Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 37, pp. 1–17.

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

EXISTENCE OF SOLUTIONS FOR IMPLICIT OBSTACLE

PROBLEMS INVOLVING NONHOMOGENEOUS PARTIAL

DIFFERENTIAL OPERATORS AND MULTIVALUED TERMS

SHENGDA ZENG, YUNRU BAI, LESZEK GASIŃSKI, IRENEUSZ KRECH

Abstract. In this article, we study an implicit obstacle problem with a non-

linear nonhomogeneous partial differential operator and a multivalued operator
which is described by a generalized gradient. Under quite general assumptions

on the data, and employing Kluge’s fixed point principle for multivalued op-

erators, Minty technique and a surjectivity theorem, we prove that the set of
weak solutions to the problem is nonempty, bounded and weakly closed.

1. Introduction

Let Ω ⊆ RN be a bounded domain with a C1,α-boundary ∂Ω for some 0 < α < 1.
In this paper, we study the following implicit obstacle problem with a nonlinear
nonhomogeneous partial differential operator and a multivalued operator which is
described by a generalized gradient, namely

−div a(x,∇u(x)) + ∂j(x, u(x)) 3 f(x) in Ω,

u = 0 on ∂Ω,

T (u) ≤ U(u).

(1.1)

In the above f : Ω → R and j : Ω × R → R are given two functions, such that
f ∈ Lp

′
(Ω) (where 1 < p < ∞ and 1

p + 1
p′ = 1) and j is locally Lipschitz with

respect to the second variable. By ∂j(x, u(x)) we denote the Clarke’s generalized

gradient of j with respect to the last variable. Finally T,U : W 1,p
0 (Ω)→ R are two

given functions, which satisfy appropriate assumptions listed in Section 3.
In this article we prove that the set of weak solutions to the problem is nonempty,

bounded and weakly closed. In particular we obtain the existence of at least one
weak solution to problem (1.1). The main tools used in the proof are the surjectiv-
ity theorem for multivalued mappings due to Le [33], Kluge’s fixed point principle
as well as some techniques of nonsmooth analysis. Problem (1.1) combines several
interesting phenomena like a nonhomogeneous operator of p-Laplacian type, a mul-
tivalued mapping provided by the Clarke generalized subdifferential and an implicit
obstacle inequality. The latter means that any solution u ∈W 1,p

0 (Ω) of (1.1) has to

2010 Mathematics Subject Classification. 35J20, 35J25, 35J60.
Key words and phrases. Implicit obstacle problem; Clarke generalized gradient;

nonhomogeneous partial differential operator; fixed point theorem; surjectivity theorem.
c©2021 Texas State University.

Submitted April 1, 2020. Published May 6, 2021.

1



2 S. ZENG, Y. BAI, L. GASIŃSKI, I. KRECH EJDE-2021/37

belong to K(u), which is the image of the multivalued map K : W 1,p
0 (Ω)→ 2W

1,p
0 (Ω)

defined by
K(u) := {v ∈W 1,p

0 (Ω) : T (v)− U(u) ≤ 0},
for some obstacles given by the functions T : W 1,p

0 (Ω) → R and U : W 1,p
0 (Ω) →

(0,+∞).
For the problems with a nonhomogeneous operator of p-Laplacian type we refer

to Bai-Gasińnski-Papageorgiou [2], Candito-Gasiński-Livrea [6], Gasiński-O’Regan-
Papageorgiou [20, 21], Gasiński-Papageorgiou [27, 28], Marino-Winkert [35, 36],
Papageorgiou-Winkert [39], Papageorgiou-Rǎdulescu [40], Papageorgiou-Rǎdulescu-
Repovš [41, 42]. In all the aforementioned papers, we find different types of non-
homogeneous operators and boundary value conditions, but we do not have multi-
valued terms as well as they do not deal with obstacle problems. For the problems
dealing with multivalued terms modeled by Clarke’s subdifferential we refer to
the papers of Averna-Marano-Motreanu [1], Denkowski-Gasiński-Papageorgiou [10,
11, 12, 13], Filippakis-Gasiński-Papageorgiou [15, 16], Gasiński [17, 18], Gasiński-
Motreanu-Papageorgiou [19], Gasiński-Papageorgiou [23, 24], Kalita-Kowalski [30],
Papageorgiou-Vetro-Vetro [43, 44], Zeng-Liu-Migórski [45]. None of them deals with
nonhomogeneous operators and obstacle problems. Finally, for the problems deal-
ing with obstacle problems we refer to the papers of Caffarelli-Salsa-Silvestre [4],
Caffarelli-Ros-Oton-Serra [5], Choe [8], Choe-Lewis [9], Feehan-Pop [14], Ober-
man [38]. As for the paper combining both nonhomogeneous operator and mul-
tivalued term provided by a subdifferential we refer to the paper of Gasiński-
Papageorgiou [25], although their approach is different from ours and is based on
the nonsmooth critical point theory.

This article is organized as follows. In Section 2 we recall some definitions of
function spaces needed in the sequel as well as the formulations of the main tools
needed for our proofs, in particular the surjectivity results of Le [33] and Kluge’s
fixed point theorem. In Section 3 we provide the list of assumptions on the data of
problem (1.1) and give the definition of the weak solution. In Section 4 we consider
an auxiliary problem (see (4.2)) and indicate some properties of its solution set.
Finally, in Section 5, we state and prove the main result of the paper (Theorem
5.1), which says that the solution set of (1.1) is a nonempty, bounded and weakly

closed subset of W 1,p
0 (Ω).

2. Preliminaries

For a bounded domain Ω ⊆ R and 1 ≤ r ≤ ∞, in what follows, by Lr(Ω) and
Lr(Ω;RN ) we denote the usual Lebesgue spaces endowed with the norms denoted

by ‖ · ‖r. Moreover, W 1,r
0 (Ω) stands for the Sobolev space endowed with the norm

‖u‖ = ‖∇u‖r for all u ∈W 1,r
0 (Ω).

