Yaghoub JALILIAN Department of Mathematics,Razi University,Kermanshah,Iran E-mail:y.jalilian@razi.ac.ir

FRACTIONAL INTEGRAL INEQUALITIES AND THEIR APPLICATIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS∗

Yaghoub JALILIAN Department of Mathematics,Razi University,Kermanshah,Iran E-mail:y.jalilian@razi.ac.ir

In this paper,first we obtain some new fractional integral inequalities.Then using these inequalities and fixed point theorems,we prove the existence of solutions for two different classes of functional fractional differential equations.

fractional integral inequality;existence of solution;Caputo fractional derivative;fractional differential equation;fixed point 2010 MR Subject Classification34A80;35A23

1　Introduction

Fractional differential equations appear in various fields of engineering and science such as viscoelasticity,electrochemistry,control,electromagnetic,porous media,etc.For example in [1–5],we can see applications of fractional differential equations in signal processing,complex dynamics in biological tissues,viscoelastic materials,thermal systems and heat conduction. One can see the application of fractional differential equations in complex physical systems, physical systems description and control,in[6–8].In the books of Mainardi[9]and Tarasov [8],there are applications of fractional calculus in complex physical systems and dynamics of viscoelastic.To see some developments of fractional calculus,related to special functions,we refer the reader to the book by Kiryakova[1o](also see[11–13]).Also in the monograph of Klafter et al.[14],we can find the latest developments in the field of fractional dynamics.To see some recent results on the existence of solutions for fractional differential equations,we refer the reader to[15–25].In the monographs of Kilbas et al.[26],Podlubny[27],Diethelm[28] and Samko et al.[29]there are some basic information and existence results for various type of fractional differential equations.

Integral inequalities are very useful in the study of ordinary differential and integral equations.For example the Gronwall-Bellman inequality and its generalizations play an important role in the discussion of existence,uniqueness,boundedness,and qualitative behavior of solutions(see[3o,31]).Motivated by applications of fractional integral inequalities(see[32,33]),

we study the following fractional integral inequalities

where n∈ℕ,αi>o,o<µi≤1 for i=1,2,···,n,o≤a

Theorem 1.1Let u be a nonnegative continuous function defined on I=[a,b],and let p(t):I→(o,∞)be a nondecreasing continuous function.Assume that q(t):I→[o,∞)is a nondecreasing continuous function.If u satisfies inequality(1.1),then for k∈ℕ such that (k+1)min{α1,α2,···,αn}>1,

Theorem 1.2Let u be a nonnegative continuous function defined on J=[o,b],and let p(t):J→(o,∞)be a nondecreasing continuous function.Assume that q(t):J→[o,∞)is a nondecreasing continuous function.If u satisfies inequality(1.2),then for k∈ℕ such that (k+1)min{α1,α2,···,αn}>1,

and Hk+1(t,s)is defined by(1.5).

The Lipschitz continuity of the nonlinear part of a fractional differential equation is a basic assumption considered in many papers[19,2o,22,32,34,35].By Theorem 1.1 and a nonlinear alternative of Leray-Schauder type[36],we prove the existence of solution for the following functional fractional differential equation without using the Lipschitz continuity

where f:[a,b]×ℝ×ℝ×ℝ→ℝ is a continuous function and o<α1<α2<α<1.Also,as an application of Theorem 1.2,we prove the existence of solution for the fractional pantograph equation

under a condition weaker than the Lipschitz continuity,here f:[o,b]×ℝ×ℝ×ℝ→ℝ is continuous and o<α,µi<1 for i=1,2.

The organization of the paper is as follows.In Section 2,we give some basic definitions and results concerning the Riemann-Liouville fractional integral and derivative.Some generalizations of the Gronwall-Bellman inequality and a version of nonlinear alternative of Leray-Schauder fixed point theorem is also given in this section.In Section 3,we present our main results and some singular integral inequalities.In Section 4,we prove the existence of solution for problems(1.9)and(1.1o).Finally in Section 5 some examples are given to illustrate our main results.

