Existence of positive solutions of the Cauchy problem for a second order differential equation

(1)

R E S E A R C H

Open Access

Existence of positive solutions of the Cauchy

problem for a second-order differential

equation

Toshiharu Kawasaki

1*

_{and Masashi Toyoda}

2

*_{Correspondence:}

[email protected]

1_{College of Engineering, Nihon}

University, Fukushima, 963-8642, Japan

Full list of author information is available at the end of the article

Abstract

In this paper we consider the equationu(t) =f(t,u(t),u(t)) and prove the unique solvability of the Cauchy problemu(0) = 0,u(0) =

λ

with

λ

> 0.

1 Introduction

In [], Knežević-Miljanović considered the Cauchy problem

u(t) =P(t)ta_u₍_t₎σ_, _t_∈_{(, ],}

u() = , u() =λ, ()

wherePis continuous,a,σ,λ∈Rwithσ<  andλ> , and_|P(t)|ta+σ_dt_<_∞_{. Moreover,}

in [], Kawasaki and Toyoda considered the Cauchy problem

u(t) =f(t,u(t)), for almost everyt∈[, ],

u() = , u() =λ, ()

wheref is a mapping from [, ]×(,∞) intoRandλ∈Rwithλ> . They proved the unique solvability of Cauchy problem () using the Banach ﬁxed point theorem. The the-orem in [] is as follows.

Theorem Suppose that a mapping f from[, ]×[,∞)intoRsatisﬁes the following.

(a) The mappingt−→f(t,u)is measurable for anyu∈(,∞),and the mapping

u−→f(t,u)is continuous for almost everyt∈[, ].

(b) |f(t,u)| ≥ |f(t,u)|for almost everyt∈[, ]and for anyu,u∈[,∞)with u≤u.

(c) There existsα∈Rwith <α<λsuch that

 

f(t,αt)dt<∞.

(d) There existsβ∈Rwithβ> such that

_∂u∂f(t,u)≤β|f(t,u)|

u

for almost everyt∈[, ]and for anyu∈(,∞).

(2)

Then there exists h∈Rwith <h≤such that Cauchy problem()has a unique solution in X,where X is a subset

X=

u

u∈C[,h],u() = ,u() =λ andαt≤u(t)for any t∈[,h]

of C[,h],which is the class of continuous mappings from[,h]intoR.

The case thatf(t,u(t)) =P(t)ta_u₍_t₎σ _{in the above theorem is the theorem of}

Knežević-Miljanović [].

In this paper, we consider the Cauchy problem

u(t) =f(t,u(t),u(t)), for almost everyt∈[, ],

u() = , u() =λ,

wheref is a mapping from [, ]×(,∞)×RintoRandλ∈Rwithλ> . We prove the unique solvability of this Cauchy problem using the Banach ﬁxed point theorem.

In Section , we consider the following four cases foruandv.

(I) Decreasing foruinf(t,u,v)(b) and decreasing forvinf(t,u,v)(b). (II) Decreasing foruinf(t,u,v)(b) and increasing forvinf(t,u,v)(b). (III) Increasing foruinf(t,u,v)(b) and decreasing forvinf(t,u,v)(b). (IV) Increasing foruinf(t,u,v)(b) and increasing forvinf(t,u,v)(b).

Theorems ., ., . and . are the cases of (I), (II), (III) and (IV), respectively.

2 Main results

In this section, we consider the Cauchy problem

u(t) =f(t,u(t),u(t)), for almost everyt∈[, ],

u() = , u() =λ, ()

wheref is a mapping from [, ]×(,∞)×RintoRandλ∈Rwithλ> . First, we consider the case of (I).

Theorem . Letλbe a real number withλ> .Suppose that a mapping f from[, ]× (,∞)×RintoRsatisﬁes the following:

(a) The mappingt−→f(t,u,v)is measurable for any(u,v)∈(,∞)×R,and the mapping(u,v)−→f(t,u,v)is continuous for almost everyt∈[, ];

(b) |f(t,u,v)| ≥ |f(t,u,v)|for almost everyt∈[, ],for anyu,u∈(,∞)with u≤uand for anyv∈R;

(b) |f(t,u,v)| ≥ |f(t,u,v)|for almost everyt∈[, ],for anyu∈(,∞)and for any v,v∈Rwithv≤v;

