Startseite Multiplicity of k-convex solutions for a singular k-Hessian system
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Multiplicity of k-convex solutions for a singular k-Hessian system

  • Zedong Yang und Zhanbing Bai EMAIL logo
Veröffentlicht/Copyright: 9. Dezember 2024
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Demonstratio Mathematica
Aus der Zeitschrift Demonstratio Mathematica Band 57 Heft 1

Abstract

In this article, we study the following nonlinear k -Hessian system with singular weights

S k 1 k ( σ ( D 2 u 1 ) ) = λ b ( ∣ x ∣ ) f ( − u 1 , − u 2 ) , in Ω , S k 1 k ( σ ( D 2 u 2 ) ) = λ h ( ∣ x ∣ ) g ( − u 1 , − u 2 ) , in Ω , u 1 = u 2 = 0 , on ∂ Ω ,

where λ > 0 , 1 ≤ k ≤ N is an integer, Ω stands for the open unit ball in R N , and S k ( σ ( D 2 u ) ) is the k -Hessian operator of u . By using the fixed point index theory, we prove the existence and nonexistence of negative k -convex radial solutions. Furthermore, we establish the multiplicity result of negative k -convex radial solutions based on a priori estimate achieved. More precisely, there exists a constant λ ∗ > 0 such that the system admits at least two negative k -convex radial solutions for λ ∈ ( 0 , λ ∗ ) .

Keywords: k-Hessian system; k-convex solution; multiplicity; upper and lower solution
MSC 2010: 35J60; 35J75; 35B09; 35A16

1 Introduction

In this article, we deal with a class of nonlinear singular k -Hessian system with the 0-Dirichlet boundary value condition

(1.1) S k 1 k ( σ ( D 2 u 1 ) ) = λ b ( ∣ x ∣ ) f ( − u 1 , − u 2 ) , in Ω , S k 1 k ( σ ( D 2 u 2 ) ) = λ h ( ∣ x ∣ ) g ( − u 1 , − u 2 ) , in Ω , u 1 = u 2 = 0 , on ∂ Ω ,

where λ > 0 , Ω = { x ∈ R N : ∣ x ∣ < 1 } , 1 ≤ k ≤ N is an integer, the nonlinearities f , g and the weights b , h satisfy the following conditions:

(A1): f , g ∈ C ( [ 0 , ∞ ) × [ 0 , ∞ ) , ( 0 , ∞ ) ) .

(A2): b , h ∈ C ( [ 0 , 1 ) , [ 0 , ∞ ) ) are singular at 1, and not vanishing identically on any subinterval of [ 0 , 1 ) , ∫ 0 1 b k ( s ) d s < + ∞ and ∫ 0 1 h k ( s ) d s < + ∞ .

(A3):

f ∞ = lim s + t → ∞ f ( s , t ) s + t = ∞ , g ∞ = lim s + t → ∞ g ( s , t ) s + t = ∞ .

For k = 1 , 2 , … , N and u ∈ C 2 ( Ω ) , S k ( σ ( D 2 u ) ) is the k -Hessian operator of u , and given by

S k ( σ ( D 2 u ) ) = S k ( σ 1 , σ 2 , … , σ N ) = ∑ 1 ≤ j 1 < ⋅ ⋅ ⋅ < j k ≤ N σ j 1 ⋅ σ j 2 … σ j k ,

where σ 1 , σ 2 , … , σ N are the eigenvalues of the Hessian matrix ( D 2 u ) of second derivatives of u . The k -Hessian equation was initiated by Caffarelli et al. [1], and treated as well by Ivochkina [2] in 1985. A natural class of functions for the solutions to the k -Hessian equations is the k -convex function, which is discussed by Trudinger and Wang [3]. A function u ∈ C 2 ( Ω ) is called k -convex if and only if σ ( D 2 u ( x ) ) ∈ Γ ¯ k , where

Γ k ≔ { σ ( D 2 u ) ∈ R N : S l ( σ ( D 2 u ) ) > 0 , 1 ≤ l ≤ k } .

When k = 1 , the k -Hessian operator is the Laplace operator; while for k = N , it corresponds to the Monge-Ampère operator, namely,

Laplace operator Monge-Ampère operator
S 1 ( σ ( D 2 u ( x ) ) ) = ∑ i = 1 N σ i = Δ u , S N ( σ ( D 2 u ( x ) ) ) = ∏ i = 1 N σ i = det D 2 u

For 2 ≤ k ≤ N , it is a fully nonlinear partial differential operator [4]. In recent years, more attention has been paid to the study of the k -Hessian equations motivated by application to differential geometry, fluid dynamics, physics, and other applied subjects. In [5], it is indicated that Monge-Ampère problems have a connection with a reflector shape design or Weingarten curvature. It is noted in [6,7] that these equations are related to theoretical condensed matter physics and renormalization group and are used to describe some phenomena in statistical physics. There are many excellent results about the existence, uniqueness, and multiplicity of the solutions for the k -Hessian equations by means of different methods [8–30].

In 2016, by applying a Pohozaev-type identity and the monotone separation techniques, Wei [12] studied the uniqueness of negative radial solutions for the k -Hessian equation:

(1.2) S k ( σ ( D 2 u ) ) = f ( − u ) , in Ω , u = 0 , on ∂ Ω ,

where f ∈ C 2 ( [ 0 , ∞ ) ) , f ( 0 ) = 0 , f ( s ) > 0 on ( 0 , ∞ ) and satisfies − s f ′ ( − s ) > k f ( − s ) , or − s f ′ ( − s ) < k f ( − s ) for s < 0 .

