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International Journal of Differential Equations
Volume 2011 (2011), Article ID 949547, 12 pages
Oscillatory Solutions of Neutral Equations with Polynomial Nonlinearities
1Department of Mathematics, University of Mining and Geology “St. I. Rilski”, 1700 Sofia, Bulgaria
2Department of Mathematics and Physics, Higher School of Civil Engineering “L. Karavelov”, 1373 Sofia, Bulgaria
Received 1 June 2011; Accepted 31 August 2011
Academic Editor: Elena Braverman
Copyright © 2011 Vasil G. Angelov and Dafinka Tz. Angelova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Existence uniqueness of an oscillatory solution for nonlinear neutral equations by fixed point method is proved.
In [1, 2], we have considered a lossless transmission line terminated by a nonlinear resistive load and parallel connected capacitance (cf. Figure 1). The nonlinear boundary condition is caused by the polynomial type V-I characteristics of the nonlinear load at the second end of the transmission line (cf. Figure 1).
The voltage and current , of the lossless transmission line can be found by solving the following mixed problem for the hyperbolic partial differential system: where and are prescribed initial functions, is the length of the line, is the per-unit length capacitance, and is per-unit length inductance (cf. [3–10]). Here, the V-I characteristic of the nonlinear resistive load is , where are real numbers, is parallel connected capacitance, is the source voltage, is the source resistance, and is the line characteristic impedance.
The above formulated mixed problem can be reduced (cf. [1, 2, 11]) to an equivalent initial value problem for a neutral functional differential equation (cf. ). Here, we consider the problem of an existence uniqueness of oscillatory solutions of the equation where , . In fact, (1.4) is differential difference equation, and the initial function should be prescribed on an interval with length 2T. Let us note that the initial function can be obtained shifting the initial function from (1.3) along the characteristics on and along the characteristics . on (cf. [1, 2]). So, we obtain an initial function on .
Now, we are able to formulate the main problem: to find a solution of (1.4) with advanced prescribed zeros on the interval .
Let be the set of zeros of the initial function; that is, such that .
Let be a strictly increasing sequence of real numbers satisfying the following conditions (C):(C1),(C2),(C3)for every there is such that where .
Introduce the sets: consisting of all continuous and bounded functions differentiable with bounded derivatives on every interval (the derivatives at do not necessary exist), , , where are positive constants prescribed below.
We assume that , ().
The set turns out into a complete uniform space with respect to the family of pseudometrics , (), where , .
One can verify that is closed subset of with respect to the above metric.
Remark 1.1. The functions from are not necessary differentiable at (). That is why we consider a space with a countable family of pseudometrics, and then, we have to apply the fixed point theory from .
Define the operator by where and is M. A. Krasnoselskii operator (cf. ).
Remark 1.2. The operator is well defined, because the initial function is defined on the interval . We notice that maps into itself. Indeed, consider the set consisting of all continuous and bounded functions differentiable with bounded derivatives on every interval . Introduce the set . Then, assigns to every function the function translated to the right on the interval . So, the function coincides with on . Besides , and then that is, .
2. Main Results
Lemma 2.1. If , problem (1.4) has a solution iff the operator has a fixed point in , that is,
Proof. Let be a solution of (1.4). Then, integrating (1.4) on the interval (), we obtain , and then,
that is, is a fixed point of B.
Conversely, let be a solution of ; that is,
Then, introducing , we obtain
Let us assume that . We have just obtained that . Then, for sufficiently large (and sufficiently small , one can reach the inequality . Consequently, . It follows that and, after a differentiation, we obtain (1.4).
Lemma 2.1 is thus proved.
Theorem 2.2. Let be the set of zeros of the initial function; that is, and . If , , then, there exists a unique oscillatory solution of the initial value problem (1.4), belonging to .
Proof. We show that maps into itself; that is, .
Indeed, for every , the function is continuous on and differentiable on every . We have also and .
We show that , . (The last inequalities imply that is bounded because .)
We notice that , . For sufficiently large , we obtain for We have
Therefore, for sufficiently large , we obtain
Consequently, the operator maps into itself.
We show that B is a contractive operator. Indeed,
It remains to estimate the derivative of B.
It follows .
Consequently, where does not depend on and .
We have to verify that is j-bounded. Indeed, since j is an identity mapping,
Therefore, in view of the fixed point theorem for contractive mappings in uniform spaces (cf. ), the operator B has a unique fixed point, and it is an oscillatory solution of (1.4).
Theorem 2.2 is thus proved.
3. Numerical Example
Finally, we summarize all inequalities needed for the applications:
Consider a line with the following specific parameters:
Let us check the propagation of millimeter waves . We have If we choose , then , and .
Consequently, , and .
Since , then the above inequalities (omitting the second one) become
If the V-I characteristic of the nonlinear resistive element is , then ; . It follows that .
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