Second Result

LetU =C([0,1],E)denotes the Banach space of all continuous functions fromu: J→E with

kuk_∞=sup{|u(t)|: t∈J}.

Then the product spaceC :=U ×V defined byC ={(u,v):u∈U,v∈V}is Banach space under the norm

k(u,v)k_C =kuk_∞+kvk_∞. We further will use the following hypotheses.

(H1) For anyi=1,2, f_i: J×E²→Esatisfie the Caratheodory conditions.

(H2) There exist p_i,q_i∈C(J,R⁺), such that,

kf(t,x)k ≤ p_i(t)kx₁k+q_i(t)kx₁k, fort∈J and eachx_i∈E,i=1,2.

(H3) For anyt∈J and each bounded measurable setsB_i⊂E, i=1,2, we have

h→0lim⁺µ(f(J_t,h×B₁,B₂),0)≤p₁(t)µ(B₁) +q₁(t)µ(B₂) and

h→0lim⁺µ(0,f(J_t,h×B₁,B₂))≤p₂(t)µ(B₁) +q₂(t)µ(B₂) whereµ is the Kuratowski measure of compactness and J_t,h= [t−h,t]∩J.

Set

p^∗_i =sup

t∈J

p_i(t) and q^∗_i =sup

t∈J

q_i(t),i=1,2.

Theorem 6.4. Assume that conditions (H1)-(H3) hold. If

M<1 (6.12)

With

M: =^P2_i=1(M_i)

=^P2_i=1

ß 2(p^∗_i+q^∗_i)R

Γq(αi+βi+1)+ ^2|λi|R

Γq(αi+1)

™

then the problem(6.2)has at least one solution onJ.

Proof. Transform the problem (6.2) into a fixed point problem. Consider the operator F: C →C defined by the formula

Fx_i(t) =I_q^αi+βih(t)−λ_iI_q^αih(t) +t^(αi)nη_i−γ_i−I_q^αi+βih(1) +λ_iI_q^αih(1)^o+γ_i (6.13)

2. A. Boutiara, M. Benbachir, K. Guerbati, Existence Theory for a Langevin Fractionalq-Difference System in Banach Space, (submitted).

6.4. EXISTENCE THEORY FOR A NONLINEAR LANGEVIN FRACTIONALQ-DIFFERENCE SYSTEM IN BANACH SPACE

Clearly, the fixed points of the operatorFare solutions of the problem (6.2). Let R≥ ηi

1−M, i=1,2. (6.14)

and consider

D_R={x_i∈C,i=1,2 :k(x₁,x₂)k ≤R}.

Clearly, the subset D_R is closed, bounded and convex. We shall show that F satisfies the assumptions of Mönch’s fixed point theorem. The proof will be given in three steps.

Step 1 :First we show thatF_iis sequentially continuous :

Let{x_1,n,x_2,n}_nbe a sequence such that(x_1,n,x_2,n)→(x₁,x₂)inC. Then for anyt∈J , k(F_ix_i,n)(t)−(F_ix_i)(t)k ≤I_q^αi+βikf(s,x_1,n(s),x_2,n(s))− f(s,x₁(s),x₂(s))k(t)

+|λ_i|I_q^αikx_i,n(s)−x_i(s)k(t)

+t^(αi)I_q^αi+βikf(s,x_1,n(s),x_2,n(s))−f(s,x₁(s),x₂(s))k(1) +t^(αi)|λ_i|I_q^αikx_i,n(s)−x_i(s)k(1)

≤ⁿ2I_q^αi+βi(1)(t)^okf(s,x_1,n(s),x_2,n(s))−f(s,x₁(s),x₂(s))k +^¶2|λ_i|I_q^αi(1)(t)^©kx_i,n(s)−x_i(s)k.

Since for anyi=1,2, the function f_isatisfies assumptions (H1), we have f_i(t,x_1,n(t),x_2,n(t)) converge uniformly to f_i(t,x₁(t),x₂(t)).