In 2020, by applying the monotone iterative method, Wang et al. [17] showed the existence of entire positive bounded and blow-up radial solution for the following problem with the k -Hessian operator

(1.3) G S k 1 k ( σ ( D 2 u 1 ) ) S k 1 k ( σ ( D 2 u 1 ) ) = b ( ∣ x ∣ ) f ( u 1 , u 2 ) , x ∈ R N , G S k 1 k ( σ ( D 2 u 2 ) ) S k 1 k ( σ ( D 2 u 2 ) ) = h ( ∣ x ∣ ) g ( u 1 , u 2 ) , x ∈ R N ,

where G is a nonlinear operator, f , g ∈ C ( [ 0 , ∞ ) × [ 0 , ∞ ) ) are positive and increasing for every variables.

In 2022, by using Guo-Krasnosel’skii fixed-point theorem [31,32], we [23] considered the k -Hessian problem

(1.4) S k ( σ ( D 2 u 1 ) ) = λ 1 b ( ∣ x ∣ ) f ( − u 1 , − u 2 ) , in Ω , S k ( σ ( D 2 u 2 ) ) = λ 2 h ( ∣ x ∣ ) g ( − u 1 , − u 2 ) , in Ω , u 1 = u 2 = 0 , on ∂ Ω ,

under the assumption that f and g are k -sublinear or k -superlinear growth at 0 or ∞ . The problem (1.4) has at least one radial solution for all λ i > 0 , and at least two radial solutions for the case λ i ≥ μ 1 k > 0 or λ i ≤ μ 2 k , where λ i , μ i > 0 , i = 1 , 2 .

Recently, Zhang [24] used the index theory of fixed points for compact maps [33] to deal with the k i -Hessian problem

(1.5) ( − 1 ) k i S k i 1 k i ( σ ( D 2 u i ) ) = b i ( ∣ x ∣ ) f ( u 1 , u 2 , … , u m ) , in Ω , u = 0 , on ∂ Ω ,

where k i ∈ { 1 , 2 , … , N } , N < 2 k i , i ∈ { 1 , 2 , … , m } , f ∈ C ( [ 0 , ∞ ) m , [ 0 , ∞ ) ) . He showed that problem (1.5) has one, two, and many positive radial solutions under various hypotheses on the function f .

Inspired by the above works, by using the fixed point index theory [31,32], we present new results on the existence and nonexistence of negative k -convex radial solutions for the singular k -Hessian problem (1.1) in a ball Ω . Furthermore, by establishing a prior estimate and combining upper and lower solution method, the multiplicity of negative k -convex radial solutions for the system (1.1) is also obtained. Compared with the works [21–24], our results consider not only the existence and multiplicity of radial solutions but also the nonexistence of radial solutions, which complement and improve the previous research works.

2 Preliminaries

In this section, we first present a lemma of the property of radial functions with respect to S k ( σ ( D 2 u ) ) , which will be used in the proof of the main results.

Lemma 2.1

[8] Suppose that v ( r ) ∈ C 2 [ 0 , R ) and v ′ ( 0 ) = 0 , then the function u ( x ) = v ( r ) with r = ∣ x ∣ < R belongs to C 2 ( B R ) , and

σ ( D 2 u ) = v ″ ( r ) , v ′ ( r ) r , … , v ′ ( r ) r , r ∈ ( 0 , R ) , ( v ″ ( 0 ) , v ″ ( 0 ) , … , v ″ ( 0 ) ) , r = 0 , S k ( σ ( D 2 u ) ) = C N − 1 k − 1 v ″ ( r ) v ′ ( r ) r k − 1 + C N − 1 k v ′ ( r ) r k , r ∈ ( 0 , R ) , C N k ( v ″ ( 0 ) ) k , r = 0 ,

where B R = { x ∈ R N : ∣ x ∣ < R } and C N k = N ! k ! ( N − k ) ! .

Lemma 2.2

[8] Let f ( t ) be a continuous function defined on R and satisfying that f ( t ) > 0 is monotone non-decreasing in ( 0 , + ∞ ) and f ( t ) = 0 in ( − ∞ , 0 ] . For any positive number a, assume v ( r ) ∈ C 0 [ 0 , R ) ⋂ C 1 ( 0 , R ) is a solution of the Cauchy problem

v ′ ( r ) = k C N − 1 k − 1 r k − N ∫ 0 r s N − 1 f k ( v ( s ) ) d s 1 k , r > 0 , v ( 0 ) = a .

Then v ( r ) ∈ C 2 [ 0 , R ) , and it satisfies equation

C N − 1 k − 1 v ″ ( r ) v ′ ( r ) r k − 1 + C N − 1 k v ′ ( r ) r k = f k ( v ( r ) ) ,

with v ′ ( 0 ) = 0 and

σ r ≔ v ″ ( r ) , v ′ ( r ) r , … , v ′ ( r ) r ∈ Γ k ,

for 0 ≤ r < R .

By using the analogous proof of Lemma 2.2, we have the following Lemma.

