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Existence and uniqueness of positive solutions to boundary value problem with increasing homeomorphism and positive homomorphism operator

Abstract

In this paper, we consider the following nonlinear boundary value problem: ( φ ( u ( t ) ) ) +a(t)f(u(t))=0, 0<t<1, u(0)= i = 1 m 2 α i u( ξ i ), u (1)=0, where φ:RR is an increasing homeomorphism and positive homomorphism with φ(0)=0. By using a fixed-point theorem on partially ordered sets, we obtain sufficient conditions for the existence and uniqueness of positive and nondecreasing solutions to the above boundary value problem.

MSC:34B18, 34B27.

1 Introduction

In this paper, we consider the existence and uniqueness of a positive and nondecreasing solution to the following boundary value problem:

( φ ( u ( t ) ) ) +a(t)f ( u ( t ) ) =0,0<t<1,
(1.1)
u(0)= i = 1 m 2 α i u( ξ i ), u (1)=0,
(1.2)

where φ:RR is an increasing homeomorphism and positive homomorphism with φ(0)=0. Here ξ i (0,1) with 0< ξ 1 < ξ 2 << ξ m 2 <1 and α i satisfy α i [0,+), 0< i = 1 m 2 α i <1.

A projection φ:RR is called an increasing homeomorphism and positive homomorphism, if the following conditions are satisfied:

  1. (1)

    φ(x)φ(y), for all x,yR with xy;

  2. (2)

    φ is a continuous bijection and its inverse mapping is also continuous;

  3. (3)

    φ(xy)=φ(x)φ(y), for all x,y R + .

In the above definition, we can replace the condition (3) by the following stronger condition:

  1. (4)

    φ(xy)=φ(x)φ(y), for all x,yR, where R=(,+).

Remark 1.1 If conditions (1), (2), and (4) hold, then it implies that φ is homogeneous generating a p-Laplace operator, i.e. φ(x)= | x | p 2 x, for some p>1.

Recently, the existence and multiplicity of positive solutions for the p-Laplacian operator, i.e., φ(x)= | x | p 2 x, for some p>1, have received wide attention, see [13] and references therein. We know that the oddness of a p-Laplacian operator is key to the proof. However, in this paper we define a new operator, which improves and generates a p-Laplacian operator for some p>1, and φ is not necessarily odd. Moreover research of increasing homeomorphisms and positive homomorphism operators has proceeded very slowly, see [4, 5].

In [4], Liu and Zhang studied the existence of positive solutions of quasilinear differential equation

( φ ( x ) ) + a ( t ) f ( x ( t ) ) = 0 , 0 < t < 1 , x ( 0 ) β x ( 0 ) = 0 , x ( 1 ) + δ x ( 1 ) = 0 ,

where φ:RR is an increasing homeomorphism and positive homomorphism and φ(0)=0. They obtain the existence of one or two positive solutions by using a fixed-point index theorem in cones. But the uniqueness of the solution is not treated.

In [5], the authors showed that there exist countably many positive solutions by using the fixed-point index theory and a new fixed-point theorem in cones. They also assumed that the operator φ:RR is an increasing homeomorphism and a positive homomorphism, and φ(0)=0.

In [6], the authors established the existence and uniqueness of a positive and nondecreasing solution to a singular boundary value problem of a class of nonlinear fractional differential equation. Their analysis relies on a fixed-point theorem in partially ordered sets. The existence of a fixed point in partially ordered sets has been considered recently in [610].

But whether or not we can obtain the existence and uniqueness of a positive and nondecreasing solution to the boundary value problem (1.1)-(1.2) still remains unknown. So, motivated by all the works above, we will prove the existence and uniqueness of a positive and nondecreasing solution for the boundary value problems (1.1)-(1.2) by using a fixed-point theorem on partially ordered sets.

2 Some definitions and fixed-point theorems

Definition 2.1 Let (E,) be a real Banach space. A nonempty, closed, convex set PE is said to be a cone provided the following are satisfied:

  1. (a)

    if yP and λ0, then λyP;

  2. (b)

    if yP and yP, then y=0.

If PE is a cone, we denote the order induced by P on E by ≤, that is, xy if and only if yxP.

The following fixed-point theorems in partially ordered sets are fundamental and important to the proofs of our main results.

