Some fixed point theorems and existence of positive solutions of two-point boundary-value problems

https://doi.org/10.1016/j.na.2007.11.040Get rights and content

Abstract

Three theorems are obtained for the existence of at least one or three fixed points for a completely continuous mapping, which extend the Krasnoselskii’s compression–expansion theorem in cones. Based on them two theorems for the existence of positive solutions of two-point boundary-value problems are proved under a quite relaxed condition compared with the existing literature.

Introduction

In the study of boundary-value problems (BVPs), the existence of positive solutions has been attracting much attention of mathematicians [1], [2], [3], [4], [5], [6]. Since most results are obtained by the use of the Krasnoselskii’s compression–expansion theorem [7] in cones, the nonlinear item f(t,x) or a(t)f(x) that appeared in their differential equations is required to be nonnegative or at least bounded below. Among them, V. Anuhadha, D.D. Hai and R. Shivaji researched the two-point BVP {(p(t)u)+λf(t,u)=0,r<t<R,au(r)bp(r)u(r)=0,cu(R)+dp(R)u(R)=0, under the assumptions

  • (A1)

    pC([r,R],(0,));

  • (A2)

    a,b,c,d[0,) with ac+ad+bc>0;

  • (A3)

    fC([r,R]×R+,R) and there exists an M>0 such that f(t,u)M for each t[r,R],u0; and

  • (A4)

    limuf(t,u)u= uniformly on a compact subinterval [α,β] of [r,R].

They proved the following

Theorem A Theorem 2.1 in [2]

Suppose (A1)–(A4hold. Then(1.1)has a positive solution for λ>0 sufficiently small.

R.P. Agarwal, Huei-Lin Hong and Cheh-Chib yeh [4]studied a similar BVP in the form{(p(t)u)+λf(t,u)=0,0<t<1,α1u(0)β1p(0)u(0)=0,α2u(1)+β2p(1)u(1)=0,with the conditions

(B1pC([0,1],(0,)) ;

(B2λ>0,αi,βi0 for i=1,2 and α1α2+α1β2+α2β1>0 ;

(B3fC([0,1]×[0,),R) and there exists a positive constant M such that f(t,u)M for every t[0,1],u>0 .

Letγ=(β1+α101drp(r))(β2+α201drp(r))α1β2+α2β1+α1α201drp(r).They proved

Theorem B

Let (B1) (B3) hold. Assume that there exist a function h:[0,1]R+ and a positive constant k such thatf(t,u)M+h(t)fort[14,34],0uγk(M+1),1434G(12,s)h(s)dsγ(M+1)andlimumax0t1f(t,u)u=c1[0,Rk).Then BVP(1.2)has at least one positive solution for λ(0,k] .

Theorem C

Let (B1)–(B3hold. Assume that there exist a function h:[0,1]R+ and a positive constant k such thatf(t,u)M+h(t),u[0,γk(M+1)],01G(s,s)h(s)dsγ(M+1).(a) If limumint[14,34]f(t,u)u=,then BVP(1.2)has at least one positive solution for λ[0,k] .

(b) If k>1 and limumint[14,34]f(t,u)u=c2(Qσ,) , then BVP(1.2)has at least one positive solution for λ[1,k].

Ge and Ren [6] researched {(p(t)u(t))+λa(t)f(t,u(t))=0,α1u(0)β1p(0)u(0)=0=α2u(1)+β2p(1)u(1), with the assumptions

(C1) and (C2) are the same as (B1) and (B2);

(C3aC((0,1),R+),0<01G(t,s)a(s)ds< for t(0,1), where G(t,s)=1ρ{(β1+α10tdrp(r))(β2+α2s1drp(r)),0ts1(β1+α10sdrp(r))(β2+α2t1drp(r)),0st1. The existence of positive solutions in [6] is established essentially in a bounded area Ω={xC([0,1],R):Mw(t)x(t)r(orR)}, where w(t)=1ρ[(β1+α10t1rp(r)dr)(β2+α2t1rp(r)dr)α1α20trp(r)drt11rp(r)dr] is the unique solution of {(p(t)u(t))+1=0α1u(0)β1u(0)=0=α2u(1)+β2u(1) with ρ=α1β2+α2β1+α1α201drp(r). Therefore the restriction f(t,u)M in (A3) or (B3) is removed.

In this paper we will study the BVP (1.2) under the conditions

(H1pC([0,1],(0,));

(H2λ>0,αi,βi0 for i=1,2 and ρ=α1β2+α2β1+α1α201drp(r)>0. Different from [4] we give the existence of positive solutions of BVP (1.2) in the cone K=C([0,1],R+) without the restriction that f is bounded below.

To this end we establish some fixed point theorems in the cone at first.

Section snippets

Fixed point theorems

The Krasnoselskii’s compression–expansion theorem is a useful tool in proving the existence of positive solutions to BVPs. The theorem says

Theorem D

Krasnoselskii, [7]

Let X be a Banach space and KX a cone. Assume that Ω1,Ω2 are two open bounded sets in X with 0Ω1Ω¯1Ω2 andT:K(Ω¯2Ω1)Kis a completely continuous operator such that

(i) Tuu,uKΩ1 , and Tuu,uKΩ2

or

(ii) Tuu,uKΩ1 , and Tuu,uKΩ2 .

Then T has a fixed point in K(Ω¯2Ω1) .

On the other hand, the condition (i) or (ii) inTheorem Dis a

Positive solutions of BVP (1.2)

Let X=C([0,1],R) with x=max0x1|x(t)| and K={xX:x(t)0}. It is well known that the existence of positive solutions to BVP (1.2) is equivalent to the existence of solutions of the abstract equation u=Tu,uK, where (Tu)(t)=λ01G(t,s)f(s,u(s))ds and G(t,s) is given in (1.5). Obviously T:KX is a completely continuous mapping. Let η=1, then both (2.1), (2.2) hold.

Lemma 3.1

[6], Lemma 2.2

Define Θ:TKK by (Θy)(t)=max{y(t),0} for yTK . Then ΘT:KK is also a completely continuous operator.

For fC([0,1]×R+,R) and pC([0,1]

References (7)

There are more references available in the full text version of this article.

Cited by (17)

  • The three-solutions theorem for p-Laplacian boundary value problems

    2012, Nonlinear Analysis, Theory, Methods and Applications
  • Positive solutions for a system of higher-order multi-point boundary value problems

    2011, Computers and Mathematics with Applications
    Citation Excerpt :

    In recent years, the multi-point boundary value problems for second-order or higher-order differential or difference equations/systems have been investigated by many authors (see also [16–38]), by using different methods such us fixed point theorems in cones, the Leray–Schauder continuation theorem and its nonlinear alternatives and the coincidence degree theory.

  • Positive solutions of even order system periodic boundary value problems

    2010, Nonlinear Analysis, Theory, Methods and Applications
    Citation Excerpt :

    The main approaches are based on the fixed point theory on cones, the upper and lower solution method, and the variational method. See, for example, [1–9] for recent developments in this area. The existence of nodal solutions of BVPs has also been studied using the shooting method and a bifurcation approach, see [10–14] for some work.

View all citing articles on Scopus

Sponsored by the National Natural Foundation of China (No. 10671023).

View full text