Lemma 2.3

Assume that (A1) and (A2) are satisfied and ( v 1 ( r ) , v 2 ( r ) ) ∈ ( C 0 [ 0 , 1 ] ⋂ C 1 ( 0 , 1 ] ) 2 is a solution of the Cauchy problem

v 1 ′ ( r ) = − k C N − 1 k − 1 r k − N ∫ 0 r s N − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k , 0 < r < 1 , v 2 ′ ( r ) = − k C N − 1 k − 1 r k − N ∫ 0 r s N − 1 [ λ h ( s ) g ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k , 0 < r < 1 , v 1 ( 1 ) = 0 , v 2 ( 1 ) = 0 .

Then ( v 1 ( r ) , v 2 ( r ) ) ∈ C 2 [ 0 , 1 ] × C 2 [ 0 , 1 ] , and it satisfies the system

C N − 1 k − 1 ( − v 1 ) ″ ( r ) − v 1 ′ ( r ) r k − 1 + C N − 1 k − v 1 ′ ( r ) r k = [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k , C N − 1 k − 1 ( − v 2 ) ″ ( r ) − v 2 ′ ( r ) r k − 1 + C N − 1 k − v 2 ′ ( r ) r k = [ λ h ( s ) g ( v 1 ( s ) , v 2 ( s ) ) ] k ,

with v i ′ ( 0 ) = 0 and

σ i , r ≔ − v i ″ ( r ) , − v i ′ ( r ) r , … , − v i ′ ( r ) r ∈ Γ ¯ k ,

for 0 ≤ r < 1 , where i = 1 , 2 .

By the standard derivation, one can convert the singular problem (1.1) with the 0-Dirichlet boundary value condition u 1 ∣ ∂ Ω = u 2 ∣ ∂ Ω = 0 to the following boundary value problem. An analogous proof process can also be found in [21,23].

Lemma 2.4

( u 1 ( x ) , u 2 ( x ) ) = ( − v 1 ( r ) , − v 2 ( r ) ) is a radial solution of the singular problem (1.1) if and only if ( v 1 ( r ) , v 2 ( r ) ) is a solution of the following boundary value problem

(2.1) r N − k k [ ( − v 1 ( r ) ) ′ ] k ′ = r N − 1 C N − 1 k − 1 [ λ b ( r ) f ( v 1 ( r ) , v 2 ( r ) ) ] k , 0 < r < 1 , r N − k k [ ( − v 2 ( r ) ) ′ ] k ′ = r N − 1 C N − 1 k − 1 [ λ h ( r ) g ( v 1 ( r ) , v 2 ( r ) ) ] k , 0 < r < 1 , v 1 ′ ( 0 ) = 0 , v 2 ′ ( 0 ) = 0 , v 1 ( 1 ) = 0 , v 2 ( 1 ) = 0 .

Next, we consider the following equivalent integral system of (2.1).

(2.2) v 1 ( r ) = ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t , r ∈ [ 0 , 1 ] , v 2 ( r ) = ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ h ( s ) g ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t , r ∈ [ 0 , 1 ] .

In the following lemmas, we assume that E is a real Banach space and H is a cone in E .

Lemma 2.5

[31,32] For r > 0 , define H r = { v ∈ H : ‖ v ‖ < r } . Assume that T : H ¯ r → H is a compact map such that T v ≠ v for v ∈ ∂ H r .

  1. If ‖ T v ‖ ≤ ‖ v ‖ for v ∈ ∂ H r , then i ( T , H r , H ) = 1 .

  2. If ‖ T v ‖ ≥ ‖ v ‖ for v ∈ ∂ H r , then i ( T , H r , H ) = 0 .

Lemma 2.6

[31] Let Ω ⊂ E be a bounded open set in E, 0 ∈ Ω and T : H ∩ Ω ¯ → H be condensing. Assume that T v ≠ θ v for v ∈ H ∩ ∂ Ω and θ ≥ 1 . Then i ( T , H ∩ Ω , H ) = 1 .

To apply Lemmas 2.5 and 2.6 to the system (2.2), let E = ( C [ 0 , 1 ] ) 2 be the Banach space with ‖ ( v 1 , v 2 ) ‖ = ‖ v 1 ‖ 0 + ‖ v 2 ‖ 0 , where ‖ v i ‖ 0 = max r ∈ [ 0 , 1 ] ∣ v i ( r ) ∣ , i = 1 , 2 . Define H be the positive cone in E by

H = ( v 1 , v 2 ) : ( v 1 , v 2 ) ∈ E , v 1 , v 2 ≥ 0 , min r ∈ [ 1 4 , 3 4 ] ( v 1 ( r ) + v 2 ( r ) ) ≥ 1 4 ( ‖ v 1 ‖ 0 + ‖ v 2 ‖ 0 ) .

Note that H is a nonempty set by Remark 2.1 [23].

Define

T 1 ( v 1 , v 2 ) ( r ) = ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t , T 2 ( v 1 , v 2 ) ( r ) = ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ h ( s ) g ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t ,

and

T ( v 1 , v 2 ) ( r ) = ( T 1 ( v 1 , v 2 ) ( r ) , T 2 ( v 1 , v 2 ) ( r ) ) .

In our setting, (2.2) is equivalent to

T ( v 1 , v 2 ) = ( v 1 , v 2 ) .

Lemma 2.7

Assume that (A1) and (A2) are satisfied, then T ( H ) ⊂ H and T : E → E is completely continuous.

Proof

It is a standard procedure to prove that T : E → E is completely continuous, which is divided into three steps.

