Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 63, pp. 1–25.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

SOLVABILITY OF INCLUSIONS INVOLVING PERTURBATIONS

OF POSITIVELY HOMOGENEOUS MAXIMAL

MONOTONE OPERATORS

DHRUBA R. ADHIKARI, ASHOK ARYAL, GHANSHYAM BHATT,

ISHWARI J. KUNWAR, RAJAN PURI, MIN RANABHAT

Abstract. Let X be a real reflexive Banach space and X∗ be its dual space.
Let G1 and G2 be open subsets of X such that G2 ⊂ G1, 0 ∈ G2, and G1

is bounded. Let L : X ⊃ D(L) → X∗ be a densely defined linear maximal

monotone operator, A : X ⊃ D(A) → 2X
∗

be a maximal monotone and

positively homogeneous operator of degree γ > 0, C : X ⊃ D(C) → X∗

be a bounded demicontinuous operator of type (S+) with respect to D(L),

and T : G1 → 2X
∗

be a compact and upper-semicontinuous operator whose

values are closed and convex sets in X∗. We first take L = 0 and establish

the existence of nonzero solutions of Ax + Cx + Tx 3 0 in the set G1 \ G2.
Secondly, we assume that A is bounded and establish the existence of nonzero

solutions of Lx+Ax+Cx 3 0 in G1 \G2. We remove the restrictions γ ∈ (0, 1]

for Ax+Cx+Tx 3 0 and γ = 1 for Lx+Ax+Cx 3 0 from such existing results
in the literature. We also present applications to elliptic and parabolic partial

differential equations in general divergence form satisfying Dirichlet boundary

conditions.

1. Introduction and preliminaries

Let X be a real reflexive Banach space and X∗ be its topological dual space.
The symbol 2X

∗
denotes the collection of all subsets of X∗. The norm on X is

denoted by ‖ · ‖X . When there is no risk of misunderstanding, the norms on X
and X∗ are both denoted by ‖ · ‖. The pairing 〈x∗, x〉 denotes the value of the
functional x∗ ∈ X∗ at x ∈ X. The symbols ∂Z, IntZ,Z and coZ denote the
boundary, interior, closure, and convex hull of the set Z ⊂ X, respectively. The
symbol BX(0, R) denotes the open ball of radius R > 0 with center at 0 in X. The
symbols R and R+ denote (−∞,∞) and [0,∞), respectively. For a sequence {xn}
in X and x0 ∈ X, we denote by xn → x0 and xn ⇀ x0 the strong convergence
and weak convergence, respectively. Given another real Banach Y , an operator
T : X ⊃ D(T )→ Y is said to be bounded if it maps bounded subsets of the domain
D(T ) onto bounded subsets of Y . The operator T is said to be compact if it maps
bounded subsets of D(T ) onto relatively compact subsets in Y . The operator T is
said to be demicontinuous if it is strong-to-weak continuous on D(T ).

2020 Mathematics Subject Classification. 47H14, 47H05, 47H11.
Key words and phrases. Topological degree theory; operators of type (S+);

monotone operator; duality mapping; Yosida approximant.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted April 1, 2022. Published August 30, 2022.

1



2 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

A multivalued operator A from X to X∗ is written as A : X ⊃ D(A) → 2X
∗
,

where D(A) = {x ∈ X : Ax 6= ∅} is the effective domain of A. Here, Ax means
A(x), and these notations are used interchangeably in the sequel. We denote the
graph of A by Gr(A), i.e., Gr(A) = {(x, y) : x ∈ D(A), y ∈ Ax}.

Definition 1.1. An operator A : X ⊃ D(A) → 2X
∗

is said to be positively ho-
mogeneous of degree γ > 0 if (x, y) ∈ Gr(A) implies sx ∈ D(A) for all s ≥ 0 and
(sx, sγy) ∈ Gr(A).

Remark 1.2. An equivalent condition for an operator A : X ⊃ D(A)→ 2X
∗
to be

positively homogeneous of degree γ > 0 is that x ∈ D(A) implies sx ∈ D(A) for all
s ≥ 0 and sγAx ⊂ A(sx). It follows that a positively homogeneous operator A of
degree γ > 0 satisfies 0 ∈ A(0). When A is positively homogeneous of degree γ > 0,
it can be verified that x ∈ D(A) implies sx ∈ D(A) for all s > 0 and sγAx = A(sx).
However, in general, the property sγAx = A(sx) may not be true for s = 0. For
example, let A : R ⊃ [0,∞)→ 2R be given by

Ax =

{
(−∞, 0] for x = 0

xγ for x > 0.

Clearly, A(0) = (−∞, 0] 6= {0}.

A gauge function is a strictly increasing continuous function ϕ : R+ → R+ with
ϕ(0) = 0 and ϕ(r)→∞ as r →∞. The duality mapping of X corresponding to a
gauge function ϕ is the mapping Jϕ : X ⊃ D(Jϕ)→ 2X

∗
defined by

Jϕx = {x∗ ∈ X∗ : 〈x∗, x〉 = ϕ(‖x‖)‖x‖, ‖x∗‖ = ϕ(‖x‖)}, x ∈ X.

The Hahn-Banach theorem ensures that D(Jϕ) = X, and therefore Jϕ : X → 2X
∗

is, in general, a multivalued mapping. The duality mapping corresponding to the
gauge function ϕ(r) = r is called the normalized duality mapping and denoted by
J . It is well-known that the duality mapping Jϕ satisfies

Jϕx =
ϕ(‖x‖)
‖x‖

Jx, x ∈ X \ {0}.

Since J is homogeneous of degree 1, we have

Jϕ(sx) =
ϕ(s‖x‖)
‖x‖

Jx, (s, x) ∈ R+ × (X \ {0}).

In particular, when ϕ(r) = rp−1, 1 < p < ∞, we obtain Jϕx = ‖x‖p−2Jx, x ∈
X \ {0}, which implies

Jϕ(sx) = sp−1Jϕx, (s, x) ∈ R+ ×X,
i.e., Jϕ is positively homogeneous of degree p− 1.

When X is reflexive and both X and X∗ are strictly convex, the inverse J−1
ϕ of

Jϕ is the duality mapping of X∗ with the gauge function ϕ−1(r) = rq−1, where q
is given by 1/p+ 1/q = 1. It is easy to verify that

J−1
ϕ (sx∗) = sq−1J−1

ϕ x∗, (s, x∗) ∈ R+ ×X∗. (1.1)

It is clear that Jϕ is positively homogeneous of degree γ > 0 if and only if ϕ is pos-
itively homogeneous of degree γ > 0. Additional properties of duality mappings in
connection with Banach space geometry can be found in Alber and Ryazantseva [7]
and Cioranescu [19].



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 3

Definition 1.3. An operator A : X ⊃ D(A) → 2X
∗

is said to be monotone if
for all (x, u), (y, v) ∈ Gr(A) we have 〈u − v, x − y〉 ≥ 0. A monotone operator
A : X ⊃ D(A) → 2X

∗
is said to be maximal monotone if Gr(A) is maximal in

X ×X∗, when X ×X∗ is partially ordered by set inclusion.

In what follows, we assume that X is reflexive and both X and X∗ are strictly
convex. It is well-known that the duality mapping Jϕ is maximal monotone. A
monotone operator A is maximal if and only if R(A+λJϕ) = X∗ for all λ ∈ (0,∞)
and all gauge functions ϕ. For a proof of this result for ϕ(r) = rp−1, 1 < p < ∞,
the reader is referred, for example, to Barbu [10, Theorem 2.3].

Definition 1.4. Let L : X ⊃ D(L) → X∗ be a densely defined linear maximal
monotone operator. An operator C : X ⊃ D(C) → X∗ is said to be of type (S+)
with respect to D(L) if for every sequence {xn} ⊂ D(L) ∩D(C) with xn ⇀ x0 in
X, Lxn ⇀ Lx0 in X∗ and

lim sup
n→∞

〈Cxn, xn − x0〉 ≤ 0,

we have xn → x0 in X. In this case, if L = 0, then C is said to be of type (S+).

Definition 1.5. A family of operators C(s) : X ⊃ G→ X∗, s ∈ [0, 1], is said to be
a homotopy of type (S+) with respect to D(L) if for every sequence {xn} ⊂ D(L)∩G
with xn ⇀ x0 in X and Lxn ⇀ Lx0 in X∗, {sn} ⊂ [0, 1] with sn → s0 and

lim sup
n→∞

〈C(sn)xn, xn − x0〉 ≤ 0,

we have xn → x0 in X, x0 ∈ G and C(sn)xn ⇀ C(s0)x0 in X∗. In this case,
if L = 0, then C(s) is said to be a homotopy of type (S+). A homotopy C(s) of
type (S+) with respect to D(L) is bounded if the set {C(s)x : s ∈ [0, 1], x ∈ G} is
bounded.

Definition 1.6. An operator T : X ⊃ D(T )→ 2X
∗

is said to be of class (P ) if

(i) it maps bounded sets to relatively compact sets;
(ii) for every x ∈ D(T ), Tx is a closed and convex subset of X∗; and

(iii) T (·) is upper-semicontinuous, i.e., for every closed set F ⊂ X∗, the set
T−(F ) = {x ∈ D(T ) : Tx ∩ F 6= ∅} is closed in X.

Hu and Papageorgiou introduced the operators of class (P ) in [21]. We recall a
compact-set valued upper-semicontinuous operator T is closed. Furthermore, given
an operator T of class P and a sequence {(xn, yn)} ⊂ Gr(T ) such that xn → x ∈
D(T ), the sequence {yn} has a cluster point in Tx.

This paper is organized as follows. In Section 2, we study variants of the stan-
dard Yosida approximants introduced in Brézis, Crandall, and Pazy [14] and their
fundamental properties. Since the topological degree theory for (S+)-operators is
employed to establish the main existence results in the later sections, we provide
several results involving variants of Yosida approximants related to the Browder
degree theory [16].

In Section 3, we first prove the existence of nonzero solutions of Ax + Cx +
Tx 3 0 by utilizing the topological degree theories developed by Browder [18] and
Skrypnik [32]. In this case, A is maximal monotone with A(0) = {0} and positively
homogeneous of degree γ > 0, C is bounded demicontinuous of type (S+), and T is
of class (P ). This result extends an analogous result for γ ∈ (0, 1] established in [2]
to an arbitrary degree of homogeneity γ > 0. Another main result established in



4 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

this section is the existence of nonzero solutions of Lx+Ax+Cx 3 0, where L, C
are as above, and A is a bounded maximal monotone and positively homogeneous
of degree γ > 0. This result extends an analogous result for γ = 1 established in
[2] to an arbitrary degree of homogeneity γ > 0.