Hence, the Lebesgue dominated convergence theorem implies that(F_i(x_1,n,x_2,n))(t)converges uniformly to(F_i(x₁,x_2,))(t)Thus(F(x_1,n,x_2,n))→(F(x₁,x_2,)). HenceF:D_R→D_Ris sequen-tially continuous.

Step 2 :Second we show thatF_imapsD_Rinto itself :

Takex_i∈D_R, i=1,2, by (H2), we have, for eacht∈J and assume that(F_i(x_i))(t)6=0, i=1,2.

k(F_ix_i)(t)k ≤I_q^αi+βikf(s,x₁(s),x₂(s))k(t)−λI_q^αkx_ik(t)

+t^(αi)nηi−γi−I_q^αi+βikf(s,x₁(s),x₂(s))k(1) +λiI_q^αkx_ik(1)^o+γi

≤I_q^αi+βi[kx₁kp_i(s) +kx₂kq_i(s)] (t)−λ_iI_q^αikx_ik(t)

+t^(αi)nηi−γi−I_q^αi+βi[kx₁kp_i(s) +kx₂kq_i(s)] (1) +λiI_q^αikx_ik(1)^o+γi

≤(p^∗_i +q^∗_i)RⁿI_q^αi+βi(1)(t) +t^(αi)I_q^αi+βi(1)(1)^o

+Rⁿ|λ_i|I_q^αi(1)(t) +t^(αi)|λ_i|I_q^αi(1)(1)^o+t^(αi)(η_i−γ_i) +γ_i

≤

® 2(p^∗_i +q^∗_i)R Γ_q(α_i+β_i+1)

´

+

® 2|λ_i|R Γ_q(α_i+1)

´

+ηi

=RM_i+ηi. Hence we get

k(F(x₁,x₂))k_C ≤

2 X i=1

(RM_i+η_i)≤R. (6.15)

6.4. EXISTENCE THEORY FOR A NONLINEAR LANGEVIN FRACTIONALQ-DIFFERENCE SYSTEM IN BANACH SPACE

Step 3 :we show thatF_i(D_R)is equicontinuous :

By Step 2, it is obvious that F(D_R)⊂C is bounded. For the equicontinuity of F(D_R), let t₁,t₂∈J ,t₁<t₂andx∈D_R, soFx(t₂)−Fx(t₁)6=0. Then

kFx_i(t₂)−Fx_i(t₁)k ≤ |I_q^αi+βif(s,x₁(s),x₂(s))(t₂)−I_q^αi+βif(s,x₁(s),x₂(s))(t₁)|

+|λ_i|(I_q^αi|x_i(s)|(t₂)−I_q^αi|x_i(s)|(t₂))

+ (t₂^(αi)−t₂^(αi))ⁿη_i−γ_i−I_q^αi+βi|f(s,x₁(s),x₂(s))|(1) +λ_iI_q^αi|x_i|(1)^o

≤(p^∗_i +q^∗_i)R|I_q^αi+βi(1)(t₂)−I_q^αi+βi(1)(t₁)|+R|λ_i|(I_q^α|(1)|(t₂)−I_q^α|(1)|(t₁)) + (t₂^(αi)−t₁^(αi))ⁿηi−γi−I_q^αi+βi|f(s,x₁(s),x₂(s))|(1) +λiI_q^αi|x_i|(1)^o

≤ R(p^∗_i +q^∗_i) Γq(α_i+βi+1)

n(t₂^αi+βi−t₁^αiv) +2(t₂−t₁)^αi+βio + R|λ_i|

Γ_q(α_i+1)

¶(t₂^αi−t₁^αi) +2(t₂−t₁)^αi©

+ (t₂^(αi)−t₁^(αi))ⁿη_i−γ_i−I_q^αi+βi|f(s,x₁(s),x₂(s))|(1) +λ_iI_q^αi|x_i|(1)^o Ast₁→t₂, the right hand side of the above inequality tends to zero.