  1. The operator T is continuous in view of the continuity of b , h , f , g .

  2. Let Ω l be a bounded subset of H , i.e.,

    Ω l ≔ { ( v 1 , v 2 ) ∈ H : ‖ ( v 1 , v 2 ) ‖ ≤ l } ,

    where l > 0 is a constant. Let

    M 1 = max ‖ ( v 1 , v 2 ) ‖ ≤ l f ( v 1 , v 2 ) , M 2 = max ‖ ( v 1 , v 2 ) ‖ ≤ l g ( v 1 , v 2 ) , c 1 = ∫ 0 1 b k ( t ) d t , c 2 = ∫ 0 1 h k ( t ) d t ,

    then for ( v 1 , v 2 ) ∈ Ω l , we have

    ‖ T 1 ( v 1 , v 2 ) ( r ) ‖ ≤ ∫ 0 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t ≤ λ k C N − 1 k − 1 1 k ∫ 0 1 t k − 1 ∫ 0 1 [ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t ≤ λ M 1 k c 1 C N − 1 k − 1 1 k ∫ 0 1 t k − 1 k d t = λ M 1 k c 1 C N − 1 k − 1 1 k k 2 k − 1

    and

    ‖ T 2 ( v 1 , v 2 ) ( r ) ‖ ≤ λ M 2 k c 2 C N − 1 k − 1 1 k k 2 k − 1 ,

    which mean that T ( Ω l ) is uniformly bounded;

  3. for ( v 1 , v 2 ) ∈ Ω l , one has

    ∣ ( T 1 ( v 1 , v 2 ) ) ′ ( t ) ∣ = k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k ≤ k C N − 1 k − 1 t k − 1 ∫ 0 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k ≤ λ M 1 k c 1 C N − 1 k − 1 1 k t k − 1 k ≤ λ M 1 k c 1 C N − 1 k − 1 1 k

    and

    ∣ ( T 2 ( v 1 , v 2 ) ) ′ ( t ) ∣ ≤ λ M 2 k c 2 C N − 1 k − 1 1 k .

    By the mean-value formula, we can see that for ( v 1 , v 2 ) ∈ Ω l and t 1 , t 2 ∈ [ 0 , 1 ] , there exists ζ ∈ [ t 1 , t 2 ] such that

    ∣ T i ( v 1 , v 2 ) ( t 1 ) − T i ( v 1 , v 2 ) ( t 2 ) ∣ = ∣ ( T i ( v 1 , v 2 ) ) ′ ( ζ ) ∣ ∣ t 1 − t 2 ∣ ≤ λ M i k c i C N − 1 k − 1 1 k ∣ t 1 − t 2 ∣ ,

    where i = 1 , 2 . When ∣ t 1 − t 2 ∣ → 0 , the right-hand side of the above inequality tends to zero. This indicates that T ( Ω l ) is equicontinuous.

By the Arzela-Ascoli theorem, we obtain that T is completely continuous.

Next, we prove that T ( H ) ⊂ H . Choosing ( v 1 , v 2 ) ∈ H , it follows from (A1) and (A2) that T i ( v 1 , v 2 ) ( r ) ≥ 0 , ( T i ( v 1 , v 2 ) ) ′ ( r ) ≤ 0 , ( T i ( v 1 , v 2 ) ) ″ ( r ) ≤ 0 . So, for every r ∈ [ 1 4 , 3 4 ] , we have

T i ( v 1 , v 2 ) ( 1 ) − T i ( v 1 , v 2 ) ( 0 ) 1 ≤ T i ( v 1 , v 2 ) ( r ) − T i ( v 1 , v 2 ) ( 0 ) r ,

i.e., T i ( v 1 , v 2 ) ( r ) − T i ( v 1 , v 2 ) ( 0 ) ≥ r ( T i ( v 1 , v 2 ) ( 1 ) − T i ( v 1 , v 2 ) ( 0 ) ) , and then

min r ∈ [ 1 4 , 3 4 ] T i ( v 1 , v 2 ) ( r ) ≥ 1 4 ‖ T i ( v 1 , v 2 ) ‖ 0 , i = 1 , 2 .

Thus,

min r ∈ [ 1 4 , 3 4 ] ( T 1 ( v 1 , v 2 ) ( r ) + T 2 ( v 1 , v 2 ) ( r ) ) = min r ∈ [ 1 4 , 3 4 ] T 1 ( v 1 , v 2 ) ( r ) + min r ∈ [ 1 4 , 3 4 ] T 2 ( v 1 , v 2 ) ( r ) ≥ 1 4 ‖ T 1 ( v 1 , v 2 ) ‖ 0 + 1 4 ‖ T 2 ( v 1 , v 2 ) ‖ 0 = 1 4 ( ‖ T 1 ( v 1 , v 2 ) ‖ 0 + ‖ T 2 ( v 1 , v 2 ) ‖ 0 ) = 1 4 ‖ ( T 1 ( v 1 , v 2 ) , T 2 ( v 1 , v 2 ) ) ‖ ,

which means that T ( H ) ⊂ H . The proof is completed.□

3 Existence and nonexistence

Denote

L 1 = k 2 k − N k C N − 1 k − 1 1 k and L 2 = k 2 k − N k C N − 1 k − 1 1 k 1 − 3 4 2 k − N k ,

where N < 2 k .