In Section 4, we present some applications of the theories developed in Section 3
to elliptic and parabolic partial differential equations, in general, divergence form
that include p-Laplacian with 1 < p <∞ and satisfy Dirichlet boundary conditions.

For additional facts and various topological degree theories related to the subject
of this paper, the reader is referred to Adhikari and Kartsatos [4, 5], Kartsatos and
Lin [22], and Kartsatos and Skrypnik [24, 26]. For further information on functional
analytic tools used herein, the reader is referred to Barbu [10], Browder [17], Pascali
and Sburlan [28], Simons [30], Skrypnik [31, 32], and Zeidler [34].

2. Variants of Yosida approximants and related properties

Let X be a strictly convex and reflexive Banach space with strictly convex X∗.
By using the duality mapping Jϕ corresponding to an arbitrary gauge function ϕ,
we study variants of the Yosida approximants in Brézis et al. [14] and resolvents
of a maximal monotone operator A : X ⊃ D(A) → 2X

∗
. For each λ > 0 and each

x ∈ X, the inclusion

0 ∈ Jϕ(xλ − x) + λAxλ (2.1)

has a unique solution xλ ∈ D(A) (see Proposition 2.1 (i)). We define Jϕλ : X →
D(A) ⊂ X and Aϕλ : X → X∗ by

Jϕλ x := xλ and Aϕλx :=
1

λ
Jϕ(x− Jϕλ x), x ∈ X. (2.2)

The operators Aϕλ and Jϕλ are variants of the standard Yosida approximant Aλ and
resolvent Jλ of A. For each x ∈ X, we have

Aϕλx ∈ A(Jϕλ x) and x = Jϕλ x+ J−1
ϕ (λAϕλx).

When ϕ(r) = rp−1, a splitting of x in terms of Aϕλ and Jϕλ is

x = Jϕλ x+ λq−1J−1
ϕ (Aϕλx), (2.3)

and therefore

Aϕλx =
(
A−1 + λq−1J−1

ϕ

)−1
x, x ∈ X. (2.4)

It is easy to verify that A = Aϕλ if and only if A = 0. In fact, if A = 0, then Jϕλ = I,
the identity operator on X. Moreover, if 0 ∈ D(A) and 0 ∈ A(0), then Aϕλ0 = 0.

The choice of an appropriate gauge function is essential for the main existence
results in this paper. The following proposition summarizes some important proper-
ties of Aϕλ and Jϕλ along the lines of analogous properties of Aλ and Jλ. A complete
proof is provided here for the reader’s convenience.

Proposition 2.1. Let X be a strictly convex and reflexive Banach space with
strictly convex dual X∗ and A : X ⊃ D(A) → 2X

∗
be a maximal monotone op-

erator. Then the following statements hold.

(i) The operator Aϕλ is single-valued, monotone, bounded on bounded subsets
of X, and demicontinuous from X to X∗.

(ii) For every x ∈ D(A) and λ > 0, we have

‖Aϕλx‖ ≤ |Ax| := inf{‖x∗‖ : x∗ ∈ Ax}.



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 5

(iii) The operator Jϕλ is bounded on bounded subsets of X, demicontinuous from
X to D(A), and

lim
λ→0

Jϕλ x = x for all x ∈ coD(A).

(iv) If λn → 0, xn ⇀ x in X, Aϕλnxn ⇀ y and

lim sup
n,m→∞

〈Aϕλnxn −A
ϕ
λm
xm, xn − xm〉 ≤ 0,

then (x, y) ∈ Gr(A) and

lim
n,m→∞

〈Aϕλnxn −A
ϕ
λm
xm, xn − xm〉 = 0.

(v) For every sequence {λn} with λn → 0, Aϕλnx ⇀ A{0}x for all x ∈ D(A). In

addition, if X∗ is uniformly convex, then Aϕλnx→ A{0}x for all x ∈ D(A).

(vi) If λn → 0 and x 6∈ D(A), then

lim
n→∞

‖Aϕλnx‖ =∞.

Proof. (i) We first show that Jϕλ is single-valued. Given x ∈ X and λ > 0, let xλ
and x̃λ be solutions of (2.1). Take y ∈ Axλ and ỹ ∈ Ax̃λ such that

Jϕ(xλ − x) + λy = 0 and Jϕ(x̃λ − x) + λỹ = 0.

This along with the monotonicity of A and Jϕ implies

〈Jϕ(xλ − x)− Jϕ(x̃λ − x), (xλ − x)− (x̃λ − x)〉 = 0. (2.5)

SinceX is strictly convex, it follows that Jϕ is strictly monotone, i.e., for u1, u2 ∈ X,
we have

〈Jϕu1 − Jϕu2, u1 − u2〉 > 0 if and only if u1 6= u2.

It follows from (2.5) that xλ = x̃λ. Thus, Jϕλ is single-valued, and therefore Aϕλ is
also single-valued. It is easy to verify the monotonicity of Aϕλ .

To show Aϕλ is bounded, let B ⊂ X be bounded. For each x ∈ B, let xλ = Jϕλ x.
Let (u, v) ∈ Gr(A). Using (2.1), it follows that

〈Jϕ(xλ − x) + λyλ, xλ − u〉 = 0,

where yλ ∈ Axλ. This implies

〈Jϕ(xλ − x), xλ − u〉 = −λ〈yλ, xλ − u〉 ≤ λ〈v, u− xλ〉.
The last inequality follows from the monotonicity of A. It then follows that

〈Jϕ(xλ − x), xλ − x〉 = 〈Jϕ(xλ − x), xλ − u〉+ 〈Jϕ(xλ − x), u− x〉
≤ λ〈v, u− xλ〉+ 〈Jϕ(xλ − x), u− x〉
= λ〈v, u− x〉+ λ〈v, x− xλ〉+ 〈Jϕ(xλ − x), u− x〉.

(2.6)

This implies

ϕ(‖xλ − x‖)‖xλ − x‖ ≤ λ‖v‖ (‖u− x‖+ ‖xλ − x‖) + ϕ(‖xλ − x‖)‖u− x‖. (2.7)

If {xλ : x ∈ B} is unbounded, the inequality (2.7) yields a contradiction. Thus, Jϕλ
is bounded on B. Since Jϕ is bounded on B, it follows from (2.2) that Aϕλ is also
bounded on B.

Let {xn} ⊂ X be such that xn → x0 ∈ X as n → ∞. Denote un = Jϕλ xn and
vn = Aϕλxn, so that

Jϕ(un − xn) + λvn = 0. (2.8)



6 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

Since Jϕλ and Aϕλ are bounded on bounded sets, both {un} and {vn} are bounded.
Since Jϕ and A are monotone, it follows from

〈Jϕ(un − xn)− Jϕ(um − xm), (un − xn)− (um − xm)〉
= −λ〈vn − vm, (un − xn)− (um − xm)〉

that

lim
n,m→∞

〈vn − vm, un − um〉 = 0,

lim
n,m→∞

〈Jϕ(un − xn)− Jϕ(um − xm), (un − xn)− (um − xm)〉 = 0.

Passing to subsequences, we may assume that un ⇀ u0 in X, vn ⇀ v0 in X∗,
and Jϕ(un − xn) ⇀ w0 in X∗ for some u0 ∈ X and some v0, w0 ∈ X∗. By [10,
Lemma 2.3], it follows that (u0, v0) ∈ Gr(A) and (u0 − x0, w0) ∈ Gr(Jϕ). Using
all these in (2.8), we obtain Jϕ(u0 − u0) + λv0 = 0, which implies u0 = Jϕλ x0 and
v0 = Aϕλx0, i.e., Jϕλ xn ⇀ Jϕλ x0 and Aϕλxn ⇀ Aϕλx0 as n → ∞. This proves the
demicontinuity of Jλ and Aλ.

(ii) Let x ∈ D(A) and λ > 0. Let y ∈ Ax and xλ = Jϕλ x. Then

0 ≤ 〈y −Aλx, x− xλ〉

= 〈y, x− xλ〉 −
1

λ
ϕ(‖x− xλ‖)‖x− xλ‖

≤ ‖y‖‖x− xλ‖ −
1

λ
ϕ(‖x− xλ‖)‖x− xλ‖,

which implies ϕ(‖x− xλ‖) ≤ λ‖y‖, and therefore

‖Aϕλx‖ =
1

λ
‖Jϕ(x− xλ)‖ ≤ ‖y‖.

Consequently, ‖Aϕλx‖ ≤ |Ax| := inf{‖y‖ : y ∈ Ax}.
(iii) The boundedness of Jϕλ on bounded subsets of X and its demicontinuity

are already proved in (i). Let x ∈ coD(A) and (u, v) ∈ Gr(A). Following the
arguments that lead to (2.7), we find that {xλ − x : λ > 0} is bounded, and
therefore {Jϕ(xλ−x) : λ > 0} is bounded. Let {λn} ⊂ (0,∞) be such that λn → 0.
Let y ∈ X∗ be such that Jϕ(xλn − x) ⇀ y in X∗. Then (2.6) yields

lim sup
n→∞

ϕ(‖xλn − x‖)‖xλn − x‖ ≤ 〈y, u− x〉.

It is clear that this argument applies to all u ∈ coD(A). Taking u = x, we obtain

lim
n→∞

ϕ(‖xλn − x‖)‖xλn − x‖ = 0.

By the homeomorphic property of the gauge function ϕ, it follows that we must
have xλn → x as n→∞. This completes the proof of (iii).

(iv) Let un = Jϕλnxn for all n. Since {Aϕλnxn} is bounded, it follows that

ϕ(‖xn − un‖) = ϕ(‖xn − Jϕλnxn‖) = ‖Jϕ(xn − Jϕλnxn)‖ = λn‖Aϕλnxn‖ → 0

as n→∞. This implies ‖xn − un‖ → 0 as n→∞. Since

〈Aϕλnxn −A
ϕ
λm
xm, xn − xm〉

= 〈Aϕλnxn −A
ϕ
λm
xm, un − um〉+ 〈Aϕλnxn −A

ϕ
λm
xm, (xn − un)− (xm − um)〉



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 7

and A is monotone, it follows as in Brézis et al. [14] that

lim
n,m→∞

〈Aϕλnxn−A
ϕ
λm
xm, xn−xm〉 = 0 and lim

n,m→∞
〈Aϕλnxn−A

ϕ
λm
xm, un−um〉 = 0.