(c) There existα∈Rwith <α<λandα∈Rwithα<λsuch that

 

(3)

(d) There existsβ∈Rwithβ> such that

∂f

∂u(t,u,v)

≤β|f(t,u,v)| u

for almost everyt∈[, ],for anyu∈(,∞)and for anyv∈R; (d) There existsβ∈Rwithβ> such that

∂f_∂v(t,u,v)≤βf(t,u,v)

for almost everyt∈[, ],for anyu∈(,∞)and for anyv∈R; (e) There exists the limit

lim

t→+



t

t



sfs,u(s),u(s)ds

for any continuously diﬀerentiable mappingufrom[, ]into[,∞); (f) Forαandα,

lim

t→+



t

t



sf(s,αs,α)ds= .

Then there exists h∈Rwith <h≤such that Cauchy problem()has a unique solution in X,where X is a subset

X=

⎧ ⎪ ⎨ ⎪ ⎩u

u∈C[,h],u() = ,u() =λ,

αt≤u(t)andα≤u(t)for any t∈[,h] and there exists the limitlimt→+tu

₍_t_)–_u₍_t₎

t

⎫ ⎪ ⎬ ⎪ ⎭

of C_[,_h_],_{which is the class of continuously diﬀerentiable mappings from}_[,_h_]_into_R_.

Proof It is noted thatC_[,_h_{] is a Banach space by the maximum norm}

u =maxmaxu(t)|t∈[,h],maxu(t)|t∈[,h]. Instead of Cauchy problem (), we consider the integral equation

u(t) =λt+

t



(t–s)fs,u(s),u(s)ds.

By condition (c), there existsh∈Rwith  <h≤ such that

h



f(t,αt,α)dt<min

λ–α,λ–α,

β α

+ β

–

.

By condition (f), there existsh∈Rwith  <h≤hsuch that

sup

t∈(,h]



t

t



sf(s,αs,α)ds≤

h



(4)

LetAbe an operator fromXintoC_[,_h_{] deﬁned by}

Au(t) =λt+

t



Since a mappingt−→λtbelongs toX,X= ∅. Moreover, we haveA(X)⊂X. Indeed, by condition (a),Au∈C_[,_h_],_Au_{() =  and}

(Au)() =

λ+

t



fs,u(s),u(s)ds

t=

=λ.

By conditions (b) and (b), we obtain that

Au(t) =λt+

t



(t–s)fs,u(s),u(s)ds

≥λt–t

h



fs,u(s),u(s)ds

≥λt–t

h



f(s,αs,α)ds

≥αt

and

(Au)(t) =λ+

t



fs,u(s),u(s)ds

≥λ–

h



fs,u(s),u(s)ds

≥λ–

h



f(s,αs,α)ds

≥α

for anyt∈[,h]. Moreover, by condition (e), there exists the limit

lim

t→+

t(Au)(t) –Au(t)

t =_tlim_→_+



t

t



sfs,u(s),u(s)ds.

We will find a fixed point ofA. Letϕbe an operator fromXintoC_[,_h_{] defined by}

ϕ[u](t) =

u(t)

t ift∈(,h],

λ ift= .

Letϕ[X] be a subset deﬁned by

ϕ[X] =ϕ[u]|u∈X. Then we have

ϕ[X] =

v

v∈C_[,_h_],_v_{() =}_λ_,

α≤v(t) andα≤v(t) +tv(t) for anyt∈[,h]

(5)

andϕ[X] is a closed subset ofC_[,_h_{]. Hence it is a complete metric space. Let}_{be an}

operator fromϕ[X] intoϕ[X] deﬁned by

ϕ[u] =ϕ[Au].

By the mean value theorem, for anyu,u∈X, there exist mappingsξ,ηsuch that

ft,u(t),u(t)

–ft,u(t),u(t)

=∂f

∂u

t,ξ(t),u_(t)u(t) –u(t)

+∂f

∂v

t,u(t),η(t)

u_(t) –u_(t) =

t∂f ∂u

t,ξ(t),u_(t)+∂f

∂v

t,u(t),η(t)

ϕ[u](t) –ϕ[u](t)

+t∂f ∂v

t,u(t),η(t)

ϕ[u](t) –ϕ[u](t)

,

minu(t),u(t)