Theorem 3.1

Assume that (A1), (A2), and (A3) are satisfied. Then,

  1. there exists a constant μ 1 > 0 such that for all λ < μ 1 , the system (1.1) admits at least one negative k-convex radial solution;

  2. for λ is sufficiently large, system (1.1) admits no negative k-convex radial solution.

Proof

(1) If q > 0 , then it follows from (A1) and (A2) that

M 1 ( q ) = L 1 max ( v 1 , v 2 ) ∈ H , ‖ ( v 1 , v 2 ) ‖ = q ∫ 0 1 s N − 1 [ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k > 0 ,

and

M 2 ( q ) = L 1 max ( v 1 , v 2 ) ∈ H , ‖ ( v 1 , v 2 ) ‖ = q ∫ 0 1 s N − 1 [ h ( s ) g ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k > 0 .

Let M ( q ) = max { M 1 ( q ) , M 2 ( q ) } . Choose a number J 1 > 0 , let μ 1 = J 1 2 M ( J 1 ) and set

H J 1 = { ( v 1 , v 2 ) : ( v 1 , v 2 ) ∈ H , ‖ ( v 1 , v 2 ) ‖ < J 1 } .

Then for λ < μ 1 and ( v 1 , v 2 ) ∈ ∂ H J 1 , we have

T 1 ( v 1 , v 2 ) ( r ) = ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t ≤ ∫ 0 1 k t N − k ∫ 0 1 s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t < μ 1 ∫ 0 1 k t N − k ∫ 0 1 s N − 1 C N − 1 k − 1 [ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t = μ 1 k 2 k − N k C N − 1 k − 1 1 k ∫ 0 1 s N − 1 [ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k = μ 1 L 1 ∫ 0 1 s N − 1 [ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k ≤ μ 1 M ( J 1 ) = J 1 2 .

In a similar way, T 2 ( v 1 , v 2 ) ( r ) < J 1 2 , which implies

‖ T ( v 1 , v 2 ) ‖ = ‖ T 1 ( v 1 , v 2 ) ‖ + ‖ T 2 ( v 1 , v 2 ) ‖ < J 1 = ‖ ( v 1 , v 2 ) ‖

for ( v 1 , v 2 ) ∈ ∂ H J 1 . By ( 1 ) of Lemma 2.5, we have i ( T , H J 1 , H ) = 1 .

Here, we shall address the case f ∞ = ∞ , and the proof for the case g ∞ = ∞ is similar. From f ∞ = ∞ , we then can choose J ^ > 0 so that f ( v 1 , v 2 ) ≥ δ ( v 1 + v 2 ) for v 1 + v 2 ≥ J ^ , where δ > 0 is chosen so that

λ δ L 2 4 ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k > 1 .

Let J 2 = max { 2 J 1 , 4 J ^ } and set

H J 2 = { ( v 1 , v 2 ) : ( v 1 , v 2 ) ∈ H , ‖ ( v 1 , v 2 ) ‖ < J 2 } .

If ( v 1 , v 2 ) ∈ ∂ H J 2 , then we have

min r ∈ [ 1 4 , 3 4 ] ( v 1 ( r ) + v 2 ( r ) ) ≥ 1 4 ( ‖ v 1 ‖ 0 + ‖ v 2 ‖ 0 ) = 1 4 ‖ ( v 1 , v 2 ) ‖ ≥ J ^

and

T 1 ( v 1 , v 2 ) ( r ) = ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t ≥ ∫ 3 4 1 k t N − k ∫ 1 4 3 4 s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t ≥ ∫ 3 4 1 k t N − k ∫ 1 4 3 4 s N − 1 C N − 1 k − 1 [ λ δ b ( s ) ( v 1 ( s ) + v 2 ( s ) ) ] k d s 1 k d t ≥ λ δ 4 ( ‖ v 1 ‖ 0 + ‖ v 2 ‖ 0 ) ∫ 3 4 1 k t N − k ∫ 1 4 3 4 s N − 1 C N − 1 k − 1 b k ( s ) d s 1 k d t = λ δ 4 ‖ ( v 1 , v 2 ) ‖ k 2 k − N k C N − 1 k − 1 1 k 1 − 3 4 2 k − N k ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k = λ δ 4 ‖ ( v 1 , v 2 ) ‖ L 2 ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k = ‖ ( v 1 , v 2 ) ‖ λ δ L 2 4 ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k > ‖ ( v 1 , v 2 ) ‖ .

Therefore,

‖ T ( v 1 , v 2 ) ‖ = ‖ T 1 ( v 1 , v 2 ) ‖ + ‖ T 2 ( v 1 , v 2 ) ‖ ≥ T 1 ( v 1 , v 2 ) ( r ) > ‖ ( v 1 , v 2 ) ‖

for ( v 1 , v 2 ) ∈ ∂ H J 2 . By ( 2 ) of Lemma 2.5, we have i ( T , H J 2 , H ) = 0 .

Since J 1 < J 2 , by the additivity of the fixed point index, we have i ( T , H J 2 \ H ¯ J 1 , H ) = − 1 . Consequently, T has a fixed point in H J 2 \ H ¯ J 1 , which is the positive solution of system (2.2).

(2) Since f > 0 and f ∞ = ∞ (the proof is similar if g ∞ = ∞ ), we have that the existence of a constant J ˜ > 0 such that f ( v 1 , v 2 ) ≥ J ˜ ( v 1 + v 2 ) for v 1 , v 2 ≥ 0 , where J ˜ is chosen so that

λ J ˜ L 2 4 ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k > 1 .

