Coupled fixed point theorems with applications to fractional evolution equations

R E S E A R C H

Open Access

Coupled fixed point theorems with

applications to fractional evolution equations

He Yang

1*

_{, Ravi P Agarwal}

2

_{and Hemant K Nashine}

2

*_{Correspondence:}

[email protected]

1_{College of Mathematics and}

Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

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

Abstract

In this paper, we first prove some coupled fixed point theorems in partially ordered

-orbitally complete normed linear spaces. And then apply the obtained fixed point theorems to a class of semilinear evolution systems of fractional order for proving the existence of coupled mild solutions under some weaker monotone conditions. An example is given to illustrate the application of the abstract results.

MSC: 34A08; 47H10

Keywords: coupled fixed point theorem; nonlinear fractional evolution system; equi-continuousC0-semigroup; coupled mild solution; existence

1 Introduction

LetEbe a nonempty set,:E→Ea mapping. If forx∈E, one hasx=x, thenx∈E is called a fixed point of inE. Fixed point theory plays an important role in nonlinear functional analysis. Different types of fixed point theorems have been used to prove the existence of solutions for differential and integral equations; see [–]. The Banach con-traction principle is one of the most powerful fixed point theorems in nonlinear analysis for proving the existence and uniqueness of fixed points in metric spaces. It is interesting to improve and extend the conditions of the Banach contraction principle. Recently, by weakening the requirement on the contraction in partially ordered metric spaces, a series of fixed point theorems are established for monotone mappings by Agarwalet al.[], Har-jani and Sadarangani [], Nieto and Rodriguez-López [, ], O’Regan and Petrusel [] and Ran and Reurings [].

We recall some definitions of the monotone mapping. A mapping:E→Eis mono-tone means it is monomono-tone nondecreasing or monomono-tone nonincreasing. Assume that (E,≤) is a partially ordered set and:E→E. Forx,y∈E, ifx≤yimplies(x)≤(y), is called a monotone nondecreasing mapping inE. Similarly, we can define a monotone non-increasing mapping inE. If forx,x∈E,x≤ximplies(x,y)≤(x,y) for ally∈E,

while fory,y∈E,y≤yimplies(x,y)≥(x,y) for allx∈E, then we say thatis a

mixed monotone mapping inE.

The above mentioned fixed point theorems, see [, , –], are all for the monotone mapping. In [], Bhaskar and Lakshmikantham extended the fixed point theorems ob-tained in [, , –] to the mixed monotone mapping in partially ordered metric spaces. At first, they introduced a definition of the coupled fixed point. And then some coupled

fixed point theorems were proved in partially ordered metric spaces. Recently, these cou-pled fixed point theorems were refined and improved by Lakshmikantham and Ćirić [], Luong and Thuan [] and Samet [].

In the above mentioned results, the assumptions of mixed monotone property and con-traction property of the mappingare essential. The purpose of this paper is to delete or weaken these conditions. In this paper, by utilizing a different technique, we prove some coupled fixed point theorems in a partially ordered-orbitally complete normed linear space. In our results, we neither assume that the mapping is mixed monotone, nor as-sume that it is a contraction. We divide the mappinginto two parts, and assume that every part satisfies some conditions, by using an existing Krasnoselskii-type fixed point theorem, a coupled fixed point theorem for the mappingis proved. As applications, we apply the obtained coupled fixed point theorem to a certain abstract fractional evolution systems for proving the existence of coupled mild solutions.

The rest of this paper is organized as follows. In Section , some definitions are recalled and an existing Krasnoselskii-type fixed point theorem is introduced. In Section , coupled fixed point theorems are proved. In Section , we apply the obtained coupled fixed point theorem to a certain abstract fractional evolution systems. A specific example is given in Section  to illustrate the abstract results.

2 Preliminaries

LetEbe a partially ordered normed linear space with partial order≤and the norm · E. If two elementsx,y∈Esatisfy eitherx≤yorx≥y, we say that they are comparable. IfEis complete with respect to the norm · E, we called it a partially ordered complete normed linear space.