≤ξ(t)≤maxu(t),u(t)

and

minu_(t),u_(t)≤η(t)≤maxu_(t),u_(t)

for almost everyt∈[,h]. Therefore, by conditions (b), (b), (d) and (d), we obtain that

ft,u(t),u(t)

–ft,u(t),u(t)

=

t∂f ∂u

t,ξ(t),u_(t)+∂f

∂v

t,u(t),η(t)

ϕ[u](t) –ϕ[u](t)

+t∂f ∂v

t,u(t),η(t)

ϕ[u](t) –ϕ[u](t)

≤

t∂f ∂u

t,ξ(t),u_(t)+∂f

∂v

t,u(t),η(t)

ϕ[u](t) –ϕ[u](t)

+t∂f ∂v

t,u(t),η(t)ϕ[u](t) –ϕ[u](t)

≤

β α

+β

f(t,αt,α)ϕ[u](t) –ϕ[u](t)

+βtf(t,αt,α)ϕ[u](t) –ϕ[u](t)

for almost everyt∈[,h]. Therefore we have

ϕ[u](t) –ϕ[u](t)

=

t



(t–s)fs,u(s),u(s)

–fs,u(s),u(s)

ds

≤

t



fs,u(s),u(s)

–fs,u(s),u(s)ds

≤ t  β α

+β

(6)

+βsf(s,αs,α)ϕ[u](s) –ϕ[u](s)

ds

≤

β α

+ β

h



f(s,αs,α)dsϕ[u] –ϕ[u]

for anyt∈[,h]. Moreover, we have

ϕ[u]

(t) –ϕ[u]

(t) =

t

t



sfs,u(s),u(s)

–fs,u(s),u(s)

ds

≤ 

t

t



sfs,u(s),u(s)

–fs,u(s),u(s)ds

≤ 

t

t

 s

β α

+β

f(s,αs,α)ϕ[u](s) –ϕ[u](s)

+βsf(s,αs,α)ϕ[u](s) –ϕ[u](s)

ds

≤

β α

+β

h



f(s,αs,α)ds

+β

h



f(s,αs,α)ds

ϕ[u] –ϕ[u]

for anyt∈[,h]. Hence we obtain that

ϕ[u] –ϕ[u]

≤

β α

+ β

h



f(s,αs,α)dsϕ[u] –ϕ[u].

By the Banach ﬁxed point theorem, there exists a unique mappingϕ[u]∈ϕ[X] such that

ϕ[u] =ϕ[u]. ThenAu=u.uis a solution of ().

Next, we consider the case of (II).

Theorem . Letλbe a real number withλ> .Suppose that a mapping f from[, ]× (,∞)×RintoRsatisﬁes the following:

(b) |f(t,u,v)| ≥ |f(t,u,v)|for almost everyt∈[, ],for anyu,u∈(,∞)with u≤uand for anyv∈R;

(b) |f(t,u,v)| ≤ |f(t,u,v)|for almost everyt∈[, ],for anyu∈(,∞)and for any v,v∈Rwithv≤v;

(c) There existα∈Rwith <α<λandα∈Rwithα>λsuch that

 

(7)

∂f_∂u(t,u,v)≤ β|f(t,u,v)|

u

lim

t→+



t

t



sfs,u(s),u(s)ds

for any continuously diﬀerentiable mappingufrom[, ]into[,∞); (f) Forαandα,

lim

t→+



t

t



sf(s,αs,α)ds= .

X=

⎧ ⎪ ⎨ ⎪ ⎩u

u∈C_[,_h_],_u_{() = ,}_u_{() =}_λ_,

αt≤u(t)and u(t)≤αfor any t∈[,h] and there exists the limitlimt→+tu

₍_t_)–_u₍_t₎

t

⎫ ⎪ ⎬ ⎪ ⎭

of C_[,_h_].

Proof By condition (c), there existsh∈Rwith  <h≤ such that

h



f(t,αt,α)dt<min

λ–α,α–λ,

β α

+ β

–

.

By condition (f), there existsh∈Rwith  <h≤hsuch that

sup

t∈(,h]



t

t



sf(s,αs,α)ds≤

h



f(t,αt,α)dt.

