# Existence Results for a Class of Semilinear Fractional Partial Differential Equations with Delay in Banach Spaces

Authors: Bo Zhu; Baoyan Han; Xiangyun Lin
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper, we consider a class of nonlinear time fractional partial differential equations with delay. We obtain the existence and uniqueness of the mild solutions for the problem by the theory of solution operator and the general Banach contraction mapping principle. We need not extra conditions to ensure the contraction constant0<k<1. Therefore, under some general conditions, we obtain our main results.


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## Body

## 1. Introduction

Fractional derivatives can describe the property of the memory, and they have more advantages than integer-order derivatives. Therefore, fractional differential equations have been successfully applied in many fields, such as engineering and physics. About the fractional differential equations, we refer to these papers [1–9] and the references therein. In [7, 10], the authors studied the existence results of the fractional integrodifferential equations of order 1<β≤2. In [11, 12], the authors considered a class of fractional differential equations, where the fractional derivative operator is D0Atβ with fractional order β and A is a closed densely defined operator in a Banach space. Goufo [13] studied the existence results for a class of fractional fragmentation model by theory of strongly continuous solution operators. In [14, 15], the authors investigated a class of space-time fractional diffusion equations, while in [16] the authors studied a class of linear fractional differential equations by the variational iteration method and the Adomian decomposition method. In [17–19], the authors studied the following fractional partial differential equations:(1)Dctβux,t-∂2∂x2ux,t=gt,ux,τ1t,ux,τ2t,…,ux,τlt,t∈0,T0,ux,t=0,x,t∈∂Ω×0,T0,ux,0=φx,x∈Ω.Ouyang [17] studied the existence of the local mild solutions for such problem by Leray-Schauder’s fixed theorem. Zhu et al. [18, 19] also studied the existence of the mild solutions by strict set contraction and Banach contraction mapping theorem of the problem. Li et al. [20] investigated the following fractional differential equations:(2)Dctβut=Aut+Jt1-βft,ut,t∈0,T,ut=φt,t∈-r,0,where β∈(0,1). In [20], the Lipschitz coefficient of the nonlinear function f is a constant.Inspired by the above said work, we investigate the following nonlinear fractional partial differential equations with delay:(3)Dctβux,t-∂2∂x2ux,t=Jt1-βft,ux,τ1t,…,ux,τlt,t∈0,T0,x∈Ω,ux,θ=φx,θ,x∈Ω,θ∈-r,0,where 0<β<1, Dctβ is Caputo’s fractional derivative of order β, Jt1-β is the Riemann-Liouville fractional integral of order 1-β, l is a nature number, Ω⊂Rl is a bounded domain with regular boundary ∂Ω, τi:[0,T0]→[0,T0](i=1,2,…,l) are continuous functions, and these functions satisfy 0≤τi(t)≤t(i=1,2,…,l), φ∈C([-r,0];E),E=C(Ω¯;R),Ω¯=Ω∪∂Ω.In this paper, we consider the existence results of the mild solutions of problem (3) by general Banach contraction mapping theorem. We need not extra conditions to ensure the contraction constant 0<k<1. Under some general conditions, we obtain our main results. Therefore, our results presented in this paper improve many classical results.


