New S-Type Bounds of M-Eigenvalues for Elasticity Tensors with Applications

Qi in [3, 5, 6] presented some basic studies for tensor computations and approximations. Li et al. [7–10], Bu et al. [11], Che et al. [12], and Zhao et al. [13, 14] worked on analyzing the M-eigenvalues for various elasticity tensors. .e authors in [15] proposed a tensor-based FTV model for the three-dimensional image deblurring problem, and some properties for Z-eigenvalues of tensor are given in [16–18]. Let


Introduction
Let M � 1, 2, . . . , m { } and N � 1, 2, . . . , n { }; a real tensor A � (a ijkl ) ∈ R m×n×m×n is called an elasticity tensor, if a ijkl � a kjil � a ilkj , i, k ∈ M, j, l ∈ N. (1) Consider the following optimization problem with an elasticity tensor A � (a ijkl ) [1,2]: a ijkl x i y j x k y l , Qi et al. introduced the following definition of M-eigenvalues of an elasticity tensor [3,4]. Definition 1. (see [3,4]). Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor, if there exist nonzero vectors, x ∈ R m and y � ∈ R n , and a real number λ ∈ R, such that Ayxy � λx, Axyx � λy, x T x � 1, y T y � 1, where (Ayxy) i � k∈M j,l�1 n a ijkl y j x k y l , (Axyx) l � i,k∈M j�1 n a ijkl x i y j x k . (4) en, λ is called an M-eigenvalue of A, and the nonzero vectors x and y are called the corresponding M-eigenvectors.
Qi in [3,5,6] presented some basic studies for tensor computations and approximations. Li et al. [7][8][9][10], Bu et al. [11], Che et al. [12], and Zhao et al. [13,14] worked on analyzing the M-eigenvalues for various elasticity tensors. e authors in [15] proposed a tensor-based FTV model for the three-dimensional image deblurring problem, and some properties for Z-eigenvalues of tensor are given in [16][17][18]. x i x k y T B ik y � n j,l�1 y j y l x T C jl x, (5) where B ik ∈ R n×n and C jl ∈ R m×m are symmetric matrices with entries B ik st � a iskt , C jl st � a sjtl . (6) And, assume that λ min (A) is the minimal eigenvalue of a matrix A, λ max (A) is the maximal eigenvalue of a matrix A, and ρ(A) is the spectral radius of a matrix A. In 2021, Li et al. established the following bounds for M-eigenvalues of an elasticity tensor.
Theorem 1 (see [19]). Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor and λ be an M-eigenvalue of A. en, where and Theorem 2 (see [19]). Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor and ρ M (A) be the M-spectral radius of A. en, where e following necessary and sufficient condition for strong ellipticity for general anisotropic elastic materials is presented by Han et al. [20].
Theorem 3 (see [20]). Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor. e strong ellipticity condition holds, i.e., a ijkl x i y j x k y l > 0, (12) for all nonzero vectors x ∈ R m , y ∈ R n if and only if the smallest M-eigenvalue of A is positive.
One application of the lower bound in eorem 1 is to identify the strong ellipticity condition of an elasticity tensor, and the upper bound in eorem 2 is given to accelerate convergence of the WQZ-algorithm [19]. In this paper, by breaking N into disjoint subsets S and its complement, new S-type upper bounds for the M-spectral radius of an elasticity tensor are given in Section 2. In Section 3, S-type sufficient conditions are also given to identify the strong ellipticity condition of an elasticity tensor.

