Research Article

Maximum principle plays a very important role in the study of differential equations. It gives useful information about the uniqueness, boundedness and symmetry of the solution. Maximum principles for differential equations have been discussed by Protter and Weinberger [1], R. P. Sperb [2] and in [3], [4], [5], [6]. But their discussion does not include the maximum principles for higher order derivatives. The purpose of this paper is to discuss maximum principle for higher order derivatives of a function. Let R denotes the real line and a, b  R and f is a real valued function defined on [a, b]. Maximum principle for the function gives maximum value of the function which is on the boundary of its domain or inside the domain depending on the function, domain of the function and derivative of the function. The precise definition of maximum principle is given below:

Definition

We say that the maximum principle holds for f(x) on [a, b] if f(x) ≤ max {f (a), f (b)} on [a, b]

We know that if  or  then maximum principle holds for f(x) on [a, b]. But if  for n 3 then maximum principle does not necessarily hold on [a, b]. For example, consider  and its third derivative. We present the conditions under which (x)  0 where n  3 does imply the maximum principle.

Let (I) where I = [a, b] be the set of all real valued functions on I that are n times continuously differentiable. We know that

(x) =     

Is the n^th degree Taylor’s polynomial of f (x) at a, where

f (x)   ( I ) and a  I.

Main Results

            We prove the following theorem which gives the basic result for maximum principle for a function f on [a, b]

Theorem1.1: Let f (x)   ([a, b]) for some n 2. 

                       If    0 on [a, b] then f (x)    (x)

                        on [a , b] where  (x) = (x)  +         

Proof:  Let f (x)   ([a, b]) for some n 2.

If   on [a, b]. We have by Taylor’s theorem

F (x) – P_n-2(x) = h(x)(x-a)^n-1

This gives

It follows that

It follows that

Since      on [a, b], therefore  on [a, b]

 So   h(x) h(b)   which gives    

f (x)

i.e. f(x)  on [a, b]

Now we prove the following theorem which shows that maximum principle holds under initial conditions.