## 2. Preliminaries

Let(E,·) be a Banach space and let C([0,T0];E)=u:[0,T0]→E is continuous be a Banach space with norm uC=max⁡u(t):t∈[0,T0].Definition 1 (see [21, 22]).
The Riemann-Liouville fractional integral of a functionh:(0,∞)→R of order α>0 is defined as(4)Jtαht=1Γα∫0tt-sα-1hsds,where h(t)∈L1((0,T0);E).
For convenience, we let(5)fαt=tα-1Γα,t>0,0,t≤0, and then(6)Jtαht=fα∗ut=∫0tfαt-shsds.Definition 2 (see [21, 22]).
The Riemann-Liouville fractional derivative of a functionh:(0,∞)→R of order α>0 is defined as(7)Dtαht=DtmJtm-αht, where Dtm=dm/dtm,h(t)∈L1((0,T0);E),Jtm-αh(t)∈Wm,1((0,T0);E).Definition 3 (see [21, 22]).
The Caputo fractional derivative of functionh of order α>0 is defined as(8)Dctαht=Dtαht-∑k=0m-1tkk!hk0, where h(t)∈L1((0,T0);E)∩Cm-1((0,T0);E).
For the Riemann-Liouville fractional integral operator and the Caputo fractional derivative operator, we have(9)DctαJtαht=ht,JtαDctαht=ht-∑k=0m-1tkk!hk0.Definition 4 (see [23]).
LetA be a closed linear operator with dense domain D(A) in a Banach space E; β>0. A family Sβtt≥0⊂B(E) of bounded linear operators in E is called a solution operator for the integral equation(10)ut=x+1Γβ∫0tAust-s1-βds,t≥0,x∈E, if the following conditions are satisfied:(i)
Sβ(t) is strongly continuous on R+ and Sβ(0)=I.(ii)
Sβ(t)D(A)⊂D(A) and ASβ(t)x=Sβ(t)Ax for all x∈D(A) and t≥0.(iii)
Sβ(t)x is a solution of(11)ut=x+1Γβ∫0tAust-s1-βdsfor all x∈D(A),t≥0.
We callA the infinitesimal generator of Sβ(t) or say that A generates Sβ(t).
Letu(t)=u(·,t); then the fractional partial differential equation (3) can be rewritten in the following abstract form:(12)Dctβut=Aut+Jt1-βft,uτ1t,uτ2t,…,uτlt,t∈0,T0,ut=φt,t∈-r,0,where(13)DA=u∈CΩ¯,R;u′′∈CΩ¯,R,Au=u′′,t∈-r,0, and f:[0,T0]×C([-r,0];E)×C([-r,0];E)⋯×C([-r,0];E)→E is defined by f(t,φ,φ⋯φ)(x)=f(t,φ(·,x),φ(·,x)⋯φ(·,x)) for t∈[0,T0],φ∈C([-r,0];E) and x∈Ω.
It is easy to see thatA generates a C0 semigroup Ttt≥0 on E. Theorem 3.1 in [23] means that A is the infinitesimal generator of a solution operator Sβ(t)t≥0.Definition 5 (see [24]).
Ifu∈C([-r,T0];E) is a mild solution of problem (12), then u satisfies the following integral equations:(14)ut=Sβtφ0+∫0tSβt-sfs,uτ1s,uτ2s,…,uτlsds,t∈0,T0,φt,t∈-r,0.


