What is ddns bounded

Custom - Supports both dynamic and static IP addresses. YES NO. Main Menu. Solutions Image Widgets. So, from 8 , is also bounded. Theorem 3. Let , and then, 8 has unbounded solutions. It is noted that following holds: for every solution of 8.

Then, it follows from 25 that Now, roots of are given by Since , we have that and Therefore, both roots of are positive if. Moreover, 26 can also written as Then, we see that It follows that Choose and so that It follows from this and 32 that and consequently, It follows by letting in 35 that as , and hence, it follows from this result.

Theorem 4. Let and ; then, of 8 is globally asymptotically stable. By Theorem 1 i , is a sink. Hence, it is enough to prove further that of 8 tends to. Recall that of 8 is bounded by Theorem 2. Thus, Then, from 8 , we get Now, claiming that , otherwise,.

From 37 , we obtain Since holds, then or equivalently It follows from 38 and 40 that Hence, which is impossible for. This is contradiction, and hence, the result follows. Theorem 5. Every positive solution of 8 oscillates about with semicycles of length two or three, and extreme of every semicycle occurs at the first or the second term. Let the positive solution of 8 is. First, we prove that every positive semicycle except possibly the first term has two or three terms. Assuming and for some , we obtain from 8 that If , then we have On the contrary, since , we see that So, Therefore,.

Theorem 6. Equation 8 has no periodic solution having prime period two. Let be a periodic solution of period two of 8. It follows that which implies that Substituting from 49 into 48 and after some calculation, we get From 50 , one has Obviously, is a solution of But one has to prove that this is the unique solution of Now, Thus, for.

This implies that, on , is strictly nondecreasing. We will explore dynamics of equation 6 when is a periodic sequence having period two with and. Consider and. Then, we have. By separating the even-indexed and odd-indexed terms, equation 6 now becomes where is the unique fixed point of system Theorem 7.

If , then of 54 is a sink. We consider the map on , which is described as follows: Then, Therefore, the Jacobian matrix of evaluated at is and the auxiliary equation associated with is and then, we obtain It follows by Corollary 1.

We assume that is positive bounded with for some real constants and. Theorem 8. Its proof is same as proof of Theorem 2 , and hence, it is omitted. Lemma 1. Assume 61 is satisfied, and if then. Let for , and we get Therefore, Taking the for 65 , we obtain Since is arbitrary, it follows that Similarly, We get from inequalities 67 and 68 that Since holds, we get or equivalently It follows from equation 69 that So, and one has We have from 67 , for all , Similarly, we obtain from 68 that This completes the proof.

Now, we will explore attractively of solutions of equation 6. Let represent the arbitrary positive solution of 6. Now, one can find appropriate conditions such that attracts all positive solutions of 6 , that is, Now, define : and then, equation 6 becomes.

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Warranty Information. ADJ Zone. ADJ Store. Then F T is a closed and convex subset of D. First, we will show that F T is closed. Next, we show that F T is convex. Hence p z. Theorem 3. Discrete Dynamics in Nature and Society 7 Proof. We split the proof into six steps. Step 1. By Lemma 2. This implies that Cn 1 is also closed and convex. Step 2. Step 3. Step 4. It follows from 3. From 3. Discrete Dynamics in Nature and Society 9 Step 5.

Step 6. This completes the proof. Discrete Dynamics in Nature and Society 13 Proof. Putting A 0 in Theorem 3. References 1 C. Shiau, K. Tan, and C. Shahzad and H. Bauschke, E. Genel and J. Nakajo and W. Tada and W.