This means thatF(D_R)⊂D_R. Finally we show that the implication (1.12) holds :

LetV ⊂D_R such that V =conv(F(V)∪ {(0,0)}). SinceV is bounded and equicontinuous, and therefore the functiont →v(t) =µ(V(t))is continuous on J. By assumption (H2), and the properties of the measureµ, for anyt ∈J we get.

v(t)≤µ(F(V)(t)∪ {(0,0)}))≤µ((FV)(t))

≤µ({((N₁v₁) (t),(N₂v₂) (t):(v₁,v₂)∈V})

≤2I^α1+β1µ({({(f₁(s,v₁(s),v₂(s)) (t)); 0):(v₁,v₂)∈V}) +2|λ₁|I^α1µ({(v₁(s),0):(v₁,0)∈V})

+2I^α2+β2µ({(0,f₂(s,v₁(s),v₂(s))):(v₁,v₂)∈V}) +2|λ₂|I^α2µ({(0,v₂(s)):(0,v₂)∈V})

≤2I^α1+β1[p₁(s)µ({(v₁(s),0):(v₁,0)∈V}) +q₁(s)µ({(0,v₂(s)):(0,v₂)∈V})]

+2|λ₁|I^α1µ({(v₁(s),0):(v₁,0)∈V}) +2I^α2+β2[p₂(s)µ({(v₁(s),0):(v₁,0)∈V}) +q₂(s)µ({(0,v₂(s)):(0,v₂)∈V})]

+2|λ₂|I^α2µ({(0,v₂(s)):(0,v₂)∈V})

Thus

µ(V(t))≤2I^α1+β1(p₁(s) +q₁(s))×µ(V(s)) 2|λ₁|I^α1((1)(s))×µ(V(s))

+2I^α2+β2(p₂(s) +q₂(s))×µ(V(s)) 2|λ₂|I^α2((1)(s))×µ(V(s))

6.4. EXISTENCE THEORY FOR A NONLINEAR LANGEVIN FRACTIONALQ-DIFFERENCE SYSTEM IN BANACH V(t)is relatively compact inE. In view of the Ascoli-Arzela theorem,V is relatively compact inD_R. Applying now Theorem1.58, we conclude thatFhas a fixed point, which is a solution of the problem (6.2)-(6.3).

6.4.2 Example

In this section, we present some examples to illustrate our results.

LetE=l¹={x= (x₁,x₂, ...,x_n, ...):^P∞_n=1|x_n|<∞}with the norm

Consider the following nonlinear Langevin ¹₄-fractional equation :



Clearly, the function f is continuous. For eachx∈Eandt ∈[0,1], we have

|f(t,x₁,x₂)| ≤

6.5. CONCLUSION

and

|g(t,x₁,x₂)| ≤

√2

64π|x₁|+ 1 100|x₂| Hence, the hypothesis (H2) is satisfied with p^∗₁=

√3 81, q^∗₁=

√2 49π, p^∗₂=

√2

64π andq^∗₂= ₁₀₀¹ . We shall show that condition (6.12) holds with J= [0,1]. Indeed,

® 2(p^∗₁+q^∗₁)

Γ_q(α₁+β₁+1)+ 2|λ₁| Γ_q(α₁+1)

´

+

® 2(p^∗₂+q^∗₂)

Γ_q(α₂+β₂+1)+ 2|λ₂| Γ_q(α₂+1)

´

'0.6758<1 Simple computations show that all conditions of Theorem6.4are satisfied. It follows that the coupled system (6.16) has at least one weak solution defined on J.

6.5 Conclusion

We have provided sufficient conditions for the existence of the solutions of a new class of nonlinear Langevin fractionalq-difference system with Dirichlet boundary conditions in Banach space. by using a method involving a measure of noncompactness and a fixed point theorem of Mönch type. Though the technique applied to establish the existence results for the problem at hand is a standard one, yet its exposition in the present framework is new. An illustration to the present work is also given by presenting an example.

Chapitre 7

Existence and Uniqueness Results to a Fractional q-Difference Coupled System with Integral Boundary Conditions via Topological Degree Theory

7.1 Introduction

At the present day, there are many results on the existence of solutions for fractional dif-ferential equations. In this study, we focus on which that uses the topological degree. This method is a powerful tool for the existence of solutions to BVPs of many mathematical mo-dels that arise in applied nonlinear analysis. Very recently F. Isaia [81] proved a new fixed theorem that was obtained via coincidence degree theory for condensing maps. To see more applications about the usefulness of coincidence degree theory approach for condensing maps in the study for the existence of solutions of certain integral equations, the reader can be re-ferred to [18,19,132,81,87,120,121,122,123,124,130].

