N×Zn2−m2≠4}, is compact.

Firstly, we suppose that g : [0, ￗ] × ℝ × ℝ → ℝ is 2ￗ-periodic in the second variable such that (g1)ḡ and g are superpositionally measurable, that is, ḡ(·, ·, 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 alls∈R,

where k₀ ∈ H 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))(s−η)≥α|s−η|2for almost all(x,t)∈Ωand alls,η∈R.

We define a multi-valued operator N_1,1 : H → 2^H by setting

N1,1(u):={w∈H|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 N_1,1 is bounded, upper semicontinuous, and Nu is nonempty, closed, and convex for every u ∈ H; see [16, Theorem 1.1]. Under additional condition (g3), the operator N_1,1 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 k₁ ∈ H and a positive constant c₁ such that

g1(x,t,s)≤k1(x,t)+c1|s|for almost all(x,t)∈Ωand alls∈R.

We define the Nemytskii operator N_1,2 : H → H by

N1,2(u)(x,t):=g1(x,t,u(x,t))foru∈Hand(x,t)∈Ω.

Under conditions (g4) and (g5), it is obvious that the operator N_1,2 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 k₂ ∈ H 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 w ∈ N_1,1(υ) such that

υ,ytt−yxx−w,y−N1,2(z),y=h1,yandz,ytt−yxx−4y−N2(υ,z),y=h2,y

for all y ∈ C², where C² denotes the space of twice continuously differentiable functions y : Ω̄ → ℝ such that

y ( 0 , ⋅ ) = y ( π , ⋅ ) = 0 and y ( ⋅ , 0 ) − y ( ⋅ , 2 π ) = y t ( ⋅ , 0 ) − y t ( ⋅ , 2 π ) = 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

L1υ−N1,1(υ)−N1,2(z)∋h1L2z−N2(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)−12s≤μ|s|+β1,g2(x,t,s,p)−12p≤μ|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, N_1,1 is a bounded upper semicontinuous multi-valued operator of class (S₊), and the Nemytskii operators N_1,2 and N₂ are bounded and continuous. It follows from (28) and (29) that

Lu2≥Lu,ufor allu∈D(L),

which means that 1 ∈ A, in the sense of (8). By hypotheses, we have

w+N1,2(z)−12υ≤μυ+ξ1,N2(υ,z)−12z≤μz+ξ2,

for all υ, z ∈ H, ||(υ, z)|| → ∞, and all w ∈ N_1,1 (υ), where ξ₁, ξ₂ are some constants. Applying Theorem 4.7 with ρ= 1 and α = 0, the system

L1υ−N1,1(υ)−N1,2(z)∋h1L2z−N2(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

utt−uxx∈[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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Received: 2016-12-19

Accepted: 2017-3-21

Published Online: 2017-5-20

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