Let us now consider the eigenvalue problem for the r-Laplacian with homoge-
neous Dirichlet boundary condition and 1 < r <∞ which is defined by

−∆ru = λ|u|r−2u in Ω,

u = 0 on ∂Ω.
(2.1)

A number λ ∈ R is an eigenvalue of (−∆r,W
1,r
0 (Ω)) if problem (2.1) has a non-

trivial solution u ∈ W 1,r
0 (Ω) which is called an eigenfunction corresponding to the

eigenvalue λ. We denote by σr the set of eigenvalues of (−∆r,W
1,r
0 (Ω)). From Lê



EJDE-2021/37 EXISTENCE OF SOLUTIONS FOR IMPLICIT OBSTACLE PROBLEMS 3

[34] we know that the set σr has a smallest element λ1,r which is positive, isolated,
simple and it can be variationally characterized through

λ1,r = inf
{‖∇u‖rr
‖u‖rr

: u ∈W 1,r
0 (Ω), u 6= 0

}
. (2.2)

For s > 1, we denote by s′ = s
s−1 its conjugate, the inner product in RN is

denoted by · and the norm of RN is given by | · |. Moreover, R+ = [0,+∞) and the
Lebesgue measure in RN is denoted by | · |N .

Let E be a Banach space with its topological dual E∗. A function J : E → R is
said to be locally Lipschitz at u ∈ E if there exist a neighborhood N(u) of u and a
constant Lu > 0 such that

|J(w)− J(v)| ≤ Lu‖w − v‖E for all w, v ∈ N(u).

Definition 2.1. Let J : E → R be a locally Lipschitz function and let u, v ∈ E.
The generalized directional derivative J0(u; v) of J at the point u in the direction
v is defined by

J0(u; v) := lim sup
w→u, t↓0

J(w + tv)− J(w)

t
.

The generalized gradient ∂J : E → 2E
∗

of J : E → R is defined by

∂J(u) :=
{
ξ ∈ E∗ | J0(u; v) ≥ 〈ξ, v〉E∗×E for all v ∈ E

}
for all u ∈ E.

The next proposition collects some basic results (see Migórski-Ochal-Sofonea [37,
Proposition 3.23]).

Proposition 2.2. Let J : E → R be locally Lipschitz of rank Lu > 0 at u ∈ E.
Then we have

(a) the function v 7→ J0(u; v) is positively homogeneous, subadditive, and sat-
isfies

|J0(u; v)| ≤ Lu‖v‖E for all v ∈ E.
(b) (u, v) 7→ J0(u; v) is upper semicontinuous.
(c) for each u ∈ E, ∂J(u) is a nonempty, convex, and weak∗ compact subset of

E∗ with ‖ξ‖E∗ ≤ Lu for all ξ ∈ ∂J(u).
(d) J0(u; v) = max {〈ξ, v〉E∗×E | ξ ∈ ∂J(u)} for all v ∈ E.
(e) the multivalued function E 3 u 7→ ∂J(u) ⊂ E∗ is upper semicontinuous

from E into w∗-E∗.

Next, let ϑ ∈ C1(0,∞) be any function satisfying

0 < a1 ≤
tϑ′(t)

ϑ(t)
≤ a2 and a3t

p−1 ≤ ϑ(t) ≤ a4

(
tq−1 + tp−1

)
(2.3)

for all t > 0, with some constants ai > 0, i ∈ {1, 2, 3, 4} and for 1 < q < p < ∞.
The hypotheses on a : Ω× RN → RN are listed below.

(H1) a(x, ξ) = a0 (x, |ξ|) ξ with a0 ∈ C(Ω×R+) for all ξ ∈ RN and with a0(x, t) >
0 for all x ∈ Ω, for all t > 0 and

(i) a0 ∈ C1(Ω × (0,∞)), t 7→ ta0(x, t) is strictly increasing in (0,∞),
limt→0+ ta0(x, t) = 0 for all x ∈ Ω and

lim
t→0+

ta′0(x, t)

a0(x, t)
= c > −1 for all x ∈ Ω;



4 S. ZENG, Y. BAI, L. GASIŃSKI, I. KRECH EJDE-2021/37

(ii) |∇ξa(x, ξ)| ≤ a5ϑ (|ξ|) /|ξ| for all x ∈ Ω, for all ξ ∈ RN \ {0} and some
a5 > 0;

(iii) ∇ξa(x, ξ)y · y ≥ ϑ (|ξ|) |y|2/|ξ| for all x ∈ Ω, for all ξ ∈ RN \ {0} and
all y ∈ RN .

The following lemma summarizes some properties of the function a : Ω×RN → RN .

Lemma 2.3. If hypotheses (H1) hold, then:

(i) a ∈ C(Ω × RN ;RN ) ∩ C1(Ω × (RN \ {0});RN ) and for all x ∈ Ω the map
ξ 7→ a(x, ξ) is continuous, strictly monotone and so maximal monotone as
well;

(ii) there exists a6 > 0, such that |a(x, ξ)| ≤ a6

(
1 + |ξ|p−1

)
for all x ∈ Ω and

ξ ∈ RN ;
(iii) a(x, ξ) · ξ ≥ a3

p−1 |ξ|
p for all x ∈ Ω and for all ξ ∈ RN .

Lemma 2.4. Let p ≥ 2. If the following condition holds,

(H2) t 7→ a0(t)t− catp−1 is increasing on [0,∞) with some ca > 0,

then there exists a constant ma > 0 such that

(a(x, ξ1)− a(x, ξ2), ξ1 − ξ2)RN ≥ ma|ξ1 − ξ2|p

for all ξ1, ξ2 ∈ RN and a.e. x ∈ Ω.

Proof. Since p ≥ 2, it follows from Glowinski-Marroco [29, Lemma 5.1], that there
exists a constant m(p) > 0, which depends on p only, such that

(|ξ1|p−2ξ1 − |ξ2|p−2ξ2) · (ξ1 − ξ2) ≥ m(p)|ξ1 − ξ2|p

for all ξ1, ξ2 ∈ RN .
The monotonicity of t 7→ a0(t)t− catp−1 ensures that

a0(t)t− a0(s)s ≥ ca
(
tp−1 − sp−1

)
for all t, s ∈ [0,+∞) with t ≥ s. This inequality leads to(
a(ξ1)− a(ξ2), ξ1 − ξ2

)
RN

=
(
a0(|ξ1|)ξ1 − a0(|ξ2|)ξ2, ξ1 − ξ2

)
=
[
a0(|ξ1|)|ξ1| − a0(|ξ2|)|ξ2|

][
|ξ1| − |ξ2|

]
+
[
a0(|ξ1|) + a0(|ξ2|)

][
|ξ1||ξ2| − ξ1 · ξ2

]
≥ ca

[
|ξ1|p−1 − |ξ2|p−1

][
|ξ1| − |ξ2|

]
+ ca

[
|ξ1|p−2 + |ξ2|p−2

][
|ξ1||ξ2| − ξ1 · ξ2

]
= ca

(
|ξ1|p−2ξ1 − |ξ2|p−2ξ2

)
·
(
ξ1 − ξ2

)
≥ cam(p)|ξ1 − ξ2|p

for all ξ1, ξ2 ∈ RN . This means that the desired inequality is satisfied with ma =
cam(p). �

Let us introduce the nonlinear operator A : W 1,p
0 (Ω)→W 1,p

0 (Ω)∗ as follows

〈A(u), φ〉 =

∫
Ω

(a(x,∇u(x)),∇φ(x))RNdx for all u, φ ∈W 1,p
0 (Ω), (2.4)

which possesses the following useful properties (see Gasiński-Papageorgiou [26]).