Definitions .-. can be found in [, ].

Definition . Let:E→Ebe a mapping. For anyx∈E, we define an orbit(x;) by

(x;) =x,x,x, . . . ,nx, . . ..

If for any sequence{xn} ⊂(x;),xn→x∗impliesxn→x∗for eachx∈E,is said to be-orbitally continuous inE. A normed linear space (E, · E) is called-orbitally complete if every Cauchy sequence{xn} ⊂(x;) converges to a pointx∗inE.

Definition . A functionφ:R+_→R+_{is called aD-function if it is a monotone}

nonde-creasing and upper semi-continuous function satisfyingφ() = .

Definition . A mapping:E→Eis called partially nonlinearD-Lipschitz if for all comparable elementsx,y∈E, there is aD-functionφ:R+→R+such that

x–yE≤φ

x–yE

.

Furthermore, ifφ(r) <rforr> ,is called a partially nonlinearD-contraction inE.

Definition . The norm · Eand the order relation≤on a partially ordered normed linear space (E,≤, · E) are said to be compatible if for any monotone sequence{xn}inE, subsequence{xnk}of{xn}converges tox∗implies that the whole sequence{xn}converges tox∗.

Remark . From Definition ., if the normed linear space (E, · E) is complete, then it is-orbitally complete. But the converse expression may not be true.

Remark . The D-functions and the partially nonlinear D-Lipschitz conditions are much useful in research of solutions for nonlinear differential equations via fixed point theorems; see [].

The following Krasnoselskii-type fixed point theorem was proved by Dhage in [].

Lemma . Let E be a partially ordered complete normed linear space with the partial order≤and the norm · Esuch that≤and · Eare compatible.Suppose that A,A:

E→E are two monotone nondecreasing mappings satisfying:

(a) Ais continuous and a partially nonlinearD-contraction,

(b) Ais continuous and partially compact,

(c) there is an elementv∈Esatisfyingv≤Av+Ayfor ally∈E,and

(d) every pair of elements has an upper and a lower bound inE.

Then x=Ax+Ax has a solution in E.

3 Fixed point theorems

Let (E, · E) be a-orbitally complete normed linear space. Define a positive coneKin Eby

K={x∈E:x≥}.

ThenEbecomes now a partially ordered-orbitally complete normed linear space with the partial order≤induced byK. It is well known that the partially order≤and the norm

· Eare compatible if coneKis normal.

By Lemma ., we first prove the following fixed point theorem.

Theorem . Let E be a partially ordered-orbitally complete normed linear space with the norm · Eand the partial order≤,whose positive cone K is normal,and let D be a nonempty closed subset of E.Assume that A,A:D→D are two monotone nondecreasing

mappings satisfying:

(a) Ais-orbitally continuous and

Ax–AyE<x–yE

(c) there exists an elementv∈Dsuch thatv≤Av+Ayfor ally∈D,and

(d) every pair of elements inDhas an upper and a lower bound.

Then the equation x=Ax+Ax has a solution in D.

Proof By Remark . of [] and the proof of Lemma ., if the continuity of operatorsA

andAis replaced by the-orbitally continuity in conditions (a) and (b) of Lemma .,

the conclusion of Lemma . is still true. On the other hand, sinceDis a nonempty closed subset of E, it follows thatDis-orbitally complete and has partial order ≤. For any comparable elementsx,y∈D,x≡y, by the condition (a), it follows that there isτ∈(, ) satisfying

Ax–AyE≤τx–yE<x–yE.

Letφ(r) =τr. Thenφ is a D-function andφ(r) <rfor anyr> . This implies that the condition (a) of Lemma . is satisfied. By Lemma ., we obtain the desired conclusion.

If the inequality given in assumption (c) of Lemma . is reverse, more precisely, the condition (c) of Lemma . is replaced by

(c) there isv∈Dsatisfyingv≥Av+Ayfor ally∈D.

