In this section, some existence theorems are presented.
Theorem 4.1 Let , and be nonnegative constants and . If f is continuous and satisfies
with
and
then the anti-periodic boundary problem (1.1) and (1.2) has at least one solution.
Proof Consider the family (3.1). We want to show the conditions of Theorem 3.2 hold for some positive constants M and N.
Let be a solution to (3.1) and consider the equivalent equation (3.2), that is,
(4.1)
For each , we have
Since
it follows that
(4.2)
Differentiating both sides of (4.1), we get
then
and because of
Therefore,
The rearrangement yields
(4.3)
By substituting (4.3) into (4.2) and rearranging, we obtain
So,
Hence, Theorem 3.2 holds for positive constants and . The solvability of (1.1) and (1.2) now follows. □
Theorem 4.2 Assume there exist nonnegative constants , and such that
then the anti-periodic boundary value problem (1.1) and (1.2) has at least one solution.
Proof Suppose is a solution of (3.1), and in view of (2.10), we have
Similarly,
Therefore, Theorem 3.2 holds for positive constants and . The solvability of (1.1) and (1.2) now follows. □
Example 4.1 Consider the anti-periodic boundary value problem
(4.4)
We claim (4.4) has at least one solution.
Proof Let , and in Theorem 4.2. Choose , we get for that
and
Note , we have . Thus, for ,
Then the conclusion follows from Theorem 4.2. □
Now, we reconsider the problem (1.2) and (1.3). The following result is obtained.
Theorem 4.3 Suppose is continuous. If there exist nonnegative constants , and such that
then (1.2) and (1.3) has at least one solution.
Proof The proof is similar to Theorem 4.2 and here we omit it. □
An example to highlight the Theorem 4.3 is presented.
Example 4.2 Consider the anti-periodic boundary value problem given by
(4.5)
We claim (4.5) has at least one solution.
Proof Let and see that for . For , and λ to be chosen below, see that
Thus, the conditions of Theorem 4.3 hold and the solvability follows. □
Remark 4.1 The results of [3] do not apply to the above example since grows more than linearly in . Therefore, we improve the previous results.
Finally, in order to illustrate our main results, we use the ‘bvp4c’ package in MATLAB to simulate. As shown in Figure 1(a) and (b), numerical simulations also suggest that Examples 4.1 and 4.2 with the given coefficients admit at least one solution.