JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi Publishing Corporation 838230 10.1155/2013/838230 838230 Research Article Existence of Iterative Cauchy Fractional Differential Equation Ibrahim Rabha W. Yao Jen-Chih Institute of Mathematical Sciences University Malaya, 50603 Kuala Lumpur Malaysia um.edu.my 2013 21 3 2013 2013 15 12 2012 06 02 2013 06 02 2013 2013 Copyright © 2013 Rabha W. Ibrahim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Our main aim in this paper is to use the technique of nonexpansive operators in more general iterative and noniterative fractional differential equations (Cauchy type). The noninteger case is taken in sense of the Riemann-Liouville fractional operators. Applications are illustrated.

1. Introduction

Fractional calculus and its applications (that is the theory of derivatives and integrals of any arbitrary real or complex order) are important in several widely diverse areas of mathematical, physical, and engineering sciences. It generalized the ideas of integer order differentiation and n-fold integration. Fractional derivatives introduce an excellent instrument for the description of general properties of various materials and processes. This is the main advantage of fractional derivatives in comparison with classical integer-order models, in which such effects are in fact neglected. The advantages of fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of properties of gases, liquids, rocks, and in many other fields.

Our aim in this paper is to consider the existence and uniqueness of nonlinear Cauchy problems of fractional order in sense of Riemann-Liouville operators. Also, two theorems in the analytic continuation of solutions are studied. In the fractional Cauchy problems, we replace the first-order time derivative by a fractional derivative. Fractional Cauchy problems are useful in physics. Recently, the author studied the the fractional Cauchy problems in complex domain .

One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators. The Riemann-Liouville fractional derivative could hardly pose the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. Moreover, this operator possesses advantages of fast convergence, higher stability, and higher accuracy to derive different types of numerical algorithms (see ).

Definition 1.

The fractional (arbitrary) order integral of the function f of order α>0 is defined by (1)Iaαf(t)=at(t-τ)α-1Γ(α)f(τ)dτ. When a=0, we write Iaαf(t)=f(t)*ϕα(t), where (*) denoted the convolution product (see ), ϕα(t)=tα-1/Γ(α),t>0 and ϕα(t)=0,t0 and ϕαδ(t) as α0 where δ(t) is the delta function.

Definition 2.

The fractional (arbitrary) order derivative of the function f of order 0α<1 is defined by (2)Daαf(t)=ddtat(t-τ)-αΓ(1-α)f(τ)dτ=ddtIa1-αf(t).

Remark 3.

From Definitions 1 and 2, we have (3)Dαtμ=Γ(μ+1)Γ(μ-α+1)tμ-α,μ>-1,0<α<1,Iαtμ=Γ(μ+1)Γ(μ+α+1)tμ+α,μ>-1,α>0.

Definition 4.

The Caputo fractional derivative of order μ>0 is defined, for a smooth function f(t), by (4)  cDμf(t):=1Γ(n-μ)0tf(n)(ζ)(t-ζ)μ-n+1dζ, where n=[μ]+1, (the notation [μ] stands for the largest integer not greater than μ).

Note that there is a relationship between Riemann-Liouville differential operator and the Caputo operator (5)Daμf(t)=1Γ(1-μ)f(a)(t-a)μ+cDaμf(t), and they are equivalent in a physical problem (i.e., a problem which specifies the initial conditions, that is, if f(0)=0, then the Riemann-Liouville derivative and the Caputo derivative of order α(0,1) coincide).

2. Preliminaries

We extract here the basic theory of nonexpansive mappings in order to offer the notions and results that will be needed in the next sections of the paper. Let (X,d) be a metric space. A mapping P:XX is said to be an ν-contraction if there exists ν[0,1) such that (6)d(Px,Py)νd(x,y),x,yX. In the case where ν=1, the mapping P is said to be nonexpansive. Let K be a nonempty subset of a real normed linear space E and P:KK be a map. In this setting, P is non-expansive if (7)Px-Pyx-y,      x,yK.

