An Extension to Nonlinear Sum-Difference Inequality and Applications
© W.-S.Wang and X. Zhou 2009
Received: 31 March 2009
Accepted: 17 May 2009
Published: 25 June 2009
We establish a general form of sum-difference inequality in two variables, which includes both more than two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums. We employ a technique of monotonization and use a property of stronger monotonicity to give an estimate for the unknown function. Our result enables us to solve those discrete inequalities considered in the work of W.-S. Cheung (2006). Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.
where , , and are all constants, , and are both nonnegative real-valued functions defined on a lattice in , and is a continuous nondecreasing function satisfying for all .
where . In (1.7) we replace the constant , the functions , , , and in (1.6) with a function , more general functions , and , respectively. Moreover, we consider more than two nonlinear terms and do not require the monotonicity of every . We employ a technique of monotonization to construct a sequence of functions which possesses stronger monotonicity than the previous one. Unlike the work in  for two sum terms, the maximal regions of validity for our estimate of the unknown function are decided by boundaries of more than two planar regions. Thus we have to consider the inclusion of those regions and find common regions. We demonstrate that inequalities (1.6) and other inequalities considered in  can also be solved with our result. Furthermore, we apply our result to boundary value problems of a partial difference equation for boundedness, uniqueness, and continuous dependence.
2. Main Result
Throughout this paper, let , , and , are given nonnegative integers. For any integers , let , , and . Define , and let denote the sublattice in for any .
For functions , their first-order differences are defined by and . Obviously, the linear difference equation with the initial condition has the solution . In the sequel, for convenience, we complementarily define that .
(H1) is a strictly increasing continuous function on satisfying that and for all .
(H2) All are continuous and positive functions on .
(H3) on .
(H4) All are nonnegative functions on .
which is nondecreasing in and for each fixed and and satisfies for all .
The comparison between the both sides implies that (2.8) is equivalent to the condition given in the definition of in Theorem 2.1 with .
If we choose , , , with , and and restrict to be a constant in (1.7), then we can apply Theorem 2.1 to inequality (1.6) discussed in .
3. Proof of Theorem
Thus, it means that we can take . Moreover, , .
In the following, we will use mathematical induction to prove (3.5).
that is, (3.5) is true for .
We further claim that the term is the same as , defined in (3.6), . For convenience, let . Obviously, it is that .
for all .
Considering continuities of and for as well as of in and letting , we obtain (2.4). This completes the proof.
We remark that , lie on the boundary of the lattice . In particular, (2.4) is true for all when every ( ) satisfies . Therefore, we may take , .
4. Applications to a Difference Equation
for given functions and ( ) satisfying for , and functions and satisfy that . Obviously, (4.1) is a generalization of the BVP problem considered by [26, Section 3], and the theorems of  are not able to solve it. In the following we first apply our main result to the discussion of boundedness of (4.1).
Let . Since , (4.6) is of the form (1.6). Applying our Theorem 2.1 to inequality (4.6), we obtain the estimate of as given in this corollary.
for all , then every solution of BVP (4.1) is bounded on .
Next, we discuss the uniqueness of solutions for BVP (4.1).
for and , where as assumed in the beginning of Section 2 with natural numbers and are both nonnegative functions defined on the lattice , are both nondecreasing with the nondecreasing ratio such that , for all and for and is strictly increasing odd function satisfying for . Then BVP (4.1) has at most one solution on .
Thus we conclude from (2.4) that , implying that for all since is strictly increasing. It proves the uniqueness.
If or in (4.8), the conclusion of the Corollary 4.2 also can be obtained.
where is strictly increasing odd function satisfying for , , and are functions satisfying .
are all sufficiently small. Then solution of BVP (4.14) is sufficiently close to the solution of BVP (4.1).
By (4.10) we see that ( ) as . It follows from (4.18) that and hence depends continuously on and .
Our requirement of the small difference in Corollary 4.4 is stronger than the condition (iii) in [26, Theorem ], but it is easier to check than the condition of them.
The authors thank Professor Weinian Zhang (Sichuan University) for his valuable discussion. The authors also thank the referees for their helpful comments and suggestions. This work is supported by the Natural Science Foundation of Guangxi Autonomous Region (200991265), by Scientific Research Foundation of the Education Department of Guangxi Autonomous Region of China (200707MS112) and by Foundation of Natural Science of Hechi University (2006A-N001) and Key Discipline of Applied Mathematics of Hechi University of China (200725).
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