Proposition 2.5. If (H1) hold and the operator A : W 1,p
0 (Ω)→W 1,p

0 (Ω)∗ is defined
by (2.4), then A is bounded, monotone, continuous, hence maximal monotone and
of type (S+). Moreover, if the function t 7→ a0(t)t− catp−1 is increasing on [0,∞)



EJDE-2021/37 EXISTENCE OF SOLUTIONS FOR IMPLICIT OBSTACLE PROBLEMS 5

with some ca > 0, then A is strongly monotone with constant ma > 0, where ma is
given in Lemma 2.4.

The following examples present some operators fitting in our setting.

Example 2.6. In the definitions of the operators a, we drop the dependence on x
just for simplicity. All the following maps satisfy (H1):

(i) If a(ξ) = |ξ|p−2ξ with 1 < p < ∞, then the corresponding operator is the
classical p-Laplacian

∆pu = div(|∇u|p−2∇u) for all u ∈W 1,p(Ω).

(ii) If a(ξ) = |ξ|p−2ξ + µ|ξ|q−2ξ with 1 < q < p < ∞ and µ > 0, then the
corresponding operator is the so called weighted (p, q)-Laplacian defined
by ∆pu+ µ∆qu for all u ∈W 1,p(Ω).

(iii) If a(ξ) = (1 + |ξ|2)
p−2
2 ξ with 1 < p < ∞, then this map represents the

generalized p-mean curvature differential operator defined by

div
[
(1 + |∇u|2)

p−2
2 ∇u

]
for all u ∈W 1,p(Ω).

Besides, we recall the notion of pseudomonotonicity for multivalued operators
(see e.g., Gasiński-Papageorgiou [22, Definition 1.4.8]).

Definition 2.7. Let X be a real reflexive Banach space. The operator A : X → 2X
∗

is called pseudomonotone if the following conditions hold:

(i) the set A(u) is nonempty, bounded, closed and convex for all u ∈ X.
(ii) A is upper semicontinuous from each finite-dimensional subspace of X to

the weak topology on X∗.
(iii) if {un} ⊂ X with un ⇀ u in X and if u∗n ∈ A(un) is such that

lim sup
n→∞

〈u∗n, un − u〉X∗×X ≤ 0,

then to each element v ∈ X, exists u∗(v) ∈ A(u) with

〈u∗(v), u− v〉X∗×X ≤ lim inf
n→∞

〈u∗n, un − v〉X∗×X .

Furthermore, we will state the surjectivity theorem for multivalued mappings
which are defined as the sum of a maximal monotone multivalued operator and a
bounded multivalued pseudomonotone mapping. This theorem was proved in Le
[33, Theorem 2.2]. We use the notation BR(0) = {u ∈ X : ‖u‖X < R}.

Theorem 2.8. Let X be a real reflexive Banach space, let F : D(F ) ⊂ X → 2X
∗

be
a maximal monotone operator, let G : D(G) = X → 2X

∗
be a bounded multivalued

pseudomonotone operator and let L ∈ X∗. Assume that there exist u0 ∈ X and
R ≥ ‖u0‖X such that D(F ) ∩BR(0) 6= ∅ and

〈ξ + η − L, u− u0〉X∗×X > 0

for all u ∈ D(F ) with ‖u‖X = R, all ξ ∈ F (u) and all η ∈ G(u). Then there exists
u ∈ D(F ) ∩D(G) such that

F (u) +G(u) 3 L.

Finally, we recall the fixed point theorem of Kluge [32] which will be used in the
proof of our main existence result.



6 S. ZENG, Y. BAI, L. GASIŃSKI, I. KRECH EJDE-2021/37

Theorem 2.9. Let Z be a reflexive Banach space and let C ⊂ Z be a nonempty,
closed and convex set. Assume that Ψ: C → 2C is a multivalued mapping such that
for every u ∈ C, the set Ψ(u) is nonempty, closed, and convex, and Gr Ψ (the graph
of Ψ) is sequentially weakly closed. If either C is bounded or Ψ(C) is bounded, then
the map Ψ has at least one fixed point in C.

3. Assumptions and Data Properties

To obtain the existence of solutions for problem (1.1), we need the following
assumptions for the data of problem (1.1).

(H3) f ∈ Lp′(Ω),
(H4) j : Ω× R→ R is a function such that

(i) x 7→ j(x, r) is measurable on Ω for all r ∈ R and there exists a function
l ∈ Lq1(Ω) with q1 ∈ (1, p∗) such that the function x 7→ j(x, l(x))
belongs to L1(Ω).

(ii) r 7→ j(x, r) is locally Lipschitz continuous for a.e. x ∈ Ω.
(iii) there exist θ ≥ 1 with θ ≤ min{q1, p}, αj ≥ 0 with αjλ1,p <

a3
p−1δθ,

and βj ∈ L1
+(Ω) such that for all r ∈ R and a.e. x ∈ Ω it holds

j0(x, r;−r) ≤ αj |r|θ + βj(x),

where

δθ =

{
1 if θ = p,

+∞ otherwise.

(iv) there exist cj ≥ 0 and γj ∈ L
q1

q1−1

+ (Ω) such that

|ξ| ≤ cj |r|q1−1 + γj(x) for all ξ ∈ ∂j(x, r), all r ∈ R and a.e. x ∈ Ω,

where ∂j(x, r) stands for the generalized gradient of j with respect to
the variable r.

(v) there exists a constant mj ≥ 0 such that for all r1, r2 ∈ R and a.e.
x ∈ Ω the inequality is satisfied

(ξ1 − ξ2)(r1 − r2) ≥ −mj |r1 − r2|p

whenever ξ1 ∈ ∂j(x, r1) and ξ2 ∈ ∂j(x, r2).