The following result is a fixed point theorem for non expansive mappings, according to Berinde; see for example .

Theorem 5.

Let K be a nonempty closed convex and bounded subset of a uniformly Banach space E. Then any nonexpansive mapping P:KK has at least a fixed point.

Definition 6.

Let K be a convex subset of a normed linear space E and let P:KK be a self-mapping. Given an x0K and a real number λ[0,1], the sequence xn defined by the formula (8)xn+1=(1-λ)xn+λPxn,n=0,1,2, is usually called Krasnoselskij iteration or Krasnoselskij-Mann iteration.

Definition 7.

Let K be a convex subset of a normed linear space E and let P:KK be a self-mapping. Given an x0K and a real number λn[0,1], the sequence xn defined by the formula (9)xn+1=(1-λn)xn+λnPxn,n=0,1,2, is usually called Mann iteration.

Edelstein  proved that strict convexity of E suffices for the Krasnoselskij iteration converge to a fixed point of P. While, Egri and Rus  proved that for any subset of E, the Mann iteration converge to a fixed point of P when P is a non-expansive mapping.

We need the following results, which can be found in .

Lemma 8.

Let K be a convex and compact subset of a Banach space E and let P:KK be a non-expansive mapping. If the Mann iteration process xn satisfies the assumptions

xnK for all positive integers n,

0λnb<1 for all positive integers n,

n=0λn=.

Then xn converges strongly to a fixed point of P.

Lemma 9.

Let K be a closed bounded convex subset of a real normed space E and P:KK a non-expansive mapping. If I-P maps closed bounded subset of E into closed subset of E and xn is the Mann iteration, with λn satisfying assumptions (a)–(c) in Lemma 8, then xn converges strongly to a fixed point of P in K.

3. Existence Theorems and Approximation of Solutions

For most of the differential and integral equations with deviating arguments that appear in recent literature, the deviation of the argument usually involves only the time itself. However, another case, in which the deviating arguments depend on both the state variable u and the time t, is of importance in theory and practice. Equations of the form (10)u(t)=f(t,u(u(t))) are called iterative differential equations. These equations are important in the study of infection models and are related to the study of the motion of charged particles with retarded interaction (see [3, 8, 9]).

In this section, we establish the existence and uniqueness results for the fractional differential equation (11)Dα(u(t)-μtα)=f(t,u(u(t))), with initial condition u(0)=0, where t,μΓ(α+1):=u0J:=[0,T] and fC(J×J,J). For tJ denote (12)Mt=max{t,T-t}CL,α={LΓ(α+1)u:|u(t1)-u(t2)|LΓ(α+1)|t1-t2|α,t1,t2J},    L>0. It is clear that CL,α is a nonempty convex and compact subset of the Banach space (C[J],·), where x=suptJ|x(t)|.

Theorem 10.

Assume that the following conditions are satisfied for the initial value problem (11):

fC[J×J,J];

>0:|f(t,u)-f(t,v)||u-v|, for all t,u,vJ;

if L is the Lipschitz constant such that |u(t1)-u(t2)|(L/Γ(α+1))|t1-t2|α, then (13)M=max{|f(t,u)|:(t,u)J×J}L2;

one of the following conditions holds:

M(Tα/Γ(α+1))Mu0, where Mu0=max{u0,T-u0};

u0=0,M(Tα/Γ(α+1))T-u0,f(t,u)0, for all t,uJ;

u0=T,M(Tα/Γ(α+1))u0,f(t,u)0, for all t,uJ.

If (14)(L~+1)TαΓ(α+1)1, then there exists at least one solution of problem (11) in CL,α which can be approximated by the Krasnoselskij iteration (15)un+1=(1-λ)un+λu0+λ0t(t-τ)α-1Γ(α)f(τ,u(u(τ)))dτ, where λ(0,1) and u1CL,α are arbitrary.

Proof.

Consider the integral operator P:CL,αC(J)(16)Pu(t)=u0+0t(t-τ)α-1Γ(α)f(τ,u(u(λτ)))dτ,tJ,uCL,α.