2　Preliminaries

In this section,we recall some preliminaries which will be needed in this paper.Let[a,b]⊂ℝbe a finite interval and assume that α,β,γ∈ℂ and R(z)=Real(z)for z∈ℂ.The Riemann-Liouville fractional integral and derivative of order α∈ℂ are defined by

respectively,where n=[R(α)]+1 when α/∈ℕo={o,1,···}([α]denotes the integer part of α) [26].The Caputo fractional derivative of order α on[a,b]is defined by

Let C[a,b]be the space of continuous functions f on[a,b]with the norm

Also denote by Cγ[a,b]the space of functions f given on(a,b]such that(x−a)γf(x)∈C[a,b] with the norm

Notice that for γ=o,Cγ[a,b]=C[a,b].The following lemma(see[26],Lemma 2.8(a)) is concerning with the continuity of the fractional integration operatorfrom the space Cγ[a,b]into C[a,b].

Lemma 2.1Let R(α)>o and o≤R(γ)≤1.If R(γ)≤R(α),then the fractional

The semigroup property of the fractional integration operatorand the composition relation between the fractional integration operatorand the fractional differentiation operatorare given by the following lemma(see[26],Lemma 2.9).

Lemma 2.2Let R(α),R(β)>o and f(x)∈C[a,b].Then for any x∈[a,b]the following assertions are true

(c)If R(α)>R(β),then

(d)Let n=[R(α)]+1 for R(α)/∈ℕ and let∈Cn[a,b].Then

The following result(see[26],Lemma 2.21,part(a))mentions that the Caputo fractional differentiation operatoris the left inverse of the Riemann-Liouville fractional integration operatorwhen R(α)/∈ℕoor α∈ℕ.

Lemma 2.3Let α∈ℂ with R(α)>o and let y(x)∈C[a,b].If R(α)/∈ℕ or α∈ℕ,then

The next lemma[32]is concerning with the composition of the Caputo fractional differentiation operatorwith the fractional integration operator

Lemma 2.4Let m−1<α

In the following,we recall a generalization of the Gronwall-Bellman inequality proved by Bellman[37].

Theorem 2.5Let u be continuous function defined on I=[a,b],and let n(t)be a positive continuous and nondecreasing function.Suppose the function K(t,s)is continuous and nonnegative on triangle△:a≤s≤t≤b and nondecreasing in t for each s∈I.If

To study the existence of solution for problems(1.9)and(1.1o)we need the following fixed point theorem[36].

Theorem 2.6Let X be a normed linear space,K be a convex subset of X,O be an open subset of K and θ∈O(θ is the zero element of X).Suppose that NK is a continuous and compact operator whereis closure of O.Then either

(ii)there exists u∈∂O such that u=λTu for some λ∈(o,1)where∂O is the boundary of O in K.

3　Proof of Main Results

We start this section with some auxiliary results.First we need some notations.Define

Using change of variable vs=z,we have

for any v,δ>o.Then using Lemma 2.2(a),forµ,α>o,we have

Lemma 3.1Let u∈C[a,b],and αi>o for i=1,2,···,n.Then for any k,n∈ℕ,

Then the proof of this lemma is an adoption of the proof of multinomial theorem.

Lemma 3.2Let u∈C[a,b]be a nonnegative function,k∈ℕ,αi>o and o<µi≤1 for i=1,2,···,k.Then

where α=max{α1,α2,···,αk}andµ=min{µ1,µ2,···,µk}.

ProofThe proof is straightforward by(3.2)and induction on k.

Lemma 3.3Let u∈C[a,b]be a nonnegative function,k,n∈ℕ,αi>o and o<µi≤1 for i=1,2,···,n.Then

where α=max{α1,···,αn}andµ=min{µ1,···,µn}.

ProofFirst consider a term ofall 1≤m≤n.It is clear thatWe denote the mentioned term byNote that exactly jmelements of the set{β1,···,βk},are equal to αmand jmelements of{v1,···,vk},are equal toµmfor all 1≤m≤n.Hence by Lemma 3.2 we have

where α=max{α1,···,αn}andµ=min{µ1,···,µn}.Also we know that there are exactlyways to obtain terms like(t)by using jmcopies offor all 1≤ m≤n.Then by inequality(3.6)we obtain inequality(3.5).