LetU =C([0,1],R)the Banach space of all continuous functions fromu: J→Rwith kuk_∞=sup{|u(t)|: t∈J}.

Then the product spaceC :=U ×V defined byC ={(u,v):u∈U,v∈V}is Banach space under the norm

k(u,v)k_C =kuk_∞+kvk_∞.

This section is mainly concerned with the existence results for the following fractional q-difference system of the form







D^q_q¹u₁(τ) =F1(τ,u₁(τ),u₂(τ)),

,τ∈J := [0,1], D^q_q²u₂(τ) =F2(τ,u₁(τ),u₂(τ)),

(7.1)

7.2. EXISTANCE RESULT

with the fractional boundary conditions



7.2 Existance Result

¹

For the existence of solutions for the problem (7.1)-(7.15), we need the following auxi-liary lemmas.

Lemma 7.1. LetFi: J×R²→Rbe a continuous function for each i=1,2. Then problem (7.1)-(7.15)is equivalent to the problem of obtaining the solutions of the integral equation u_i(τ) =I_q^qiFui(τ) + (Λ_1,i−Λ4,iτ)I_q^qi+βiFui(η_i) + (Λ_2,i+Λ3i,τ)^Äb_iI_q^qi+αiFui(σ_i)−I_q^qiFui(1)^ä

(7.3) if and only if u_i, i=1,2is a solution of the fractional boundary-value problem



1. A. Boutiara, M. Benbachir, K. Guerbati, Existence and Uniqueness Results to a Fractional q-Difference Coupled System with integral Boundary conditions via Topological Degree Theory, (submitted).

7.2. EXISTANCE RESULT

Proof. For some constantsc_0,i,c_1,i∈Rand 1<q_i≤2,the general solution ofDq^qiu_i(τ) = Fui(τ)can be written as

u_i(τ) =I_q^qiFui(τ) +c_0,i+c_1,iτ. (7.7) Using the boundary conditions (7.5) in (7.7) we may obtain

Ñ which, on solving, yields

c_0,i= 1

Substituting the value ofc_0,i,c_1,iin (7.7) we get (7.3), which completes the proof.

We use the following sufficient assumptions in the proofs of our main results.

(H1) There exist constantsLi>0,i=1,2 such that forτ ∈Jand eachu_i,v_i∈C,i=1,2. In the following, we set an abbreviated notation for the fractional q-integral of the Caputo type of orderq_i>0, for a function with two variables as

I_q^qiFui(τ) = 1 Γ(q_i)

Z _τ

0 (τ−qs)^α−1F(s,u₁(s),u₂(s))ds.

Moreover, for computational convenience we put ω_i=

7.2. EXISTANCE RESULT

and

ω¯_i=







|Λ_4,i| η_i^qi+βi

Γ(q_i+β_i+1)+|Λ_3,i| |b_i|σ_i^qi+αi

Γ(q_i+α_i+1)+ 1 Γ(q_i+1)

!





. (7.12) By Lemma7.1, we consider two operatorsT,S :C −→C as follows :

T u_i(τ) =I_q^qiFu_i(τ), τ∈J, and

Su_i(τ) = (Λ_1,i−Λ4,iτ)I_q^qi+βiFui(η_i) + (Λ_2,i+Λ3,iτ)^Äb_iI_q^qi+αiFui(σ_i)−I_q^qiFui(1)^ä, τ ∈J.

Then the integral equation (7.3) in Lemma7.1can be written as an operator equation K u_i(τ) =T u_i(τ) +Su_i(τ), τ∈J.

The continuity ofFi, i=1,2, shows that the operator K :C →C is well define and fixed points of the operator equation are solutions of the integral equations (7.3) in Lemma7.1.