Our aim is to show that P has a fixed point in CL,α. We proceed to apply Schauder fixed point theorem or Banach fixed point theorem.

First we show that CL,α is invariant set with respect to P, that is, T(CL,α)CL,α. In virtue of condition (A4(a)) and for all tJ,  uCL,α, we have (17)|Pu(t)|u0+|0t(t-τ)α-1Γ(α)f(τ,u(u(λτ)))dτ|u0+MTαΓ(α+1)T,|Pu(t)|u0-|0t(t-τ)α-1Γ(α)f(τ,u(u(λτ)))dτ|u0-MTαΓ(α+1)u0-Mu00.

Thus (Pu)(t)J,tJ. In the similar manner of (A4(a)), we treat the cases (A4(b)) and (A4(c)). Now for every t1,t2J, by (A3), we obtain (18)|(Pu)(t1)-(Pu)(t2)|=|0t1(t-τ)α-1Γ(α)f(τ,u(u(λτ)))dτ-0t2(t-τ)α-1Γ(α)f(τ,u(u(λτ)))dτ|M|t1α-t2α+2(t1-t2)α|Γ(α+1)2M|t1-t2|αΓ(α+1)L|t1-t2|αΓ(α+1).

Hence (Pu)CL,α whenever uCL,α. Therefore, P:CL,αCL,α (i.e., P is a self-mapping of CL,α). Let u,vCL,α and tJ, by employing (A2), we have (19)|(Pu)(t)-(Pv)(t)|=|0t(t-τ)α-1Γ(α)f(τ,u(u(λτ)))dτ-0t(t-τ)α-1Γ(α)f(τ,v(v(λτ)))dτ|0t(t-τ)α-1Γ(α)|f(τ,u(u(λτ)))  -f(τ,v(v(λτ)))|dτ0t(t-τ)α-1Γ(α)|u(u(λτ))-v(v(λτ))|dτ=0t(t-τ)α-1Γ(α)|u(u(λτ))-u(v(λτ))+u(v(λτ))-v(v(λτ))|dτTαΓ(α+1)[L~+1]u-v,

where (20)L~=maxLΓ(α+1).

Now, by taking the supremum in the last assertion, we get (21)(Pu)-(Pv)(L~+1)TαΓ(α+1)u-v.

If (L~+1)Tα/Γ(α+1)<1, then P is a contraction mapping and hence in view of Banach fixed point theorem, (11) has a unique solution. Now if (22)(L~+1)TαΓ(α+1)=1,

then P is non-expansive and, hence, continuous; thus Schauder fixed point theorem implies that (11) has a solution in CL,α. Finally, in view of Lemmas 8 and 9, we obtain the second part of the theorem.

Next we establish the solution of (11) in a subset of CL,α defined by (23)CL,α,δ={uCL,α:u(t)δtαΓ(α+1),tJ},  δ(0,1).

It is clear that CL,α,δ is nonempty, convex, and compact subset in C[J].

Theorem 11.

Assume that the following conditions are satisfied.

u0(δt0α/2Γ(α+1)),  t0(0)J.

If L is the Lipschitz constant such that |u(t1)-u(t2)|(L/Γ(α+1))|t1-t2|α, then Mmin{(δ/2),(L/2)}.

There exists a τ¯>0 such that τ¯>-(ln(1-δ)/δ(T-t0)),Tt0 and (24)Tα-1Γ(α)τ¯(1δ+L~)max{eτ¯t0-1,1-eτ¯(t0-T)}1.

If (A2), (A4) hold then there exists at least one solution of problem (11) in CL,α,δ which can be approximated by the Krasnoselskij iteration (25)un+1=(1-λ)un+λu0+λ0t(t-τ)α-1Γ(α)f(τ,u(u(τ)))dτ, where λ(0,1) and u1CL,α,δ are arbitrary.

Proof.

We assume the Banach space C[J] endowed with Bielecki’s norm is given by the formula (26)uB=maxtJ|u(t)|e-s(t-t0),      s>0,t>t0(t,s,t0J=[0,T],T<).

