# On a Time-Fractional Integrodifferential Equation via Three-Point Boundary Value Conditions

Authors: Dumitru Baleanu; Shahram Rezapour; Sina Etemad; Ahmed Alsaedi
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

The existence and the uniqueness theorems play a crucial role prior to finding the numerical solutions of the fractional differential equations describing the models corresponding to the real world applications. In this paper, we study the existence of solutions for a time-fractional integrodifferential equation via three-point boundary value conditions.


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## Body

## 1. Introduction

There is no doubt that the fractional calculus has various important applications in many fields as mathematics (see the following monographs [1, 2] and the references therein), physics [3], and economics [4], as well as in many other branches of science and engineering [3, 5, 6]. Recently the numerical methods were applied intensively to solve complicated fractional differential equations (FDE) which describe the real world applications. As an example, we mention that several physical processes exhibit fractional order behavior varying with time and/or space [7]. Thus, a class of phenomena can be obtained by analyzing this most general model. A fundamental step is to prove the existence and uniqueness of the proposed fractional nonlinear differential equations for these models. There are many works on fractional differential equations and inclusions. The existence of solutions for integrodifferential equations of fractional order with nonlocal three-point fractional boundary conditions was developed in [8] while the importance of antiperiodic type integral boundary conditions was discussed in [9]. The existence of solutions for fractional differential inclusions in the presence of the separated boundary conditions in Banach space was developed in [10]. We recall that the existence and multiplicity of positive solutions for singular fractional boundary value problems were analyzed in [11]. The existence and multiplicity of positive solutions for singular fractional boundary value problems were the topic debated in [12]. Making use of the fixed point results on cones, the existence and uniqueness of positive solutions for some nonlinear fractional differential equations were developed in [13]. Further results on positive solutions of a boundary value problem for nonlinear fractional differential equations were reported in [14] and new results on the existence results on nonlinear fractional differential equations were reported in [15]. For other related results we suggest for the readers [16, 17] as well as the references therein. We recall that the existence of solutions for some fractional partial differential equations was investigated in [18, 19] and the references therein.Letn be a natural number, n-1<α<n, a,b∈R, and u∈C([a,b]×[a,b],R). The Riemann-Liouville time-fractional order integral of the function u is defined by I∗tαu(x,t)=1/Γ(α)∫0t‍u(x,τ)/t-τ1-αdτ for all x,t∈[a,b] whenever the integral exists (more details regarding the basic definitions of the fractional calculus can be seen in [1]). Also, the Caputo derivative of time-fractional of order α for the function u is defined by Dc∗tαu(x,t)=1/Γ(n-α)∫0t‍t-τn-α-1∂nu(x,τ)/∂τndτ (see for more details [1]). It has been proved that the general solution of the time-fractional differential equation Dc∗tαu(x,t)=0 is given by ux,t=c0+c1t+c2t2+⋯+cn-1tn-1, where c0,…,cn-1 are real constants and n=[α]+1 (see [20]). Also, for each T>0 and u∈C([0,T]×[0,T]) we have I∗tαDc∗tαu(x,t)=u(x,t)+c0+c1t+c2t2+⋯+cn-1tn-1, where c0,…,cn-1 are real constants and n=[α]+1 ([20]). Let M be a bounded set in a metric space (X,d). Then the Kuratowski measure of noncompactness is defined by (see [21])(1)αM=inf⁡ϵ:Mcoveredbyfinitelymanysetsinwhichthediameterofeachsetislessthanorequaltoϵ.Let Φ:D(Φ)⊆X→X be a bounded and continuous operator on a Banach space X. Then Φ is called a condensing map whenever α(Φ(B))<α(B) for all bounded sets B⊂D(Φ), where α denotes the Kuratowski measure of noncompactness (see [21]).In this paper, we study the existence of solutions for a time-fractional integrodifferential equation via three-point boundary value conditions under certain conditions. We mention that the investigated equation generalized a huge class of classical ordinary differential equations which can be found in applications in engineering and physics. The boundary conditions used in this paper are as general as possible and for particular cases of the parameters we recover several cases of fractional nonlinear differential equations. In this way, we use the next Lemma and Sadovskii’s fixed point theorem for condensing maps (see [22]).Lemma 1 (see [23]).
LetX be a Banach space, D⊆X, k∈[0,1), K:D→X a k-contraction, and C:D→X a compact operator. Then K+C is a k-set contraction and so a condensing map.Theorem 2 (see [22]).
LetB be a convex, bounded, and closed subset of a Banach space X and Φ:B→B a condensing map. Then Φ has a fixed point.


