Multiple positive solutions of nonlinear BVPs for differential systems involving integral conditions
© Zhang; licensee Springer. 2014
Received: 8 December 2013
Accepted: 6 March 2014
Published: 21 March 2014
In this paper, we consider the following system of nonlinear third-order nonlocal boundary value problems (BVPs for short):
where , and are linear functionals on given by Riemann-Stieltjes integrals and are not necessarily positive functionals; a, b, c, d are nonnegative constants with . By using the Guo-Krasnoselskii fixed point theorem, some sufficient conditions are obtained for the existence of at least one or two positive solutions and nonexistence of positive solutions to the above problem. Two examples are also included to illustrate the main results.
The theory of BVPs with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. Moreover, BVPs with Riemann-Stieltjes integral boundary condition (BC for short) have been considered recently as both multipoint and Riemann integral type BCs are treated in a single framework. For more comments on Stieltjes integral BC and its importance, we refer the reader to the papers by Webb and Infante [1–3] and their other related works.
by applying monotone iterative techniques, where , and are linear functionals on given by Riemann-Stieltjes integrals.
by applying the fixed point index theory.
by using fixed point index theory in a cone.
The result was quite general and covered a wide class of systems of BVPs. Here was of the form involving positive Riemann-Stieltjes measures.
where , and are linear functionals on given by Riemann-Stieltjes integrals with signed measures; a, b, c, d are nonnegative constants with . To the best of our knowledge, the study of existence of positive solutions of third-order differential systems (1.6) has not been done.
A vector is said to be a positive solution of BVP (1.6) if and only if u, v satisfy BVP (1.6) and u, v are positive on . The proof of our main results is based on the well-known Guo-Krasnoselskii fixed point theorem, which we present now.
for and for ; or
for and for .
Then A has a fixed point in .
2 Preliminary lemmas
In the remainder of this paper, we always assume that
(H1) , and ;
in our case. The kernel is Green’s function corresponding to BVP (2.1). By Lemma 2.1 and Lemma 2.2 in , we can get the following properties of Green’s function.
where , .
where , .
so , , , which together with (2.3) shows that holds. □
Let equipped with the norm , where is the usual supremum norm in . Similar to Lemma 2.1 in , we can get the following lemma.
Lemma 2.4 If , then . And so, E is a Banach space when it is endowed with the norm .
Then it is easy to verify that K is a cone in E.
Lemma 2.5 .
which shows that . □
Similar to the proof of Lemma 2.4 in , we can get the following lemma.
Lemma 2.6 is completely continuous.
3 Main results
Theorem 3.1 Assume that and . Then BVP (1.6) has at least one positive solution.
Theorem 3.2 Assume that and . Then BVP (1.6) has at least one positive solution.
Proof The proof is similar to Theorem 3.1 and therefore omitted. □
Then BVP (1.6) has at least two positive solutions.
which together with (3.13), (3.14) shows that T has at least two fixed points in and . □
Similarly, we can get the following theorem.
Then BVP (1.6) has at least two positive solutions.
Theorem 3.5 If and for and , then BVP (1.6) has no monotone positive solution.
which shows that . This is a contradiction. □
Similarly, we can prove the following theorem.
Theorem 3.6 If and for and , then BVP (1.6) has no monotone positive solution.
In this section, we give an example to illustrate our main results.
Let , then , , , .
It is easy to compute that , , , , which show that and . So, it follows from Theorem 3.1 that BVP (4.1) has at least one positive solution.
It is easy to compute that , , , and , which show that , , , and .
So, it follows from Theorem 3.3 that BVP (4.1) has at least two positive solutions.
The work is supported by the Science and Technology Foundation of Hebei Province (Z2013016) and the Science and Technology Plan Foundation of Tangshan (12110233b) and the Scientific Research Foundation of Tangshan College (13011B). The author would like to thank the anonymous referees very much for helpful comments and suggestions which led to the improvement of presentation and quality of the work.
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