Let P be defined as in the proof of Theorem 10. By assumptions (A2), (A4), and (A6), it follows that (27)P(CL,α,δ)CL,α,δ.

Now we prove that CL,α,δ is an invariant set with respect to the operator P. Indeed, if uCL,α,δ and tJ then in view of (A5) and (A6), we have (28)Pu(t)u0+MtαΓ(α+1)=u0+M(tα-t0α)+t0αΓ(α+1)δt0α2Γ(α+1)+δtα2Γ(α+1)-δt0α2Γ(α+1)+δt0α2Γ(α+1)δtαΓ(α+1),t>t0,

that is, PuCL,α,δ.

Let u,vCL,α,δ and tJ, we have (29)|(Pu)(t)-(Pv)(t)|=|0t(t-τ)α-1Γ(α)f(τ,u(u(λτ)))dτ-0t(t-τ)α-1Γ(α)f(τ,v(v(λτ)))dτ|Tα-1Γ(α)|0t(L~|u(τ)-v(τ)|+|u(v(λτ))-v(v(λτ))|L~|u(τ)-v(τ)|)dτ0t|Tα-1Γ(α)(|0tL~es(τ-t0)dτ|+|0tes(δτ-t0)dτ|)u-vBTα-1Γ(α)(|L~s(es(t-t0)-1)|+1δs|es(δt-t0)-es(δt0-t0)||L~s(es(t-t0)-1)|)u-vB.

This yields (30)|(Pu)(t)-(Pv)(t)|e-s(τ-t0)Tα-1sΓ(α)(1δL~|1-e-s(t-t0)|+1δ|es(δ-1)t-es(δt0-t)|)u-vB:=L(t)u-vB,

where L(t) is a continuous function. Then there exists a constant L^>0 such that (31)maxtJ|L(t)|L^.

Thus we have (32)Pu-PvBL^u-vB,

which shows that P is Lipschitzian, hence continuous. By Schauder’s fixed point theorem, it follows that T has at least one fixed point which is actually a solution of the initial value problem (11).

We proceed to show that P is nonexpansive function. The function (33)g(t)=1-e-s(t-t0),      s>0,t>t0

is strictly increasing on J and g(t0)=0; furthermore, (34)maxtJg(t)=max{eτ~t0-1,1-eτ~(t0-T)}.

Similarly for the function (35)h(t)=es(δ-1)t-es(δt0-t)

then (36)h(t)=ses(δ-1)t[(δ-1)+esδ(t-t0)].

Now the function (37)k(t)=(δ-1)+esδ(t-t0)

is strictly decreasing on J; hence, (38)k(t)k(T)=(δ-1)+esδ(T-t0).

For δ(0,1) and Tt0 then by the assumption (A7), there exists a τ¯>0 such that (39)τ¯>-ln(1-δ)δ(T-t0),  Tt0

which implies that k(T)>0 and hence h is strictly increasing on J. If we put s=τ¯, we have (40)maxtJ|h(t)|=max{|1-esδt0|,|es(δ-1)T-es(δt0-T)|}.

But since δ(0,1) thus we get (41)|es(δ-1)T-es(δt0-T)|=es(δ-1)T|1-esδ(t0-T)|1-esδ(t0-T),

for sufficient s,  δ,  T, and t0. Moreover, we have (42)|1-esδt0|esδt0-1.

This shows that P is non-expansive.

Similar argument holds when T=t0, in (38) we have k(T)=δ>0, hence h is strictly increasing on J. Finally, one can use Lemmas 8 and 9 to obtain the second part of the theorem. This completes the proof.

Example 12.

Consider the following initial value problem associated to an fractional iterative differential equation (44)D0.5u(t)=-13+14u(u(t)),      t[0,1],u(0)=13, where uC1([0,1],[0,1]). We are focused on the solutions uC1([0,1],[0,1]) belonging to the set (45)C1,0.5={1Γ(3/2)u:|u(t1)-u(t2)|1Γ(3/2)|t1-t2|0.5,t1,t2[0,1]}={10.886u:|u(t1)-u(t2)|10.886|t1-t2|,t1,t2[0,1]}={|t1-t2|u:|u(t1)-u(t2)|1.1|t1-t2|,t1,t2[0,1]}.