## 2. Main Results

Suppose thatJ=[0,1]×[0,1], 1≤q<2, and X={u:u,∂q/∂tqu∈C(J,R)} endowed via the norm(2)u=sup⁡(x,t)∈Jux,t+sup⁡(x,t)∈J∂q∂tqux,t,where ∂q/∂tqu(x,t) denotes the standard Caputo time-fractional derivative. Let 2≤α<3, 0<β<1, 1≤δ<2, 0≤η≤1, and λ,μ∈R, and f:J×X2→X and g:J×X→X are continuous functions. We investigate the nonlinear time-fractional integrodifferential equation as follows:∗-∂α∂tαux,t=λfx,t,ux,t,∂q∂tqux,t+μI∗tβg(x,t,u(x,t))via the three-point boundary value conditions ∂δ+1/∂tδ+1u(x,0)=∂δ+1/∂tδ+1u(x,1), ∂δ/∂tδu(x,1)-∂δ/∂tδu(x,η)=0, and u(x,0)=0. In this way, we give first next result.Lemma 3.
An elementu0 in X is a solution for the problem ∗ via the three-point boundary value conditions if and only if u0 is a solution for the time-fractional integral equation(3)ux,t=∫01‍Gxt,τλfx,τ,ux,τ,∂q∂τqux,τ+μI∗tβgx,τ,ux,τ∫01‍dτ,where(4)Gx(t,τ)=-t-τα-1Γα+2tΓ(2-δ)(1-η2-δ)+t2η1-δ-1Γ3-δ1-τα-δ-2×2Γα-δ-1η1-δ2-δ+η2-δδ-1-1-1+[t2(1-δ)Γ(3-δ)-2tΓ(3-δ)]1-τα-δ-12Γ(α-δ)[η1-δ(2-δ)+η2-δ(δ-1)-1]+[2tΓ(3-δ)-t2(1-δ)Γ(3-δ)]η-τα-δ-1Γ(α-δ)2[η1-δ(2-δ)+η2-δ(δ-1)-1],whenever 0≤τ≤η≤t≤1,(5)Gx(t,τ)=-t-τα-1Γα+2tΓ(2-δ)(1-η2-δ)+t2(η1-δ-1)Γ(3-δ)1-τα-δ-2×2Γα-δ-1η1-δ2-δ+η2-δδ-1-1-1+[t2(1-δ)Γ(3-δ)-2tΓ(3-δ)]1-τα-δ-12Γ(α-δ)[η1-δ(2-δ)+η2-δ(δ-1)-1],whenever 0≤η≤τ≤t≤1,(6)Gx(t,τ)=2tΓ(2-δ)(1-η2-δ)+t2(η1-δ-1)Γ(3-δ)1-τα-δ-2×2Γα-δ-1η1-δ2-δ+η2-δδ-1-1-1+[t2(1-δ)Γ(3-δ)-2tΓ(3-δ)]1-τα-δ-12Γ(α-δ)[η1-δ(2-δ)+η2-δ(δ-1)-1],whenever 0≤η≤t≤τ≤1,(7)Gx(t,τ)=2tΓ(2-δ)(1-η2-δ)+t2(η1-δ-1)Γ(3-δ)1-τα-δ-2×2Γα-δ-1η1-δ2-δ+η2-δδ-1-1-1+[t2(1-δ)Γ(3-δ)-2tΓ(3-δ)]1-τα-δ-12Γ(α-δ)[η1-δ(2-δ)+η2-δ(δ-1)-1],whenever 0≤t≤η≤τ≤1,(8)Gx(t,τ)=2tΓ(2-δ)(1-η2-δ)+t2(η1-δ-1)Γ(3-δ)1-τα-δ-2×2Γα-δ-1η1-δ2-δ+η2-δδ-1-1-1+[t2(1-δ)Γ(3-δ)-2tΓ(3-δ)]1-τα-δ-12Γ(α-δ)[η1-δ(2-δ)+η2-δ(δ-1)-1]+[2tΓ(3-δ)-t2(1-δ)Γ(3-δ)]η-τα-δ-1Γ(α-δ)2[η1-δ(2-δ)+η2-δ(δ-1)-1],whenever 0≤t≤τ≤η≤1, and(9)Gx(t,τ)=-t-τα-1Γ(α)+2tΓ(2-δ)(1-η2-δ)+t2(η1-δ-1)Γ(3-δ)1-τα-δ-2×2Γα-δ-1η1-δ2-δ+η2-δδ-1-1-1+[t2(1-δ)Γ(3-δ)-2tΓ(3-δ)]1-τα-δ-12Γ(α-δ)[η1-δ(2-δ)+η2-δ(δ-1)-1]+[2tΓ(3-δ)-t2(1-δ)Γ(3-δ)]η-τα-δ-1Γ(α-δ)2[η1-δ(2-δ)+η2-δ(δ-1)-1],whenever 0≤τ≤t≤η≤1.Proof.