The conclusion of (iv) now follows from [10, Lemma 2.3].

(v) Let x ∈ D(A). Since X∗ is reflexive and strictly convex and Ax is a closed and
convex subset of X∗, it follows that there exists a unique element of Ax, denoted
by A{0}x, such that ‖A{0}x‖ = inf{‖x∗‖ : x∗ ∈ Ax}. Let {λn} ⊂ (0,∞) be such
that λn → 0 and Aϕλnx ⇀ y in X∗ as n→∞. As in the proof of (iv), with xn = x,

we have y ∈ Ax. In view of part (ii), it follows that

‖y‖ ≤ lim inf
n→∞

‖Aϕλnx‖ ≤ lim sup
n→∞

‖Aϕλnx‖ ≤ ‖A
{0}x‖,

and therefore we must have y = A{0}x and Aϕλnx ⇀ A{0}x in X∗. Moreover, if X∗

is uniformly convex, then, by [10, Lemma 1.1], we obtain Aϕλnx→ A{0}x in X∗.

(vi) Suppose, on the contrary, that there is a sequence {λn} with λn → 0 and

an element x 6∈ D(A) such that {‖Aϕλnx‖} is bounded. Let R > 0 be such that

‖Aϕλnx‖ ≤ R for all n. Then, by (2.2), we have

ϕ(‖x− Jϕλnx‖) = ‖Jϕ(x− Jϕλnx)‖ ≤ Rλn.

Since ϕ−1 is also a gauge function, we obtain Jϕλnx → x as n → ∞. This implies

x ∈ D(A), a contradiction. �

A proof of the following lemma for ϕ(r) = r can be found in Boubakari and
Kartsatos [13]. Since we are dealing here with an arbitrary gauge function ϕ, we
provide a complete proof.

Lemma 2.2. Let A : X ⊃ D(A) → 2X
∗

be maximal monotone and G ⊂ X be
bounded. Let 0 < λ1 < λ2. Then there exists a constant K, independent of λ, such
that

‖Aϕλx‖ ≤ K
for all x ∈ G and λ ∈ [λ1, λ2].

Proof. For every x ∈ X, we have

Aϕλx =
1

λ
Jϕ(x− xλ),

where xλ = Jϕλ x. Let (u, v) ∈ Gr(A). In view of (2.7) in the proof of (i) in
Proposition 2.1, we have

ϕ(‖xλ − x‖)‖xλ − x‖ ≤ λ‖v‖ (‖u− x‖+ ‖xλ − x‖) + ϕ(‖xλ − x‖)‖u− x‖
≤ λ2‖v‖ (‖u− x‖+ ‖xλ − x‖) + ϕ(‖xλ − x‖)‖u− x‖.

By the properties of the gauge function ϕ, it follows that ϕ(‖xλ − x‖) must be
bounded, i.e., there exists a constant K0 > 0 such that

ϕ(‖xλ − x‖) ≤ K0

for all x ∈ G and all λ ∈ [λ1, λ2]. Consequently, we have

‖Aϕλx‖ =
1

λ
ϕ(‖xλ − x‖) ≤

1

λ1
K0 =: K

for all x ∈ G and all λ ∈ [λ1, λ2]. �



8 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

By a well-known renorming theorem due to Troyanski [33], a reflexive Banach
space X can be renormed with an equivalent norm with respect to which both X
and X∗ become locally uniformly convex (therefore strictly convex). With such a
renorming, the duality mapping Jϕ is a homeomorphism from X onto X∗. Hence-
forth, we assume that both X and X∗ are reflexive and locally uniformly convex.

The following lemma involving Aϕλ and Jϕλ plays an important role in the sequel.
Its proof is omitted here because of its similarity to [6, Lemma 1], except that, for
the general ϕ here, we must make use of

x = Jϕλ x+ J−1
ϕ (λAϕλx) and 〈Aϕλx, J

−1
ϕ (λAϕλx)〉 = ϕ−1(λ‖Aϕλx‖)‖A

ϕ
λx‖, x ∈ X.

The lemma for Aλ and Jλ is essentially due to Brézis et al. [14].

Lemma 2.3. Let A : X ⊃ D(A) → 2X
∗

and S : X ⊃ D(S) → 2X
∗

be maximal
monotone operators such that 0 ∈ D(A) ∩D(S) and 0 ∈ S(0) ∩A(0). Assume that
A + S is maximal monotone and that there is a sequence {λn} ⊂ (0,∞) such that
λn → 0, and a sequence {xn} ⊂ D(S) such that xn ⇀ x0 ∈ X and Aϕλnxn + w∗n ⇀
y∗0 ∈ X∗, where w∗n ∈ Sxn. Then the following statements are true.

(i) The inequality

lim
n→∞

〈Aϕλnxn + w∗n, xn − x0〉 < 0 (2.9)

is impossible.
(ii) If

lim
n→∞

〈Aϕλnxn + w∗n, xn − x0〉 = 0, (2.10)

then x0 ∈ D(A+ S) and y∗0 ∈ (A+ S)x0.

Definition 2.4. An operator A : X ⊃ D(A) → 2X
∗

is said to be strongly quasi-
bounded if for every S > 0 there exists K(S) > 0 such that ‖x‖ ≤ S and 〈x∗, x〉 ≤ S
for some x∗ ∈ Ax imply ‖x∗‖ ≤ K(S).

It is obvious that a bounded operator is strongly quasibounded. With regard to
possibly unbounded operators, Browder and Hess [18] and Pascali and Sburlan [28]
have shown that a monotone operatorA with 0 ∈ IntD(A) is strongly quasibounded.
The following lemma with the particular case ϕ(r) = r addressed in Kartsatos and
Quarcoo[23, Lemma D] is needed in the sequel.

Lemma 2.5. Let A : X ⊃ D(A) → 2X
∗

be a strongly quasibounded maximal
monotone operator such that 0 ∈ A(0). Let {λn} ⊂ (0,∞) and {xn} ⊂ X be such
that

‖xn‖ ≤ S and 〈Aϕλnxn, xn〉 ≤ S1 for all n,

where S, S1 are positive constants. Then there exists a number K > 0 such that
‖Aϕλnxn‖ ≤ K for all n.

Proof. Denote wn = Aϕλnxn and un = Jϕλnxn for all n. Then we have

wn ∈ Aun and xn = un + J−1
ϕ (λnwn).

In view of 0 ∈ A(0), we obtain

0 ≤ 〈wn, un〉 = 〈wn, xn − J−1
ϕ (λnwn)〉

= 〈wn, xn〉 − 〈wn, J−1
ϕ (λnwn)〉

= 〈wn, xn〉 − ϕ−1(λn‖wn‖)‖wn‖



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 9

≤ S1 − ϕ−1(λn‖wn‖)‖wn‖.

This yields 〈wn, un〉 ≤ S1 and ϕ−1(λn‖wn‖)‖wn‖ ≤ S1 for all n. Suppose {wn} is
unbounded. Then there exists a subsequence, denoted again by {wn}, such that
‖wn‖ → ∞ and 1 ≤ ‖wn‖ for all n. Consequently, ϕ−1(λn‖wn‖) ≤ S1 for all n,
and since xn = un + J−1

ϕ (λnwn), it follows that

λn‖wn‖ = ‖Jϕ(xn − un)‖ = ϕ(‖xn − un‖).
This implies ‖xn − un‖ = ϕ−1(λn‖wn‖) ≤ S1 for all n. Since {xn} is bounded,
we obtain the boundedness of {un} and {〈wn, un〉}, which contradicts the strong
quasiboundedness of A. Consequently, {wn} is bounded. �

For the rest of this paper, we take the gauge function ϕ(r) = rp−1, p > 1. For
the special case ϕ(r) = r, the reader can find proofs of Lemma 2.6 in Kartsatos and
Skrypnik [25] when 0 ∈ A(0) and in Asfaw and Kartsatos [8], without the condition
0 ∈ A(0). We note that Zhang and Chen in [35, Lemma 2.7] proved the continuity
of x 7→ Aλx on D(A) for each λ > 0, also without the condition 0 ∈ A(0). In
[8, Lemma 6], however, the continuity of x 7→ Aλx on X is used with no mention
of its validity. We next provide a detailed proof of the continuity of the mapping
(λ, x) 7→ Aϕλx on (0,∞)×X.

Lemma 2.6. Let A : X ⊃ D(A) → 2X
∗

be a maximal monotone operator. Then
the mapping (λ, x) 7→ Aϕλx is continuous on (0,∞)×X.

Proof. We first prove the continuity of x 7→ Aϕλ0
x on X for each fixed λ0 > 0. To

this end, let {xn} ⊂ X be such that xn → x0 ∈ X. By Lemma 2.2, we have the
boundedness of {Aϕλ0

xn}, and therefore

lim
n→∞

〈Aϕλ0
xn −Aϕλ0

x0, xn − x0〉 = 0. (2.11)

We know that

xn = Jϕλ0
xn + λq−1

0 J−1
ϕ (Aϕλ0

xn) and x0 = Jϕλ0
x0 + λq−1

0 J−1
ϕ (Aϕλ0

x0). (2.12)

Since Aϕλ0
xn ∈ A(Jϕλ0

xn) and Aϕλ0
x0 ∈ A(Jϕλ0

x0), the monotonicity of A together

with (2.11) and (2.12) yields

lim
n→∞

〈Aϕλ0
xn −Aϕλ0

x0, J
−1
ϕ (Aϕλ0

xn)− J−1
ϕ (Aϕλ0

x0)〉 = 0. (2.13)

Since J−1
ϕ is a duality mapping from X∗ to X, it follows, in view of [19, Proposi-

tion 2.17], that
Aϕλ0

xn → Aϕλ0
x0 as n→∞.

This proves the continuity of Aϕλ0
on X.

