2.3 Weak Solutions for fractional Langevin equations involving two fractional
2.3.1 Introduction
The Langevin equation (first formulated by Langevin in 1908 to give an elaborate descrip-tion of Brownian modescrip-tion) is found to be an effective tool to describe the evoludescrip-tion of physical phenomena in fluctuating environments [48]. Although the existing literature on solutions of fractional Langevin equations is quite wide (see, for example, [6, 14,15, 31,101, 132,133, 134, 143]). But, to the best of the author’s knowledge, there is no literature to research the existence of weak solutions for fractional Langevin equations involving two fractional orders in Banach Spaces, so the research of this paper is new.
In this section, we study the existence of weak solutions for an initial value problem, posed in a given Banach space. More specifically, we pose the following fractional Langevin equations involving two fractional orders with initial value problems
cDβ0+(cDα0++γ)x(t) = f(t,x(t)), t∈I:= [0,1], x(k)(0) =µk, 0≤k<l,
x(α+k)(0) =νk, 0≤k<n.
(2.14)
WherecDα0+,cDβ
0+ are the Caputo fractional derivativesm−1<α ≤m, n−1<β <n,l = max{m,n},m,n∈N,γ ∈R, f :[0,1]×X −→X is a given function satisfying some assump-tions that will be specified later,X is a Banach space with normk · k,µk,νk∈X.
This problem was studied recently in [134] in the scalar case using Banach contraction principal and the nonlinear alternative of Leray–Schauder. Here we extend the results of [134]
to cover the abstract case.
2.3.2 Existence of solutions
2First of all, we define what we mean by a weak solution for the initial value problem (2.14).
Definition 2.5. By a weak solution of (2.14), we mean a functionx:I−→X such that the weak fractional derivativecDα0+,cDβ0+ exists and are weakly continuous and satisfies problem (2.14).
For the existence of weak solutions for the initial value problem (2.14), we need the following auxiliary lemma.
Lemma 2.6 ([134]). x(t) is a solution of the initial problem (2.14) if and only if x(t) is a solution of the integral equation :
x(t) =I0α++β f(t,x(t))−γI0α+x(t) +Q(t). (2.15)
2. C. Derbazi, H. Hammouche, M. Benchohra, J. Henderson, Weak Solutions for fractional Langevin equations involving two fractional orders with initial value problems in Banach Spaces.(submitted)
2.3. WEAK SOLUTIONS FOR FRACTIONAL LANGEVIN EQUATIONS INVOLVING TWO FRACTIONAL ORDERS WITH INITIAL VALUE PROBLEMS IN BANACH SPACES.
Where
For simplicity of presentation, we give some notations and list some conditions as fol-lows :
(H1) For eacht∈I,the function f(t,·)is weakly sequentially continuous ; (H2) For eachx∈C(I,X), the function f(·,x(·))is Pettis integrable onI; (H3) There existpf ∈L∞(I,R+)such that
kf(t,x)k ≤pf(t)kxk,∀(t,x)∈I×X.
(H4) For each bounded setD⊂X, and eacht∈I, the following inequality holds β(f(t,D))≤pf(t)β(D).
Now we are in able to establish the main results.
Theorem 2.7. Assume that assumptions (H1)-(H4) hold. If
Mpf <1. (2.16)
Then the initial value problem(2.14)has at least one solution.
Proof.
Transform the integral equation (2.15) into a fixed point equation. Consider the operator N :C(I,X)−→C(I,X)defined by : integrable (Proposition1.5) and thus, the operatorN is well defined.
Let
R≥ Q∗ 1−Mpf
, (2.18)
2.3. WEAK SOLUTIONS FOR FRACTIONAL LANGEVIN EQUATIONS INVOLVING TWO FRACTIONAL ORDERS WITH INITIAL VALUE PROBLEMS IN BANACH SPACES.
and consider the set
D={x∈C(I,X):kxk∞≤R,kx(t2)−x(t1)k ≤L|t2−t1|},
Notice thatDis a closed, convex, bounded, and equicontinuous subset ofC(I,X).We shall show that the operatorN satisfies all the assumptions of Theorem 1.49. The proof will be given in several steps.
Step 1 :We will show that the operatorN mapsDintoD.
Takex∈D,t∈Iand assume thatN x(t)6=0.Then there existsϕ∈X∗such thatkN x(t)k= ϕ(N x(t)).Thus
kN x(t)k=ϕ
I0α+β+ f(t,x(t))−γI0α+x(t) +Q(t)
≤I0α+β+ ϕ(f(t,x(t))) +|γ|I0α+ϕ(x(t))) +ϕ(Q(t)).
From (H3) we get
kN x(t)k ≤ kxknI0α++βpf(t) +|γ|I0α+(1)(t)o+kQ(t)k
≤R
® kpfkL∞
Γ(α+β+1)+ |γ| Γ(α+1)
´
+Q∗,
≤RMpf +Q∗≤R.
Next, lett1,t2∈I be such thatt1<t2and letx∈Dbe such that N x(t2)−N x(t1)6=0.
Then there existsϕ ∈X∗such that
kN (x)(t2)−N (x)(t1)k=ϕ(N (x)(t2)−N (x)(t1)).
