Part-Metric and Its Applications to Cyclic Discrete Dynamic Systems

and Applied Analysis 3 Based on these properties and Theorem 2.1, Kruse and Nesemann in 9 obtained the following result. Lemma 2.2 see 9, Corollary 2 . Let T : n → n be a continuous mapping with a unique equilibrium x∗ ∈ n . Suppose that for the discrete dynamic system 2.1 there is some k ∈ such that for the part-metric p inequality p Tx, x∗ < p x, x∗ holds for all x / x∗. Then, x∗ is globally asymptotically stable. Our idea is to adapt the part-metric to matrices. For any two matrices with positive entries A aij m×n ∈ m×n and B bij m×n ∈ m×n , we define the part-metric in the following natural way: P A,B log2 max 1≤i≤m,1≤j≤n { aij bij , bij aij } . 2.4 Note that anm × n matrix aij m×n is equivalent to a vector with mn elements, such as ( aij ) m×n ←→ a1,1, a2,1, . . . , am,1, a1,2, a2,2, . . . , am,2, . . . , a1,n, a2,n, . . . , am,n T . 2.5 Thus, for the above matrices A and B, we have that P A,B p (( A1 , A T 2 , . . . , A T n )T , ( B 1 , B T 2 , . . . , B T n )T) max 1≤j≤n { p ( Aj, Bj )} , 2.6 where Aj a1j , a2j , . . . , amj T , Bj b1j , b2j , . . . , bmj T , j 1, 2, . . . , n. From this and the above-mentioned properties for the part-metric we have the following: 1 the part-metric P is a continuous metric on m×n , 2 m×n ,P is a complete metric space, 3 the distances induced by the part-metric P and the Euclidean norm are equivalent on m×n . From this and by Lemma 2.2, we have that the next result holds. Theorem 2.3. Let T : m×n → m×n be a continuous mapping with the unique equilibrium C ∈ m×n . Suppose that for the discrete dynamic system Xn 1 TXn, n ∈ 0 , 2.7 there is a k ∈ such that for metric P, the inequality P TX,C < P X,C holds for each X/ C. Then, C is globally asymptotically stable. Remark 2.4. Note that if we do not assume in Theorem 2.3 that C is the unique equilibrium of 2.7 , then if C̃ is another equilibrium it must be P ( C̃,C ) P ( TC̃,C ) < P ( C̃,C ) , 2.8 which is impossible. Hence, there is only one equilibrium of 2.7 . 4 Abstract and Applied Analysis 3. Main Result Let Y Y1, Y2, . . . , Yq be a square q × q matrix, where Yi y 1 i , y 2 i , . . . , y q i T , i 1, 2, . . . , q, and Φ is defined by Φ Y Φ ( Y1, Y2, . . . , Yq ) ⎛ ⎜⎜⎜⎜⎜⎜⎝ φ ( y 1 1 , y 2 2 , . . . , y q q ) φ ( y 2 1 , y 3 2 , . . . , y 1 q ) .. φ ( y q 1 , y 1 2 , . . . , y q−1 q ) ⎞ ⎟⎟⎟⎟⎟⎟⎠ , 3.1 where φ : q → is a continuous mapping. Clearly, Φ is a continuous mapping and our system becomes