Letu0 be a solution for the time-fractional integrodifferential equation ∗ via the boundary value conditions. Put y0(x,t)=λf(x,t,u0(x,t),∂q/∂tqu0(x,t))+μI∗tβg(x,t,u0(x,t)). Choose c0,c1,c2∈R such that u0(x,t)=-∫0t‍t-τα-1/Γ(α)y0(x,τ)dτ+c0+c1t+c2t2. Thus, we get(10)∂δ∂tδu0x,t=-∫0t‍t-τα-δ-1Γ(α-δ)y0(x,τ)dτ+c1t1-δΓ(2-δ)+c22t2-δΓ(3-δ)and (∂δ+1/∂tδ+1)u0(x,t) = -∫0t‍((t-τ)α-δ-2/Γα-δ-1)y0(x,τ)dτ + c1(t-δ/Γ1-δ)+c2(2t1-δ/Γ(2-δ)). Now by using the boundary value conditions, we obtain c0=0,(11)c1=Γ(2-δ)(1-η2-δ)[η1-δ(2-δ)+η2-δ(δ-1)-1]×∫01‍1-τα-δ-2Γ(α-δ-1)y0(x,τ)dτ-Γ(3-δ)[η1-δ(2-δ)+η2-δ(δ-1)-1]×∫01‍1-τα-δ-1Γ(α-δ)y0(x,τ)dτ+Γ(3-δ)[η1-δ(2-δ)+η2-δ(δ-1)-1]×∫0η‍η-τα-δ-1Γ(α-δ)y0(x,τ)dτ,c2=(1-δ)Γ(3-δ)2[η1-δ(2-δ)+η2-δ(δ-1)-1]×∫01‍1-τα-δ-1Γ(α-δ)y0(x,τ)dτ-(1-δ)Γ(3-δ)2[η1-δ(2-δ)+η2-δ(δ-1)-1]×∫0η‍η-τα-δ-1Γ(α-δ)y0(x,τ)dτ+(η1-δ-1)Γ(3-δ)2[η1-δ(2-δ)+η2-δ(δ-1)-1]×∫01‍1-τα-δ-2Γ(α-δ-1)y0(x,τ)dτ.Hence,(12)u0x,t=-∫0t‍t-τα-1Γ(α)y0(x,τ)dτ+2tΓ(2-δ)(1-η2-δ)+t2(η1-δ-1)Γ(3-δ)2[η1-δ(2-δ)+η2-δ(δ-1)-1]×∫01‍1-τα-δ-2Γ(α-δ-1)y0(x,τ)dτ+t2(1-δ)Γ(3-δ)-2tΓ(3-δ)2[η1-δ(2-δ)+η2-δ(δ-1)-1]×∫01‍1-τα-δ-1Γ(α-δ)y0(x,τ)dτ+2tΓ(3-δ)-t2(1-δ)Γ(3-δ)2[η1-δ(2-δ)+η2-δ(δ-1)-1]×∫0η‍η-τα-δ-1Γ(α-δ)y0(x,τ)dτ=∫01‍Gx(t,τ)y0(x,τ)dτ.Thus, u0 is a solution for the time-fractional integral equation. It is obvious that u0 is a solution for the time-fractional integrodifferential equation ∗ whenever u0 is a solution for the time-fractional integral equation. This completes the proof.By considering the proof of last result, one can get that the solution of the time-fractional integrodifferential equation∗ is in the form(13)ux,t=-λ∫0t‍t-τα-1Γαfx,τ,ux,τ,∂q∂tqux,τdτ-μ∫0t‍t-τα+β-1Γ(α+β)g(x,τ,u(x,τ))dτ+tQ+t2R2Nλ×∫01‍1-τα-δ-2Γ(α-δ-1)f(x,τ,u(x,τ),∂q∂tqu(x,τ))dτ+tQ+t2R2Nμ×∫01‍1-τα+β-δ-2Γ(α+β-δ-1)g(x,τ,u(x,τ))dτ+t2S-tM2Nλ×∫01‍1-τα-δ-1Γ(α-δ)f(x,τ,u(x,τ),∂q∂tqu(x,τ))dτ+t2S-tM2Nμ∫01‍1-τα+β-δ-1Γ(α+β-δ)g(x,τ,u(x,τ))dτ+tM-t2S2Nλ×∫0η‍η-τα-δ-1Γ(α-δ)f(x,τ,u(x,τ),∂q∂tqu(x,τ))dτ+tM-t2S2Nμ∫0η‍η-τα+β-δ-1Γ(α+β-δ)g(x,τ,u(x,τ))dτ,where Q=2Γ(2-δ)(1-η2-δ), R=(η1-δ-1)Γ(3-δ), S=(1-δ)Γ(3-δ), M=2Γ(3-δ), and N=[η1-δ(2-δ)+η2-δ(δ-1)-1] with η1-δ,η2-δ≠1.Theorem 4.
