Solution of fractional differential equations via \(\alpha \psi\) Geraghty type mappings

R E V I E W

Open Access

Solution of fractional differential

equations via

α

–

ψ

-Geraghty type mappings

Hojjat Afshari

1*

_{, Sabileh Kalantari}

1

_{and Dumitru Baleanu}

2

*_{Correspondence:} [email protected] 1_{Department of Mathematics, Basic} Science Faculty, University of Bonab, Bonab, Iran

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

Abstract

Using fixed point results of

α

–

ψ-Geraghty contractive type mappings, we examine

the existence of solutions for some fractional differential equations inb-metric spaces. By some concrete examples we illustrate the obtained results.

Keywords: Fractional differential equation; Normal cone;

α

–

ψ-Geraghty

contractive type mapping

1 Introduction

In 2012, Samet et al. [11] presented the concepet ofα-admissible mappings, which was expanded by several authors (see [5,6,9]). Baleanu, Rezapour, and Mohammadi [3] studied the existence of a solution for problemDν_{w(ξ) =h(ξ,w(ξ)) (ξ ∈[0, 1], 1 <ν≤2). Afshari,}

Aydi, and Karapinar [1,2] considered generalizedα–ψ-Geraghty contractive mappings inb-metric spaces.

We investigate the existence of solutions for some fractional differential equations in

b-metric spaces. We denoteI= [0, 1].

Definition 1.1([7,10]) The Caputo derivative of orderν of a continuous functionh: [0,∞)→Ris defined by

c_Dν

h(ξ) = 1 (n–ν)

ξ

0

(ξ–ζ)n–ν–1h(n)(ζ)dζ,

wheren– 1 <ν<n,n= [ν] + 1, [ν] is the integer part ofν, and

(z) = ∞

0

xz–1_e–x_{dx. (1)}

Definition 1.2([7,10]) The Riemann–Liouville derivative of a continuous functionhis defined by

Dνh(ξ) = 1 (n–ν)

d dξ

n ξ

0

h(ζ) (ξ–ζ)ν–n–1dζ

n= [ν] + 1,

where the right-hand side is defined on (0,∞).

Let be the set of all increasing continuous functionsψ : [0,∞)→[0,∞) such that ψ(λx)≤λψ(x)≤λx forλ> 1, and letBbe the family of nondecreasing functions γ : [0,∞)→[0,_s12) for somes≥1.

Definition 1.3([1]) Let (X,d) be ab-metric space (with constants). A functiong:X→X

is a generalizedα–ψ-Geraghty contraction if there existsα:X×X→[0,∞) such that

α(z,t)ψs3d(gz,gt)≤γψd(z,t)ψd(z,t) (2)

for allz,t∈X, whereγ ∈Bandψ∈.

Definition 1.4([11]) Letg:X→Xandα:X×X→[0,∞) be given. Theng is called α-admissible if forz,t∈X,

α(z,t)≥1 ⇒ α(gz,gt)≥1. (3)

Theorem 1.5([1]) Let(X,d)be a complete b-metric space,and let f :X→X be a gener-alizedα–ψ-Geraghty contraction such that

(i) f isα-admissible;

(ii) there existsu0∈Xsuch thatα(u0,fu0)≥1;

(iii) if{un} ⊆X,un→uinX,andα(un,un+1)≥1,thenα(un,u)≥1.

Then f has a fixed point.

2 Main result

ByX=C(I) we denote the set of continuous functions. Letd:X×X→[0,∞) be given by

d(y,z) =(y–z)2_∞=sup ξ∈I

y(ξ) –z(ξ)2. (4)

Evidently, (X,d) is a completeb-metric space withs= 2 but is not a metric space. Now we study the problem

Dν

Dξw(ξ) =h

ξ,w(ξ), ξ∈I, 3 <ν≤4, (5)

under the conditions

w(0) =w(0) =w(1) =w(1) = 0, (6)

whereDν_{is the Riemann–Liouville derivative, andh:I×X→Ris continuous.}

Lemma 2.1([13]) Given h∈C(I×X,R)and3 <ν≤4,the unique solution of

Dν

Dξw(ξ) =h

ξ,w(ξ), ξ∈I, 3 <ν≤4, (7)

where

is given by w(ξ) =₀1G(ξ,ζ)h(s,w(s))ds,where

Then problem(7)has at least one solution.

Hence, fory,z∈C(I) andξ∈Iwithθ(y(ξ),z(ξ))≥0, we have

(Ay–Az)2∞≤1

8

(ψ((y–z)2

∞))2

4(y–z)2_{∞+ 1} .

Letα:C(I)×C(I)→[0,∞) be defined by

α(y,z) = ⎧ ⎨ ⎩

1, θ(y(ξ),z(ξ))≥0,ξ∈I,

0 otherwise.

Defineγ : [0,∞)→[0,1₄) byγ(q) = _4q+1q ands= 2. So

α(y,z)ψ8d(Ay,Az)≤8α(y,z)ψd(Ay,Az)≤(ψ(d(y,z))) 2 4d(y,z) + 1

≤ (ψ(d(y,z)))2 4ψ(d(y,z)) + 1

= 1

γ(ψ(d(y,z)))γ

ψd(y,z) (ψ(d(y,z))) 2 4ψ(d(y,z)) + 1

≤γψd(y,z)ψd(y,z), γ ∈B.

ThenAis anα–ψ–contractive mapping. From (iii) and the definition ofαwe have

α(y,z)≥1 ⇒ θy(ξ),z(ξ)≥0

⇒ θA(y),A(z)≥0 ⇒ αA(y),A(z)≥1,

fory,z∈C(I). Thus,Aisα-admissible. By (ii) there existsy0∈C(I) such thatα(y0,Ay0)≥1. By (iv) and Theorem1.5there isy∗∈C(I) such thaty∗=Ay∗. Hencey∗is a solution of the

problem.

