Multiple Solutions for Boundary Value Problems of th-Order Nonlinear Integrodifferential Equations in Banach Spaces
The boundary value problems of a class of th-order nonlinear integrodifferential equations of mixed type in Banach space are considered, and the existence of three solutions is obtained by using the fixed-point index theory.
Guo  considered the initial value problems of a class of integrodifferential equations of Volterra type and obtained the existence of maximal and minimal solutions by establishing a comparison result. In , the author and Qin investigated a first-order impulsive singular integrodifferential equation on the half line in a Banach space and proved the existence of two positive solutions by means of the fixed-point theorem of cone expansion and compression with norm type. For other results related to integrodifferential equations in Banach spaces please see also [3–6] and the references therein. It is worth pointing out that the nonlinear terms involved in the equations they considered are either sublinear or superlinear globally.
In this paper, by using fixed-point index theory (for details please see ), we consider the th-order integrodifferential equations with nonlinear terms neither sublinear nor superlinear globally and prove the existence of three solutions.
Let be a real Banach space and a cone in which defines a partial ordering in by if and only if . is said to be normal if there exists a positive constant such that implies , where denotes the zero element of and the smallest is called the normal constant of . If and , we write . is said to be solid if its interior is not empty; that is, . In case of , we write . For details on cone theory, please see .
We consider the following boundary value problem (BVP for short) in : where , , denotes the zero element of , and with , , , and the set of all nonnegative numbers. Let
Denote that is a map from into and is continuous on . It is clear that is a Banach space with norm defined by Let It is obvious that and are two cones in and , respectively.
Lemma 1. is the solution of problem (1) if and only if is the fixed point of operator defined by
Proof. For , Taylor's formula with the integral remainder term gives
Taking , we have
into (8), we get
Let be the solution of BVP (1). Then (10) implies
Comparing this with (6), we have , which means that is the fixed point of the operator in .
On the other hand, let be the fixed point of the operator . By (6), where . It follows by taking and in (12) that It is also clear from (12) that Hence, . Then (13)–(14) imply that is the solution for BVP (1) in .
Remark 2. By and , one can see that is neither sublinear nor superlinear globally.
Lemma 3 (see ). Let be a bounded set of . Then where .
Lemma 5. Let hold. Then operator defined by (6) is a strict set contraction from into .
Proof. It is easy to see that and is a bounded operator by (6), (12), and .
Now we check that operator is continuous from into . Let , , and For any , by (6), Then the Lebesgue dominated convergence theorem gives Hence, Similarly, in view of (12), we get Then Consequently, the continuity of operator is proved.
Let be bounded. Then is bounded. We prove that is equicontinuous on . In fact, , by (12), According to the absolute continuity of Lebesgue integral, is equicontinuous on . Therefore, Lemma 4 implies that where ( is fixed, ). By (6), we see that where , It follows from (31) and that which implies, according to Lemma 3, that where in view of (15).
Similarly, we have Thus, we get by (34) and (35). Noticing that is bounded and continuous, the conclusion follows.
Theorem 6. Let be a normal solid cone and let , , and hold. Then BVP (1) has at least three solutions in .
Proof. Condition implies that there exist and , such that, for ,
Choose . Let
For , we have , , and . So, it follows from (6), (12), and (36) that
Hence, . Thus, we have shown that
Similarly, by (18), it is easy to get that there is a number such that and
where and is the normal constant of .
Let It is easy to see that , , and are all nonempty bounded open convex sets of , and As the proof of (38), for , by , On the other hand, according to , for , , , and , we get by (12) that Condition also implies that Consequently, in view of (43) and (45), we have shown that It follows from (39), (40), (42), (46), and Lemma 5 that where denotes the fixed-point index . Therefore, has three fixed points , , and. By Lemma 1, BVP (1) has at least three solutions in .
An application of Theorem 6 is as follows.
Example 7. Consider
where and .
Obviously, () is the trivial solution of BVP (48).
Conclusion. BVP (48) has at least two nontrivial nonnegative solutions.
Proof. Let , -dimensional space, with norm and
Then is a normal and solid cone in and (48) can be regarded as a BVP of the form (1), where
Obviously, and is satisfied for since is finite-dimensional.
One can see that Then (51) implies that Therefore, Hence, On the other hand, it is easy to see that Thus, (55) and (56) imply that is satisfied.
Now, we check . Let , and , . Obviously, and, for , , , , , and (i.e., , , , , , , , ). Then (51) implies that where . So, we have . Hence, is satisfied. And, finally, the conclusion follows from Theorem 6.
The author is grateful to Professor Guo Dajun and two anonymous referees for their valuable suggestions and comments.
D. Guo, “Positive solutions of an infinite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 5, pp. 2078–2090, 2009.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.View at: MathSciNet
D. Guo, V. Lakshmikantham, and X. Liu, Nonlinear Integral Equations in Abstract Spaces, vol. 373 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.View at: MathSciNet