Let ( v 1 , v 2 ) ∈ E be a positive solution of system (2.2). By Lemma 2.7, we have ( v 1 , v 2 ) ∈ H and

min r ∈ [ 1 4 , 3 4 ] ( v 1 ( r ) + v 2 ( r ) ) ≥ 1 4 ( ‖ v 1 ‖ 0 + ‖ v 2 ‖ 0 ) .

For λ is large enough, we have

v 1 ( r ) = ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t ≥ ∫ 3 4 1 k t N − k ∫ 1 4 3 4 s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t ≥ ∫ 3 4 1 k t N − k ∫ 1 4 3 4 s N − 1 C N − 1 k − 1 [ λ J ˜ b ( s ) ( v 1 ( s ) + v 2 ( s ) ) ] k d s 1 k d t ≥ λ J ˜ 4 ( ‖ v 1 ‖ 0 + ‖ v 2 ‖ 0 ) ∫ 3 4 1 k t N − k ∫ 1 4 3 4 s N − 1 C N − 1 k − 1 b k ( s ) d s 1 k d t = λ J ˜ 4 ‖ ( v 1 , v 2 ) ‖ k 2 k − N k C N − 1 k − 1 1 k 1 − 3 4 2 k − N k ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k = λ J ˜ 4 ‖ ( v 1 , v 2 ) ‖ L 2 ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k = ‖ ( v 1 , v 2 ) ‖ λ J ˜ L 2 4 ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k > ‖ ( v 1 , v 2 ) ‖ ,

which is a contradiction. This completes the proof.□

4 Multiplicity result

Now let us list an assumption on f and g , which is to be used in Theorem 4.1.

(A4): f ( s 1 , t 1 ) ≤ f ( s 2 , t 2 ) and g ( s 1 , t 1 ) ≤ g ( s 2 , t 2 ) for 0 ≤ s 1 ≤ s 2 , 0 ≤ t 1 ≤ t 2 .

In order to obtain the multiplicity result, we first need the following auxiliary lemmas.

Lemma 4.1

Assume that conditions (A1), (A2), and (A3) are satisfied, and λ belongs to a compact subset I of ( 0 , ∞ ) . Then there exists a constant α I > 0 such that ‖ ( v 1 , v 2 ) ‖ ≤ α I for all possible positive solutions ( v 1 , v 2 ) of the system (2.2).

Proof

Assume that there exists a function sequence { ( v 1 ( m ) , v 2 ( m ) ) } m ≥ 1 of positive solutions of the system (2.2), with corresponding { λ m } m ≥ 1 ⊂ I , such that

lim m → ∞ ∥ ( v 1 ( m ) , v 2 ( m ) ) ∥ = ∞ .

By Lemma 2.7, we have ( v 1 ( m ) , v 2 ( m ) ) ∈ H and

min r ∈ [ 1 4 , 3 4 ] ( v 1 ( m ) ( r ) + v 2 ( m ) ( r ) ) ≥ 1 4 ( ‖ v 1 ( m ) ‖ 0 + ‖ v 2 ( m ) ‖ 0 ) .

As before, here, we only consider the case f ∞ = ∞ , and the proof for the case g ∞ = ∞ is similar. From f ∞ = ∞ , we may choose J ^ > 0 so that f ( v 1 , v 2 ) ≥ δ ( v 1 + v 2 ) for v 1 + v 2 ≥ J ^ , where δ > 0 is chosen so that

λ m δ L 2 4 ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k > 1 .

Choosing m large enough so that 1 4 ( ‖ v 1 ( m ) ‖ 0 + ‖ v 2 ( m ) ‖ 0 ) ≥ J ^ , we have

‖ v 1 ( m ) ‖ 0 ≥ v 1 ( m ) ( r ) = ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ m b ( s ) f ( v 1 ( m ) ( s ) , v 2 ( m ) ( s ) ) ] k d s 1 k d t ≥ ∫ 3 4 1 k t N − k ∫ 1 4 3 4 s N − 1 C N − 1 k − 1 [ λ m b ( s ) f ( v 1 ( m ) ( s ) , v 2 ( m ) ( s ) ) ] k d s 1 k d t ≥ ∫ 3 4 1 k t N − k ∫ 1 4 3 4 s N − 1 C N − 1 k − 1 [ λ m δ b ( s ) ( v 1 ( m ) ( s ) + v 2 ( m ) ( s ) ) ] k d s 1 k d t ≥ λ m δ 4 ( ‖ v 1 ( m ) ‖ 0 + ‖ v 2 ( m ) ‖ 0 ) ∫ 3 4 1 k t N − k ∫ 1 4 3 4 s N − 1 C N − 1 k − 1 b k ( s ) d s 1 k d t = λ m δ 4 ( ‖ v 1 ( m ) ‖ 0 + ‖ v 2 ( m ) ‖ 0 ) k 2 k − N k C N − 1 k − 1 1 k 1 − 3 4 2 k − N k ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k = ( ‖ v 1 ( m ) ‖ 0 + ‖ v 2 ( m ) ‖ 0 ) λ m δ L 2 4 ∫ 1 4 3 4 s N − 1 b k ( s ) d s 1 k > ‖ v 1 ( m ) ‖ 0 + ‖ v 2 ( m ) ‖ 0 .