By Theorem . of [], the conclusion of Lemma . is still true. Hence we can obtain the following fixed point theorem. Since the proof is similar to Theorem ., we omit the details here.

Theorem . Let E be a partially ordered-orbitally complete normed linear space with the norm · Eand the partial order≤,whose positive cone K is normal,and let D be a nonempty closed subset of E.Assume that A,A:D→D are two monotone nondecreasing

mappings satisfying the conditions(a), (b), (c) and(d).Then x=Ax+Ax has a solution

in D.

LetE:=E×E. Define a sum and a scalar multiplication inEby

w+v= (w,w) + (v,v) = (w+v,w+v),

λw=λ(w,w) = (λw,λw)

for allw= (w,w),v= (v,v)∈Eandλ∈R. And define a positive cone and a norm inE

by

KE=

w= (w,w)∈E:w,w∈K

,

vE=(v,v)E=vE+vE, v= (v,v)∈E.

LetQ:E→E. A pair of elements (x,y)∈Eis called a coupled fixed point ofQ:E→E if and only if it satisfies

x=Q(x,y), y=Q(y,x).

By Theorem ., the following coupled fixed point theorem is obtained.

Theorem . Let E be a partially ordered-orbitally complete normed linear space with the partial order≤and the norm · E,whose positive cone K is normal,and let D be a nonempty closed subset of E.Assume that A,A:D→D are two monotone nondecreasing

mappings satisfying

(i) Ais-orbitally continuous and a partially nonlinearD-contraction,

(ii) Ais-orbitally continuous and partially compact,

(iii) there is an elementv∈Dsatisfyingv≤Av+Ayfor ally∈D,and

(iv) every pair of elements inDhas an upper and a lower bound.

Then Q(x,y) =Ax+Ay has a coupled fixed point inE.

Proof Since (E,≤, · E) is a partially ordered-orbitally complete normed linear space, and positive coneKis normal, it follows that (E,≤, · E) is a partially ordered-orbitally complete normed linear space and positive coneKE is normal. SinceDis a nonempty closed subset ofE, it follows thatD×Dis a nonempty closed subset ofE.

LetD=D×D. Define two operatorsA,A:D→Dby

Au= (Ax,Ay), Au= (Ay,Ax), ∀u= (x,y)∈D.

If the operator equationu=Au+Auhas a solutionu= (x,y)∈D, namely,

(x,y) =u=Au+Au=A(x,y) +A(x,y),

then we obtain

(x,y) =Au+Au

= (Ax,Ay) + (Ay,Ax)

= (Ax+Ay,Ay+Ax)

=Q(x,y),Q(y,x).

This implies that the operatorQ(x,y) has a coupled fixed point inE. We will apply Theo-rem . to prove that the operator equationu=Au+Auhas a solution inE. The proof

will be given in several steps.

Step I.Ais-orbitally continuous and

Aw–AvE<w–vE,

SinceA is-orbitally continuous, by the definition ofA, it is easy to see thatA is

-orbitally continuous.

For all comparable elementsw= (w,w),v= (v,v)∈Dwithw≡v, we have

Aw–AvE=(Aw,Aw) – (Av,Av)E =(Aw–Av), (Aw–Av)E =Aw–AvE+Aw–AvE

≤ϕw–vE

+ϕw–vE

<w–vE+w–vE =(w–v,w–v)E =(w,w) – (v,v)E =w–vE.

Hence, we obtain

Aw–AvE<w–vE

for all comparable elementsw,v∈Dwithw≡v.

Step II.Ais-orbitally continuous and partially compact.

SinceAis-orbitally continuous, by the definition ofA, it is easy to see thatA is

-orbitally continuous.