Theorem 2.1 ([7])

Let (E,) be a partially ordered set and suppose that there exists a metric d in E such that (E,d) is a complete metric space. Assume that E satisfies the following condition:

if { x n } is a nondecreasing sequence in E such that  x n x, then  x n x,nN.
(2.1)

Let T:EE be a nondecreasing mapping such that

d(Tx,Ty)d(x,y)ψ ( d ( x , y ) ) ,for xy,

where ψ:[0,+)[0,+) is a continuous and nondecreasing function such that ψ is positive in (0,+), ψ(0)=0 and lim t ψ(t)=. If there exists x 0 E with x 0 T( x 0 ), then T has a fixed point.

If we consider that (E,) satisfies the following condition:

for x,yE there exists zE which is comparable to x and y,
(2.2)

then we have the following result.

Theorem 2.2 ([8])

Adding condition (2.2) to the hypotheses of Theorem  2.1, we obtain uniqueness of the fixed point.

3 Main results

The basic space used in this paper is E=C[0,1]. Then E is a real Banach space with the norm u= max 0 t 1 |u(t)|. Note that this space can be equipped with a partial order given by

x,yC[0,1],xyx(t)y(t),t[0,1].

In [8] it is proved that (C[0,1],) with the classic metric given by

d(x,y)= sup 0 t 1 { | x ( t ) y ( t ) | }

satisfies condition (2.1) of Theorem 2.1. Moreover, for x,yC[0,1] as the function max{x,y}C[0,1], (C[0,1],) satisfies condition (2.2).

The main result of this paper is the following.

Theorem 3.1 The boundary value problem (1.1)-(1.2) has a unique positive solution u(t) which is strictly increasing if the following conditions are satisfied:

  1. (A)

    a(t) is a nonnegative measurable function defined in [0,1] and a(t) does not identically vanish on any subinterval of [0,1] and

    0< 0 1 a(t)dt<+;

(f1) f:[0,+)[0,+) is continuous and nondecreasing respect to u and f(u(t))0 for tZ[0,1] with μ(Z)>0 (μ denotes the Lebesgue measure);

(f2) there exists 1<λ+1< 1 i = 1 m 2 α i φ 1 ( 0 1 a ( τ ) d τ ) such that for u,v[0,+) with uv and t[0,1]

φ ( ln ( v + 2 ) ) f(v)f(u)φ ( ln ( u + 2 ) ( u v + 1 ) λ ) .

Proof Consider the cone

K= { u C [ 0 , 1 ] : u 0 } .

As K is a closed set of C[0,1], K is a complete metric space with the distance given by d(u,v)= sup t [ 0 , 1 ] |u(t)v(t)|.

Now, we consider the operator T defined by

Tu(t)= 0 t φ 1 ( s 1 a ( τ ) f ( u ( τ ) ) d τ ) ds+ i = 1 m 2 α i 0 ξ i φ 1 ( s 1 a ( τ ) f ( u ( τ ) ) d τ ) d s 1 i = 1 m 2 α i .

By conditions (A), (f1), we have T(K)K.

We now show that all the conditions of Theorem 2.1 and Theorem 2.2 are satisfied.

Firstly, by condition (f1), for u,vK and uv, we have

T u ( t ) = 0 t φ 1 ( s 1 a ( τ ) f ( u ( τ ) ) d τ ) d s + i = 1 m 2 α i 0 ξ i φ 1 ( s 1 a ( τ ) f ( u ( τ ) ) d τ ) d s 1 i = 1 m 2 α i 0 t φ 1 ( s 1 a ( τ ) f ( v ( τ ) ) d τ ) d s + i = 1 m 2 α i 0 ξ i φ 1 ( s 1 a ( τ ) f ( v ( τ ) ) d τ ) d s 1 i = 1 m 2 α i = T v ( t ) .