We now proceed to prove the continuity of (λ, x) 7→ Aϕλx on (0,∞) × X. Let
{λn} ⊂ (0,∞) and {xn} ⊂ X be such that λn → λ0 ∈ (0,∞) and xn → x0 ∈ X as
n→∞. Let G ⊂ X be a bounded set that contains xn for all n. Rename λ1, λ2 > 0
such that λn ∈ [λ1, λ2] for all n. Since

Jϕλnxn ∈ A
−1(Aϕλnxn) and xn = Jϕλnxn + λq−1

n J−1
ϕ (Aϕλnxn),

it follows that

Jϕλnxn + λq−1
0 J−1

ϕ (Aϕλnxn) ∈ A−1(Aϕλnxn) + λq−1
0 J−1

ϕ (Aϕλnxn)

=
(
A−1 + λq−1

0 J−1
ϕ

)
(Aϕλnxn).



10 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

This implies

Aϕλnxn =
(
A−1 + λq−1

0 J−1
ϕ

)−1 (
Jϕλnxn + λq−1

0 J−1
ϕ (Aϕλnxn)

)
= Aϕλ0

(
Jϕλnxn + λq−1

0 J−1
ϕ (Aϕλnxn)

)
= Aϕλ0

(
xn + (λq−1

0 − λq−1
n )J−1

ϕ (Aϕλnxn)
)
.

By Lemma 2.2, {Aϕλnxn} is bounded, and so is {J−1
ϕ (Aϕλnxn)}. Since λn → λ0,

we have (λq−1
0 − λq−1

n )J−1
ϕ (Aϕλnxn) → 0 as n → ∞. The continuity of Aϕλ0

implies

Aϕλnxn → Aϕλ0
x0 as n→∞. This completes the proof. �

Remark 2.7. We anticipate that Lemma 2.6 holds for any gauge function ϕ. Since
the formula (2.4) may not hold for Aϕλ with a general ϕ, the above proof does not
go through and this subject may be of independent research interest.

Let G be an open and bounded subset of X. Let L : X ⊃ D(L)→ X∗ be densely
defined linear maximal monotone, A : X ⊃ D(A) → 2X

∗
maximal monotone, and

C(s) : X ⊃ G → X∗, s ∈ [0, 1], a bounded homotopy of type (S+) with respect
to D(L). Since Gr(L) is closed in X × X∗, the space Y = D(L) associated with
the graph norm ‖x‖Y = ‖x‖X + ‖Lx‖X∗ , x ∈ Y , becomes a real reflexive Banach
space. We may assume that Y and its dual Y ∗ are locally uniformly convex.

Let j : Y → X be the natural embedding and j∗ : X∗ → Y ∗ its adjoint.
Since j : Y → X is continuous, we have D(j∗) = X∗. This implies that j∗

is also continuous. Since j−1 is not necessarily bounded, we have, in general,
j∗(X∗) 6= Y ∗. Moreover, j−1(G) = G ∩D(L) is closed and j−1(G) = G ∩D(L) is

open, j−1(G) ⊂ j−1(G), and ∂(j−1(G)) ⊂ j−1(∂G).
We define M : Y → Y ∗ by (Mx, y) := 〈Ly, J−1(Lx)〉, x, y ∈ Y , where the duality

pairing (·, ·) is in Y ∗ × Y , and J−1 is the inverse of the duality map J : X → X∗

and is identified with the duality map from X∗ to X∗∗ = X. Also, for every x ∈ Y
such that Mx ∈ j∗(X∗), we have J−1(Lx) ∈ D(L∗), Mx = j∗ ◦ L∗ ◦ J−1(Lx), and

(Mx−My, x− y) = 〈Lx− Ly, J−1(Lx)− J−1(Ly)〉 ≥ 0

for all y ∈ Y such that My ∈ j∗(X∗). Moreover, it is easy to see that M is
continuous on Y , and therefore M is maximal monotone.

We now define L̂ : Y → Y ∗ and Ĉ(s) : j−1(G) → Y ∗ by L̂ = j∗ ◦ L ◦ j and

Ĉ(s) = j∗ ◦ C(s) ◦ j, respectively, and for each t > 0, we also define Âϕt : Y → Y ∗

by Âϕt = j∗ ◦ Aϕt ◦ j, where Aϕt is the Yosida approximant of A corresponding to
the gauge function ϕ.

The next lemma employs Lemma 2.5 and follows as in [5, Lemma 5], and there-
fore its proof is omitted.

Lemma 2.8. Let G ⊂ X be open and bounded. Assume the following:

(i) L : X ⊃ D(L)→ X∗ is linear, maximal monotone with D(L) = X;
(ii) A : X ⊃ D(A) → 2X

∗
is strongly quasibounded, maximal monotone with

0 ∈ A(0); and
(iii) C(t) : X ⊃ G → X∗ is a bounded homotopy of type (S+) with respect to

D(L).

Then, for a continuous curve f(s), 0 ≤ s ≤ 1, in X∗, the set

F =
{
x ∈ j−1(G) : L̂+ Âϕt + Ĉ(s) + tMx = j∗f(s) for some t > 0, s ∈ [0, 1]

}



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 11

is bounded in Y .

The next two propositions are essential for the existence results in Section 2 and
Section 3.

Proposition 2.9. Let A : X ⊃ D(A) → 2X
∗

be maximal monotone and C : X ⊃
D(C)→ X∗ be bounded, demicontinuous and of type (S+). Suppose that G ⊂ X is
open and bounded such that 0 ∈ A(0), p ∈ X∗, and

p 6∈ (A+ C)x

for all x ∈ ∂G ∩D(A) ∩D(C). Then the following statements hold.

(i) There exists t0 > 0 such that

Aϕt x+ Cx 6= p

for all x ∈ ∂G ∩D(C) and t < t0.
(ii) For fixed t1, t2 > 0, define q(t) := tt1 + (1 − t)t2, t ∈ [0, 1]. Then the

operator

H(t, x) = Aϕq(t)x+ Cx, (t, x) ∈ [0, 1]×G

is a homotopy of type (S+).
(iii) For every sequence {tn} ⊂ (0,∞) such that tn → 0, limn→∞ dS+

(Aϕtn +
C,G, p) exists and does not depend on the choice of {tn}.

Proof. (i) Without loss of generality, we assume that p = 0. In fact, if p 6= 0, then
we replace C with C−p. Suppose that (iii) is false. Then there exist {tn} ⊂ (0,∞)
and {xn} ⊂ ∂G such that tn → 0 and

Aϕtnxn + Cxn = 0 (2.14)

for all n. Since C is bounded, {Cxn} is bounded. This implies that {Aϕtnxn} is also
bounded. We may assume that there exist x0 ∈ X and w0 ∈ X∗ such that xn ⇀ x0

in X and Aϕtnxn ⇀ w0 in X∗. If

lim sup
n→∞

〈Cxn, xn − x0〉 > 0,

we find a subsequence of {xn}, denoted again by itself, such that

lim
n→∞

〈Cxn, xn − x0〉 > 0.

In view of (2.14), we obtain

lim
n→∞

〈Aϕtnxn, xn − x0〉 < 0;

however, this is impossible by (i) of Lemma 2.3. We then must have

lim sup
n→∞

〈Cxn, xn − x0〉 ≤ 0.

By the (S+)−property of C, we have xn → x0, and consequently

lim
n→∞

〈Aϕtnxn, xn − x0〉 = 0.

By (ii) of Lemma 2.3, we obtain x0 ∈ D(A) and w0 ∈ Ax0. Since C is demicon-
tinuous, Cxn ⇀ Cx0 in X∗. This implies w0 = −Cx0, i.e., 0 ∈ (A + C)(∂G),
contradicting 0 /∈ (A+ C)(∂G).



12 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

(ii) Let t1, t2 ∈ (0, t0] be such that t1 < t2. Consider the following one-parameter
family of operators:

H(t, x) := Aϕq(t)x+ Cx, (t, x) ∈ [0, 1]×G.

We prove that H(t, ·) is a bounded homotopy of type (S+). The boundedness of
H(·, ·) follows from Lemma 2.2 and the boundedness of C. Let {tn} ⊂ [0, 1] and
{xn} ⊂ G satisfy tn → t0 and xn ⇀ x0 in X, and

lim sup
n→∞

〈Aϕq(tn)xn + Cxn, xn − x0〉 ≤ 0. (2.15)

Using the monotonicity of Aϕq(t) in (2.15), we obtain

lim sup
n→∞

〈Aϕq(tn)x0 + Cxn, xn − x0〉 ≤ 0. (2.16)

By Lemma 2.6, we have Aϕq(tn)x0 → Aϕq(t0)x0, and therefore (2.16) yields

lim sup
n→∞

〈Cxn, xn − x0〉 ≤ 0.

Since C is demicontinuous and of type S+, it follows that xn → x0 in X and
Cxn ⇀ Cx0 in X∗. Consequently, we have

Aϕq(tn)xn + Cxn ⇀ Aϕq(t0)x0 + Cx0

as n→∞. This proves that H(t, ·), t ∈ [0, 1], is a homotopy of type (S+).

(iii) By the invariance of the degree, dS+
, for (S+)-mappings under the homo-

topies of type (S+), we have

dS+(Aϕt1 , G, 0) = dS+(H(0, ·), G, 0) = dS+(H(1, ·), G, 0) = dS+(Aϕt2 , G, 0).

It follows that dS+(Aϕt , G, 0) exists and is independent of t ∈ (0, t0]. �

Remark 2.10. Let A, C, G, and p be the same as in Proposition 2.9. When we
define a degree mapping of A+ C, denoted by D(A+ C,G, p), by

D(A+ C,G, p) = lim
t→0+

dS+(Aϕt , G, p),

we can verify that the degree mapping D has the same four basic properties as the
Browder degree in [16]. By the uniqueness of the Browder degree established by
Berkovits and Miettunen [12], the degree D coincides with the Browder degree for
A+ C.

By replacing T̂t everywhere in [5, Lemma 5, Lemma 6, and Lemma 8] with Âϕt
with the gauge function ϕ(r) = rp−1 and by following the methodology used in
[5] in conjunction with Lemmas2.3, 2.5,2.6, and 2.8, we obtain Proposition 2.11
below. Its proof is omitted here because the method of proof is similar to that in
[5] and Proposition 2.9, except for having to deal with Âϕt . For further properties
of L + A + C in relation to the following proposition for ϕ(r) = r, the reader is
referred to Addou and Mermri [1] and Adhikari and Kartsatos[5].