LetC⊂Dbe a bounded chain. SinceAis partially compact inD, it follows thatA(C) is

equi-continuous and uniformly bounded inD. SetC=C×C. ThenCis a bounded chain inD. Next, we claim thatA(C) is equi-continuous and uniformly bounded inD

SinceA(C)⊂Dis uniformly bounded, there is a constantM>  satisfyingAzE≤M for anyz∈C. For anyZ∈A(C), there arex,y∈Csatisfyingu= (x,y)∈Csuch thatZ=

Au. Thus,

ZE=AuE=(Ay,Ax)E =AyE+AxE

≤M.

This implies thatA(C) is uniformly bounded inD.

SinceA(C) is equi-continuous inD, for anyz∈Candt>t, we have

(Az)(t) – (Az)(t)_E→

ast–t→. Hence for anyZ=Au=A(x,y)∈A(C), we have

Z(t) –Z(t)E=(Au)(t) – (Au)(t)E

=(Ay)(t), (Ax)(t)

=(Ay)(t) – (Ay)(t)_E+(Ax)(t) – (Ax)(t)_E

→

ast–t→. This implies thatA(C) is equi-continuous inD.

Therefore, by the Arzela-Ascoli theorem, A(C)⊂ D is relatively compact.

Conse-quently,A:D→Dis partially compact.

Step III. There is an elementu∈Dsatisfying

u≤Au+Au

for allu∈D.

Letu= (v,v). For anyu= (x,y)∈D, by the condition (iii), we have

Au+Au= (Av,Av) + (Ay,Ax)

= (Av+Ay,Av+Ax)

≥(v,v)

=u.

Hence we obtain the desired conclusion.

Step IV. Every pair of elements inDhas an upper and a lower bound.

For every pair of elementsw= (w,w),v= (v,v)∈D, by condition (iv), there exist

z,z,z,z∈Dsuch that

z≤w, z≤v, z≥w, z≥v,

z≤w, z≤v, z≥w, z≥v.

Thus, we have

(z,z)≤(w,w)≤(z,z),

(z,z)≤(v,v)≤(z,z).

Consequently, every pair of elementsw,v∈Dhas an upper and a lower bound.

Therefore, by Theorem ., the operator equationu=Au+Auhas a solution inE.

By Theorems . and ., the following coupled fixed point theorem is obtained. Because its proof is similar to Theorem ., we omit the details.

Theorem . Let E be a partially ordered-orbitally complete normed linear space with the norm·Eand the order relation≤,positive cone K be normal,and let D be a nonempty closed subset of E.Assume that A,A:D→D are two monotone nondecreasing mappings

satisfying

(i) Ais-orbitally continuous and a partially nonlinearD-contraction,

(iii) there is an elementv∈Dsatisfyingv≥Av+Ayfor ally∈D,and

(iv) every pair of elements inDhas an upper and a lower bound.

Then Q(x,y) =Ax+Ay has a coupled fixed point inE.

Remark . The hypothesis (iv) of Theorems . and . holds if the partially ordered setEis a lattice. We known that the setC(J,X) is a lattice, whereC(J,X) is the set of all continuousX-valued functions onJ∈R,Xis a partially ordered set. For anyx,y∈C(J,X), max{x,y}andmin{x,y}are the upper and lower bounds, respectively.

Remark . The assumptions of mixed monotone property and contractive property of the mappingQare essential in [, , , ]. But in Theorems . and ., we neither assume that the mappingQis mixed monotone, nor assume that the mappingQis a contraction. We only suppose that the mappingQis a nondecreasing mapping and a part ofQ(namely, the operatorA) is a partially nonlinearD-contraction. Plus with other assumptions we

obtain the coupled fixed point theorems. Hence Theorems . and . extend the main results of [, , , ].