To satisfy (A4(a)), we have ML/21/2, M1/3=max{1/3,2/3}=2/3=0.666 and (46)MTαΓ(α+1)=12×10.886=0.56<0.666.

Hence (A4(a)) is satisfied. The function f(t)=-(1/3)+(1/4)u is Lipschitzian with the Lipschitz constant =1/4. This shows that (47)(L~+1)TαΓ(α+1)=2.1×0.250.886=0.592<1.

Therefore, by Theorem 10, we obtain information on the existence and approximation for the solutions of the initial value problem (44).

If we consider the function f(t)=-(1/3)+(422/1000)u in Example 12, then we obtain (48)(L~+1)TαΓ(α+1)=2.1×0.4220.8861.

Therefore, again by Theorem 10 we pose the existence and approximation of the solutions of the initial value problem (44).

Again, we consider the problem (44) on the interval [(3/4),1] for =0.025, where uC1([(3/4),1],[(3/4),1]). We are interested in the solutions uC1([3/4,1],[3/4,1]) belonging to the set (49)C1,(1/2),(3/4)={δtαΓ(α+1)uC1,(1/2):u(t)δtαΓ(α+1),tJ},δ(0,1)={u:u(t)(3/4)t(1/2)Γ(3/2),t[(3/4),1]}={u:u(t)0.846t,t[34,1]}. Our aim is to satisfy the assumptions of Theorem 11. (A2) and (A4) are valid. Since u0=1/3 and t0=3/4, we have (50)u0δt0α2Γ(α+1)13<38; hence (A5) is satisfied. Moreover, a computation gives (51)Mmin{δ2,L2}=min{δ2,L2}={38,12}=38 thus (A6) is satisfied. Now we proceed to satisfy (A7), since (52)-ln(1-δ)δ(T-t0)=-ln(1/4)×163=6.933,max{eτ¯t0-1,1-eτ¯(t0-T)}=max{189.5,.826} then for τ¯=7, we impose (53)Tα-1Γ(α)τ¯(1δ+L~)max{eτ¯t0-1,1-eτ¯(t0-T)}=0.17537.17max{189.5,.826}=0.758<1.

Hence in view of Theorem 11, problem (44) has a solution in the set C1,(1/2),(3/4)  .

We can observe that problem (44) has not a solution on the set C1,(1/2),(1/2) over the interval [(1/2),1]: (54)C1,(1/2),(1/2)={uC1,(1/2):u(t)δtαΓ(α+1),tJ},δ(0,1)={u:u(t)(1/2)t(1/2)Γ(3/2),t[12,1]}={u:u(t)0.564t,t[12,1]}. For u0=(1/3), t0=(1/2), α=(1/2), δ=(1/2), a calculation poses (55)u0δt0α2Γ(α+1)13>0.351.772; therefore, condition (A5) dose not satisfy.

Finally, problem (44) has not a solution on the set C1,(1/2),(1/2) over the interval [(3/4),1]: (in view of Theorem 11) (56)C1,(1/2),(1/2)={uC1,(1/2):u(t)δtαΓ(α+1),tJ},δ(0,1)={u:u(t)(1/2)t(1/2)Γ(3/2),t[34,1]}={u:u(t)0.5t,t[34,1]}. For u0=(1/3),  t0=(3/4),  α=(1/2),  δ=(1/2), a calculation yields (57)u0δt0α2Γ(α+1)13>14; therefore, condition (A5) dose not satisfy.

As such iterative fractional differential equations are used to generalize the model infective disease processes, pattern formation in the plane, and are important in investigations of dynamical systems, future works will be also devoted to them.

Acknowledgment

The author is thankful to the anonymous referee for his/her helpful suggestions for the improvement of this paper.

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