S-Type Upper Bounds
In this section, we give S-type upper bounds for the largest M-eigenvalues of an elasticity tensor, and the relationship between the S-type upper bounds and existed upper bounds is also established. e sets S m , S m , S n , and S n are defined by M � S m ⋃ S m and S m ∩ S m � ∅, N � S n ⋃ S n , and S n ∩ S n � ∅. where Proof. Let λ be an M-eigenvalue of A with the M-eigenvectors x, y, Obviously, at least one of |x p | and |x s | is nonzero.
Let |y q | � max j∈S n |y j | and |y t | � max j∈S n |y j | , from the q-th equation of λy � Axyx, we have and similarly, we can get □ We compare the S-type upper bounds in eorem 4 with the results in [19], which shows that our new S-type upper bounds are always tighter than the results in [19].
Theorem 5. Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor. en, Journal of Mathematics We only proof the following case, and the other case can be proved similarly. If from the proof of eorem 4, From inequalities (20) or (21), there is an and therefore, ρ M (A) ≤ c.

□
In 2009, the following WQZ-algorithm was presented to compute the largest M-eigenvalue of an elasticity tensor [4].
e following example in [4] is taken to show that the tighter upper bound can accelerate convergence of the WQZ-algorithm.
In [4], υ is taken as follows: In Figure 1, we can find that, when taking υ � 211.4729, the sequence generated in the WQZ-algorithm converges to the largest M-eigenvalue more rapidly than taking υ � 1998.6000 and υ � 462.2316.

S-Type M-Eigenvalue Inclusion Sets and Strong Ellipticity Conditions
In this section, based on the S-type M-eigenvalue inclusion sets of an elasticity tensor, S-type sufficient conditions for strong ellipticity conditions are given. Let (Ax 2 ) jl � n j,l�1 a ijkl x i x k and (Ay 2 ) ik � m i,k�1 a ijkl y j y l , we need the following lemma.
Lemma 1 (see [23]). Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor. en, the strong ellipticity condition holds if and only if the matrix Ax 2 ∈ R n×n (or Ay 2 ∈ R m×m ) is positive definite for each nonzero x ∈ R m (or y ∈ R n ). Theorem 6. Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor and λ be an M-eigenvalue of A with the M-eigenvectors x, y. en, where Journal of Mathematics 5 Proof. Let λ be an M-eigenvalue of A with the M-eigenvectors x and y, Obviously, at least one of |x p | and |x s | is nonzero.  Step 0: given a tensor A � (a ijkl ), vectors x 0 ∈ R m and y 0 ∈ R n . Set t � 0 and A � υI + A, where I � (e ijkl ) ∈ R m×n×m×n with the entries as follows: Step 1: compute Output x * , y * .
Let |y q | � max j∈S n |y j | and |y t | � max j∈S n |y j | , from the q-th equation of λy � Axyx, similarly we can get λ ∈ Δ 1 (A). □ Theorem 7. Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor. If there exists S m or S n such that or then the strong ellipticity condition holds.
If λ ∈ Δ 1 (B), the second conclusion can be obtained similarly. □ e following sufficient conditions for strong ellipticity are given by Li et al. [19].
or λ min C ll > g 1 (l), for all l ∈ N, then the strong ellipticity condition holds.
Based on the above theorems, we introduce the definitions strictly diagonally dominated (M-SDD) and S-type strictly diagonally dominated (M-SSDD) elasticity tensors, which are based on the eigenvalues of matrices of B ik and C jl . Definition 2. Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor. If or λ min C ll > g 1 (l), for all l ∈ N, then the elasticity tensor A is called strictly diagonally dominated(M-SDD).
Definition 3. Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor. If there exists S m or S n such that or then the elasticity tensor A is called S-type strictly diagonally dominated(M-SSDD).
Next, we give the relationships between the M-SDD elasticity tensor and the M-SSDD elasticity tensor. Theorem 9. Let A � (a ijkl ) ∈ R m×n×m×n be an elasticity tensor. If A is an M-SDD elasticity tensor, then A is an M-SSDD elasticity tensor.
Proof. If A is an M-SDD elasticity tensor, we only prove the following case; the other case can be proved similarly. For all i ∈ M, en, for all i ∈ S m and k ∈ S m , which imply that and then A is an M-SSDD elasticity tensor.
□ Now, the following example is explored to show the efficiency of the results in eorems 8 and 9.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.