Obviously, this is contradictory. This completes the proof.□

For the convenience of writing, we will write ( v 1 , v 2 ) ≤ ( v 3 , v 4 ) if v 1 ( r ) ≤ v 3 ( r ) and v 2 ( r ) ≤ v 4 ( r ) hold for all r ∈ [ 0 , 1 ] .

Definition 4.1

If the function pair ( v ¯ 1 , v ¯ 2 ) ∈ C [ 0 , 1 ] × C [ 0 , 1 ] satisfies

v ¯ 1 ( r ) ≥ ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v ¯ 1 ( s ) , v ¯ 2 ( s ) ) ] k d s 1 k d t , v ¯ 2 ( r ) ≥ ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ h ( s ) g ( v ¯ 1 ( s ) , v ¯ 2 ( s ) ) ] k d s 1 k d t ,

we say ( v ¯ 1 , v ¯ 2 ) is an upper solution of (2.2).

Definition 4.2

If the function pair ( v ̲ 1 , v ̲ 2 ) ∈ C [ 0 , 1 ] × C [ 0 , 1 ] satisfies

v ̲ 1 ( r ) ≤ ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ̲ ( s ) , v 2 ̲ ( s ) ) ] k d s 1 k d t , v ̲ 2 ( r ) ≤ ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ h ( s ) g ( v 1 ̲ ( s ) , v 2 ̲ ( s ) ) ] k d s 1 k d t ,

we say ( v ¯ 1 , v ¯ 2 ) is a lower solution of (2.2).

Lemma 4.2

Assume that (A1), (A2), and (A4) are satisfied. Let ( v ¯ 1 , v ¯ 2 ) is an upper solution of system (2.2) such that ( 0 , 0 ) ≤ ( v ¯ 1 , v ¯ 2 ) , then there exists a positive solution ( v 1 , v 2 ) of the system (2.2) with ( 0 , 0 ) ≤ ( v 1 , v 2 ) ≤ ( v ¯ 1 , v ¯ 2 ) .

Proof

On view of the monotonicity conditions (A4) on f and g , we first note that ( 0 , 0 ) is a lower solution of the system (2.2), and it is clear that the result can be proved by the usual monotonic iterative scheme. This completes the proof.□

Denote

Γ = { λ > 0 : ( 2.2 ) has a positive solution }

and λ ∗ = sup Γ . By Theorem 3.1, Γ is nonempty and bounded, and thus, λ ∗ ∈ ( 0 , ∞ ) . We claim that λ ∗ ∈ Γ . In face, let λ m → λ ∗ , where λ m ∈ Γ . Because { λ m } m ≥ 1 are bounded, by Lemma 4.1, one see that the corresponding positive solutions ( v 1 ( m ) , v 2 ( m ) ) are bounded. It follows from the compactness of the integral operators T 1 and T 2 that λ ∗ ∈ Γ . Let ( v 1 ∗ , v 2 ∗ ) be a positive solution of system (2.2) corresponding to λ ∗ .

Lemma 4.3

Assume that (A1), (A2), and (A4) are satisfied. Let 0 < λ < λ ∗ . Then there exists ε ∗ > 0 such that for 0 < ε ≤ ε ∗ , ( v 1 ∗ + ε , v 2 ∗ + ε ) is an upper solution of system (2.2).

Proof

It follows from (A1), (A4), and ( v 1 ∗ , v 2 ∗ ) ≥ ( 0 , 0 ) that there exists a constant c > 0 so that f ( v 1 ∗ ( r ) , v 2 ∗ ( r ) ) ≥ c > 0 and g ( v 1 ∗ ( r ) , v 2 ∗ ( r ) ) ≥ c for all r ∈ [ 0 , 1 ] . By uniform continuity, there exists ε ∗ > 0 such that

∣ f ( v 1 ∗ ( r ) + ε , v 2 ∗ ( r ) + ε ) − f ( v 1 ∗ ( r ) , v 2 ∗ ( r ) ) ∣ < c ( λ ∗ − λ ) λ ,

and

∣ g ( v 1 ∗ ( r ) + ε , v 2 ∗ ( r ) + ε ) − g ( v 1 ∗ ( r ) , v 2 ∗ ( r ) ) ∣ < c ( λ ∗ − λ ) λ ,

for all t ∈ [ 0 , 1 ] and 0 < ε ≤ ε ∗ . Further, one has

v 1 ∗ ( r ) + ε > ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ ∗ b ( s ) f ( v 1 ∗ ( s ) , v 2 ∗ ( s ) ) ] k d s 1 k d t

= ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ∗ ( s ) + ε , v 2 ∗ ( s ) + ε ) ] k d s 1 k d t − ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) ( f ( v 1 ∗ ( s ) + ε , v 2 ∗ ( s ) + ε ) − f ( v 1 ∗ ( s ) , v 2 ∗ ( s ) ) ) ] k d s 1 k d t + ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ ( λ ∗ − λ ) b ( s ) f ( v 1 ∗ ( s ) , v 2 ∗ ( s ) ) ] k d s 1 k d t , ≥ ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ∗ ( s ) + ε , v 2 ∗ ( s ) + ε ) ] k d s 1 k d t .

Similarly,

v 2 ∗ ( r ) + ε > ∫ r 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ h ( s ) g ( v 1 ∗ ( s ) + ε , v 2 ∗ ( s ) + ε ) ] k d s 1 k d t .