Proposition 2.11. Let G ⊂ X be open and bounded. Assume that L : X ⊃
D(L)→ X∗ is linear, maximal monotone with D(L) = X; A : X ⊃ D(A)→ 2X

∗
is

strongly quasibounded, maximal monotone with 0 ∈ A(0); and C(t) : X ⊃ G→ X∗,
t ∈ [0, 1], is a bounded homotopy of type (S+) with respect to D(L). Suppose that

0 6∈ (L+A+ C(t))x

for all x ∈ ∂G ∩D(L) ∩D(A). Then the following statements hold.



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 13

(i) There exists t0 > 0 such that

L̂x+ Âϕt x+ Ĉ(t)x+ tMx 6= 0

for all (t, x) ∈ [0, 1]× (∂G ∩D(L)) and t < t0.
(ii) For fixed numbers t1, t2 > 0, define q(t) := tt1 + (1− t)t2, t ∈ [0, 1]. Then

the operator

Ĥ(t, x) = L̂x+ Âϕq(t)x+ Ĉ(t)x+ s(t)Mx,

with (t, x) ∈ [0, 1]× (G ∩D(L)), is a homotopy of type (S+).
(iii) For every sequence {tn} ⊂ (0,∞) such that tn → 0,

lim
n→∞

dS+(L̂+ Âϕtn + Ĉ(t) + tnM,G, 0)

exists and does not depend on the choice of {tn}.

3. Existence of nontrivial solutions

Hu and Papageorgiou generalized in [21] the Browder degree theory [16] to the
mappings of the form A+C+T , where A : X ⊃ D(A)→ 2X

∗
is maximal monotone

with 0 ∈ A(0), C : X ⊃ D(C)→ X∗ is bounded demicontinuous of type (S+), and
T is of class (P ). With an application of the (S+)-degree developed by Browder
[16] and Skrypnik [32], we prove in Theorem 3.3 the existence of nonzero solutions
of Ax + Cx + Tx 3 0 when A + C + T satisfies certain boundary conditions, and
the operator A, in addition, is positively homogeneous of degree γ > 0. This result
extends the existence result for γ ∈ (0, 1] in [2] to γ > 0 (see also [6, Theorem 6]
for γ = 1).

The following lemma, which is crucial to the existence results in this section,
shows that positively homogeneous maximal monotone operators transmit the ho-
mogeneity into their Yosida approximants corresponding to Jϕ with ϕ(r) = rp−1,
p > 1, and a suitable value of p.

Lemma 3.1. Let A : X ⊃ D(A) → 2X
∗

be maximal monotone and positively
homogeneous of degree γ > 0. Then, for each t > 0, the Yosida approximant Aϕt
corresponding to the gauge function ϕ(r) = rp−1, p > 1, satisfies

Aϕt (sx) =

{
sγAϕtsγ+1−px for (s, x) ∈ (R+ \ {0})×X
0 for (s, x) ∈ {0} ×X.

(3.1)

Consequently, if p = γ + 1, then Aϕt is positively homogeneous of degree γ, i.e.,

Aϕt (sx) = sγAϕt x for all (s, x) ∈ R+ ×X.

Proof. Let t > 0 be fixed. The case s = 0 is trivial. Assume s > 0, and let

y = Aϕt (sx) = (A−1 + tq−1J−1
ϕ )−1(sx), x ∈ X,

where q satisfies 1/p+ 1/q = 1. Then

y ∈ A(−tq−1J−1
ϕ y + sx) = A

(
s
(
−tq−1s−1J−1

ϕ y + x
))
.

This means (
s
(
−tq−1s−1J−1

ϕ y + x
)
, y
)
∈ Gr(A).

Since A is positively homogeneous of degree γ > 0, we obtain(
−tq−1s−1J−1

ϕ y + x, s−γy
)
∈ Gr(A),



14 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

i.e.,
s−γy ∈ A

(
−tq−1s−1J−1

ϕ y + x
)
.

In view of (1.1), we have

s−γ(1−q)J−1
ϕ (s−γy) = J−1

ϕ y,

and therefore

s−γy ∈ A
(
−tq−1sγ(q−1)−1J−1

ϕ (s−γy) + x
)
.

This implies

x ∈
(
A−1 + tq−1sγ(q−1)−1J−1

ϕ

)
(s−γy).

Using

tq−1sγ(q−1)−1 = (tsγ)
q−1 (

s1−p)q−1
=
(
tsγ+1−p)q−1

,

we obtain

y = sγ
(
A−1 +

(
tsγ+1−p)q−1

J−1
ϕ

)−1

x = sγAϕtsγ+1−px.

Thus, we have
Aϕt (sx) = sγAϕtsγ+1−px.

Clearly, Aϕt is positively homogeneous of degree γ if p = γ + 1. �

Remark 3.2. In the settings of Lemma 3.1 with p = γ + 1, it follows from (2.3)
that the resolvent Jϕt is positively homogeneous of degree 1 in the following sense:
for each t > 0, we have Jϕt (sx) = sJϕt x for all x ∈ X and all s ≥ 0.

Theorem 3.3. Assume that G1, G2 ⊂ X are open, bounded with 0 ∈ G2 and G2 ⊂
G1. Let A : X ⊃ D(A) → 2X

∗
be maximal monotone and positively homogeneous

of degree γ > 0 with A(0) = {0}; C : G1 → X∗ bounded, demicontinuous and of
type (S+); and T : G1 → 2X

∗
of class (P ). Assume, further, that

(H1) there exists v∗0 ∈ X∗ \ {0} such that Ax + Cx + Tx 63 λv∗0 for all (λ, x) ∈
R+ × (D(A) ∩ ∂G1), and

(H2) Ax+ Cx+ Tx+ λJx 63 0 for all (λ, x) ∈ R+ × (D(A) ∩ ∂G2).

Then the inclusion Ax+Cx+Tx 3 0 has a nonzero solution x ∈ D(A)∩ (G1 \G2).

Proof. To study the solvability of the inclusion

Ax+ Cx+ Tx 3 0, x ∈ G1,

we consider the associated approximate equation

Aϕt x+ Cx+ qεx = 0, t > 0, x ∈ G1, ε > 0. (3.2)

Here, the gauge function is taken to be ϕ(r) = rp−1, 1 < p <∞ so that γ = p− 1,
and qε : G1 → X∗ is an approximate continuous Cellina-selection as in [9, Lemma 6]

and [21] satisfying qεx ∈ T (Bε(x)∩G1)+Bε(0) for all x ∈ G1 and qε(G1) ⊂ coT (G1).
We show that the equation (3.2) has a solution xt,ε in G1 \G2 for all sufficiently

small t and ε. To this end, we first show that there exist τ0 > 0, t0 > 0 and ε0 > 0
such that the equation

Aϕt x+ Cx+ qεx = τv∗0 (3.3)

has no solution in G1 for every τ ≥ τ0, t ∈ (0, t0] and ε ∈ (0, ε0].
Assuming the contrary, let {τn} ⊂ (0,∞), {tn} ⊂ (0,∞), {εn} ⊂ (0,∞) and

{xn} ⊂ G1 be such that τn →∞, tn → 0, εn → 0 and

Aϕtnxn + Cxn + qεnxn = τnv
∗
0 . (3.4)



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 15

We can assume that qεnxn → g∗ ∈ X∗ in view of the properties of T . Then
‖Aϕtnxn‖ → ∞ as ‖τnv∗0‖ → ∞ and {Cxn} is bounded. Thus, from (3.4), we obtain

Aϕtnxn

‖Aϕtnxn‖
+

Cxn
‖Aϕtnxn‖

+
qεnxn
‖Aϕtnxn‖

=
τn

‖Aϕtnxn‖
v∗0 . (3.5)

This implies

τn‖v∗0‖
‖Aϕtnxn‖

→ 1 so that
τn

‖Aϕtnxn‖
→ 1

‖v∗0‖
as n→∞. (3.6)

Since p − 1 = γ, by Lemma 3.1, Aϕt is also homogeneous of degree γ = p − 1, and
therefore we obtain

Aϕtnxn

‖Aϕtnxn‖
= Aϕtn

( xn
‖Aϕtnxn‖1/γ

)
. (3.7)

Let un = xn/‖Aϕtnxn‖
1/γ . It is clear that un → 0. In view of (3.5), (3.6), and (3.7),

we obtain Aϕtnun → h with h = v∗0/‖v∗0‖. This implies

lim
n→∞

〈Aϕtnun, un〉 = 〈h, 0〉 = 0.

Since tn → 0, by (ii) of Lemma 2.3 with S = 0 we obtain 0 ∈ D(A) and h ∈ A(0) =
{0}, a contradiction to ‖h‖ = 1.

We now consider the homotopy mapping

H1(s, x, t, ε) = Aϕt x+ Cx+ qεx− sτ0v∗0 , s ∈ [0, 1], x ∈ G1, (3.8)

where t ∈ (0, t0] and ε ∈ (0, ε0] are fixed. By following the arguments as in [2,
Theorem 3.1], we can show that, for every s ∈ [0, 1] the operator x 7→ Cx −
sτ0v

∗
0 is bounded, demicontinuous and of type (S+) on G1, and that the equation

H1(s, x, t, ε) = 0 has no solution in ∂G1 for all sufficiently small t ∈ (0, t0], ε ∈ (0, ε0]
and all s ∈ [0, 1]. In doing this, we need to use Lemma 2.3. The details are omitted.

It follows from Proposition 2.9 that the mapping H1(s, x, t, ε) is an admissible
homotopy for the degree, dS+

, of (S+)-mappings, and dS+
(H1(s, ·, t, ε), G1, 0) is

well-defined and is a constant for all s ∈ [0, 1] and for all t ∈ (0, t0], ε ∈ (0, ε0].
Assume that

dS+
(H1(1, ·, t1, ε1), G1, 0) 6= 0,

for some sufficiently small t1 ∈ (0, t0] and ε1 ∈ (0, ε0]. Then the equation

Aϕt1x+ Cx+ gε1x = τ0v
∗
0

has a solution in G1. However, this contradicts our choice of the number τ0 in (3.3).
Consequently,

dS+(Aϕt + C + qε, G1, 0) = dS+(H1(0, ·, t, ε), G1, 0) = 0, t ∈ (0, t0], ε ∈ (0, ε0].

