Positive Solutions to Nonlinear First-Order Nonlocal BVPs with Parameter on Time Scales
© C. Gao and H. Luo. 2011
Received: 4 May 2010
Accepted: 3 June 2010
Published: 27 June 2011
We discuss the existence of solutions for the first-order multipoint BVPs on time scale : , , , where is a parameter, is a fixed number, , is continuous, is regressive and rd-continuous, , , , , and . For suitable , some existence, multiplicity, and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index.
where is a fixed number, , is continuous, is regressive and rd-continuous, , and , is defined in its standard form; see [1, page 59] for details.
The multipoint boundary value problems arise in a variety of different areas of applied mathematics and physics. For example, the vibrations of a guy wire of a uniform cross-section and composed of parts of different densities can be set up as a multipoint boundary value problem ; also many problems in the theory of elastic stability can be handled by a multipoint problem . So, the existence of solutions to multipoint boundary value problems have been studied by many authors; see [4–13] and the reference therein. Especially, in recent years the existence of positive solutions to multipoint boundary value problems on time scales has caught considerable attention; see [10–14]. For other background on dynamic equations on time scales, one can see [1, 15–18].
Motivated by the above results, by using the well-known fixed-point index theory [16, 19], we attempt to obtain some existence, multiplicity and nonexistence criteria of positive solutions to (1.1) for suitable .
The rest of this paper is arranged as follows. Some preliminary results including Green's function are given in Section 2. In Section 3, we obtain some useful lemmas for the proof of the main result. In Section 4, some existence and multiplicity results are established. At last, some nonexistence results are given in Section 5.
Throughout the rest of this paper, we make the following assumptions:
Our main tool is the well-known results from the fixed-point index, which we state here for the convenience of the reader.
Theorem 2.1 (see ).
This proof is similar to [13, Lemma ], so we omit it.
Similar to the proof of [13, Lemma ], we can see that is completely continuous. By the above discussions, its not difficult to see that being a solution of BVP (1.1) equals the solution that is a fixed point of the operator .
3. Some Lemmas
4. Some Existence and Multiplicity Results
Similar to the proof of Theorem 4.1, we have the following results.
Similar to the proof of Theorem 4.3, we have the following results.
By making some minor modifications to the proof of Theorem 4.5, we can obtain the existence of at least one positive solution, if one of the following conditions is satisfied:
5. Some Nonexistence Results
which is a contradiction.
which is a contradiction.
This work was supported by the NSFC Young Item (no. 70901016), HSSF of Ministry of Education of China (no. 09YJA790028), Program for Innovative Research Team of Liaoning Educational Committee (no. 2008T054), the NSF of Liaoning Province (no. L09DJY065), and NWNU-LKQN-09-3
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