- Open Access
Multiple positive solutions of a semipositone singular boundary value problem on time scales
© Dahal; licensee Springer. 2013
- Received: 2 July 2013
- Accepted: 28 October 2013
- Published: 21 November 2013
In this paper, we study the following singular semipositone boundary value problem on time scales:
where and are continuous; and is Lebesgue ∇-integrable. By constructing a special cone and using a fixed point theorem, we establish some sufficient conditions for the existence of multiple positive solutions. Two examples are given at the end of the paper to demonstrate our result.
MSC:39A10, 39A13, 34B16, 34B18, 34N05.
- time scales
- positive solutions
- fixed point theorem
- Green’s function
where is a time scale; and are continuous; and is Lebesgue ∇-integrable. Note that denotes the time scale interval, . The study of analysis on time scales was initiated by Stefan Hilger in 1988, and the first paper appearing in this field by Hilger was . An excellent introduction to time scales calculus can be found in Chapter 1 of  and in . Chapters 7 and 9 of the text  deal with finding positive solutions of several boundary value problems on time scales by various contemporary authors. We also want to guide the readers to take a look at early research papers on time scales [4–6]. Note that in our problem, q may change its sign, so we call this type of problem semipositone. The study of nonlinear, singular boundary value problems is not new but the consideration of a semipositone case is relatively new even in differential equations. Semipositone problems arise in many physical and chemical processes such as in chemical reactor theory . In applications one is interested in finding positive solutions. In recent years, several authors studied semipositone BVPs on time scales, and we want to mention some papers in literature [8–14], and the references therein. Among several other papers in difference equations, Bai and Xu  recently studied semipositone problems in difference equations. All the above mentioned papers are concerned with the existence of only one positive solution. Anderson and Wong in  and Bai and Xu in  also require that the nonlinear term of the equation has a finite lower bound. But we do not require that in this paper and, in fact, the nonlinear term is allowed to decrease without bound. Thus this paper fills the gap in literature on time scales calculus providing the existence of multiple positive solutions and allowing the nonlinear term decrease without bound at the same time. To our best knowledge, this result is new in the time scales setting and it covers the results not only for ordinary differential equations but also difference equations, q-difference equations, and other exotic time scales. The nabla derivative was introduced in . As a special case when , this result includes those of . Let a and b be such that , and has at least two points.
Next we want to construct a cone in which we will look for positive solutions.
Let , with , and define .
Then one can easily verify that X is a real Banach space, and P is a cone in X.
Now we state the well-known fixed point theorem that we will use later in this paper.
Theorem 1 
for all and for all , or
for all and for all .
Then A has a fixed point in .
It is easy to see that finding solutions of BVP (9) is equivalent to finding fixed points of the operator T on P.
Now we state and prove a lemma that connects singular positone BVP (9) to main BVP (1).
Lemma 2 If is the unique positive solution of singular positone BVP (9) such that then BVP (1) has a positive solution .
Now we impose the following conditions for the rest of the paper:
(H1) and are continuous.
(H2) is Lebesgue ∇-integrable such that
(H3) There exists such that for ,
(H4) There exists such that for ,
(H5) uniformly for t in any closed subinterval of .
To apply Theorem 1, we first prove the following lemmas.
Lemma 3 Assume that (H1) and (H2) hold. Then is completely continuous.
Therefore is uniformly bounded.
By standard arguments (see ) using the Arzela-Ascoli theorem and the Lebesgue dominated convergence theorem, we can easily see that T is a completely continuous operator. □
Lemma 4 Assume that (H1)-(H3) hold, and set and . Then for all , where is as given in (H3).
Thus we have for all . □
Lemma 5 Assume that (H1)-(H5) hold, and set . Then for all , where is as given in (H4).
Thus we have , . □
Take . Then we have .
Lemma 6 Assume that (H1)-(H5) hold, and set . Then , .
Thus for all . □
Theorem 7 Suppose that (H1)-(H5) hold. Then T has two fixed points and such that , where , from (H3), and , as defined in (H2).
are two positive solutions of BVP (1). □
In this section we give two examples as applications of Theorem 7.
then for any , we have .
It is clear that .
then for any , we have .
Note it is clear that .
The sole author has made all contributions.
The author would like to thank the anonymous referee for very helpful comments and suggestions.
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