We next consider the homotopy mapping

H2(s, x, t, ε) = s(Aϕt x+ Cx+ qεx) + (1− s)Jx, (s, x) ∈ [0, 1]×G2. (3.9)

We claim that there exist t1 ∈ (0, t0] and ε1 ∈ (0, ε0] such that H2(s, x, t, ε) = 0
has no solution on ∂G2 for any s ∈ [0, 1], any t ∈ (0, t1] and any ε ∈ (0, ε1]. To
prove the claim, we assume the contrary and then follow the argument used in [2,
Theorem 3.1] along with the properties of Aϕt established in Lemma 2.3 to arrive
at a contradiction to (H2). For the sake of convenience, we assume that t0 and ε0
are sufficiently small so that we may take t1 = t0 and ε1 = ε0.



16 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

It follows from Proposition 2.9 that H2(s, x, t, ε) is an admissible homotopy for
the degree of (S+)-mappings and dS+(H2(s, ·, t, ε), G2, 0) is well-defined and con-
stant for all s ∈ [0, 1], all t ∈ (0, t0] and all ε ∈ (0, ε0]. By the invariance of the
(S+)-degree, for all t ∈ (0, t0] and ε ∈ (0, ε0], we have

dS+
(H2(1, ·, t, ε), G2, 0) = dS+

(Aϕt + C + qε, G2, 0)

= dS+
(H2(0, ·, t, ε), G2, 0)

= dS+(J,G2, 0) = 1.

Thus, for all t ∈ (0, t0], ε ∈ (0, ε0], we have

dS+
(Aϕt + C + qε, G1, 0) 6= dS+

(Aϕt + C + qε, G2, 0).

Using the excision property of the (S+)-degree, which is an easy consequence of
its finite-dimensional approximations, for every t ∈ (0, t0] and every ε ∈ (0, ε0],
there exists a solution xt,ε ∈ G1 \ G2 of Aϕt x + Cx + qεx = 0. Let tn ∈ (0, t0] and
εn ∈ (0, ε0] be such that tn → 0, εn → 0 and let xn ∈ G1 \G2 be the corresponding
solutions of Aϕt x+ Cx+ qεx = 0, i.e.,

Aϕtnxn + Cxn + qεnxn = 0.

We may assume that xn ⇀ x0 in X and qεnxn → g∗ ∈ X∗. We observe that

〈Aϕtnxn, xn − x0〉 = −〈Cxn + qεnxn, xn − x0〉.

If

lim sup
n→∞

〈Cxn + qεnxn, xn − x0〉 > 0,

then we obtain a contradiction from (i) of Lemma 2.3 with S = 0 there. Conse-
quently,

lim sup
n→∞

〈Cxn + qεnxn, xn − x0〉 ≤ 0,

and hence

lim sup
n→∞

〈Cxn, xn − x0〉 ≤ 0.

By the (S+)-property of C, we obtain xn → x0 ∈ G1 \G2. Then Cxn ⇀ Cx0 and
Aϕtnxn ⇀ −Cx0 − g∗. Using this in (ii) of Lemma 2.3 with S = 0 there, we obtain
x0 ∈ D(A) and −Cx0 − g∗ ∈ Ax0. By a property of the selection qεnxn as in Hu
and Papageorgiou [21], we have g∗ ∈ Tx0, and therefore Ax0 +Cx0 +Tx0 3 0. We
also have

x0 ∈ G1 \G2 = (G1 \G2) ∪ ∂(G1 \G2) ⊂ (G1 \G2) ∪ ∂G1 ∪ ∂G2.

By (H1) and (H2), we have x0 /∈ ∂G1 ∪ ∂G2, and hence x0 ∈ D(A)∩ (G1 \G2). �

Remark 3.4. We point out that the condition A(0) = {0} on the homogeneous
maximal monotone operator A used in Theorem 3.3 is rather mild in view of Rock-
afellar’s result [29] which says that a monotone map is locally bounded at every
point in the interior of its domain.

The existence of nonzero solutions of Lx + Ax + Cx 3 0, where the maximal
monotone operator A is strongly quasibounded and positively homogeneous of de-
gree γ = 1, is established in [2]. In the following theorem, we extend this result
to an arbitrary degree γ > 0 for the same combination of operators in the spirit
of the Berkovits-Mustonen theory in [11] and the theories developed in [6]. We



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 17

recall that the maximal monotone operator A investigated in [6] is strongly quasi-
bounded. However, by a result of Hess [20], a strongly quasibounded and positively
homogeneous operator of degree γ > 0 is necessarily bounded. Therefore, in the
following theorem, we assume that the maximal monotone operator A is bounded.

Theorem 3.5. Assume that G1, G2 ⊂ X are open, bounded with 0 ∈ G2 and
G2 ⊂ G1. Let L : X ⊃ D(L) → X∗ be linear maximal monotone with D(L) = X,
and A : X ⊃ D(A)→ 2X

∗
bounded, maximal monotone and positively homogeneous

of degree γ > 0. Also, let C : G1 → X∗ be bounded, demicontinuous and of type
(S+) with respect to D(L). Moreover, assume that

(H3) there exists v∗ ∈ X∗ \ {0} such that Lx + Ax + Cx 63 λv∗ for all (λ, x) ∈
R+ × (D(L) ∩D(A) ∩ ∂G1), and

(H4) Lx+Ax+ Cx+ λJx 63 0 for all (λ, x) ∈ R+ × (D(L) ∩D(A) ∩ ∂G2).

Then the inclusion Lx+Ax+Cx 3 0 has a solution x ∈ D(L)∩D(A)∩ (G1 \G2).

Proof. We begin by observing that a positively homogeneous and bounded maximal
monotone operator A of degree γ > 0 satisfies 0 ∈ D(A) and A(0) = {0}. To solve
the inclusion

Lx+Ax+ Cx 3 0, x ∈ G1, (3.10)

let us consider the associated equation

L̂x+ Âϕt x+ Ĉx+ tMx = 0, t ∈ (0,∞), x ∈ j−1(G1). (3.11)

Here, the gauge function is ϕ(r) = rp−1, 1 < p <∞, and γ = p−1. We can show as
in [5, Lemma 5] that there exists R > 0 such that the open ball BY (0, R) contains
all the solutions of (3.11). We recall that Y = D(L).

We shall prove that (3.11) has a solution xt ∈ j−1(G1 \ G2) for all sufficiently
small t > 0. We first claim that there exist τ0 > 0 , t0 > 0 such that

L̂x+ Âϕt x+ Ĉx+ tMx = τj∗v∗ (3.12)

has no solution in G1
R(Y ) := j−1(G1)∩BY (0, R) for all t ∈ (0, t0] and all τ ∈ [τ0,∞).

Assume the contrary and let {τn} ⊂ (0,∞), {tn} ⊂ (0, 1) and {xn} ⊂ G1
R(Y ) such

that τn →∞, tn → 0 and

L̂xn + Âϕtnxn + Ĉxn + tnMxn = τnj
∗v∗. (3.13)

We note that j∗ is one-to-one because j(Y ) = Y , which is dense in X. This implies
that j∗v∗ is nonzero, and therefore ‖τnj∗v∗‖Y ∗ → +∞. Also, the sequence {xn}
is bounded in Y and so we may assume that xn ⇀ x0 in X and Lxn ⇀ Lx0

in X∗. In particular, {Lxn} is bounded in X∗. Since Mxn ∈ j∗(X∗), we have
J−1(Lu) ∈ D(L∗) and

Mxn = j∗L∗J−1(Lxn).

Since j∗, L∗, J−1 are bounded, we have the boundedness of {Mxn}. It is clear that

L̂xn and Ĉxn are bounded in Y ∗, and therefore (3.13) implies that ‖Âϕtnxn‖Y ∗ →
∞. Since A is positively homogeneous of degree γ, applying Lemma 3.1 for γ = p−1
shows that each Aϕtn is also positively homogeneous of γ = p− 1. Consequently,

Âϕtnxn

‖Âϕtnxn‖Y ∗
= Âϕtn

(
xn

‖Âϕtnxn‖
1/γ
Y ∗

)
(3.14)



18 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

for all n. Define βn := 1/‖Âϕtnxn‖Y ∗ and δn := β
1/γ
n . Since ‖Âϕtnxn‖Y ∗ → ∞, it

follows that βnxn → 0 and δnxn → 0 in X as n→∞. From (3.13) and (3.14), we
find

L̂(βnxn) + Âϕtn(δnxn) + βnĈxn + tnβnMxn = τnβnj
∗v∗. (3.15)

Because ‖Âϕtn(δnxn)‖Y ∗ = 1 and the remaining terms on the left in (3.15) con-
verge to 0 in X∗ as n → ∞, we obtain τnβn → 1/‖j∗v∗‖Y ∗ , and therefore

Âϕtn(δnxn) → y0, where y0 = j∗v∗/‖j∗v∗‖Y ∗ . Since un := δnxn → 0 as n → ∞,

we have 〈Âϕtnun, un〉 → 〈y0, 0〉 = 0 as n → ∞. By Lemma 2.3, (ii), we have
y0 ∈ A(0) = {0}, which is a contradiction to ‖y0‖Y ∗ = 1.

We now consider the homotopy H : [0, 1]× Y → Y ∗ defined by

H(s, x) = L̂x+ Âϕt x+ Ĉx+ tMx− sτ0j∗v∗, s ∈ [0, 1], x ∈ j−1(G1), (3.16)

where t ∈ (0, t0] is fixed. It can be easily seen that C − sτ0v∗ is bounded demicon-
tinuous on G1 and of type (S+) with respect to D(L).