Lemma 7.2. The operatorT :C →C is Lipschitz with constant ^P2_i=1`_Fi=^P2_i=1Γ(q^Li

i+1). Moreover,T satisfies the growth condition given below

kT(u₁,u₂)k ≤

2 X i=1

1

Γ(α+1)(K_iku₁k^p+M_iku₂k^p+N_i), for every u_i∈C.

Proof.

To show that the operatorT is Lipschitz. Letu_i,v_i∈C ,i=1,2, then we have

|T u_i(τ)−Tv_i(τ)|=I_q^qiFi,ui−I_q^qiFi,vi

≤I_q^qi|Fi,u_i−Fi,v_i|(τ)

≤I_q^qi(1)Li 2 X i=1

(ku_i−v_ik)

= Li

Γ(q_i+1)

2 X i=1

(ku_i−v_ik). For allτ∈J, we obtain

kTu_i−Tv_ik ≤ Li

Γ(q_i+1)

2 X i=1

(ku_i−v_ik).

Hence,T :C −→C is a Lipschitzian onC with Lipschitz constant`_Fi = _Γ(q^Li

i+1). By Pro-position1.67,T isκ–Lipschitz with constant`_Fi. Moreover, for growth condition, we have

|Tu_i(τ)| ≤I_q^qi|Fui|(τ)

≤(K_iku₁k^p+M_iku₂k^p+N_i)I_q^α(1)

= 1

Γ(q_i+1)(K_iku₁k^p+M_iku₂k^p+N_i).

7.2. EXISTANCE RESULT

Hence it follows that

kTu_ik ≤ 1

Γ(q_i+1)(K_iku₁k^p+M_iku₂k^p+N_i), which implies that

kT(u₁,u₂)k ≤

2 X i=1

1

Γ(α+1)(K_iku₁k^p+M_iku₂k^p+N_i).

Lemma 7.3. S is continuous and satisfies the growth condition given as below, kSu_ik ≤(K_iku₁k^p+M_iku₂k^p+N_i)ω_i, for every u_i∈C, whereω_iis given by(7.11).

Proof. Choose a bounded subset D_r ={(u₁,u₂)∈C :k(u₁,u₂)k ≤r} ⊂C and consider a sequence^¶z_n= (u_1,n,u_2,n)^©∈D_r such that z_n→z= (u₁,u₂)asn→∞inD_r.We need to show thatkSz_n−Sz| →0,n→∞.From the continuity ofFi,u, it follows thatFi,un→Fi,u, asn→∞.In view of(H₂),we obtain the following relations :

(τ−sq)^qi−1kFi,un−Fi,uk ≤(K_iku₁k^p+M_iku₂k^p+N_i) (τ−sq)^qi−1,i=1,2.

(η_i−sq)^qi+βi−17→(K_iku₁k^p+M_iku₂k^p+N_i) (η_i−sq)^qi+βi−1,i=1,2, (σ_i−sq)^qi+αi−17→(K_iku₁k^p+M_iku₂k^p+N_i) (σ_i−sq)^qi+αi−1,i=1,2, (1−sq)^qi−17→(K_iku₁k^p+M_iku₂k^p+N_i) (1−sq)^qi−1,i=1,2,

which implies that each term on the left is integrable. By Lebesgue Dominated convergent theorem, we obtain

I_q^qi+βi|Fi,u_n−Fi,u|(η_i)→0 as n→+∞, I_q^qi+αi|Fi,un−Fi,u|(σ_i)→0 as n→+∞, I_q^qi|Fi,u_n−Fi,u|(1)→0 as n→+∞.

It follows thatkSz_n−Szk →0 as n→+∞. Which implies the continuity of the operator S.