Corollary 2.4 Suppose that there existθ:R2_{→Randψ∈such that}

h(ξ,c) –h(ξ,d)≤ 10 3 4√8

ψ(|c–d|2₎

4(c–d)2_{∞+ 1} (10)

forξ∈I and c,d∈Rwithθ(c,d)≥0.Also,suppose that conditions(ii)–(iv)from Theorem 2.3hold for h,where G(ξ,ζ)is given in(9).Then the problem

D72

Dξw(ξ) =h

ξ,w(ξ), ξ∈I, (11)

where

w(0) =w(0) =w(1) =w(1) = 0,

Proof By Lemma2.2

min

1

0

G(ξ,ζ)dζ= 10–5 and max

1

0

G(ξ,ζ)dζ= 4×10–3. (12)

Using (10) and (12), by Theorem2.3we obtain

_Ay(ξ) –Az(ξ)2≤1 8

(ψ(|y–z|2₎₎2 4(y–z)2_{∞+ 1}.

The rest of the proof is according to Theorem2.3.

Lemma 2.5([8]) If h∈C(I×X,R)and h(ξ,w(ξ))≤0,then the problem

–Dν₀₊w(ξ) =hξ,w(ξ), (0 <ξ< 1, 3 <ν≤4),

w(0) =w(0) =w(0) =w(1) = 0

(13)

has a unique positive solution

w(ξ) = 1

0

G(ξ,ζ)hζ,w(ζ)dζ,

where G(ξ,ζ)is given by

G(ξ,ζ) = 1 (ν)

⎧ ⎨ ⎩

ξν–1_{(1 –ζ)}ν–3_{– (ξ–ζ)}ν–1_{, 0≤ζ≤ξ≤1,}

ξν–1_{(1 –ζ)}ν–3_{, 0≤ξ≤ζ≤1.} (14)

Lemma 2.6([12]) The function G(ξ,ζ)in Lemma2.5has the following property:

1

(ν)ζ(2 –ζ)(1 –ζ)

ν–3_ξν–1_{≤G(ξ,ζ)≤} 1

(ν)(1 –ζ)

ν–3_ξν–1_,

whereξ,ζ∈I and3 <ν≤4.

Based on Theorem2.3, we get the following result.

Corollary 2.7 Assume that there existθ:R2→Randψ∈such that

_h(ξ,c) –h(ξ,d)≤ 1 2√2M

ψ(|c–d|2₎

4(c–d)2_{∞+ 1},

where M=sup_ξ∈I01G(ξ,ζ)dζ.Also,suppose that conditions(ii)–(iv)from Theorem2.3 are satisfied,where G(ξ,ζ)is given in(14).Then problem(13)has at least one solution.

Proof By Lemma2.5 y∈C(I) is a solution of (13) if and only if a solution of y(ξ) = 1

0 G(ξ,ζ)h(ζ,y(ζ))dζ. DefineA:C(I)→C(I) byAy(ξ) = 1

Lemma2.6we get

+ 2ξ (2 –η2_)(ν)

0 0

|ζ–n|ν–1dn dζ

≤1 8γ

ψy–z2_∞ψy–z2_∞

for ally,z∈C(I) withθ(y(ξ),z(ξ))≥0,ξ∈I, so that

(Ay–Az)2_∞≤1 8γ

ψy–z2_∞ψy–z2_∞.

Letα:C(I)×C(I)→[0,∞) be defined by

α(y,z) = ⎧ ⎨ ⎩

1 θ(y(ξ),z(ξ))≥0,ξ∈I,

0 otherwise.

Then

α(y,z)ψ8d(Ay,Az)≤8α(y,z)ψd(Ay,Az)

≤α(y,z)ψγψd(y,z)ψd(y,z) ≤γψd(y,z)ψd(y,z)

for ally,z∈C(I), and thusAis anα–ψ–contractive mapping. From Theorem1.5, based

on the proof of Theorem2.3, we can deduce the proof of Theorem2.8.

Here we find a positive solution for

c_Dν

Dξ w(ξ) =h

ξ,w(ξ), 0 <ν≤1,ξ∈I, (16)

where

w(0) + 1

0

w(ζ)dζ=w(1).

Note thatc_Dν_{is the Caputo derivative of orderν. We consider the Banach space of}

con-tinuous functions onIendowed with the sup norm. We have the following lemma.

Lemma 2.9([4]) Let0 <ν≤1and h∈C([0,T]×X,R)be given.Then the equation

c_Dν

w(ξ) =hξ,w(ξ) ξ∈[0,T],T≥1

with

w(0) + T

0

w(ζ)dζ=w(T)

has a unique solution given by

w(ξ) = T

0

where G(ξ,ζ)is defined by

By Lemma2.9and Theorem2.4we get the following conclusion.

Corollary 2.10 Assume that there existθ:R2_{→Randψ∈such that}

rem2.3we conclude the desired result.

Example2.11 Letψ(r) =r,θ(x,z) =xz, andyn(ξ) = _n2ξ₊₁. We considerh:I×[–2, 2]→ [–2, 2] and the periodic boundary value problem

D72

Acknowledgements

Not applicable.

Funding

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Basic Science Faculty, University of Bonab, Bonab, Iran.}2_{Department of Mathematics,} Cankaya University, Ankara, Turkey.

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Received: 5 June 2018 Accepted: 18 September 2018

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