Letp1∈(0,1), p2∈(0,α-δ), p=min⁡{p1,p2}, L,m∈L1/p([0,1],R+), and ψ:R+→R+ be a nondecreasing bounded function. Suppose that f:J×X2→X and g:J×X→X are continuous functions such that |f(x,t,u,w)-f(x,t,v,z)|≤L(t)(|u-v|+|w-z|) and |g(x,t,u(x,t)|≤m(t)ψ(u) for all (x,t)∈J and u,v,w,z∈X. Put(14)γ:=λ1Γα1-pα-p1-p+1Γα-q1-pα-q-p1-p+Q+R2N1Γα-δ-11-pα-δ-p-11-p+2-qQ+2R2NΓ3-q1Γα-δ-1×1-pα-δ-p-11-p+S2N1+ηα-δ-pΓα-δ1-pα-δ-p1-p+2S-2-qM2NΓ3-q1+ηα-δ-pΓα-δ×1-pα-δ-p1-pL1/p.If γ<1, then the time-fractional integrodifferential equation ∗ has at least one solution.Proof.
First, put(15)w=λσ+μb×1Γ(α+β)1-pα+β-p1-p+1Γ(α+β-q)×1-pα+β-q-p1-p+Q+R2N1Γ(α+β-δ-1)×1-pα+β-δ-p-11-p+2-qQ+2R2NΓ3-q×1Γ(α+β-δ-1)1-pα+β-δ-p-11-p+S2N1+ηα+β-δ-pΓα+β-δ1-pα+β-δ-p1-p+2S-2-qM2NΓ3-q1+ηα+β-δ-pΓα+β-δ×1-pα+β-δ-p1-pm1/p,where(16)σ=K2S-2-qM2NΓ3-q1Γ(α+1)+1Γ(α-q+1)+Q+R2N1Γ(α-δ)+2-qQ+2R2NΓ3-q1Γ(α-δ)+S2N1+ηα-δΓα-δ+1+2S-2-qM2NΓ3-q1+ηα-δΓα-δ+1.Now, choose r≥w/1-γ and consider the closed, convex, and bounded subset Br={u∈X:u≤r} of X. Define the operator Φ:Br→X by(17)(Φu)(x,t)=-λ∫0t‍t-τα-1Γαfx,τ,ux,τ,∂q∂tqux,τdτ-μ∫0t‍t-τα+β-1Γα+βgx,τ,ux,τdτ+tQ+t2R2N×λ∫01‍1-τα-δ-2Γα-δ-1fx,τ,ux,τ,∂q∂tqux,τdτ+μ∫01‍1-τα+β-δ-2Γα+β-δ-1gx,τ,ux,τdτ+t2S-tM2N×λ∫01‍1-τα-δ-1Γα-δfx,τ,ux,τ,∂q∂tqux,τdτ+μ∫01‍1-τα+β-δ-1Γα+β-δgx,τ,ux,τdτ-λ∫0η‍η-τα-δ-1Γα-δfx,τ,ux,τ,∂q∂tqux,τdτ-μ∫0η‍η-τα+β-δ-1Γα+β-δgx,τ,ux,τdτ,for all (x,t)∈J. Now, decompose Φ by Φ=Φ1+Φ2, where(18)(Φ1u)(x,t)=-λ∫0t‍t-τα-1Γαfx,τ,ux,τ,∂q∂tqux,τdτ+tQ+t2R2Nλ×∫01‍1-τα-δ-2Γα-δ-1fx,τ,ux,τ,∂q∂tqux,τdτ+t2S-tM2Nλ×∫01‍1-τα-δ-1Γα-δfx,τ,ux,τ,∂q∂tqux,τdτi-∫0η‍η-τα-δ-1Γα-δfx,τ,ux,τ,∂q∂tqux,τdτ,(Φ2u)(x,t)=-μ∫0t‍t-τα+β-1Γα+βgx,τ,ux,τdτ+tQ+t2R2Nμ∫01‍1-τα+β-δ-2Γα+β-δ-1gx,τ,ux,τdτ+t2S-tM2Nμ∫01‍1-τα+β-δ-1Γα+β-δgx,τ,ux,τdτi-μ∫0η‍η-τα+β-δ-1Γα+β-δgx,τ,ux,τdτ,for all (x,t)∈J. Now, we show that Φ(Br)⊂Br. Let sup⁡x,t∈[0,1]|f(x,t,0,0)|≤K, (x,t)∈J, and u∈Br. Then, we