In this section, we extend some results in the previous section to the following equation:
(4.1)
where and are linear operators defined as
where and for , , , .
Consider the boundary value problem for the linear differential equation:
(4.2)
where is sufficiently large, . We introduce the following assumptions.
(P1) The condition implies that boundary value problem (4.2) has a unique solution .
(P2) The condition implies that on J.
(P3) The condition () implies that on J.
(P4) The condition () implies that on J.
We say that the boundary condition satisfies P123 if (4.2) satisfies conditions (P1), (P2), (P3), and P134 if (4.2) satisfies conditions (P1), (P3), (P4).
Theorem 4.1
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(1)
Assume that the boundary condition satisfies P123 and (H1) holds. Then (4.1) has at least one solution x with .
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(2)
Assume that the boundary condition satisfies P134 and (H2) holds. Then (4.1) has at least one solution x with .
The proof of Theorem 4.1 is similar to that of Theorem 3.1 and we omit it.
Remark 4.1 The solution obtained in Corollary 3.2 or (2) of Theorem 4.1 may be trivial. Further suppose that
Then the solution obtained is nonnegative and nontrivial.
Remark 4.2 The boundary condition satisfies P134 if it satisfies P123.
Remark 4.3 Consider the two-point boundary conditions:
(4.3)
(4.4)
One can easily check that boundary conditions (4.3), (4.4) satisfy P123, and conditions (4.5), (4.6), (4.7) satisfy P134.
Next, we consider the boundary conditions
(4.8)
where α, β, λ, μ, (), () are constants and , , .
Theorem 4.2 Set , .
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(1)
If , , then the boundary condition satisfies P123.
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(2)
If , or , , then the boundary condition satisfies P134.
Proof Without loss of generality, we assume that . For any sufficiently large and , the linear differential equation
(4.9)
has a unique solution .
Suppose that on J; if is not true, by the maximum principle, we get that or . If , then . From the boundary conditions, we have
By the maximum principle, we obtain that , which is a contradiction. If , then . From the boundary conditions, we have
By the maximum principle, we obtain that , a contradiction.
If and , , then .
Now suppose that on J and .
If there is such that , noting that , we have
If , from the boundary conditions, we obtain that , , which implies that . The case has been discussed. If , from the boundary conditions, we obtain that , , which implies that . The case has also been discussed. The proof is complete. □
Example 4.1 Consider the differential equation
(4.10)
where , are constants.
Let and . Set , . If n is a sufficiently large, positive integer, then for any , ,
By Theorems 4.1 and 4.2, (4.10) has a solution . Hence, (4.10) has infinitely many solutions.
Example 4.2 Consider the differential equation
(4.11)
where , , are constants.
In fact, and . The boundary conditions in (4.11) satisfy P134 for and P123 for .
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(1)
Equation (4.11) has a solution and , for all . Set , then for all . By Theorems 4.1 and 4.2, (4.11) has a solution . Now we show that for . Assume that there exists with . Since , is minimum value and , . On the other hand, , a contradiction. If , then . This is impossible.
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(2)
Equation (4.11) has infinitely many solutions for . Set , , where is an integer. Since for all , (4.11) has a solution . Hence, (4.11) has infinitely many solutions.