For the growth condition, using the assumption (H2) we have

|Su_i(τ)| ≤(|Λ_1,i|+|Λ_4i|)I_q^qi+βiFui(η_i) + (|Λ_2,i|+|Λ_3,i|)^Ä|b|I_q^qi+αiFui(σ_i) +I_q^qiFui(1)^ä,

≤(K_iku₁k^p+M_iku₂k^p+N_i) (|Λ_1,i|+|Λ_4i|)I_q^qi+βi(1)(η_i)

+ (K_iku₁k^p+M_iku₂k^p+N_i) (|Λ_2,i|+|Λ_3,i|)^Ä|b_i|I_q^qi+αi(1)(σ_i) +I_q^qi(1)^ä

≤(K_iku₁k^p+M_iku₂k^p+N_i)







(|Λ_1,i|+|Λ_4i|) η_i^qi+βi Γ(q_i+β_i+1) + (|Λ_2,i|+|Λ_3,i|) |b_i| σ_i^qi+αi

Γ(q_i+αi+1)+ 1 Γ(q_i+1)

!)

= (K_iku₁k^p+M_iku₂k^p+N_i)ω_i.

7.2. EXISTANCE RESULT

Which implies that,

kS(u₁,u₂)k ≤

2 X i=1

(K_iku₁k^p+M_iku₂k^p+N_i)ω_i,i=1,2. (7.13) whereω_i, i=1,2 is given by (7.11). This completes the proof of Lemma7.3.

Lemma 7.4. The operator S :C −→C is compact. Consequently,S is κ-Lipschitz with zero constant.

Proof. In order to show thatS is compact. Let us take a bounded setΩ⊂Br, i=1,2. We are required to show thatS(Ω)is relatively compact inC. For arbitraryu_i∈Ω⊂Br, then with the help of the estimates (7.13) we can obtain

kSuk ≤(K_ir^p+M_ir^p+N_i)ω_i,

whereωiis given by (7.11), which shows thatS(Ω)is uniformly bounded.

Now, for equi-continuity ofS takeτ₁,τ₂∈Jwithτ₁<τ₂, and letu_i∈Ω. Thus, we get

|Su_i(τ₂)−Su_i(τ₁)| ≤ |Λ_4,i|(τ₂−τ1)I_q^qi+βiFu_i(η_i)

+|Λ_3,i|(τ₂−τ1)^Äb_iI_q^qi+αiFu_i(σ_i)−I_q^qiFu_i(1)^ä

≤ω¯_i(K_iku_ik^p+M_ikv_ik^p+N_i) (τ₂−τ₁).

Which implies that,

|S(u₁,u₂)(τ₂)−S(u₁,u₂)(τ₁)| ≤

2 X i=1

¯

ω_i(K_iku_ik^p+M_ikv_ik^p+N_i) (τ₂−τ₁).

where ¯ω_iis given by (7.12). From the last estimate, we deduce that

kS(u₁,u₂)(τ₂)−S(u₁,u₂)(τ₁)k →0 when τ2→τ1.

Therefore,S is equicontinuous. Thus, by Ascoli–Arzelà theorem, the operatorS is compact and hence by Proposition1.66.S isκ–Lipschitz with zero constant.

Theorem 7.5. Suppose that (H1)–(H2) are satisfied, then the BVP (7.1) has at least one solution (u₁,u₂)∈C, provided that ^P2_i=1`_Fi <1, i=1,2, and the set of the solutions is bounded inC.

Proof. LetT ,S,K are the operators defined in the start of this section. These operators are continuous and bounded. Moreover, by Lemma7.2,T isκ–Lipschitz and by Lemma7.4, S isκ–Lipschitz with constant 0. Thus, K isκ–Lipschitz with constant `<sub>Fi</sub>. HenceK is strictκ–contraction with constant`_Fi. Since^P2_i=1`_Fi<1, soK isκ-condensing.

Now consider the following set

Θ={(u₁,u₂)∈C : there existξ ∈[0,1]such thatu_i=ξK u_i,i=1,2}.

7.2. EXISTANCE RESULT

whereω_iis given by (7.11). From the above inequalities, we conclude thatΘ is bounded in C. If it is not bounded, then dividing the above inequality bya:=ku_ikand lettinga→∞, we

which is a contradiction. Thus the setΘis bounded inC and the operatorK has at least one fixed point which represent the solution of BVP (7.1).

To end this section, we give an existence and uniqueness result.

Theorem 7.6. Under assumption (H1) the BVP(7.1)has a unique solution if

2

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