(H5) T : W 1,p
0 (Ω) → R is a positively homogeneous (i.e., T (tu) = tT (u) for all

t > 0 and u ∈W 1,p
0 (Ω)) and subadditive function such that

T (u) ≤ lim sup
n→∞

T (un) (3.1)

whenever {un} ⊂ W 1,p
0 (Ω) is a sequence such that un ⇀ u in W 1,p

0 (Ω), as

n→∞, for some u ∈W 1,p
0 (Ω).

(H6) U : W 1,p
0 (Ω)→ (0,+∞) is weakly continuous, i.e., for any sequence {un} ⊂

W 1,p
0 (Ω) such that un ⇀ u, as n→∞, for some u ∈W 1,p

0 (Ω), we have

U(un)→ U(u), as n→∞. (3.2)

Remark 3.1. Assumption (H4)(v) is usually called relaxed monotonicity condition
(see e.g. Migórski-Ochal-Sofonea [37]) for the locally Lipschitz function r 7→ j(x, r).
It is equivalent to the inequality

j0(x, s1; s2 − s1) + j0(x, s2; s1 − s2) ≤ mj |s1 − s2|p



EJDE-2021/37 EXISTENCE OF SOLUTIONS FOR IMPLICIT OBSTACLE PROBLEMS 7

for all s1, s2 ∈ R and for a.e. x ∈ Ω.
Indeed, positive homogeneity and subadditivity of T confirm that T is also a

convex function. On the other hand, it is not difficult to see that if T : W 1,p
0 (Ω)→ R

is lower semicontinuous, then inequality (3.1) holds automatically.

Let us introduce a multivalued map K : W 1,p
0 (Ω)→ 2W

1,p
0 (Ω) defined by

K(u) =
{
v ∈W 1,p

0 (Ω) : T (v)− U(u) ≤ 0
}

(3.3)

for all u ∈W 1,p
0 (Ω).

Lemma 3.2. Assume that T : W 1,p
0 (Ω)→ R satisfies H(T ) and let U : W 1,p

0 (Ω)→
(0,+∞) be any map. Then the map K defined by (3.3) has nonempty, closed and
convex values.

Proof. Let u ∈ W 1,p
0 (Ω) be fixed. It follows from the positive homogeneity of T

and U(u) > 0, that T (0) = 0 < U(u), namely, 0 ∈ K(u) 6= ∅ for each u ∈W 1,p
0 (Ω).

Let {vn} ⊂ K(u) be a sequence such that vn → v in W 1,p
0 (Ω) as n → ∞ for

some v ∈W 1,p
0 (Ω). Then, for each n ∈ N, we have

T (vn) ≤ U(u).

Passing to the upper limit as n → ∞ in the above inequality and using (3.1) we
deduce that

T (v) ≤ lim sup
n→∞

T (vn) ≤ U(u).

This means that v ∈ K(u), i.e., the set K(u) is closed.
For any v1, v2 ∈ K(u) and t ∈ (0, 1) fixed, let us set vt = tv1 + (1 − t)v2.

Therefore, T (vi) ≤ U(u) for i = 1, 2. However, the convexity of T (see Remark 3.1)
guarantees

T (vt) ≤ tT (v1) + (1− t)T (v2) ≤ tU(u) + (1− t)U(u) = U(u),

which gives that vt ∈ K(u). Therefore, we conclude that the set K(u) is convex in

W 1,p
0 (Ω). �

The weak solutions for problem (1.1) are understood in the following sense.

Definition 3.3. We say that u ∈ W 1,p
0 (Ω) is a weak solution of problem (1.1) if

u ∈ K(u) and∫
Ω

(a(x,∇u(x)),∇v(x)−∇u(x))RN dx+

∫
Ω

j0(x, u(x); v(x)− u(x)) dx

≥
∫

Ω

f(x)
[
v(x)− u(x)

]
dx

for all v ∈ K(u), where the multivalued function K is given by (3.3).

Consider the function J : Lq1(Ω)→ R defined by

J(u) =

∫
Ω

j(x, u(x)) dx for all u ∈ Lq1(Ω). (3.4)

On account of hypotheses (H4) and the definition of J (see (3.4)), the next lemma
is a direct consequence of Migórski-Ochal-Sofonea [37, Theorem 3.47].

Lemma 3.4. Under assumptions (H4)(i)–(iv), we have

(i) J : Lq1(Ω)→ R is locally Lipschitz continuous;



8 S. ZENG, Y. BAI, L. GASIŃSKI, I. KRECH EJDE-2021/37

(ii) we have

J0(u; v) ≤
∫

Ω

j0(x, u(x); v(x)) dx,

J0(u;−u) ≤ αj‖u‖θθ + ‖βj‖1
for all u, v ∈ Lq1(Ω);

(iii) for each u ∈ Lq1(Ω), we have

∂J(u) ⊂
∫

Ω

∂j(x, u(x)) dx,

‖ξ‖q′1 ≤ cJ(1 + ‖u‖q1−1
q1 ) for all ξ ∈ ∂J(u),

with some cJ > 0.

Moreover, if condition (H4)(v) holds, then

J0(u; v − u) + J0(v;u− v) ≤ mj‖u− v‖pp (3.5)

for all u, v ∈W 1,p
0 (Ω).

4. Auxiliary problems

Employing Lemma 3.4(ii) we know that if u ∈ W 1,p
0 (Ω) solves the following

problem: Find u ∈W 1,p
0 (Ω) such that u ∈ K(u) and∫

Ω

(a(x,∇u(x)),∇v(x)−∇u(x))RNdx+ J0(u; v − u)

≥
∫

Ω

f(x)
[
v(x)− u(x)

]
dx

(4.1)

for all v ∈ K(u), then u is a weak solution to problem (1.1) as well. Using this fact,
we will prove that problem (4.1) is solvable. To this end, first we investigate the
following inequality problem:

Given w ∈W 1,p
0 (Ω), find u ∈ K(w) such that∫
Ω

(a(x,∇u(x)),∇v(x)−∇u(x))RN dx+ J0(u; v − u)

≥
∫

Ω

f(x)
[
v(x)− u(x)

]
dx

(4.2)

for all v ∈ K(w). Additionally, consider the multivalued map Γ: W 1,p
0 (Ω) →

2W
1,p
0 (Ω) given by

Γ(w) =
{
u ∈W 1,p

0 (Ω) : u solves problem (4.2) associated with w
}

(4.3)

for all w ∈W 1,p
0 (Ω). Indeed, it is not difficult to verify that u ∈W 1,p

0 (Ω) is a fixed
point of Γ, if and only if u solves problem (4.1). Motivated by this fact, we shall
employ Kluge’s fixed point theorem (see Theorem 2.9), to show that the fixed point
set of Γ is nonempty.