4 Existence results for fractional evolution systems

Let (X, · ) be a-orbitally complete normed linear space. Define its positive cone as K={x∈X:x≥}. ThenX becomes a partially ordered-orbitally complete normed linear space with the norm · and the partial order≤induced by the coneK. In this section, we always assume that K is normal. Investigate the existence of coupled mild solutions to the initial value problem of the fractional hybrid evolution system

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

C_Dσ

tx(t) +Ax(t) =f(t,x(t)) +h(t,y(t)),

C_Dσ

ty(t) +Ay(t) =f(t,y(t)) +h(t,x(t)), t∈J, x() =τ,y() =τ∈X,

(.)

whereJ= [,b],b>  is a constant,C_Dσ

t denotes theσ ∈(, ) order Caputo fractional derivative, –Agenerates aC-semigroupS(t) (t≥) of uniformly bounded linear operator

inX,f andhare given functions.

For theC-semigroupS(t) (t≥), ifS(t)x≥ for allx≥, it is called a positiveC

-semigroup. Throughout this section, we always assume that –Agenerates a positiveC

-semigroupS(t) (t≥) of uniformly bounded linear operator inX. Namely, there is a con-stantM>  such thatS(t) ≤Mfor allt≥.

Definitions . and . can be found in [, , ].

Definition . The fractional integral of orderσ>  with the lower limits zero for a func-tionf∈L(J,E) is defined by

Iσf(t) =  (σ)

t



f(s)

(t–s)–σ ds, t> ,

Definition . The Riemann-Liouville derivative of ordern–  <σ<n,n∈Nwith the

The Caputo fractional derivative of order  <σ<  with the lower limits zero for a func-tionf∈L_{(J,E) can be defined as}

Proof (i) and (ii) can be found in reference [, ]. (iii) is easily seen from the definitions ofUσ(t) andVσ(t). So, we omit the details here.

LetC(J,X) be a set of all continuousX-valued functions on the intervalJand let

KC=

x∈C(J,X) :x(t)∈K,t∈J.

Then C(J,X) is a partially ordered -orbitally complete normed linear space with the normxC:=sup{x(t):t≥}and the partial order≤induced byKC. It is clear that KCis normal becauseKis normal.

(Ax)(t) = (.) if and only if it satisfies the following system of operator equations:

⎧ (H) There is an elementx∈Esatisfying

⎧

virtue of the assumptions (H)-(H), we have the following lemmas.

Lemma . Assume that the hypothesis(H)holds.Then the operator A:D→D is

-orbitally continuous,nondecreasing and a partially nonlinearD-contraction in E.

≤Mτ+ This implies thatAmapsDinto itself.

SinceS(t) (t≥) is a positiveC-semigroup, by (H) and Lemma ., it follows that

This implies thatA:D→Dis-orbitally continuous.

For any comparable elementsx,y∈D, without loss of generality, we assume thatx≥y. By (H), for anyt∈J, we have

orbitally continuous,nondecreasing and partially compact in E.

Proof By the assumption (H), (.) and (.), for anyx∈EwithxC≤r∗, we have

SinceS(t) (t≥) is a positiveC-semigroup, by (H) and Lemma ., a similar proof as

in Lemma . shows thatA:D→Dis-orbitally continuous and nondecreasing.

It follows thatZC≤MMb σ

(σ+). HenceA(C) is uniformly bounded inE. By the Arzela-Ascoli

theorem,A:D→Dis partially compact.

Theorem . Let the hypotheses(H)-(H)hold.Then the fractional evolution system(.) has a coupled mild solution on J.

Proof Define two operatorsA andAas in (.) and (.). By Lemmas . and ., we

deduce thatA:D→Dis-orbitally continuous, nondecreasing and a partially

nonlin-earD-contraction as well asA:D→Dis-orbitally continuous, nondecreasing and

partially compact.

