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 . 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
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 .
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
In the following, we will use mathematical induction to prove (3.5).
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).
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 .
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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