Proof of Theorem 1.1Since q(t)is nondecreasing and using inequality(1.1),we have

After k times using inequality(1.1),we get

By Lemma 3.1 and inequality(3.7),we obtain

Since p(t)is nondecreasing we have

where Pk(t)is defined by(1.4).Let k∈ℕ such that(k+1)min{α1,α2,···,αn}>1.Then j1α1+···+jnαn−1>o for j1+···+jn=k+1.Therefore,Hk+1(t,s)defined by(1.5)is continuous and nonnegative on△:a≤s≤t≤b,and nondecreasing in t for each s∈[a,b]. Using inequalities(3.8)and(3.1o),we obtain

Applying Theorem 2.5 we obtain inequality(1.3).

Proof of Theorem 1.2Similar to the proof of Theorem 1.1,since q(t)is nondecreasing and after k times using inequality(1.2),we have

By Lemma 3.3 and inequality(3.12),we obtain

where α=max{α1,···,αn}andµ=min{µ1,···,µn}.Similar to(3.9)we have

If we define Pk(t)as(1.7),then from(3.14)we get

Now,we consider k∈ℕ such that(k+1)min{α1,α2,···,αn}>1.Then by the assumption o<µi≤1 for i=1,2,···,n,we have

where Hk+1(t,s)is defined by(1.5).If we definethen Hk+1(t,s)satisfies all conditions of Theorem(2.5),and by using inequalities(3.13),(3.15)and (3.16)we get

The above inequality and Theorem 2.5 yield inequality(1.6).

4　Applications

In this section,we use Theorems 1.1,1.2 and a nonlinear alternative of Leray-Schauder type, to prove the existence of solutions for problems(1.9)and(1.1o).First we consider problem (1.9).

Theorem 4.1Let f:[a,b]×ℝ×ℝ×ℝ→ℝ be a continuous function and o<α1< α2<α<1.The function u(t)∈C[a,b]withsatisfies problem(1.9)if and only if

where v(t)∈C[a,b]satisfies the integral equation

ProofLet u(t)∈C[a,b]be a solution of(1.9)such thatu∈C[a,b].Then Lemma 2.4(b1)implies∈C[a,b].Using Lemma 2.2(d),the definition of the Caputo fractional derivative and by applying the operatorto both sides of equation(1.9),we have

Using(4.5)and(4.6)we can rewrite(4.4)as follows Now,let v(t)∈C[a,b]satisfies(4.2)and let u(t)be defined by(4.1).Then u(t)∈C[a,b]and using Lemma 2.3,v(t)both sides of(4.2)and using Lemma 2.2 (a),we obtain that u satisfies equation(4.3).Applyingto both sides of(4.3)and using Lemma 2.3 we conclude that u is a solution of(1.9).Since f,u,are continuous functions,u is continuous on[a,b].

Now we consider problem(1.9)under the following assumption.

(h)f:[a,b]×ℝ×ℝ×ℝ→ℝ is a continuous function and there are a constant co>o and a nonnegative function

In the next lemma,using Theorem 1.1,we give a prior bound for solutions of integral equation(4.2)which is important in the proof of the existence of solution for problem(1.9).

Theorem 4.2Let(h)hold and let v(t)∈C[a,b]satisfy the integral equation

Then for k∈ℕ such that(k+1)(α−α2)>1,we have

ProofWe follow the argument given in[33].Let v(t)∈C[a,b]satisfy integral equation (4.8).By assumption(h),Lemma 2.2(a)and the H¨older inequality,we have

where pλ(t)is defined by(4.7).By inequality(4.1o)and Theorem 1.1 for n=3,the conclusion follows.

Now,we prove the existence of solution for equation(1.9).

Theorem 4.3Suppose assumption(h)holds.Then problem(1.9)has at least one solution u(t)∈C[a,b]which

ProofBy Theorem 4.1 it is enough to show that integral equation(4.2)has at least one solution in C[a,b].To do this we define the operator F on C[a,b]as follows We show that the operator F satisfies assumptions of Theorem 2.6.Let k∈ℕ such that

In the following steps we prove that F:Br→C[a,b]is a continuous and compact operator where Bris the ball centered at o with radius r.

Step 1F(Br)is a bounded subset of C[a,b].

Step 2F:Br→C[a,b]is continuous.

Fix ε>o and take arbitrarily v,uThen for t∈[a,b],we

The uniform continuity of f on[a,b]×I1×I2×I3yields that ω(f,ε)→o as ε→o.Hence by inequality(4.13),the operator F is continuous on

Step 3F:Br→C[a,b]is compact.