To apply Theorem ., it remains to prove that there is an elementx∈Dsatisfyingx≤ Ax+Ayfor ally∈D. By (H), there is an elementsx∈Dsatisfying

⎧ ⎨ ⎩

C_Dσ

tx(t) +Ax(t)≤f(t,x(t)) +h(t,y(t)), t∈J, x()≤τ∈X,

for ally∈D. LetF(t) =Dσ_{x(t) +Ax(t),t∈J. Then we have}

x(t) =Uσ(t)x() +

t



(t–s)σ–Vσ(t–s)F(s)ds ≤Uσ(t)τ+

t



(t–s)σ–Vσ(t–s)

ft,x(t)+ht,y(t)ds

= (Ax)(t) + (Ay)(t)

for ally∈Dandt∈J. Hence all the conditions of Theorem . are satisfied. By Theo-rem ., the system (.) has a coupled fixed point inE. Therefore, the fractional evolution

system (.) has a coupled mild solution inE.

By Theorem ., we can obtain the following corollaries easily.

Corollary . Let the hypotheses(H), (H)and(H)hold.In addition,the following con-dition is satisfied:

(H) The functionf :J×X→X is continuous inxfor allt∈Jand there is a constant β∈(, _Mb(σ+)σ )such that

≤f(t,u) –f(t,v)≤β(u–v), ∀u≥v,t∈J.

Then the fractional evolution system(.)has a coupled mild solution on J.

Corollary . Let the hypotheses(H), (H)and(H)hold.In addition,the following con-dition is satisfied:

(H) The functionf :J×X→Xis continuous inxfor allt∈J and there is a constant γ∈(, _Mb(σ+)σ )such that

≤f(t,u) –f(t,v)≤ γu–v

 +u–v, ∀u≥v,t∈J.

5 Applications

In this section, we apply the obtained abstract results to the following fractional hybrid dynamic system:

This implies that the condition (H) holds.

Theorem . Suppose that the following conditions are satisfied:

(P) The functionf :I×I×R→Ris continuous and there exist a constantρ∈(,

Then the fractional hybrid dynamic system(.)has a coupled mild solution.

Then S(t) (t ≥) is an equi-continuous C-semigroup, but it is not compact, and

sup_t∈IS(t) ≤. This implies that the condition (H) holds. Let

x(t)(z) =x(z,t),

y(t)(z) =y(z,t),

ft,x(t)(z) =fz,t,x(z,t),

ht,x(t)(z) =hz,t,x(z,t).

Then the fractional hybrid dynamic system (.) can be rewritten into the abstract frac-tional evolution system (.).

By the assumptions (P) and (P), the conditions (H) and (H) hold. Hence by Theo-rem ., the abstract fractional evolution system (.) has a coupled mild solution, which is also the coupled mild solution of the fractional hybrid dynamic system (.).

Similarly, using Corollaries . and ., we can obtain the following theorems.

Theorem . Assume that the condition(P)and the following condition are satisfied:

(P) The functionf :I×I×R→Ris continuous and there exist a constantβ∈(, √

π

 )

such that

≤fz,t,u(z,t)–fz,t,v(z,t)≤βu(z,t) –v(z,t)

for all(z,t)∈I×Iandu,v∈C(I×I,R)withu≥v.

Then the fractional hybrid dynamic system(.)has a coupled mild solution.

Theorem . Let the condition(P)and the following condition hold:

(P) The functionf :I×I×R→Ris continuous and there exist a constantγ ∈(, √

π

 )

such that

≤fz,t,u(z,t)–fz,t,v(z,t)≤ γu–vC  +u–vC

for all(z,t)∈I×Iandu,v∈C(I×I,R)withu≥v.

Then the fractional hybrid dynamic system(.)has a coupled mild solution.

Acknowledgements

The first author is thankful to the NSF (No. 11661071) and the third author is thankful to the United States-India Education Foundation, New Delhi, India and IIE/CIES, Washington, DC, USA for Fulbright-Nehru PDF Award (No. 2052/FNPDR/2015).

Competing interests

None of the authors have any competing interests in the manuscript.

Authors’ contributions

All authors contributed equally in writing this paper. All authors read and approved the final manuscript.

Author details

1_{College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China.}2_{Department of}

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Received: 24 January 2017 Accepted: 13 July 2017 References

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