This completes the proof.□

Theorem 4.1

Assume that (A1), (A2), (A3), and (A4) are satisfied.

Then there exists a constant λ ∗ > 0 such that the system (1.1) admits at least two negative k-convex radial solutions for λ ∈ ( 0 , λ ∗ ) , at least one negative k-convex radial solution for λ = λ ∗ and no solution for λ ∈ ( λ ∗ , + ∞ ) .

Proof

Let 0 < λ < λ ∗ . Because ( v 1 ∗ , v 2 ∗ ) is an upper solution of system (2.2), by Lemma 4.2, we see that there is a positive solution ( v ^ 1 , v ^ 2 ) of the system (2.2) with ( 0 , 0 ) ≤ ( v ^ 1 , v ^ 2 ) ≤ ( v 1 ∗ , v 2 ∗ ) . Thus, there exists a positive solution for 0 < λ ≤ λ ∗ , and a positive solution does not exist for λ > λ ∗ . Next, we find another positive solution of the system (2.2) for 0 < λ < λ ∗ . Set

Ω 1 = { ( v 1 , v 2 ) : ( v 1 , v 2 ) ∈ E , − ε < v i ( r ) < v i ∗ ( r ) + ε , r ∈ [ 0 , 1 ] , i = 1 , 2 } ,

where ε > 0 . It is clear that Ω 1 is bounded and open in E and ( 0 , 0 ) ∈ Ω 1 . Because T is completely continuous, T : H ⋂ Ω 1 ¯ → H is condensing. Furthermore, ( v ^ 1 , v ^ 2 ) ∈ Ω 1 for 0 < λ ≤ λ ∗ .

Let ( v 1 , v 2 ) ∈ H ⋂ ∂ Ω 1 . Then there exists r 0 ∈ [ 0 , 1 ] such that v i ( r 0 ) = v i ∗ ( r 0 ) + ε ( i = 1 , 2 ) . Here, assume that v 1 ( r 0 ) = v 1 ∗ ( r 0 ) + ε , by ( A 4 ) , we obtain for θ ≥ 1

T 1 ( v 1 , v 2 ) ( r 0 ) = ∫ r 0 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ( s ) , v 2 ( s ) ) ] k d s 1 k d t ≤ ∫ r 0 1 k t N − k ∫ 0 t s N − 1 C N − 1 k − 1 [ λ b ( s ) f ( v 1 ∗ ( s ) + ε , v 2 ∗ ( s ) + ε ) ] k d s 1 k d t < v 1 ∗ ( r 0 ) + ε = v 1 ( r 0 ) ≤ θ v 1 ( r 0 ) .

Similarly, T 1 ( v 1 , v 2 ) ( r 0 ) < θ v 2 ( r 0 ) . Thus, for ( v 1 , v 2 ) ∈ H ∩ ∂ Ω 1 and θ ≥ 1 , T ( v 1 , v 2 ) ≠ θ ( v 1 , v 2 ) . By Lemma 2.6, one has

i ( T , H ∩ Ω 1 , H ) = 1 .

Let J = max { α I + 1 , J 2 , ‖ ( v 1 ∗ + ε , v 2 ∗ + ε ) ‖ } , where α I and J 2 have been defined in Lemma 4.1 and Theorem 3.1, respectively. Set

H J = { ( v 1 , v 2 ) : ( v 1 , v 2 ) ∈ H , ‖ ( v 1 , v 2 ) ‖ < J } .

From Lemma 4.1, we know that T ( v 1 , v 2 ) ≠ ( v 1 , v 2 ) for ( v 1 , v 2 ) ∈ ∂ H J . If ( v 1 , v 2 ) ∈ ∂ H J , then, by the proof of Theorem 3.1, we can obtain that ‖ T ( v 1 , v 2 ) ‖ ≥ ‖ ( v 1 , v 2 ) ‖ . Consequently, by Lemma 2.5, one has

i ( T , H J , H ) = 0 .

It follows from the additivity of the fixed point index that

i ( T , H J \ H ∩ Ω 1 ¯ , H ) = − 1 .

Thus, T has another fixed point in H J \ H ∩ Ω 1 ¯ , which establishes that the system (2.2) admits at least two positive solutions for λ ∈ ( 0 , λ ∗ ) . This completes the proof.□

Remark 4.1

What we need to note is that when f , g ∈ C ( [ 0 , ∞ ) × [ 0 , ∞ ) ) satisfy f ( 0 , 0 ) = g ( 0 , 0 ) = 0 , f ( s , t ) > 0 , g ( s , t ) > 0 for s , t ∈ ( 0 , ∞ ) , that is weaker than the condition (A1), Theorems 3.1 and 4.1 are still valid.

Acknowledgement

The authors would like to thank the handling editor and the referees for their helpful comments and suggestions.

  1. Funding information: This work was sponsored by the National Natural Science Foundation of China (12371173), the Shandong Natural Science Foundation of China (ZR2021MA064 and ZR2020MA017), and the Taishan Scholar project of China.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animals use.

  5. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2023-06-20
Revised: 2024-06-20
Accepted: 2024-08-21
Published Online: 2024-12-09

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0066
Schlagwörter für diesen Artikel
k-Hessian system; k-convex solution; multiplicity; upper and lower solution
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Artikel in diesem Heft

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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