2.3. WEAK SOLUTIONS FOR FRACTIONAL LANGEVIN EQUATIONS INVOLVING TWO FRACTIONAL ORDERS WITH INITIAL VALUE PROBLEMS IN BANACH SPACES.
Then, we have
Step 2 : We will show that the operatorN has a weakly sequentially continuous.
Let(xn)be a sequence inDand letxn(t)→x(t)in(E,w)for eacht∈I. Fixt ∈J. Since f satisfies assumption (H1), we have f(t,xn(t)) converges weakly uniformly to f(t,x(t)).
Hence the Lebesgue Dominated Convergence theorem for Pettis integral implies N xn(t) converges weakly uniformly toN x(t) in (X,w). We do it for eacht ∈I so N xn →N x.
ThenN :D−→Dis weakly sequentially continuous.
Step 3 : The implication(1.7)holds.
Now letV be a subset ofDsuch thatV =conv(N (V)∪ {0}).
Clearly,
V(t)⊂conv(N (V(t))∪ {0}),t∈I.
Further, asV is bounded and equicontinuous, the functiont→v(t) =β(V(t))is continuous onI. By assumption (H5), and the properties of the measureβ,for anyt∈I, we have
2.3. WEAK SOLUTIONS FOR FRACTIONAL LANGEVIN EQUATIONS INVOLVING TWO FRACTIONAL ORDERS WITH INITIAL VALUE PROBLEMS IN BANACH SPACES.
v(t)≤β(conv(N (V)(t)∪ {0}))≤β(N (V)(t))
≤β(I0α++βf(t,V(t)) +γI0α+V(t) +Q(t))
≤I0α++ββ(f(t,V(t))) +|γ|I0α+β(V(t))
≤I0α++β(pf(t)v(t)) +|γ|I0α+(v(t))
≤ kvk∞
® kpfkL∞
Γ(α+β+1)+ |γ|
Γ(α+1)
´
,
≤ kvk∞Mpf. which gives
kvk∞≤ kvk∞Mpf. This means that
kvk∞(1−Mpf)≤0.
By (2.16) it follows thatkvk∞=0, that is v(t) =0 for eacht∈I, and thenV(t)is relatively weakly compact inX. Applying Theorem1.49we conclude thatN has a fixed point which is a solution of the problem (2.14).
2.3.3 An Example
In this section we give an example to illustrate the usefulness of our main result. Let X =c0={x= (x1,x2, . . . ,xn, . . .):xn→0(n→∞)},
be the Banach space of real sequences converging to zero, endowed its usual norm kxk∞=sup
n≥1
|xn|.
Example 2.8. Consider the following fractional Langevin problem posed inc0:
cD
1 4
0+(cD
1 2
0++101)x(t) = f(t,x(t)), t∈I:= [0,1]
x(0) =µ0= (0,0, . . . ,0, . . .) x(12)(0) =ν0=Ä12,14, . . . ,21n, . . .ä.
(2.19)
Note that, this problem is a particular case of IVP (2.14), where α= 1
2,β = 1
4,γ = 1 10, and f :J×c0−→c0given by
f(t,x) = 1 (t2+2)2
ß n
n+1ln(|xn|+1)
™
n≥1, fort ∈I,x={xn}n≥1∈c0.
2.3. WEAK SOLUTIONS FOR FRACTIONAL LANGEVIN EQUATIONS INVOLVING TWO FRACTIONAL ORDERS WITH INITIAL VALUE PROBLEMS IN BANACH SPACES.
It is clear that condition (H1) and (H2) holds, and as kf(t,x)k∞= 1
(t2+2)2
n
n+1ln(|xn|+1)
≤ 1
(t2+2)2kxk∞,
for eacht∈Jandx∈c0,condition (H3) follows with pf(t) =(t2+2)1 2,t∈I.
On the other hand, for any bounded setD⊂c0, we have β(f(t,D))≤ 1
(t2+2)2β(D), for eacht∈I.
Hence (H4) is satisfied.
We shall check that condition (2.11) is satisfied. Indeed Mpf =0.3849<1.
and
Q∗ 1−Mpf
=0.9172.
Thenr can be chosen as r=1≥0.9172. Consequently, Theorem 2.7 implies that problem (2.19) has at least one solutionx∈C(I,c0).
Chapitre 3
Fractional differential equations in Banach algebras
3.1 Introduction
The aim of this chapter is to prove the existence of solutions for a class of hybrid frac-tional differential equations in the Banach algebra of all continuous functions on a bounded interval. We also present examples to show the validity of conditions and efficiency of our results. This exposition is divided into three parts. The first one deals with the existence of solutions for fractional hybrid differential equations with three-point boundary hybrid condi-tions. The second one deals with results connecting the existence of solution fractional hybrid differential equations with p-Laplacian operator. Our approach mainly depends on a hybrid fixed point theorem for three operators in a Banach algebra due to Dhage [59], while the last part of this chapter is devoted to the existence of solutions for fractional hybrid differential equations with deviating arguments under hybrid conditions with the help of a technique as-sociated with the measure of noncompactness and generalized Darbo fixed point theorem in a Banach algebra. Moreover, an example is given at the end of each section to illustrate the validity of our main results. The chapter is inspired in the papers [56,57,58].