Theorem 4.1. Let U : W 1,p
0 (Ω) → (0,+∞). Under the assumptions (H1), (H3),

(H4)(i)–(iv) and (H5), we have

(i) for each w ∈ W 1,p
0 (Ω), the set of solutions to problem (4.2) is nonempty,

bounded, and closed in W 1,p
0 (Ω), i.e., Γ has nonempty, bounded, and closed

values.



EJDE-2021/37 EXISTENCE OF SOLUTIONS FOR IMPLICIT OBSTACLE PROBLEMS 9

(ii) if p ≥ 2, hypotheses (H4)(v), H(0), and the smallness condition

mjλ
−1
1,p ≤ ma, (4.4)

are fulfilled, then for each w ∈ W 1,p
0 (Ω), the set of solutions to problem

(4.2) is convex, namely, Γ(w) is convex.

Proof. (i) Let w ∈W 1,p
0 (Ω) be fixed and IK(w) : W 1,p

0 (Ω)→ R = R∪ {+∞} be the
indicator function of K(w), i.e.,

IK(w)(u) =

{
0 if u ∈ K(w),

+∞ otherwise.

Keeping in mind that f ∈ Lp′(Ω) ⊂W 1,p
0 (Ω)∗, problem (4.2) can be rewritten equiv-

alently to the following variational-hemivariational inequality: Find u ∈ W 1,p
0 (Ω)

such that

〈Au, v − u〉+ J0(u; v − u) + IK(w)(v)− IK(w)(u) ≥ 〈f, v − u〉 (4.5)

for all v ∈ W 1,p
0 (Ω), where A : W 1,p

0 (Ω) → W 1,p
0 (Ω)∗ is given by (2.4). However,

by the Hahn-Banach Theorem, see e.g. Brezis [3, Theorem 1.6 (the first geometric
form)], it is not difficult to prove that problem (4.5) is equivalent to the following

inclusion problem: Find u ∈W 1,p
0 (Ω) such that

Au+ ∂J(u) + ∂CIK(w)(u) 3 f, (4.6)

where the notation ∂CIK(w) stands for the subdifferential of IK(w) in the sense of
convex analysis.

We shall use the surjectivity result (see Theorem 2.8), to show that problem

(4.6) is solvable in W 1,p
0 (Ω). For this reason, we start with the following claim.

Claim 1. A+∂J : W 1,p
0 (Ω)→ 2W

1,p
0 (Ω)∗ is a bounded pseudomonotone multivalued

operator such that for each u ∈W 1,p
0 (Ω), the set A(u) +∂J(u) is closed and convex

in W 1,p
0 (Ω)∗.

Directly from Proposition 2.2 and Lemma 3.4 we see the set A(u) + ∂J(u) is

closed and convex in W 1,p
0 (Ω)∗ for each u ∈ W 1,p

0 (Ω). Moreover, Proposition 2.5,

Lemma 3.4(iii) and the fact q1 < p∗ indicate that W 1,p
0 (Ω) 3 u 7→ A(u) + ∂J(u) ⊂

W 1,p
0 (Ω)∗ is a bounded map.

Next, we assert that W 1,p
0 (Ω) 3 u 7→ A(u) + ∂J(u) ⊂ W 1,p

0 (Ω)∗ is upper

semicontinuous from W 1,p
0 (Ω) to W 1,p

0 (Ω)∗ with weak topology. By f Migórski-
Ochal-Sofonea [37, Proposition 3.8], it is sufficient to show that for any weakly

closed subset D in W 1,p
0 (Ω)∗, the set (A + ∂J)−(D) is closed in W 1,p

0 (Ω). Let
{un} ⊂ (A+ ∂J)−(D) be a sequence such that

un → u in W 1,p
0 (Ω) as n→∞, for some u ∈W 1,p

0 (Ω). (4.7)

So, for each n ∈ N, we are able to find ξn ∈ ∂J(un) such that

u∗n = Aun + ξn ∈ (A(un) + ∂J(un)) ∩D.

But, the continuity of A (see Proposition 2.5) ensures that A(un) → A(u) in

W 1,p
0 (Ω)∗, as n → ∞. Taking into account Lemma 3.4(iii) and convergence (4.7),

we conclude that the sequence {ξn} is bounded in W 1,p
0 (Ω)∗, so, without any

loss of generality, we may assume that ξn ⇀ ξ in W 1,p
0 (Ω)∗, as n → ∞, with



10 S. ZENG, Y. BAI, L. GASIŃSKI, I. KRECH EJDE-2021/37

some ξ ∈ W 1,p
0 (Ω)∗. Notice that ∂J is upper semicontinuous from W 1,p

0 (Ω) to w-

W 1,p
0 (Ω)∗ and has bounded, convex, closed values (see Proposition 2.2(d)), so, it

has a closed graph in W 1,p
0 (Ω)×w−W 1,p

0 (Ω)∗ (see Kamenskii-Obukhovskii-Zecca
[31, Theorem 1.1.4]). But, thanks to the weak closedness of D, we derive that
A(u) + ξ ∈ D and ξ ∈ ∂J(u), which provides that u ∈ (A + ∂J)−(D). Conse-

quently, A + ∂J is upper semicontinuous from W 1,p
0 (Ω) to W 1,p

0 (Ω)∗ with weak
topology.

Next, we show that A+∂J is pseudomonotone. Let {un} and {u∗n} be sequences
such that

un ⇀ u in W 1,p
0 (Ω), (4.8)

u∗n ∈ A(un) + ∂J(un) with lim sup
n→∞

〈u∗n, un − u〉 ≤ 0. (4.9)

Our goal is to show that for each v ∈ W 1,p
0 (Ω) there exists an element u∗(v) ∈

A(u) + ∂J(u) such that

lim inf
n→∞

〈u∗n, un − v〉 ≥ 〈u∗(v), u− v〉. (4.10)

From (4.9), we are able to find a sequence {ξn} ⊂ W 1,p
0 (Ω)∗ such that for each

n ∈ N, ξn ∈ ∂J(un) and

u∗n = A(un) + ξn.