Au(t) =λt+

t



Since a mappingt−→λtbelongs toX,X= ∅. Moreover, we haveA(X)⊂X. Indeed, by condition (a),Au∈C_[,_h_],_Au_{() =  and}

(Au)() =

λ+

t



fs,u(s),u(s)ds

t=

(8)

By conditions (b) and (b), we obtain that

Au(t) =λt+

t



≥λt–t

h



_f

s,u(s),u(s)ds

≥λt–t

h



f(s,αs,α)ds

≥αt

and

(Au)(t) =λ+

t



fs,u(s),u(s)ds

≤λ+

h



fs,u(s),u(s)ds

≤λ+

h



f(s,αs,α)ds

≤α

for anyt∈[,h]. Moreover, by condition (e), there exists the limit

lim

t→+

t(Au)(t) –Au(t)

t =_tlim_→_+



t

t



sfs,u(s),u(s)ds.

ϕ[u](t) =

u(t)

t ift∈(,h],

λ ift= ,

and

ϕ[X] =ϕ[u]|u∈X

=

v

v∈C_[,_h_],_v_{() =}_λ_,

α≤v(t) andv(t) +tv(t)≤αfor anyt∈[,h]

.

Thenϕ[X] is a closed subset ofC_[,_h_{] and hence it is a complete metric space. Let}_be

an operator fromϕ[X] intoϕ[X] deﬁned by

ϕ[u] =ϕ[Au].

Then we can show, just like Theorem ., that by the Banach ﬁxed point theorem there exists a unique mappingϕ[u]∈ϕ[X] such thatϕ[u] =ϕ[u] and henceAu=u.

(9)

Theorem . Letλbe a real number withλ> .Suppose that a mapping f from[, ]× (,∞)×RintoRsatisﬁes the following:

(b) |f(t,u,v)| ≤ |f(t,u,v)|for almost everyt∈[, ],for anyu,u∈(,∞)with u≤uand for anyv∈R;

(b) |f(t,u,v)| ≥ |f(t,u,v)|for almost everyt∈[, ],for anyu∈(,∞)and for any v,v∈Rwithv≤v;

(c) There existα∈Rwith <α<λandα∈Rwithα<λsuch that

 

ft, (λ–α)t,αdt<∞;

_∂u∂f(t,u,v)≤β|f(t,u,v)|

u

lim

t→+



t

t



sfs,u(s),u(s)ds

for any continuously diﬀerentiable mappingufrom[, ]into[,∞); (f) Forαandα,

lim

t→+



t

t



sfs, (λ–α)s,αds= .

X=

⎧ ⎪ ⎨ ⎪ ⎩u

u∈C[,h],u() = ,u() =λ,

αt≤u(t)≤(λ–α)t andα≤u(t)for any t∈[,h] and there exists the limitlimt→+tu

₍_t_)–_u₍_t₎

t

⎫ ⎪ ⎬ ⎪ ⎭

of C_[,_h_].

Proof By condition (c), there existsh∈Rwith  <h≤ such that

h



ft, (λ–α)t,αdt<min

λ–α,λ–α,

β α

+ β

–

(10)

By condition (f), there existsh∈Rwith  <h≤hsuch that

sup

t∈(,h]



t

t



sfs, (λ–α)s,αds≤

h



ft, (λ–α)t,αdt.

Au(t) =λt+

t



Since a mappingt−→λtbelongs toX,X=∅. Moreover,A(X)⊂X. Indeed, by condition (a),Au∈C[,h],Au() = ,

(Au)() =

λ+

t



fs,u(s),u(s)ds

t=

=λ,

by conditions (b) and (b),

Au(t) =λt+

t



≥λt–t

h



fs,u(s),u(s)ds

≥λt–t

h



fs, (λ–α)s,αds

≥αt,

Au(t) =λt+

t



≤λt+t

h



_f

s,u(s),u(s)ds

≤λt+t

h



≤(λ–α)t,

(Au)(t) =λ+

t



fs,u(s),u(s)ds

≥λ–

h



_f

s,u(s),u(s)ds

≥λ–

h



≥α

for anyt∈[,h], and by condition (e), there exists the limit

lim

t→+

t(Au)(t) –Au(t)

t =_tlim_→_+



t

t



(11)

ϕ[u](t) =

u(t)

t ift∈(,h],

λ ift= ,

and

ϕ[X] =ϕ[u]|u∈X

=

v

v∈C[,h],v() =λ,

α≤v(t)≤λ–αandα≤v(t) +tv(t) for anyt∈[,h]

.

