Xiuli Lin,　Zengqin Zhao(School of Mathematical Sciences,Qufu Normal University,Shandong 273165,PR China)

ITERATIVE POSITIVE SOLUTIONS FOR SINGULAR RIEMANN-STIELTJES INTEGRAL BOUNDARY VALUE PROBLEM??

Xiuli Lin?,Zengqin Zhao
(School of Mathematical Sciences,Qufu Normal University,Shandong 273165,PR China)

By applying iterative technique,we obtain the existence of positive solutions for a singular Riemann-Stieltjes integral boundary value problem in the case that f (t,u) is non-increasing respect to u.

Riemann-Stieltjes integral boundary value problems;positive solution;non-increasing;iterative technique

2000 Mathematics Subject Classification 26A42

1　Introduction

Problems with boundary conditions,especially Riemann-Stieltjes integral boundary condition,have been studied in many papers (see[1-8]and the references therein) .In[3],by applying monotone iterative technique,Mao and Zhao established a sufficient condition for the existence of positive solutions for problem (1.1) :

where A is right continuous on[0,1) ,left continuous at t=1,and nondecreasing ondenotes the Riemann-Stieltjes integral of u with respect to A.k is a constant and f (t,u) is increasing with respect to u.

In this paper,we consider the case that f (t,u) is non-increasing with respect to u, and f (t,u) may be singular at u=0,t=0 (and/or t=1) .By searching an iterative initial element,we construct a non-monotonic iterative sequence which has nondecreasing and non-increasing subsequence to obtain the existence and uniqueness of positive solutions in some set Q.Meanwhile,we also give an error estimate.

2　Preliminaries

The following conditions are assumed in this paper:

From (2.1) ,it is easy to see that if τ∈[1,+∞) ,then

(S3) There exists a k＞0 such that sinh

Lemma 2.1[1]Assume that h∈C (0,1) and (S3) holds.Then the following linear boundary value proble m

has a unique positive solution u expressed in the following form

Remark 2.1 Assume that (S1) , (S2) and (S3) hold.Then solutions for (1.1) are equivalent to continuous solutions of the integral equation

where F (t,s) is defined by (2.4) .

Lemma 2.2[3]For any t,s∈[0,1],there exist constants c1,c2＞0 such that

3　Main Results

In this section,we state and prove our main result.

Let E be a Banach space C[0,1],and define

Theorem 3.1 Let (S1) - (S3) hold and assume that

Then the BVP (1.1) has a unique positive solution x?in Q.Moreover,for any initialthere exists a sequence of functionsdefined by

uniformly converges to the unique positive solution x?(t) for the BVP (1.1) .And for n＞1,we have the error estimation

where k is a constant with 0＜k＜1 and determined by x0.

Proof Define an operator K:E→E by

where F (t,s) is defined by (2.4) .

For any u∈Q,there exists a c∈ (0,1) such that

which,together with (S2) , (2.1) and (2.2) ,implies that

Thus from (3.1) , (3.4) and Lemma 2.2 we can obtain

Thus,we obtain K:Q→Q.By the standard argument,K:Q→Q is completely continuous.

From (S2) ,it is easy to see that K is non-increasing and K2is non-decreasing with respect to u.

Next,we prove there exists an iterative sequence {xn} satisfying

Since Ke∈Q,there exists a constant ce∈ (0,1) such that

For ceas in (3.8) ,there exists a sufficiently large constant z0such that

By (2.1) , (2.2) and (3.8) - (3.10) and that K is non-increasing with respect to u,we have

Note again that K is non-increasing with respect to u.By (3.10) , (3.14) , (3.15) ,it follows that

From (3.9) - (3.15) ,there exists an iterative sequence {xn} satisfying

In what follows,we prove there exists an x?∈Q such that

For τ∈ (0,1]and noting that K is non-increasing with respect to u,from (3.4) we have

Then from (3.18) and that K2is non-decreasing,we have

In view of (3.7) , (3.19) ,we have for any n,p∈N,

By (3.20) , (3.21) ,we can see that (3.17) holds.Letting n?→∞in (3.10) ,we obtain x?(t) =Kx?(t) ,and x?(t) is a positive solution for the BVP (1.1) .

Now,we prove the uniqueness of x?.Let y?∈Q be another positive solution for the BVP (1.1) ,and then there exists a constant cy?with 0＜cy?＜1 such that

Letting n?→∞in (3.22) ,we have y?=x?.Hence,the positive solution for the BVP (1.1) is unique in Q.

For any x0∈Q,there exist constants 0＜cx0,cKx0＜1,such that

For the above cx0cKx0,let z0large enough such that

Then we can similarly have an iterative sequence {xn} satisfying

and for any numbers n,p,

So,the sequence {xn(t) } uniformly converges to the positive solution x?(t) for the BVP (1.1) on[0,1].

Let p→∞in (3.23) , (3.24) ,then we can have the error estimation

where k is a constant with 0＜k＜1 and determined by x0.The proof is completed.

Example 3.1 Consider

Acknowledgements The authors would like to thank the referee for his/her careful reading and kind suggestions.

References

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(edited by Liangwei Huang)

?This work was supported by Program for Scientific research innovation team in Colleges and universities of Shandong Province,the Doctoral Program Foundation of Education Ministry of China (20133705110003) ,the Natural Science Foundation of Shandong Province of China (ZR2014AM007) ,the National Natural Science Foundation of China (11571197) .

?Manuscript October 6,2015;Revised March 17,2016

?.E-mail:lin-xiuli78@163.com