The latter combined with the inequality in (4.9) implies

lim sup
n→∞

〈Aun, un − u〉+ lim inf
n→∞

〈ξn, un − u〉 ≤ 0. (4.11)

Applying (4.8) and the compactness of the embedding of W 1,p
0 (Ω) into Lq1(Ω), gives

un → u in Lq1(Ω), as n→∞.
On the other hand, employing Chang [7, Theorem 2.2], we have

∂(J |W 1,p
0 (Ω))(u) ⊂ ∂(J |Lq1 (Ω))(u) for all u ∈W 1,p

0 (Ω),

which implies that
〈ξn, un − u〉 = 〈ξn, un − u〉Lq1 (Ω). (4.12)

Additionally, Lemma 3.4(iii) and the boundedness of the sequence {un} in W 1,p
0 (Ω)

implies that the sequence {ξn} is contained in Lq1(Ω). Then, passing to the limit
in (4.12) as n→∞ to obtain

lim
n→∞

〈ξn, un − u〉 = lim
n→∞

〈ξn, un − u〉Lq1 (Ω) = 0.

Inserting the above equality into (4.11) yields

lim sup
n→∞

〈Aun, un − u〉 = lim sup
n→∞

〈Aun, un − u〉+ lim inf
n→∞

〈ξn, un − u〉 ≤ 0.

The latter combined with Proposition 2.5 (i.e., the fact that A is type of (S+))

and (4.8) finds that un → u in W 1,p
0 (Ω), as n → ∞. Moreover, the reflexivity of

W 1,p
0 (Ω)∗ and boundedness of {ξn} ⊂W 1,p

0 (Ω)∗ permit us to conclude that

ξn ⇀ ξ in W 1,p
0 (Ω)∗ for some ξ ∈W 1,p

0 (Ω)∗.

Now we can assert that ξ ∈ ∂J(u) (see, e.g., Kamenskii-Obukhovskii-Zecca [31,
Theorem 1.1.4]). Now, because

lim inf
n→∞

〈u∗n, un − v〉 = lim inf
n→∞

〈A(un) + ξn, un − v〉 = 〈A(u) + ξ, u− v〉,



EJDE-2021/37 EXISTENCE OF SOLUTIONS FOR IMPLICIT OBSTACLE PROBLEMS 11

it is clear that (4.10) holds with u∗ = A(u) + ξ ∈ A(u) + ∂J(u). Therefore, A+ ∂J
is pseudomonotone. This proves Claim 1.

Next, we prove that there exists R > 0 such that

〈Au+ ξ + η − f, u〉 > 0 (4.13)

for all u ∈ K(w) with ‖u‖ = R, all ξ ∈ ∂J(u) and all η ∈ ∂C(IK(w))(u).

For this purpose, let u ∈ W 1,p
0 (Ω) be fixed. For any ξ ∈ ∂J(u) and η ∈

∂C(IK(w))(u), since 0 ∈ K(w) and f ∈ Lp′(Ω) ⊂W 1,p
0 (Ω)∗, we have

〈Au+ ξ + η − f, u〉

≥
∫

Ω

(a(x,∇u(x)),∇u(x))RN dx+

∫
Ω

ξ(x)u(x) dx+ IK(w)(u)− IK(w)(0)

− ‖f‖W 1,p
0 (Ω)∗‖u‖

≥ a3

p− 1
‖∇u‖pp −

∫
Ω

ξ(x)[−u(x)] dx+ IK(w)(u)− ‖f‖W 1,p
0 (Ω)∗‖u‖

≥ a3

p− 1
‖∇u‖pp − J0(u;−u) + IK(w)(u)− ‖f‖W 1,p

0 (Ω)∗‖u‖,

(4.14)

where we have used Lemma 2.3(iii). Notice that IK(w) : W 1,p
0 (Ω)→ R is a proper,

convex and lower semicontinuous function, so we now apply Gasiński-Papageorgiou
[22, Proposition 1.3.1], for finding aK(w), bK(w) ≥ 0 such that

IK(w)(v) ≥ −aK(w)‖v‖ − bK(w) for all v ∈W 1,p
0 (Ω). (4.15)

Additionally, Lemma 3.4(ii) implies that

J0(u;−u) ≤ αj‖u‖θθ + ‖βj‖1. (4.16)

We now distinguish two cases: θ < p and θ = p. When θ < p, let c(θ) > 0 be
such that

‖u‖θ ≤ c(θ)‖u‖ for all u ∈W 1,p
0 (Ω) (4.17)

(its existence is follows from the continuity of the embedding from W 1,p
0 (Ω) to

Lr(Ω) for any r ∈ (1, p∗)). Inserting (4.15) and (4.16) into (4.14) and using (4.17),
we have

〈Au+ ξ + η − f, u〉 ≥ a3

p− 1
‖∇u‖pp − αj‖u‖θθ − ‖βj‖1 − aK(w)‖u‖

− bK(w) − ‖f‖W 1,p
0 (Ω)∗‖u‖

≥ a3

p− 1
‖u‖p − αjc(θ)θ‖u‖θ − ‖βj‖1 − aK(w)‖u‖

− bK(w) − ‖f‖W 1,p
0 (Ω)∗‖u‖.

(4.18)

Since θ < p, we can find a constant R0 > 0 large enough such that

a3

p− 1
Rp0 − αjc(θ)θRθ0 − ‖βj‖1 − aK(w)R0 − bK(w) − ‖f‖W 1,p

0 (Ω)∗R0 > 0.

Therefore, for each R ≥ R0 fixed, the desired inequality (4.13) holds.



12 S. ZENG, Y. BAI, L. GASIŃSKI, I. KRECH EJDE-2021/37

Next, if θ = p, using variational characterization of λ1,p (see (2.2)), we deduce
that

〈Au+ ξ + η − f, u〉

≥ a3

p− 1
‖∇u‖pp − αj‖u‖pp − ‖βj‖1 − aK(w)‖u‖ − bK(w) − ‖f‖W 1,p

0 (Ω)∗‖u‖

≥ (
a3

p− 1
− αjλ−1

1,p)‖∇u‖pp − ‖βj‖1 − aK(w)‖u‖ − bK(w) − ‖f‖W 1,p
0 (Ω)∗‖u‖.

(4.19)

As 1 < p and αjλ
−1
1,p <

a3
p−1 , we can take R0 > 0 large enough such that for all

R ≥ R0 it holds( a3

p− 1
− αjλ−1

1,p

)
Rp − ‖βj‖1 − aK(w)R− bK(w) − ‖f‖W 1,P

0 (Ω)∗R > 0.

Therefore, the inequality (4.13) holds.

Recall that IK(w) : W 1,p
0 (Ω) → R is a proper, convex and lower semicontinuous

function, so, ∂CIK(w) : W 1,p
0 (Ω) → 2W

1,p
0 (Ω)∗ is maximal monotone. The latter to-

gether with Theorem 2.8 implies that there exists uw ∈W 1,p
0 (Ω) resolving inclusion

(4.6). Thus, Γ(w) 6= ∅ for each w ∈W 1,p
0 (Ω).