## 3. Main Results

We define the operatorF:C([-r,T0];E)→C([-r,T0];E) by(15)Fut=Sβtφ0+∫0tSβt-sfs,uτ1s,uτ2s,…,uτlsds,t∈0,T0,φt,t∈-r,0.Theorem 6.
Assume that the conditions(H1) and (H2) hold.
( H 1 ) There exists a real number M>0 such that Sβ(t)≤M,t∈0,T0.
( H 2 ) The function f:[0,T0]×C([-r,0];E)×C([-r,0];E)⋯×C([-r,0];E)→E is continuous in t on [0,T0], and there exist nonnegative Lebesgue integrable functions Li∈L1(J,R+)(i=1,2,…,l) such that(16)ft,u1,u2,…,ul-ft,v1,v2,…,vl≤∑ilLitui-vi,where t∈[0,T0],ui,vi∈C([-r,0];E).
Then problem (12) has a unique mild solution u∈C([-r,T0];E), which means that (3) has a unique mild solution.Proof.
For any0<ε<1, by the property of the Lebesgue integrable function, there exists a continuous function ϕ(s) such that ∫0T0Ls-ϕsds<ε, where L(s)=M∑ilLi(s). From conditions (H1)-(H2) and (15), for any u,v∈C([-r,T0];E), we obtain(17)Fut-Fvt≤∫0tSβt-sfs,uτ1s,…,uτls-fs,vτ1s,…,vτlsds≤∫0tM∑ilLisuτis-vτisds≤∫0tLsdsu-vC-r,T0;E≤∫0tLs-ϕsds+∫0tϕsdsu-vC-r,T0;E≤ε+Ntu-vC-r,T0;E=C10ε1+C11Nt11!u-vC-r,T0;Ewhere N=maxt∈Jϕt. Next, for any nature number n, we will prove the following inequality:(18)Fnut-Fnvt≤Cn0εn+Cn1εn-1Nt11!+⋯+Cnnεn-nNtnn!u-vC-r,T0;E.Obviously, for n=1, (18) holds. Assume that n=k and (18) holds; that is,(19)Fkut-Fkvt≤Ck0εk+Ck1εk-1Nt11!+⋯+Ckkεk-kNtkk!u-vC-r,T0;E.By (15), (H1)-(H2), and formula Ck+1m=Ckm+Ckm-1, we have(20)Fk+1ut-Fk+1vt≤∫0tMfs,Fkuτ1s,…,Fkuτls-fs,Fkvτ1s,…,Fkvτlsds≤∫0tLsFkuτis-Fkvτisds≤∫0tLs-ϕsCk0εk+Ck1εk-1Ns11!+⋯+Ckkεk-kNskk!dsu-vC-r,T0;E+∫0tϕsCk0εk+Ck1εk-1Ns11!+⋯+Ckkεk-kNskk!dsu-vC-r,T0;E≤εCk0εk+Ck1εk-1Nt11!+⋯+Ckkεk-kNtkk!u-vC-r,T0;E+N∫0tCk0εk+Ck1εk-1Ns11!+⋯+Ckkεk-kNskk!dsu-vC-r,T0;E≤εCk0εk+Ck1εk-1Nt11!+⋯+Ckkεk-kNtkk!+Ck0εkNt11!+Ck1εk-1Nt22!+⋯+Ckkεk-kNtk+1k+1!u-vC-r,T0;E≤Ck+10εk+1+Ck+11εkNt11!+⋯+Ck+1k+1εk+1-k+1Ntk+1k+1!u-vC-r,T0;E. By mathematical induction, we obtain that (18) holds for n=k+1. Therefore, for any n, we have(21)Fnu-FnvC-r,T0;E≤Cn0εn+Cn1εn-1d11!+⋯+Cnnεn-ndnn!u-vC-r,T0;E,where d=NT0. It is easy to see that(22)limk→∞⁡εk-1kkk-1k-11/k=ε<1. Therefore, we choose sufficiently large nature number K>2 such that(23)εK-1KKK-1K-11/K≡α<1.For any nature number n>2K, such that n=Kh+p(0≤p<K). Obviously, h=[n/K]<[n/2]. For any sufficiently large positive integer n>2K; by the Stirling formula(24)n!=nen2πn1+O1n and by (23), we have(25)S1≡Cn0εn+Cn1εn-1d11!+⋯+Cnhεn-hdhh!≤εn-hCnh1+d+d22!+···+dhh!=εn-hCnhO1=O1εn-hnn2πn1+O1/hhh2πh2πn-hn-hn-h=OKhhbKK-1n-hh=ObK-1KK/K-1K-1hh=OαKhh=Oαnn.On the other hand, without loss of generality, we assume that d=NT0>1. By Stirling formula and Cn[n/2]=O(2n/n), we get(26)S2≡Cnh+1εn-h-1dh+1h+1!+⋯+Cnnεn-ndnh!≤1h+1!Cnn/2εn-h-1dh+1+bn-ndn=bn-hCnhO1=O2n/neh+1εn-h-1dh+1+εn-ndn2πh+1h+1h+11+O1/h+1≤O2n/neh+11+d+d2+⋯+dh+1+⋯+dn2πh+1h+1h+1≤O12neh+1dn+1nh+1h+1h+1≤O12neh+1dn+1hh+2=o1hλ+1=o1nλ+1n→+∞,where λ>0. By (21), (25), and (26), we get(27)Fnu-FnvC-r,T0;E≤S1+S2u-vC-r,T0;E=Oαnn+o1hλ+1u-vC-r,T0;E=o1nλ+1u-vC-r,T0;E,n→+∞. Therefore, for fixed constant λ>0, there exists a positive integer n0 such that, for any u,v∈C([-r,T0];E), we have(28)Fnu-FnvC-r,T0;E≤1nr+1u-vC-r,T0;E,∀n>n0. By general Banach contraction mapping principle, for operator F there exists a unique fixed point u∈C([-r,T0];E), which means that problem (3) has a unique mild solution.Remark 7.