This proves that T is a nondecreasing operator. On the other hand, for uv and by (f2) we have

d ( T u , T v ) = sup 0 t 1 | ( T u ) ( t ) ( T v ) ( t ) | = sup 0 t 1 ( ( T u ) ( t ) ( T v ) ( t ) ) sup 0 t 1 0 t [ φ 1 ( s 1 a ( τ ) f ( u ( τ ) ) d τ ) φ 1 ( s 1 a ( τ ) f ( v ( τ ) ) d τ ) ] d s + i = 1 m 2 α i 0 ξ i [ φ 1 ( s 1 a ( τ ) f ( u ( τ ) ) d τ ) φ 1 ( s 1 a ( τ ) f ( v ( τ ) ) d τ ) ] d s 1 i = 1 m 2 α i φ 1 ( 0 1 a ( τ ) d τ ) ( ln ( u + 2 ) ( u v + 1 ) λ ln ( v + 2 ) ) + i = 1 m 2 α i ξ i φ 1 ( 0 1 a ( τ ) d τ ) 1 i = 1 m 2 α i ( ln ( u + 2 ) ( u v + 1 ) λ ln ( v + 2 ) ) [ φ 1 ( 0 1 a ( τ ) d τ ) + i = 1 m 2 α i ξ i φ 1 ( 0 1 a ( τ ) d τ ) 1 i = 1 m 2 α i ] ( ln ( u + 2 ) ( u v + 1 ) λ v + 2 ) ( λ + 1 ) ln ( u v + 1 ) φ 1 ( 0 1 a ( τ ) d τ ) 1 i = 1 m 2 α i .

Since the function h(x)=ln(x+1) is nondecreasing, and condition (f2), then we have

d ( T u , T v ) ( λ + 1 ) ln ( u v + 1 ) φ 1 ( 0 1 a ( τ ) d τ ) 1 i = 1 m 2 α i < ln ( u v + 1 ) = u v ( u v ln ( u v + 1 ) ) .

Let ψ(x)=xln(x+1). Obviously ψ:[0,+)[0,+) is continuous, nondecreasing, positive in (0,+), ψ(0)=0, and lim x + ψ(x)=+. Thus, for uv, we have

d(Tu,Tv)d(u,v)ψ ( d ( u , v ) ) .

By conditions (A) and (f1), we know that

( T 0 ) ( t ) = 0 t φ 1 ( s 1 a ( τ ) f ( 0 ) d τ ) d s + i = 1 m 2 α i 0 ξ i φ 1 ( s 1 a ( τ ) f ( 0 ) d τ ) d s 1 i = 1 m 2 α i 0 .

Therefore, by Theorem 2.1 we know that problem (1.1)-(1.2) has at least one nonnegative solution. As (K,) satisfies condition (2.2), thus, Theorem 2.2 implies the uniqueness of the solution. By definition of T and conditions (A), (f1), it is easy to prove that this solution u(t) is strictly increasing. □

4 Example

Example 4.1 Consider the boundary value problem

{ ( φ ( u ( t ) ) ) + 1 5 t 4 f ( u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = 1 4 u ( 1 4 ) + 1 4 u ( 1 2 ) , u ( 1 ) = 0 ,
(4.1)

where

φ(u)={ u 3 1 + u 2 , u 0 , u 2 , u > 0 ,

a(t)= 1 5 t 4 and f(x)= [ ln ( x + 2 ) ] 2 for x[0,+).

Proof Note that f is a continuous function and f(x)>0. Moreover, f is nondecreasing with respect to x since f x = 2 x + 2 ln(x+2)>0. On the other hand, for uv, we have

φ ( ln ( v + 2 ) ) = [ ln ( v + 2 ) ] 2 = f ( v ) f ( u ) = [ ln ( u + 2 ) ] 2 ( ln ( u + 2 ) ( u v + 1 ) ) 2 = φ ( ln ( u + 2 ) ( u v + 1 ) ) .

In this case, λ=1 because 1<λ+1< 1 i = 1 m 2 α i φ 1 ( 0 1 a ( τ ) d τ ) = 5 2 . Thus Theorem 3.1 implies that the boundary value problem (4.1) has a unique positive solution which is strictly increasing. □

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Acknowledgements

The authors would like to express sincere thanks to the anonymous referee for his/her carefully reading the manuscript and valuable comments and suggestions. The authors are supported by the National Natural Science Foundation of China (Grant No. 11301038), Research Foundation during the 12st Five-Year Plan Period of Department of Education of Jilin Province, China (Grant [2013] No. 252), Youth Foundation for Science and Technology Department of Jilin Province (20130522100JH).

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Miao, F., Zhou, C. & Song, Y. Existence and uniqueness of positive solutions to boundary value problem with increasing homeomorphism and positive homomorphism operator. Adv Differ Equ 2014, 20 (2014). https://doi.org/10.1186/1687-1847-2014-20

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Keywords

  • partially ordered sets
  • fixed-point theorem
  • positive solution