have(19)Φ1ux,t≤λ∫0tt-τα-1Γα×fx,τ,ux,τ,∂q∂tqux,τ-fx,τ,0,0+fx,τ,0,0x,τ,ux,τ,∂q∂tqux,τdτ∫0tt-τα-1Γα+Q+R2Nλ×∫011-τα-δ-2Γα-δ-1fx,τ,ux,τ,∂q∂tqux,τ-fx,τ,0,0+fx,τ,0,0x,τ,ux,τ,∂q∂tqux,τdτ∫011-τα-δ-2Γα-δ-1fx,τ,ux,τ,∂q∂tqux,τ+S2Nλ∫011-τα-δ-1Γα-δ×fx,τ,ux,τ,∂q∂tqux,τ-fx,τ,0,0+fx,τ,0,0x,τ,ux,τ,∂q∂tqux,τdτ∫011-τα-δ-1Γα-δ+S2Nλ∫0ηη-τα-δ-1Γα-δ×fx,τ,ux,τ,∂q∂tqux,τ-fx,τ,0,0+fx,τ,0,0x,τ,ux,τ,∂q∂tqux,τdτ∫011-τα-δ-1Γα-δ≤λr∫0t‍t-τα-1ΓαLτdτ+λKΓα+1+Q+R2Nλr∫01‍1-τα-δ-2Γα-δ-1Lτdτ+Q+R2NλKΓα-δ+S2Nλr∫01‍1-τα-δ-1Γα-δLτdτ+S2NλKΓα-δ+1+S2Nλr∫0η‍η-τα-δ-1Γα-δLτdτ+S2NλKηα-δΓα-δ+1≤λrΓα∫0t‍t-τα-1/1-pdτ1-p∫01‍L1/pτdτp+Q+R2NλrΓα-δ-1∫01‍1-τα-δ-2/1-pdτ1-p×∫01‍L1/pτdτp+S2NλrΓα-δ∫01‍1-τα-δ-1/1-pdτ1-p×∫01‍L1/pτdτp+S2NλrΓα-δ∫0η‍η-τα-δ-1/1-pdτ1-p×∫01‍L1/pτdτp+λK1Γα+1+Q+R2N1Γα-δ+S2N1+ηα-δΓα-δ+1≤λr1Γ(α)1-pα-p1-p+Q+R2N1Γ(α-δ-1)1-pα-δ-p-11-p+S2N1Γ(α-δ)1-pα-δ-p1-p+S2Nηα-δ-pΓ(α-δ)1-pα-δ-p1-pL1/p+λK1Γα+1+Q+R2N1Γα-δ+S2N1+ηα-δΓα-δ+1and similarly(20)∂q∂tqΦ1ux,t≤λ∫0tt-τα-q-1Γα-qfx,τ,ux,τ,∂q∂tqux,τ-fx,τ,0,0+fx,τ,0,0t-τα-q-1Γα-qdτ+2-qQ+2R2NΓ3-qλ×∫011-τα-δ-2Γα-δ-1fx,τ,ux,τ,∂q∂tqux,τ-fx,τ,0,0+fx,τ,0,01-τα-δ-2Γα-δ-1dτ+2S-2-qM2NΓ3-qλ×∫01‍1-τα-δ-1Γα-δfx,τ,ux,τ,∂q∂tqux,τ-fx,τ,0,0+fx,τ,0,01-τα-δ-1Γα-δdτ+2S-2-qM2NΓ3-qλ×∫0η‍η-τα-δ-1Γα-δfx,τ,ux,τ,∂q∂tqux,τ-fx,τ,0,0+fx,τ,0,0x,τ,ux,τ,∂q∂tqux,τdτ≤λrΓα-q∫0t‍t-τα-q-1/1-pdτ1-p×∫01‍L1/pτdτp+2-qQ+2R2NΓ3-qλrΓα-δ-1×∫01‍1-τα-δ-2/1-pdτ1-p∫01‍L1/pτdτp+2S-2-qM2NΓ3-qλrΓα-δ×∫01‍1-τα-δ-1/1-pdτ1-p∫01‍L1/pτdτp+2S-2-qM2NΓ3-qλrΓα-δ×∫0η‍η-τα-δ-1/1-pdτ1-p∫01‍L1/pτdτp+λK1Γα-q+1+2-qQ+2R2NΓ3-q1Γα-δ+2S-2-qM2NΓ3-q1+ηα-δΓα-δ+1≤λr1Γα-q1-pα-q-p1-p+2-qQ+2R2NΓ3-q×1Γ(α-δ-1)1-pα-δ-p-11-p+2S-2-qM2NΓ3-q1Γ(α-δ)1-pα-δ-p1-p+2S-2-qM2NΓ3-qηα-δ-pΓ(α-δ)1-pα-δ-p1-p×L1/p+λK1Γα-q+1+2-qQ+2R2NΓ3-q1Γα-δ+2S-2-qM2NΓ3-q1+ηα-δΓα-δ+1.Thus, we get(21)Φ1u(x,t)≤λr1Γ(α)1-pα-p1-p+1Γ(α-q)1-pα-q-p1-pi+Q+R2N1Γα-δ-11-pα-δ-p-11-pi+2-qQ+2R2NΓ3-q1Γα-δ-1i×1-pα-δ-p-11-pi+S2N1+ηα-δ-pΓα-δ1-pα-δ-p1-pi+2S-2-qM2NΓ3-q1+ηα-δ-pΓα-δi×1-pα-δ-p1-pL1/p+λK1Γα+1+1Γα-q+1+Q+R2N1Γα-δ+2-qQ+2R2NΓ3-q1Γα-δ+S2N1+ηα-δΓα-δ+1+2S-2-qM2NΓ3-q1+ηα-δΓα-δ+1.If