First we show that F(Br)is equicontinuous subset of C[a,b].Let t1,t2∈[a,b],t1

The right hand side of(4.14)tends to zero when t1→t2.Thenis equicontinuous.

Step 1,inequality(4.14)and the Arzela-Ascoli theorem imply that F:Br→C[a,b]is a compact operator.

Step 4The equation v=λFv doesn’t have any solution in∂Brfor λ∈(o,1).

Let v∈∂Brand v=λFv.Then v is a solution of integral equation(4.8).Hence by Theorem 4.2,v satisfies inequality(4.9)and consequently‖v‖C

By the above steps and Theorem 2.6,F has a fixed point inwhich implies equation (4.2)has a solution in

In the sequel,we prove the existence of solution for problem(1.1o)under the following assumption.

(h1)f:[o,b]×ℝ×ℝ×ℝ→ℝ is a continuous function and there are a constant co>o and a nonnegative function c(t)∈Lq[o,b]for 1

Lemma 4.4Let f:[o,b]×ℝ×ℝ×ℝ→ℝ be a continuous function and o<α,µi<1 for i=1,2.Then u∈C[o,b]is a solution of problem(1.1o)if and only if u is a solution of the integral equation

ProofThe proof is straight forward and we omit it.

Similar to the proof of Theorem 4.3,we need a prior bound for solutions of the following integral equation

where λ∈(o,1].Put

Theorem 4.5Let(h1)hold and let u(t)∈C[o,b]satisfy integral equation(4.16).Then for k∈ℕ such that(k+1)α>1,we have

ProofLet u(t)∈C[o,b]be a solution of(4.16).Using(3.1),the H¨older inequality and assumption(h1)we have

Using Theorem 1.2,for n=3 and inequality(4.18),we obtain inequality(4.17).

Theorem 4.6Let assumption(h1)hold.Then problem(1.1o)has at least one solution u(t)∈C[o,b].

ProofDefine the operator F on C[o,b]as

and let k∈ℕ such that(k+1)α>1.Also,putSince f is continuous and by Lemma 2.1,for any u∈C[o,b],Fu∈C[o,b].Now,we show that→C[o,b]is continuous.Fix ε>o and take arbitrarily v,u∈such that‖v−u‖C≤ε. Then for t∈[a,b],we have

The rest of the proof is similar to the proof of Theorem 4.3.

5　Examples

In this section,we give some examples to demonstrate the applicability of our results.

Example 5.1We consider the following nonlinear fractional differential equation

Therefore,assumption(h)is satisfied and using Theorem 4.3,problem(5.1)has at least one solution in C[1,3].

Example 5.2Consider the fractional pantograph equation

Hence assumption(h1)is satisfied and by using Theorem 4.6 the fractional pantograph equation (5.2)has at least one solution in C[o,1].

References

[1]Magin R,Ortigueira M,Podlubny I,Trujillo J J.On the fractional signals and systems.Signal Processing, 2011,91:350–371

[2]Magin R.Fractional calculus models of complex dynamics in biological tissues.Comput Math Appl,2010, 59:1586–1593

[3]Merala F C,Roystona T J,Magin R.Fractional calculus in viscoelasticity:an experimental study.Commun Nonlinear Sci Numer Simul,2010,15:939–945

[4]Sabatier J,Nguyen H C,Farges C,Deletage J Y,Moreau X,Guillemard F,Bavoux B.Fractional models for thermal modeling and temperature estimation of a transistor junction.Adv Difference Equations,2011, Article 687363

[5]Tenreiro Machado J A,Kiryakova V,Mainardi F.Recent history of fractional calculus.Commun Nonlinear Sci Numer Simul,2011,16:1140–1153

[6]Das S.Functional Fractional Calculus for System Identification and Controls.Berlin,Heidelberg:Springer-Verlag,2008

[7]Das S.Functional Fractional Calculus.Berlin,Heidelberg:Springer-Verlag,2011

[8]Tarasov V E.Fractional Dynamics:Applications of Fractional Calculus to Dynamics of Particles.Fields and Media.Berlin,Heidelberg:Springer-Verlag,2010

[9]Mainardi F.Fractional Calculus and Waves in Linear Viscoelasticity:An Introduction to Mathematical Models.London:Imperial College Press,2010