We now show that the equation H(s, x) = 0 has no solution on the boundary
∂G1

R(Y ). Here, the number R > 0 is increased, if necessary, so that the ball
BY (0, R) now also contains all the solutions x of H(s, x) = 0. To this end, assume
the contrary so that there exist {tn} ⊂ (0, t0], {sn} ⊂ [0, 1], and {xn} ⊂ ∂G1

R(Y )
such that tn → 0, sn → s0, xn ⇀ x0 in Y , Aϕtnxn ⇀ w∗ in X∗, Cxn ⇀ c∗ and

L̂xn + Âϕtnxn + Ĉxn + tnMxn = snτ0j
∗v∗. (3.17)

Here, the boundedness of {Aϕtnxn} follows as in Step I of [3, Proposition 1], except
that we now use Aϕtn in place of the operators Tsn used in [3]. Since xn ⇀ x0 in Y ,
we have xn ⇀ x0 in X and Lxn ⇀ Lx0 in X∗. Also, since xn ∈ BY (0, R) and

∂(j−1(G1) ∩BY (0, R)) ⊂ ∂(j−1(G1)) ∪ ∂BY (0, R) ⊂ j−1(∂G1) ∪ ∂BY (0, R),

we have xn ∈ j−1(∂G1) = ∂G1 ∩ Y ⊂ ∂G1. We now follow the arguments as in [2,
Theorem 2.2] in conjunction with Lemma 2.3 to arrive at

〈Lx0 + w∗ + Cx0 − s0τ0v
∗, u〉 = 0

for all u ∈ Y , where x0 ∈ D(A) and w∗ ∈ Ax0. Since Y is dense in X, we
have Lx0 + Tx0 + Cx0 3 s0τ0v

∗, which contradicts the hypothesis (H3) because
x0 ∈ D(L) ∩D(T ) ∩ ∂G1.

We shrink t0, if necessary, so that

H(s, x) = 0, s ∈ [0, 1], x ∈ G1
R(Y )

has no solution on the boundary ∂G1
R(Y ) for all t ∈ (0, t0] and all s ∈ [0, 1]. It

now follows from Proposition 2.11 that H(s, x) is an admissible homotopy for the
(S+)-degree, dS+

, and therefore dS+
(H(s, ·), G1

R(Y ), 0), is well-defined and remains
constant for all s ∈ [0, 1]. Also, by Proposition 2.11, the limit

lim
t→0+

dS+
(H(1, ·), G1

R(Y ), 0)

exists. By shrinking t0 further, if necessary, we find that dS+(H(1, ·), G1
R(Y ), 0) =

a constant for all t ∈ (0, t0]. Suppose, if possible, that

dS+(H(1, ·), G1
R(Y ), 0) 6= 0

for some t1 ∈ (0, t0]. Then there exists x0 ∈ G1
R(Y ) such that

L̂x+ Âϕt1x+ Ĉx+ t1Mx = τ0j
∗v∗.



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 19

This contradicts the choice of τ0 as stated in (3.12). Since

dS+
(H(0, ·), G1

R(Y ), 0) = dS+
(H(1, ·), G1

R(Y ), 0),

we have

dS+
(L̂+ Âϕt + Ĉ + tM,G1

R(Y ), 0) = dS+
(H(0, ·), G1

R(Y ), 0) = 0 (3.18)

for all t ∈ (0, t0].

Next, we consider the homotopy H̃ : [0, 1]× Y → Y ∗ defined by

H̃(s, x) = s(L̂x+ Âϕt x+ Ĉx) + tMx+ (1− s)Ĵx, s ∈ [0, 1], x ∈ j−1(G2).

As in [3, Step III, p. 29], it can be shown that there exists t0 > 0 (shrink it to a
smaller number if necessary) such that all the solutions of

H̃(s, x) = 0, t ∈ (0, t0], s ∈ [0, 1]

are bounded in Y . We enlarge the previous number R > 0, if necessary, so that all

solutions of H̃(s, x) = 0 as described above are contained in BY (0, R) in Y .
Again, by following arguments similar to that in [2, Theorem 2.2], we can show

the existence of t1 ∈ (0, t0] such that the equation H̃(s, x) = 0 has no solutions on
∂G2

R(Y ) for any t ∈ (0, t1] and any s ∈ [0, 1]. Here, G2
R(Y ) := j−1(G2) ∩BY (0, R).

In fact, if we assume the contrary, we can arrive at a situation that contradicts
(H4). At this point, we replace the number t0 chosen previously with t1 and call it
t0 again. Let us fix t ∈ (0, t0] and consider the homotopy equation

H̃(s, x) = s(L̂x+Âϕt x+Ĉx)+tMx+(1−s)Ĵx = 0, s ∈ [0, 1], x ∈ G2
R(Y ). (3.19)

It is already discussed that (3.19) has no solution on ∂G2
R(Y ). We note that H̃ is

an affine homotopy of bounded demicontinuous operators of type (S+) on G2
R(Y );

namely, L̂+ Âϕt + Ĉ + tM and tM + Ĵ . We also note here that tM + Ĵ is strictly

monotone. In view of Proposition 2.11, it follows that H̃(s, x) is an admissible
homotopy for the (S+)-degree, dS+ , which satisfies

dS+(H̃(1, ·), G2
R(Y ), 0) = dS+(H̃(0, ·), G2

R(Y ), 0). (3.20)

This implies

dS+
(L̂+ Âϕt + Ĉ + tM,G2

R(Y ), 0) = dS+
(tM + Ĵ , G2

R(Y ), 0) = 1 (3.21)

for all t ∈ (0, t0]. The last equality follows from [15, Theorem 3, (iv)]. From (3.18)
and (3.21), we obtain

dS+(L̂+ Âϕt + Ĉ + tM,G1
R(Y ), 0) 6= dS+

(L̂+ Âϕt + Ĉ + tM,G2
R(Y ), 0)

for all t ∈ (0, t0]. By the excision property of the (S+)-degree, for each t ∈ (0, t0],
there exists a solution xt ∈ G1

R(Y ) \G2
R(Y ) of the equation

L̂x+ Âϕt x+ Ĉx+ tMx = 0.

We now pick a sequence {tn} ⊂ (0, t0] such that tn → 0 and denote the correspond-
ing solution xt by xn, i.e.,

L̂xn + Âϕtnxn + Ĉxn + tnMxn = 0.

Since Y is reflexive, we have xn ⇀ x0 ∈ Y by passing to a subsequence. This
implies xn ⇀ x0 in X and Lxn ⇀ Lx0 in X∗. By the boundedness (therefore strong
quasiboundedness) of A, we may assume, in view of Lemma 2.5, that Aϕtnxn ⇀ w∗ ∈
X∗. By a standard argument in conjunction with Lemma 2.3 and the (S+)-property



20 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

of C with respect to D(L), we obtain xn → x0 ∈ G1
R(Y ) \G2

R(Y ). By Lemma 2.3
and the demicontinuity of C, we have x0 ∈ D(A), w∗ ∈ Ax0, and Cxn ⇀ Cx0 in
X∗. Thus, Lx0 +Ax0 + Cx0 3 0.

Finally, to show x0 ∈ G1 \G2, we note that

G1
R(Y ) \G2

R(Y ) = (G1 \G2) ∩ Y ∩BY (0, R) ⊂ G1 \G2.

Consequently, xn ∈ G1 \G2 for all n, and therefore

x0 ∈ G1 \G2 ⊂ (G1 \G2) ∪ ∂(G1 \G2) ⊂ (G1 \G2) ∪ ∂G1 ∪ ∂G2.

By (H3) and (H4), x0 6∈ ∂G1 ∪ ∂G2 and hence x0 ∈ D(L) ∩D(T ) ∩ (G1 \G2). �

3.1. Open Problem. Does Theorem 3.5 hold true if the boundedness of A is
dropped? Since a positively homogeneous operator that is strongly quasibounded
is necessarily bounded, it is desirable to determine whether Theorem 3.5 holds if
A is assumed to be “quasibounded”. An operator A : X ⊃ D(A)→ 2X

∗
is said to

be quasibounded if for every S > 0 there exists K(S) > 0 such that ‖x‖ ≤ S and
〈x∗, x〉 ≤ S‖x‖ for some x∗ ∈ Ax imply ‖x∗‖ ≤ K(S). The notions of quasibounded
and strongly quasibounded operators were introduced in Hess [20].

4. Applications

In this section, we apply Theorems 3.3 and 3.5 to elliptic and parabolic boundary
value problems in general divergence form which are obtained by modifying relevant
examples from Berkovits and Mustonen [11], Kittilä [27], and Adhikari [2].

Application 4.1. We consider the space X = Wm,p
0 (Ω) with the integer m ≥ 1,

the number p ∈ (1,∞), and the domain Ω ⊂ RN with smooth boundary. We
let N0 denote the number of all multi-indices α = (α1, . . . , αN ) such that |α| =
α1 + · · ·+ αN ≤ m. For ξ = (ξα)|α|≤m ∈ RN0 , we have a representation ξ = (η, ζ),

where η = (ηα)|α|≤m−1 ∈ RN1 , ζ = (ζα)|α|=m ∈ RN2 and N0 = N1 +N2. We let

ξ(u) = (Dαu)|α|≤m, η(u) = (Dαu)|α|≤m−1, and ζ(u) = (Dαu)|α|=m,

where Dα =
∏N
i=1

(
∂
∂xi

)αi
. We write ∇u := (Dαu)|α|=1, and when |α| = k ∈

{1, 2, . . . ,m}, we simply write Dku := (Dαu)|α|=k. Also, define q := p/(p− 1).
We now consider the partial differential expression in divergence form∑

|α|≤m

(−1)|α|DαAα(x, ξ(u)), x ∈ Ω.

The functions Aα : Ω × RN0 → R are assumed to be Carathéodory, i.e., each
Aα(x, ξ) is measurable in x for fixed ξ ∈ RN0 and continuous in ξ for almost all
x ∈ Ω. We assume the following conditions on Aα:

(H5) There exist p ∈ (1,∞), c1 > 0, and κ1 ∈ Lq(Ω) such that

|Aα(x, ξ)| ≤ c1|ξ|p−1 + κ1(x), x ∈ Ω, ξ ∈ RN0 , |α| ≤ m.

(H6) The Leray-Lions condition∑
|α|=m

[Aα(x, η, ζ1)−Aα(x, η, ζ2)](ζ1α − ζ2α) > 0

is satisfied for every x ∈ Ω, η ∈ RN1 and ζ1, ζ2 ∈ RN2 with ζ1 6= ζ2.



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 21

(H7) ∑
|α|≤m

[Aα(x, ξ1)−Aα(x, ξ2)](ξ1α − ξ2α) ≥ 0

is satisfied for every x ∈ Ω and ξ1, ξ2 ∈ RN0 .
(H8) There exist c2 > 0, κ2 ∈ L1(Ω) such that∑

|α|≤m

Aα(x, ξ)ξα ≥ c2|ξ|p − κ2(x), x ∈ Ω, ξ ∈ RN0 .