sup⁡u∈Xψ(u)≤b, then we have(22)Φ2ux,t≤μ∫0t‍t-τα+β-1Γα+βgx,τ,ux,τdτ+Q+R2Nμ∫01‍1-τα+β-δ-2Γα+β-δ-1gx,τ,ux,τdτ+S2Nμ∫01‍1-τα+β-δ-1Γα+β-δgx,τ,ux,τdτ+S2Nμ∫0η‍η-τα+β-δ-1Γα+β-δgx,τ,ux,τdτ≤μbΓα+β∫0t‍t-τα+β-1/1-pdτ1-p×∫01‍m1/pτdτp+Q+R2NμbΓα+β-δ-1×∫01‍1-τα+β-δ-2/1-pdτ1-p×∫01‍m1/pτdτp+S2NμbΓα+β-δ∫01‍1-τα+β-δ-1/1-p1-p×∫01‍m1/pτdτp+S2NμbΓα+β-δ∫0η‍η-τα+β-δ-1/1-p1-p×∫01‍m1/pτdτp≤μb1Γ(α+β)1-pα+β-p1-p+Q+R2N×1Γ(α+β-δ-1)1-pα+β-δ-p-11-p+S2N1Γ(α+β-δ)1-pα+β-δ-p1-p+S2Nηα+β-δ-pΓα+β-δ1-pα+β-δ-p1-p×m1/pand similarly(23)∂q∂tqΦ2ux,t≤μ∫0t‍t-τα+β-q-1Γα+β-qgx,τ,ux,τdτ+2-qQ+2R2NΓ3-qμ×∫01‍1-τα+β-δ-2Γα+β-δ-1gx,τ,ux,τdτ+2S-2-qM2NΓ3-qμ×∫01‍1-τα+β-δ-1Γα+β-δgx,τ,ux,τdτ+2S-2-qM2NΓ3-qμ×∫0η‍η-τα+β-δ-1Γα+β-δgx,τ,ux,τdτ≤μbΓα+β-q∫0t‍t-τα+β-q-1/1-pdτ1-p×∫01‍m1/pτdτp+2-qQ+2R2NΓ3-qμbΓα+β-δ-1×∫01‍1-τα+β-δ-2/1-pdτ1-p×∫01‍m1/pτdτp+2S-2-qM2NΓ3-q×μbΓ(α+β-δ)∫01‍1-τα+β-δ-1/1-pdτ1-p×∫01‍m1/pτdτp+2S-2-qM2NΓ3-q×μbΓ(α+β-δ)∫0η‍η-τα+β-δ-1/1-pdτ1-p×∫01‍m1/pτdτp≤μb1Γ(α+β-q)1-pα+β-q-p1-p+2-qQ+2R2NΓ3-q×1Γ(α+β-δ-1)1-pα+β-δ-p-11-p+2S-2-qM2NΓ3-q1Γ(α+β-δ)×1-pα+β-δ-p1-p+2S-2-qM2NΓ3-q×ηα+β-δ-pΓα+β-δ1-pα+β-δ-p1-pm1/p.Hence,(24)Φ2u(x,t)≤μb1Γ(α+β)1-pα+β-p1-p+1Γ(α+β-q)×1-pα+β-q-p1-p+Q+R2N×1Γ(α+β-δ-1)1-pα+β-δ-p-11-p+2-qQ+2R2NΓ3-q1Γ(α+β-δ-1)×1-pα+β-δ-p-11-p+S2N×(1+ηα+β-δ-p)Γ(α+β-δ)1-pα+β-δ-p1-p+2S-2-qM2NΓ3-q(1+ηα+β-δ-p)Γ(α+β-δ)×1-pα+β-δ-p1-pm1/pand so(25)Φu(x,t)≤Φ1u(x,t)+Φ2u(x,t)≤λr1+ηα-δ-pΓα-δ1Γ(α)1-pα-p1-p+1Γ(α-q)1-pα-q-p1-p+Q+R2N1Γ(α-δ-1)1-pα-δ-p-11-p+2-qQ+2R2NΓ3-q1Γ(α-δ-1)×1-pα-δ-p-11-p+S2N(1+ηα-δ-p)Γ(α-δ)×1-pα-δ-p1-p+2S-2-qM2NΓ3-q×1+ηα-δ-pΓα-δ1-pα-δ-p1-pL1/p+λK1Γα+1+1Γα-q+1+Q+R2N×1Γα-δ+2-qQ+2R2NΓ3-q1Γ(α-δ)+S2N1+ηα-δΓα-δ+1+2S-2-qM2NΓ3-q1+ηα-δΓα-δ+1+μb1Γ(α+β)1-pα+β-p1-p+1Γ(α+β-q)×1-pα+β-q-p1-p+Q+R2N×1Γ(α+β-δ-1)1-pα+β-δ-p-11-p+2-qQ+2R2NΓ3-q1Γ(α+β-δ-1)×1-pα+β-δ-p-11-p+S2N×(1+ηα+β-δ-p)Γ(α+β-δ)1-pα+β-δ-p1-p+2S-2-qM2NΓ3-q(1+ηα+β-δ-p)Γ(α+β-δ)×1-pα+β-δ-p1-pm1/p≤γr+w≤r.This