[10]Kiryakova V.Generalized Fractional Calculus and Applications.Harlow,New York:Longman&Wiley, 1994

[11]Kiryakova V.All the special functions are fractional differintegrals of elementary functions.J Physics A: Math&Gen,1997,30:5085–5103

[12]Kiryakova V.Some special functions related to fractional calculus and fractional(non-integer)order control systems and equations.Facta Universitatis(Sci J of University of Nis),Series:Autom Control Robot,2008, 7:79–98

[13]Machado J T,Kiryakova V,Mainardi F.Recent history of fractional calculus.Commun Nonlinear Sci Numer Simul,2011,16:1140–1153

[14]Klafter J,Lim S C,Metzler R.Fractional Dynamics in Physics:Recent Advances.Singapore:World Scientific,2011

[15]Agarwal R P,Ahmad B.Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions.Comput Math Appl,2011,62:1200–1214

[16]Agarwal R P,Benchohra M,Hamani S.A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions.Acta Appl Math,2010,109:973–1033

[17]Ahmad B,Ntouyas S K.Nonlocal fractional boundary value problems with slit-strips boundary conditions. Fractional Calculus Appl Anal,2015,18(1):261–280

[18]Balachandran K,Kiruthika S,Trujillo J J.Existence results for fractional impulsive integrodifferential equations in Banach spaces.Commun Nonlinear Sci Numer Simul,2011,16:1970–1977

[19]Balachandran K,Trujillo J J.The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces.Nonlinear Analysis:TMA,2010,72:4587–4593

[20]Balachandran K,Kiruthika S.Existence results for fractional integrodifferential equations with nonlocal condition via resolvent operators.Comput Math Appl,2011,62:1350–1358

[21]Benchohra M,Berhoun F,Nguerekata G.Bounded solutions for fractional order differential equations on the half-line.Bull Math Anal Appls,2012,4:62–71

[22]Caballero J,Harjani J,Sadarangani K.On existence and uniqueness of positive solutions to a class of fractional boundary value problems.Boundary Value Problems,2011,2011:25

[23]Hu Z,Liu W.Solvability for fractional order boundary value problems at resonance.Boundary Value Problems,2011,2011:20

[24]Rui W.Existence of solutions of nonlinear fractional differential equations at resonance.Elect J Qual Theory Differ Equ,2011,66:1–12

[25]Sun W,Wang Y.Multiple positive solutions of nonlinear fractional differential equations with integral boundary value conditions.Fractional Calculus and Applied Analysis,2014,17(3):605–616

[26]Kilbas A A,Srivastava H,Trujillo J J.Theory and Applications of Fractional Differential Equations. Amsterdam:Elsevier,2006

[27]Podlubny I.Fractional Differential Equations.New York:Academic Press,1999

[28]Diethelm K.The Analysis of Fractional Differential Equations.Berlin,Heidelberg:Springer-Verlag,2010

[29]Samko S G,Kilbas A A,Marichev O I.Fractional Integrals and Derivatives:Theory and Applications. Yverdon:Gordon and Breach,1993

[30]Bainov D D,Simeonov P.Integral Inequalities and Applications.Kluwer Academic Publishers,1992

[31]Lakshiliikantham V,Leela S.Differential and Integral Inequalities:Theory and Applications:Ordinary Differential Equations,Vol 1.New York:Academic Press,1969

[32]Aghajani A,Jalilian Y,Trujillo J J.On the existence of solutions of fractional integro-differential equations. Fract Calculus Appl Anal,2012,15(1):44–69

[33]Jalilian Y,Jalilian R.Existence of solution for delay fractional differential equations.Mediterr J Math, 2013,10:1731–1747

[34]Abbas S.Existence of solutions to fractional order ordinary and delay differential equations and applications. Elect J Differ Equ,2011,9:1–11

[35]Balachandran K,Kiruthika S,Trujillo J J.Existence of solutions of nonlinear fractional pantograph equations.Acta Math Sci,2013,33B:712–720

[36]Zeidler E.Nonlinear Functional Analysis and Applications,I:Fixed Point Theorems.New York:Springer-Verlag,1986

[37]Pachpatte B G.Inequalities for Differential and Integral Equations.New York:Academic Press,1998

∗June 15,2015;revised November 18,2015.