In Theorem6, we obtain the existence and uniqueness of the global mild solution of problem (3) under the uniform Lipschitz condition of the function f. In next Theorem 8, we assume that the function f satisfies the local Lipschitz condition.Theorem 8.
Letf:[0,T0]×C([-r,0];E)×C([-r,0];E)×⋯×C([-r,0];E)→E be continuous in t for t∈[0,T0], and f satisfies the following local Lipschitz contition: for any ϱ>0, there exist nonnegative Lebesgue integrable functions Lϱi∈L1(J,R+)(i=1,2,…,l) such that(29)ft,u1,u2,…,ul-ft,v1,v2,…,vl≤∑ilLϱitui-vi, where t∈[0,T0],ui,vi∈C([-r,0];E) with ui≤ϱ,vi≤ϱ.
Then there exists aτ>0 such that problem (3) has a unique mild solution on [-r,τ).Proof.
For all fixedξ>0, there exists δ∈[0,T0] such that Sβ(t)φ(0)-φ(0)≤ξ/2 as t∈[0,δ]. Let δ1>0 such that M∗ϱ∫0δ1∑ilLϱi(s)ds+Rδ1≤ξ/2, where R=sups∈[0,δ1]⁡f(s,0,0,…,0),Sβ(t)≤M∗,t∈[0,δ1]. Take τ=minδ,δ1. Let(30)Dξ=u∈C-r,τ;E:us=φsifs∈-r,0andsups∈0,τ⁡us-φ0≤ξ.Obviously, Dξ is a closed subset of C([-r,τ];E); thus Dξ is a Banach space. Now we consider the mapping(31)Q:Dξ→C-r,τ;E by(32)Qut=Sβtφ0+∫0tSβt-sfs,uτ1s,uτ2s,…,uτlsds,t∈0,τ,φt,t∈-r,0.Next, we will prove that Q maps Dξ into Dξ.
Letu∈Dξ; we have(33)Qut-φ0≤Sβtφ0-φ0+∫0tSβt-sfs,uτ1s,…,uτlsds≤Sβtφ0-φ0+∫0τSβt-sfs,uτ1s,…,uτls-fs,0,…,0ds+∫0tSβt-sfs,0,…,0ds≤Sβtφ0-φ0+M∗ϱ∫0τ∑ilLϱisds+M∗Rτ≤ξ.Therefore, Q maps Dξ into Dξ.
Letu,v∈Dξ and t∈[0,τ]; we have(34)Qut-Qvt≤∫0tSβt-sfs,uτ1s,…,uτls-fs,vτ1s,…,vτlsds≤M∗∫0t∑ilLϱisuτis-vτisds≤∫0tLsdsu-vC-r,τ;E≤∫0tLs-ϕsds+∫0tϕsdsu-vC-r,τ;E≤ε+Ntu-vC-r,T0;E=C10ε1+C11Nt11!u-vC-r,τ;E,where L(s)=M∗∑ilLϱi(s). The following proof of the remainder is similar to the proof of Theorem 6. Therefore, for fixed constant λ>0, there exists a positive integer n0 such that, for any u,v∈Dξ, we have(35)Qnu-QnvC-r,τ;E≤1nλ+1u-vC-r,τ;E,∀n>n0. By general Banach contraction mapping principle, for operator Q there exists a unique fixed point u∈C([-r,τ];E), which means that u∈C([-r,τ];E) is the unique mild solution of problem (12). That is, problem (3) has a unique mild solution.


## 4. An Application

Using the main results of this paper, we can solve the following time fractional partial differential equation with delay:(36)Dctβux,t=∂2∂x2ux,t+Jt1-βt1+t2ux,sint,t∈0,T0,x∈Ω,ux,θ=φx,θ,θ∈-r,0,x∈Ω,where 0<β≤1. Similar to Section 2, we can rewrite the time fractional partial differential equation (36) as the following abstract fractional evolution equation:(37)Dctβut=Aut+Jt1-βft,uτt,t∈0,T0,ut=φt,t∈-r,0.Therefore, all the conditions of Theorems 6 and 8 are satisfied; for problem (36), there exists a unique mild solution.


## 5. Conclusion

This paper considers the existence and uniqueness of the mild solutions for a class of nonlinear fractional partial differential equations with delay by general Banach contraction mapping principle. We know that the Banach contraction mapping principle needs the special conditions to ensure the contraction constant0<k<1. In this paper, we successfully overcome this condition. We need not extra conditions to ensure the contraction constant 0<k<1. Therefore, under some general conditions, we obtain the main results of this paper. Our results generalize and improve many classical results [18–20].



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