implies that ΦBr⊂Br. Now, we show that Φ1 is continuous. Let x,t∈[0,1] and {un} be a sequence with un→u. Then, we have(26)Φ1un(x,t)-Φ1u(x,t)≤λL1/pΓ(α)1-pα-p1-p+λL1/pΓ(α-q)1-pα-q-p1-p+Q+R2NλL1/pΓ(α-δ-1)1-pα-δ-p-11-p+2-qQ+2R2NΓ3-qλL1/pΓ(α-δ-1)1-pα-δ-p-11-p+S2NλL1/p(1+ηα-δ-p)Γ(α-δ)1-pα-δ-p1-p+2S-2-qM2NΓ3-qλL1/p(1+ηα-δ-p)Γ(α-δ)×1-pα-δ-p1-pun-u.Thus, Φ1 is continuous. On the other hand, Φ1 is a γ-contraction because(27)Φ1u(x,t)-Φ1v(x,t)≤λL1/pΓ(α)1-pα-p1-p+λL1/pΓ(α-q)1-pα-q-p1-p+Q+R2NλL1/pΓ(α-δ-1)1-pα-δ-p-11-p+2-qQ+2R2NΓ3-qλL1/pΓ(α-δ-1)1-pα-δ-p-11-p+S2NλL1/p(1+ηα-δ-p)Γ(α-δ)1-pα-δ-p1-p+2S-2-qM2NΓ3-qλL1/p(1+ηα-δ-p)Γ(α-δ)×1-pα-δ-p1-pu-v,for all u,v∈Br. Now, we show that Φ2 is a compact map. We showed that Φ2 is uniformly bounded. Now, we show that Φ2 maps the bounded sets into equicontinuous sets. Let (x,t1),(x,t2)∈J such that t1<t2. Then, we have(28)|Φ2u(x,t2)-Φ2u(x,t1)|≤μΓα+β∫0t1‍t2-τα+β-1-t1-τα+β-1×gx,τ,ux,τdτ+|μ|Γ(α+β)∫t1t2‍t2-τα+β-1gx,τ,ux,τdτ+t2-t1Q+t22-t12R2Nμ×∫01‍1-τα+β-δ-2Γ(α+β-δ-1)gx,τ,ux,τdτ+t22-t12S-t2-t1M2Nμ×∫01‍1-τα+β-δ-1Γα+β-δgx,τ,ux,τdτ-∫0η‍η-τα+β-δ-1Γα+β-δgx,τ,ux,τdτ≤μbΓα+β∫0t1‍t2-τα+β-1-t1-τα+β-1m(τ)dτ+μbΓα+β∫t1t2‍t2-τα+β-1m(τ)dτ+t2-t1Q+t22-t12R2Nμb×∫01‍1-τα+β-δ-2Γ(α+β-δ-1)m(τ)dτ+t22-t12S-t2-t1M2Nμb×∫01‍1-τα+β-δ-1Γ(α+β-δ)m(τ)dτt22-t12S-t2-t1M2Nt22-t12S-t2-t1M2N-∫0η‍η-τα+β-δ-1Γα+β-δmτdτ,∂q∂tqΦ2ux,t2-∂q∂tqΦ2ux,t1≤μbΓα+β-q×∫0t1‍t2-τα+β-q-1-t1-τα+β-q-1m(τ)dτ+μbΓα+β-q∫t1t2‍t2-τα+β-q-1m(τ)dτ+t21-q-t11-qΓ2-qQ+t22-q-t12-qΓ3-qR×2N-1t21-q-t11-qΓ2-qμb×∫01‍1-τα+β-δ-2Γα+β-δ-1m(τ)dτ+2t22-q-t12-qΓ3-qS-t21-q-t11-qΓ2-qM×2N-1μb×∫01‍1-τα+β-δ-1Γα+β-δmτdτt22-t12S-t2-t1M2Nt22-t12S-t2-t1M2N-∫0η‍η-τα+β-δ-1Γα+β-δmτdτ,for all u∈Br. Hence, Φ2u(x,t2)-Φ2u(x,t1)→0 as t2→t1. By using the Arzela-Ascoli