Estimation on Certain Nonlinear Discrete Inequality and Applications to Boundary Value Problem
© Wu-Sheng Wang. 2009
Received: 1 November 2008
Accepted: 14 January 2009
Published: 9 February 2009
We investigate certain sum-difference inequalities in two variables which provide explicit bounds on unknown functions. Our result enables us to solve those discrete inequalities considered by Sheng and Li (2008). Furthermore, we apply our result to a boundary value problem of a partial difference equation for estimation.
Various generalizations of the Gronwall inequality [1, 2] are fundamental tools in the study of existence, uniqueness, boundedness, stability, invariant manifolds, and other qualitative properties of solutions of differential equations and integral equation. There are a lot of papers investigating them (such as [3–8]). Along with the development of the theory of integral inequalities and the theory of difference equations, more attentions are paid to some discrete versions of Gronwall-Bellman-type inequalities (such as [9–11]). Some recent works can be found, for example, in [12–17] and some references therein.
We first introduce two lemmas which are useful in our main result.
Lemma 1.1 (the Bernoulli inequality ).
Let and , then .
Lemma 1.2 ().
where for .
where for .
Furthermore, we apply our result to a boundary value problem of a partial difference equation for estimation. Our paper gives, in some sense, an extension of a result of .
2. Main Result
Throughout this paper, let denote the set of all real numbers, let be the given subset of , and denote the set of nonnegative integers. For functions , their first-order differences are defined by , , and . We use the usual conventions that empty sums and products are taken to be 0 and 1, respectively. In what follows, we assume all functions which appear in the inequalities to be real-value, and are constants, and .
Using Lemma 1.2, the desired inequality (2.2) is obtained from (2.1), (2.3), and (2.4). This completes the proof of Lemma 2.1.
where we note the definitions of and in (2.6). From (2.6), we see is nonnegative and nonincreasing in each variable. By applying Lemma 2.1, the desired inequality (3.3) is obtained from (2.9) and (2.11). This completes the proof of Theorem 2.2.
where we note the definitions of and in (2.14) and (2.15). From (2.14) we see is nonnegative and nonincreasing in each variable. By applying Lemma 2.1, the desired inequality (2.19) is obtained from (2.9) and (2.18). This completes the proof of Theorem 2.3.
where and are defined by (2.20) and (2.21), respectively. From (2.20), we see is nonnegative and nonincreasing in each variable. By applying Lemma 2.1, the desired inequality (2.19) is obtained from (2.9) and (2.24). This completes the proof of Theorem 2.4.
3. Applications to Boundary Value Problem
where and are constants, , functions are given, and functions are nonincreasing. In what follows, we apply our main result to give an estimation of solutions of (3.1).
Let . Equation (3.6) is of the form (1.4), here . Applying our Theorem 2.2 to inequality (3.6), we obtain the estimate of as given in Corollary 3.1.
This work is supported by Scientific Research Foundation of the Education Department Guangxi Province of China (200707MS112) and by Foundation of Natural Science and Key Discipline of Applied Mathematics of Hechi University of China.
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