Thenϕ[X] is a closed subset ofC_[,_h_{], and hence it is a complete metric space. Let}_be

ϕ[u] =ϕ[Au].

Finally, we consider the case of (IV).

Theorem . Letλbe a real number withλ> .Suppose that a mapping f from[, ]× (,∞)×RintoRsatisﬁes the following:

(b) |f(t,u,v)| ≤ |f(t,u,v)|for almost everyt∈[, ],for anyu,u∈(,∞)with u≤uand for anyv∈R;

(b) |f(t,u,v)| ≤ |f(t,u,v)|for almost everyt∈[, ],for anyu∈(,∞)and for any v,v∈Rwithv≤v;

(c) There existα∈Rwith <α<λandα∈Rwithα>λsuch that

 

ft, (λ–α)t,αdt<∞;

_∂u∂f(t,u,v)≤β|f(t,u,v)|

u

(12)

(e) There exists the limit

lim

t→+



t

t



sfs,u(s),u(s)ds

for any continuously diﬀerentiable mappingufrom[, ]into[,∞); (f) Forαandα,

lim

t→+



t

t



sfs, (λ–α)s,αds= .

X=

⎧ ⎪ ⎨ ⎪ ⎩u

u∈C_[,_h_],_u_{() = ,}_u_{() =}_λ_,

αt≤u(t)≤(λ–α)t and u(t)≤αfor any t∈[,h] and there exists the limitlimt→+tu

₍_t_)–_u₍_t₎

t

⎫ ⎪ ⎬ ⎪ ⎭

of C_[,_h_].

Proof By condition (c), there existsh∈Rwith  <h≤ such that

h



ft, (λ–α)t,αdt<min

λ–α,α–λ,

β α

+ β

–

.

By condition (f), there existsh∈Rwith  <h≤hsuch that

sup

t∈(,h]



t

t



sfs, (λ–α)s,αds≤

h



ft, (λ–α)t,αdt.

Au(t) =λt+

t



Since a mappingt−→λtbelongs toX,X= ∅. Moreover,A(X)⊂X. Indeed, by condition (a),Au∈C_[,_h_],_Au_{() = ,}

(Au)() =

λ+

t



fs,u(s),u(s)ds

t=

=λ,

by conditions (b) and (b),

Au(t) =λt+

t



≥λt–t

h



fs,u(s),u(s)ds

≥λt–t

h



(13)

Au(t) =λt+

t



≤λt+t

h



fs,u(s),u(s)ds

≤λt+t

h



≤(λ–α)t,

(Au)(t) =λ+

t



fs,u(s),u(s)ds

≤λ+

h



fs,u(s),u(s)ds

≤λ+

h



f(s,αs,α)ds

≤α

for anyt∈[,h], and by condition (e), there exists the limit

lim

t→+

t(Au)(t) –Au(t)

t =_tlim_→_+



t

t



sfs,u(s),u(s)ds.

ϕ[u](t) =

u(t)

t ift∈(,h],

λ ift= ,

and

ϕ[X] =ϕ[u]|u∈X

=

v

v∈C_[,_h_],_v_{() =}_λ_,

α≤v(t)≤λ–αandv(t) +tv(t)≤αfor anyt∈[,h]

.

Thenϕ[X] is a closed subset ofC_[,_h_{] and hence it is a complete metric space. Let}_be

ϕ[u] =ϕ[Au].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

(14)

Author details

1_{College of Engineering, Nihon University, Fukushima, 963-8642, Japan.}2_{Marine Faculty of Engineering, Tamagawa}

University, Tokyo, 194-8610, Japan.

Received: 1 June 2013 Accepted: 20 September 2013 Published:07 Nov 2013 References

1. Knežević-Miljanović, J: On the Cauchy problem for an Emden-Fowler equation. Differ. Equ.45(2), 267-270 (2009) 2. Kawasaki, T, Toyoda, M: Existence of positive solution for the Cauchy problem for an ordinary differential equation. In:

Li, S, Wang, X, Okazaki, Y, Kawabe, J, Murofushi, T, Guan, L (eds.) Nonlinear Mathematics for Uncertainly and Its Applications. Advances in Intelligent and Soft Computing, vol. 100, pp. 435-441. Springer, Berlin (2011)

10.1186/1029-242X-2013-465