theorem, we get that the operator Φ2 is completely continuous and so Φ2 is compact on J. Now by using Lemma 1, the operator Φ=Φ1+Φ2 is a condensing map on Br and so Φ has a fixed point by using Theorem 2. Now by using Lemma 3, it is easy to see that the fixed point of Φ is a solution for the time-fractional integrodifferential equation ∗.Example 5.
Letα=5/2, β=3/4, q=3/2, δ=5/4, η=2/3, λ=1/1000, μ=1, and p=3/4. Now, consider the time-fractional integrodifferential equation(29)-∂5/2∂t5/2ux,t=11000tu1+u+t∂3/2/∂t3/2u1+∂3/2/∂t3/2u+I∗t3/4t2u31+u3via the boundary conditions u(x,0)=0, ∂9/4/∂t9/4u(x,0)=∂9/4/∂t9/4u(x,1), and ∂5/4/∂t5/4u(x,1)-∂5/4/∂t5/4u(x,2/3)=0. Define the maps f:J×X2→X by f(x,t,u(x,t),∂3/2/∂t3/2u(x,t))=tu/1+u+t|∂3/2/∂t3/2u|/1+∂3/2/∂t3/2u and g:J×X→X by g(x,t,u(x,t))=t2|u|3/1+|u|3. Define L(t)=t and m(t)=t2 for all t. Then, we have L1/p=t4/3=0.077. Define ψ:R+→R+ by ψ(z)=1 for all z. It is easy to check that(30)fx,t,ux,t,∂3/2∂t3/2ux,t-fx,t,vx,t,∂3/2∂t3/2vx,t≤Ltu-v+∂3/2∂t3/2u-∂3/2∂t3/2vand gx,t,ux,t≤m(t)ψ(u) for all u,v∈X and t. One can calculate that m1/p=t24/3=(3/11)3/4 and γ≃0.002579. Now by using Theorem 4, the time-fractional integrodifferential equation has a solution.


## 3. Conclusion

A time-fractional integrodifferential equation via three-point boundary value conditions was investigated and the existence of the solution was proved for three-point boundary value conditions. One example was studied in detail. For particular forms of the functionsf and g and of the investigated equation and for various values of 2≤α<3, 0<β<1, 1≤δ<2, 0≤η≤1, λ, and μ we can reobtain the forms of several nonlinear time-fractional differential equations describing the complex phenomena which arise in science and engineering.



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*Source: 785738-2015-03-26.xml*