Next, we demonstrate that Γ(w) is closed in W 1,p
0 (Ω). Let {un} ⊂ Γ(w) be such

that
un → u in W 1,p

0 (Ω) as n→∞
for some u ∈W 1,p

0 (Ω). So, for each n ∈ N, we have

〈Aun, v − un〉+ J0(un; v − un) + IK(w)(v)− IK(w)(un) ≥ 〈f, v − un〉

for all v ∈ W 1,p
0 (Ω). Passing to the upper limit as n→∞ in the above inequality,

we obtain

〈Au, v − u〉+ J0(u; v − u) + IK(w)(v)− IK(w)(u)

≥ lim sup
n→∞

[
〈Aun, v − un〉+ J0(un; v − un) + IK(w)(v)− IK(w)(un)

]
≥ lim sup

n→∞
〈f, v − un〉

= 〈f, v − u〉

for all v ∈ W 1,p
0 (Ω), where we have used the continuity of A (see Proposition 2.5),

upper semicontinuity of (u, v) 7→ J0(u; v) (see Proposition 2.2(d)) and lower semi-
continuity of IK(w). This indicates that u ∈ Γ(w), hence Γ(w) is closed.

Finally, we prove that Γ(w) is bounded. Arguing by contradiction, we suppose
that Γ(w) is unbounded. Then there exists a sequence {un} in Γ(w) such that

‖un‖ → +∞ as n→∞. (4.20)

By a simple computation (see (4.18) and (4.19)), we are able to find N0 ∈ N such
the for all n ≥ N0, it holds

0 ≥ 〈Aun, un〉 − J0(un;−un) + IK(w)(un) > 0,

where we have used the fact 0 ∈ K(w) and (4.20). This leads to a contradiction.
Therefore, Γ(w) is bounded.

(ii) Assume that hypothesis (H3)(v) holds. Let u1, u2 ∈W 1,p
0 (Ω) be two solutions

to problem (4.2). Hence

〈Aui, v − ui〉+ J0(ui; v − ui) + IK(w)(v)− IK(w)(ui) ≥ 〈f, v − ui〉



EJDE-2021/37 EXISTENCE OF SOLUTIONS FOR IMPLICIT OBSTACLE PROBLEMS 13

for all v ∈W 1,p
0 (Ω) and for i = 1, 2. But, Proposition 2.5 and Lemma 3.4 give

0 ≥ 〈Av1 −Av2, v1 − v2〉 −
(
J0(v1; v2 − v1) + J0(v2; v1 − v2)

)
≥ ma‖∇v1 −∇v2‖pp −mj‖v1 − v2‖pp
≥ (ma −mjλ

−1
1,p)‖v1 − v2‖pp ≥ 0

for all v1, v2 ∈W 1,p
0 (Ω). Hence, for i = 1, 2, we have

〈Av, v − ui〉+ J0(v; v − ui) + IK(w)(v)− IK(w)(ui) ≥ 〈f, v − ui〉

for all v ∈W 1,p
0 (Ω). Let t ∈ (0, 1) be arbitrary and let us put ut = tu1 + (1− t)u2.

Therefore, we have

〈Av, v − ut〉+ J0(v; v − ut) + IK(w)(v)− IK(w)(ut)

≥ t
[
〈Av, v − u1〉+ J0(v; v − u1) + IK(w)(v)− IK(w)(u1)

]
+ (1− t)

[
〈Av, v − u2〉+ J0(v; v − u2) + IK(w)(v)− IK(w)(u2)

]
≥ 〈f, v − ut〉

for all v ∈W 1,p
0 (Ω).

Now, employing the Minty approach we obtain that ut ∈ Γ(w). Consequently,

the set Γ(w) is convex in W 1,p
0 (Ω). �

5. Main result

Now we can state the main result of the paper. Its proof is based on Theorem 4.1
and Kluge’s fixed point theorem (see Theorem 2.9).

Theorem 5.1. Assume that (H1), (H3)–(H5), (H6) hold and p ≥ 2. If, in addition,
(H2)) and the smallness condition (4.4) are satisfied, then the set of solutions of
problem (1.1), denoted by S, is nonempty, bounded and weakly closed.

Proof. As we have already mentioned, the fixed point set of Γ (see (4.3)) is the
corresponding set of solutions to problem (4.1). Besides, Lemma 3.2 points out
that the set of solutions for problem (4.1) is a subset of the set of solutions for
problem (1.1). Consequently, it suffices to prove that the fixed point set of Γ is
nonempty.

First we show that

Gr Γ is sequentially weakly closed. (5.1)

For this purpose, let {wn}, {un} ⊂ W 1,p
0 (Ω) be two sequences such that wn ⇀ w

in W 1,p
0 (Ω) and un ∈ Γ(wn) with un ⇀ u in W 1,p

0 (Ω), as n → ∞, for some w, u ∈
W 1,p

0 (Ω). Then, for each n ∈ N, we have un ∈ K(wn) (namely, T (un) ≤ U(un))
and

〈Aun, v − un〉+ J0(un; v − un) ≥ 〈f, v − un〉 (5.2)

for all v ∈ K(wn).
However, hypotheses (H5) and (H6) imply that

T (u) ≤ lim sup
n→∞

T (un) ≤ lim sup
n→∞

U(wn) ≤ U(w).

This means u ∈ K(w).



14 S. ZENG, Y. BAI, L. GASIŃSKI, I. KRECH EJDE-2021/37

For any v ∈ K(w) fixed, owing to U(w) > 0, we now consider the sequence {vn}
constructed by

vn =
U(wn)

U(w)
v for all n ∈ N.

The non-negativity of U , positive homogeneity of T and the fact that v ∈ K(w)
(thus is, T (v) ≤ U(w)) give

T (vn) = T (
U(wn)

U(w)
v) =

U(wn)

U(w)
T (v) ≤ U(wn)U(w)

U(w)
= U(wn),

hence vn ∈ K(wn). Moreover, a simple calculating gives

lim
n→∞

‖vn − v‖ = lim
n→∞

|U(wn)− U(w)| ‖v‖
U(w)

= 0.

Thus, we obtain that vn → v, as n→∞.
Since u ∈ K(w), we can take the sequence {ūn} ⊂ W 1,p

0 (Ω) such that ūn =
U(wn)
U(w) u ∈ K(wn) for each n ∈ N and

ūn → u as n→∞.