(H9) Each Aα(x, ξ) is homogeneous of degree γ > 0 with respect to ξ.

If an operator A : Wm,p
0 (Ω)→W−m,q(Ω) is given by

〈Au, v〉 =

∫
Ω

∑
|α|≤m

Aα(x, ξ(u))Dαv, u, v ∈Wm,p
0 (Ω), (4.1)

then the conditions (H5), (H7) imply that A is bounded, continuous, and monotone
as discussed in Kittilä [27, pp. 25-26] and Pascali and Sburlan [28, pp. 274-275].
Since A is continuous, it is maximal monotone. Moreover, the condition (H9)
implies that A is positively homogeneous of degree γ > 0. For example, for m = 1,
we have |α| ≤ 1, and when

Aα(x, η, ζ) =

{
|ζ|p−2ζα for |α| = 1

0 for |α| = 0,

the operator A in (4.1) is given by A := −∆p, where ∆p is the p−Laplacian from

W 1,p
0 (Ω) to W−1,q(Ω) defined as

∆pu := div
(
|∇u|p−2∇u

)
, u ∈W 1,p

0 (Ω).

It is clear that ∆p is positively homogeneous of degree p− 1 .
Similarly, the condition (H5), with Aα replaced by Cα, implies that the operator

〈Cu, v〉 =

∫
Ω

∑
|α|≤m

Cα(x, ξ(u))Dαv, u, v ∈Wm,p
0 (Ω), (4.2)

is a bounded continuous mapping. We also know that conditions (H5), (H6), and
(H8), with Cα in place of Aα everywhere, imply that the operator C is of type (S+)
(see Kittilä [27, p. 27]).

We also consider a multifunction H : Ω× RN1 → 2R such that

(H10) H(x, r) = [ϕ(x, r), ψ(x, r)] is measurable in x and upper semicontinuous in
r, where ϕ,ψ : Ω× RN1 → R are measurable functions; and

(H11) |H(x, r)| = max[|ϕ(x, r)|, |ψ(x, r)|] ≤ a(x) + c2|r| a.e. on Ω × RN1 , where
a(·) ∈ Lq(Ω), c2 > 0.

Define T : Wm,p
0 → 2W

−m,q(Ω) by

Tu =
{
h ∈W−m,q(Ω) : ∃w ∈ Lq(Ω) such that w(x) ∈ H(x, u(x))

and 〈h, v〉 =

∫
Ω

w(x)v(x) for all v ∈Wm,p
0 (Ω)

}
.

It is well-known that T is upper-semicontinuous and compact with closed and con-
vex values (see [21, p. 254]), and therefore T is of class (P ).

We now state the following theorem as an application of Theorem 3.3.



22 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

Theorem 4.2. Assume that the operators A, C, and T are defined as above. As-
sume, further, that the rest of the conditions of Theorem 3.3 are satisfied for two
balls G1 = Bδ1(0) and G2 = Bδ2(0), where 0 < δ2 < δ1. Then the Dirichlet
boundary value problem∑

|α|≤m

(−1)|α|Dα
(
Aα(x, ξ(u)) + Cα(x, ξ(u))

)
+H(x, u) 3 0, x ∈ Ω,

Dαu(x) = 0, x ∈ ∂Ω, |α| ≤ m− 1,

has a “weak” nonzero solution u ∈ Bδ1(0) \Bδ2(0) ⊂ Wm,p
0 (Ω), which satisfies the

inclusion Au+ Cu+ Tu 3 0.

Application 4.3. Let Ω be a bounded open set in RN with smooth boundary,
m ≥ 1 an integer, and a > 0. Set Q = Ω× [0, a]. Consider differential operators of
the form

∂u

∂t
(x, t) +

∑
|α|≤m

(−1)|α|Dα
(
Aα(x, t, ξ(u(x, t)) + Cα(x, t, ξ(u(x, t))

)
(4.3)

in Q. The functions Aα = Aα(x, t, ξ) and Cα = Cα(x, t, ξ) are defined for (x, t) ∈ Q,
ξ = (ξα)|α|≤m = (η, ζ) ∈ RN0 with η = (ηγ)|α|≤m−1 ∈ RN1 , ζ = (ζα)|α|=m ∈ RN2 ,
and N1+N2 = N0. We assume that each Aα(x, t, ξ) satisfies the usual Carathéodory
condition. We consider the following conditions.

(H12) (Continuity) For some p > 1, c1 > 0, g ∈ Lq(Q) with q = p/(p − 1), we
have

|Aα(x, t, η, ζ)| ≤ c1(|ζ|p−1 + |η|p−1 + g(x, t)),

for (x, t) ∈ Q, ξ = (η, ζ) ∈ RN0 and |α| ≤ m.
(H13) (Monotonicity)∑
|α|≤m

(Aα(x, t, ξ1)−Aα(x, t, ξ2))(ξ1α − ξ2α) ≥ 0 for (x, t) ∈ Q and ξ1, ξ2 ∈ RN0 .

(H14) (Leray-Lions)∑
|α|=m

(Aα(x, t, η, ζ)−Aα(x, t, η, ζ∗))(ζα − ζ∗α) > 0,

for (x, t) ∈ Q, η ∈ RN1 and ζ, ζ∗ ∈ RN2 .
(H15) (Coercivity) There exist c0 > 0 and h ∈ L1(Q) such that∑

|α|≤m

Aα(x, t, ξ) ≥ c0|ξ|p − h(x, t), (x, t) ∈ Q and ξ ∈ RN0 .

(H16) Each Aα(x, t, ξ) is homogeneous of degree γ > 0 with respect to ξ.

Under condition (H12), the second term of (4.3) with Cα = 0 generates a con-
tinuous bounded operator A : X → X∗ defined by

〈Au, v〉 =
∑
|α|≤m

∫
Q

Aα(x, t, ξ(u(x, t)))Dαv, u, v ∈ X,

where X = Lp(0, a;V ), X∗ = Lq(0, a;V ∗), and V = Wm,p
0 (Ω). With the additional

conditions (H13) and (H16), the operator A is maximal monotone and positively
homogeneous of degree γ. Under (H12), (H14), and (H15) with Aα replaced by



EJDE-2022/63 POSITIVELY HOMOGENEOUS MAXIMAL MONOTONE OPERATORS 23

Cα and other obvious changes, the second term in (4.3) with Aα = 0 generates a
continuous, bounded operator C defined as

〈Cu, v〉 =
∑
|α|≤m

∫
Q

Cα(x, t, ξ(u(x, t)))Dαv, u, v ∈ X,

which satisfies the condition (S+) with respect to D(L), where the operator L is
defined as follows. The operator ∂/∂t generates an operator L : X ⊃ D(L)→ X∗,
where

D(L) = {v ∈ X : v′ ∈ X∗, v(0) = 0},
via the relation

〈Lu, v〉 =

∫ a

0

〈u′(t), v(t)〉V dt, u ∈ D(L), v ∈ X,

where 〈·, ·〉V is the duality pairing in V ∗ × V . The symbol u′(t) is the generalized
derivative of u(t), i.e.,∫ a

0

u′(t)ϕ(t)dt = −
∫ a

0

ϕ′(t)u(t) dt, ϕ ∈ C∞0 (0, a).

We can verify, as in Zeidler [34], that L is densely defined, linear and maximal
monotone.

Given h ∈ Lq(Q), define h∗ ∈ X∗ by

〈h∗, v〉 =

∫
Q

hv, v ∈ X.

As an application of Theorem 3.5, we obtain the following theorem.

Theorem 4.4. Assume that the operators L,A, and C are as above, with Aα
satisfying (H12), (H13), and (H16), and Cα in place of Aα satisfying (H12), (H14),
and (H15). Assume, for a given h ∈ Lq(Q), that the rest of the conditions of
Theorem 3.5 are satisfied when C is replaced with C−h∗ for two balls G1 = Bδ1(0)
and G2 = Bδ2(0) in X = Lp(0, a;V ), where 0 < δ2 < δ1 and V = Wm

0 (Ω). Then
the initial-boundary value problem

∂u

∂t
+
∑
|α|≤m

(−1)|α|Dα
(
Aα(x, t, ξ(u)) + Cα(x, t, ξ(u))

)
= h(x, t),

Dαu(x, t) = 0, (x, t) ∈ ∂Ω× [0, a], |α| ≤ m− 1,

u(x, 0) = 0, x ∈ Ω,

has a “weak” nonzero solution u ∈ Bδ1(0) \Bδ2(0) ⊂ Lp(0, a;V ) satisfying

Lu+Au+ Cu = h∗.

Acknowledgments. This research was carried out by members of the Analysis
Group of the Association of Nepalese Mathematicians in America (ANMA) within
the Collaborative Research in Mathematical Sciences program. Ghanshyam Bhatt
acknowledges Tennessee State University for supporting his engagement in this re-
search with a Non-Instructional Assignment (NIA) grant for the years 2021-2022.
The authors express their gratitude to the anonymous referees for providing feed-
back which helped to improve the article.



24 D. ADHIKARI, A. ARYAL, G. BHATT, I. KUNWAR, R. PURI, P. RANABHAT EJDE-2022/63

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Dhruba R. Adhikari
Department of Mathematics, Kennesaw State University, Marietta, GA 30060, USA

Email address: dadhikar@kennesaw.edu

Ashok Aryal

Mathematics Department, Minnesota State University Moorhead, Moorhead, MN 56563,

USA
Email address: ashok.aryal@mnstate.edu

Ghanshyam Bhatt
Department of Mathematical Sciences, Tennessee State University, Nashville, TN 37209,

USA

Email address: gbhatt@tnstate.edu

Ishwari J. Kunwar

Department of Mathematics and Computer Science, Fort Valley State University, Fort
Valley, GA 31030, USA

Email address: kunwari@fvsu.edu

Rajan Puri
Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, USA

Email address: purir@wfu.edu

Min Ranabhat

Department of Mathematical Sciences, University of Delaware, EWG 315, Newark, DE
19716, USA

Email address: ranabhat@udel.edu


	1. Introduction and preliminaries
	2. Variants of Yosida approximants and related properties
	3. Existence of nontrivial solutions
	3.1. Open Problem

	4. Applications
	Acknowledgments

	References