Inserting v = ūn into (5.2) gives

〈Aun, un − ūn〉 ≤ J0(un; ūn − un)− 〈f, ūn − un〉. (5.3)

It follows from Lemma 3.4 and the convergence un → u in Lq1(Ω), as n→∞ that

lim sup
n→∞

J0(un; ūn − un) ≤ 0.

Passing to the upper limit as n→∞ into (5.3) and using the above inequality, we
have

lim sup
n→∞

〈Aun, un − u〉 ≤ lim sup
n→∞

〈Aun, un − u〉+ lim inf
n→∞

〈Aun, u− ūn〉

≤ lim sup
n→∞

〈Aun, un − ūn〉

≤ lim sup
n→∞

J0(un; ūn − un)− lim inf
n→∞

〈f, ūn − un〉 ≤ 0.

The latter combined with Proposition 2.5 (A is of type (S+)) implies un → u in

W 1,p
0 (Ω), as n→∞.

For any v ∈ K(w) fixed, let {vn} ⊂ W 1,p
0 (Ω) be such that vn ∈ K(wn) for each

n ∈ N and vn → v in W 1,p
0 (Ω), as n → ∞. We put v = vn in (5.2) and then pass

to the upper limit as n→∞, to obtain

〈Au, v − u〉+ J0(u; v − u) ≥ lim sup
n→∞

〈Aun, vn − un〉+ lim sup
n→∞

J0(un; vn − un)

≥ lim sup
n→∞

[
〈Aun, vn − un〉+ J0(un; vn − un)

]
≥ lim sup

n→∞
〈f, vn − un〉 = 〈f, v − u〉,

where we have used the upper semicontinuity of Lq1(Ω) × Lq1(Ω) 3 (v, u) →
J0(u; v) ∈ R (see Proposition 2.2). Hence, u ∈ Γ(w). Therefore, we conclude
that Gr Γ is sequentially weakly closed. This proves (5.1).

Next we show that

the set Γ(W 1,p
0 (Ω)) is bounded in W 1,p

0 (Ω). (5.4)



EJDE-2021/37 EXISTENCE OF SOLUTIONS FOR IMPLICIT OBSTACLE PROBLEMS 15

If the above were not true, then there would exist a sequence {wn} such that

‖un‖ → ∞ as n→∞, (5.5)

where un = Γ(wn). For every n ∈ N, one has (5.2) for all v ∈ K(wn). Keeping in

mind that 0 ∈ K(w) for each w ∈ W 1,p
0 (Ω), we take v = 0 as test function in (5.2)

obtaining
〈Aun, un〉 − J0(un;−un) ≤ ‖f‖W 1,p

0 (Ω)∗‖un‖.
Using the same argument as in the proof of Theorem 4.1 (see (4.18) or (4.19)), we
could find N0 ∈ N large enough such that

0 < 〈Aun, un〉 − J0(un;−un)− ‖f‖W 1,p
0 (Ω)∗‖un‖ ≤ 0

for all n ≥ N0, this gives a contradiction. Therefore, we conclude that the set
Γ(W 1,p

0 (Ω)) is bounded in W 1,p
0 (Ω). This proves (5.4).

To conclude the proof, we need to verify the conditions of Theorem 2.9 for the
mapping Γ. Then, Γ will admit a fixed point in W 1,p

0 (Ω), which will imply that

problem (1.1) has at least one weak solution in W 1,p
0 (Ω).

Indeed, the boundedness of S can be obtained directly via using the analogous
arguments as in the proof of (5.4).

It remains to illustrate the weak closedness of S. Let {un} ⊂ S be a sequence

such that un ⇀ u in W 1,p
0 (Ω), as n → ∞, for some u ∈ W 1,p

0 (Ω). Hence, for each
n ∈ N, it is easy to see that un ∈ K(un) and

〈Av, v − un〉+

∫
Ω

j0(v(x); v(x)− un(x)) dx ≥ 〈f, v − un〉 (5.6)

for all v ∈ K(un). Because GrK is sequentially weakly closed (see the proof of

(5.1)), this implies u ∈ K(u). For any v ∈ K(u), set vn = U(un)
U(u) v. We have

vn ∈ K(un) and vn → v in W 1,p
0 (Ω), as n → ∞. Putting v = vn into (5.6) and

passing to the upper limit as n→∞, we obtain

〈Av, v − u〉+

∫
Ω

j0(v(x); v(x)− u(x)) dx ≥ 〈f, v − u〉

for all v ∈ K(u), where we have applied Fatou’s lemma. Invoking Minty approach,

we obtain u ∈ S, therefore, S is weakly closed in W 1,p
0 (Ω). �

Acknowledgment. This work was supported by the NNSF of China Grant Nos.
12001478, 12026255 and 12026256, and by the European Union’s Horizon 2020
Research and Innovation Programme under the Marie Sklodowska-Curie grant
agreement No. 823731 CONMECH, by the National Science Center of Poland un-
der Preludium Project No. 2017/25/N /ST1/00611, and by the Startup Project
of Doctor Scientific Research of Yulin Normal University No. G2020ZK07. It
was also supported by the Natural Science Foundation of Guangxi Grant No.
2020GXNSFBA297137, and by the Ministry of Science and Higher Education of
Republic of Poland under Grants Nos. 4004/GGPJIIH2020/2018/0, 3792/GGPJ/
H2020/2017/0, and 440328/PnH2/2019.

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Shengda Zeng (corresponding author)

Guangxi Colleges and Universities Key Laboratory of Complex System Optimization

and Big Data Processing, Yulin Normal University, Yulin 537000, China.
Jagiellonian University in Krakow, Faculty of Mathematics and Computer Science, ul.

Lojasiewicza 6, 30-348 Krakow, Poland
Email address: zengshengda@163.com

Yunru Bai

School of Science, Guangxi University of Science and Technology, Liuzhou, Guangxi
545006, China

Email address: yunrubai@163.com

Leszek Gasiński

Pedagogical University of Cracow, Department of Mathematics, Podchorazych 2, 30-
084 Cracow, Poland

Email address: leszek.gasinski@up.krakow.pl

Ireneusz Krech
Pedagogical University of Cracow, Department of Mathematics, Podchorazych 2, 30-

084 Cracow, Poland

Email address: ireneusz.krech@up.krakow.pl


	1. Introduction
	2. Preliminaries
	3. Assumptions and Data Properties
	4. Auxiliary problems
	5. Main result
	Acknowledgment

	References