The collision of a vehicle is often a process of strong nonlinearity and large deformation, and the finite element method is time-consuming. For this reason, a method of collision inversion research is proposed. Through the definition of the forward and inverse problems, the forward problem is inversely solved from the perspective of the inverse problem, and the collision process can be predicted quickly while the accuracy is ensured. In this paper, the idea of inversion is introduced into the collision test of the front bumper of the vehicle. First, a finite element model is established based on the geometric model of the bumper. The accuracy of the finite element model is verified by comparing the results of the drop weight test with the simulation results of the finite element model. Then, using the built simulation model, spring mass model and drop weight test, the inversion research of vehicle collision and the front bumper of a vehicle is carried out. The inversion research of the bumper first inverts the collision course in the inversion algorithm derived from the vehicle collision, and then compares the collision course derived from the inversion algorithm formula with the simulation test results. The research results show that the change trends of the time-velocity curve and the time-deformation curve obtained by the collision inversion algorithm are basically consistent with the simulation test results, indicating that the collision inversion algorithm can realize the rapid prediction of the collision course and improve derivation efficiency significantly, and it provides a new idea. Finally, under the constant energy condition of the drop weight test E = mgh, through the relationship between energy and deformation, it is concluded that the depth deformation of low-speed collision of the front bumper is greater than that of high-speed collision.
Citation: Miao Luo, Yousong Chen, Dawei Gao, Lijun Wang. Inversion study of vehicle frontal collision and front bumper collision[J]. Electronic Research Archive, 2023, 31(2): 776-792. doi: 10.3934/era.2023039
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The collision of a vehicle is often a process of strong nonlinearity and large deformation, and the finite element method is time-consuming. For this reason, a method of collision inversion research is proposed. Through the definition of the forward and inverse problems, the forward problem is inversely solved from the perspective of the inverse problem, and the collision process can be predicted quickly while the accuracy is ensured. In this paper, the idea of inversion is introduced into the collision test of the front bumper of the vehicle. First, a finite element model is established based on the geometric model of the bumper. The accuracy of the finite element model is verified by comparing the results of the drop weight test with the simulation results of the finite element model. Then, using the built simulation model, spring mass model and drop weight test, the inversion research of vehicle collision and the front bumper of a vehicle is carried out. The inversion research of the bumper first inverts the collision course in the inversion algorithm derived from the vehicle collision, and then compares the collision course derived from the inversion algorithm formula with the simulation test results. The research results show that the change trends of the time-velocity curve and the time-deformation curve obtained by the collision inversion algorithm are basically consistent with the simulation test results, indicating that the collision inversion algorithm can realize the rapid prediction of the collision course and improve derivation efficiency significantly, and it provides a new idea. Finally, under the constant energy condition of the drop weight test E = mgh, through the relationship between energy and deformation, it is concluded that the depth deformation of low-speed collision of the front bumper is greater than that of high-speed collision.
Fractional differential equations (FDEs) provide many mathematical models in physics, biology, economics, and chemistry, etc [1,2,3,4]. In fact, it consists of many integrals and derivative operators of non-integer orders, which generalize the theory of ordinary differentiation and integration. Hence, a more general approach is allowed to calculus and one can say that the aim of the FDEs is to consider various phenomena by studying derivatives and integrals of arbitrary orders. For intercalary specifics about the theory of FDEs, the readers are referred to the books of Kilbas et al.[2] and Podlubny [4]. In the literature, several concepts of fractional derivatives have been represented, consisting of Riemann-Liouville, Liouville-Caputo, generalized Caputo, Hadamard, Katugampola, and Hilfer derivatives. The Hilfer fractional derivative [5] extends both Riemann-Liouville and Caputo fractional derivatives. For applications of Hilfer fractional derivatives in mathematics and physics, etc see [6,7,8,9,10,11]. For recent results on boundary value problems for fractional differential equations and inclusions with the Hilfer fractional derivative see the survey paper by Ntouyas [12]. The ψ-Riemann-Liouville fractional integral and derivative operators are discussed in [1], while the ψ-Hilfer fractional derivative is discussed in [13]. Recently, the notion of a generalized proportional fractional derivative was introduced by Jarad et al. [14,15,16]. For some recent results on fractional differential equations with generalized proportional derivatives, see [17,18].
In [19], an existence result was proved via Krasnosel'ski˘i's fixed-point theorem for the following sequential boundary value problem of the form
{HDα,ς,ψ[HDβ,ς,ψp(w)ϕ(w,p(w))−n∑i=1Iνi;ψhi(w,p(w))]=Υ(w,p(w)),w∈[a,b],p(a)=0,HDb,ς,ψp(a)=0,p(b)=τp(ζ), | (1.1) |
where HDω,ς,ψ indicates the ψ-Hilfer fractional derivative of order ω∈{α,β}, with 0<α≤1, 1<β≤2, 0≤ς<1, Iνi;ψ is the ψ-Riemann–Liouville fractional integral of order νi>0, for i=1,2,…,n, hi∈C([0,1]×R,R), for i=1,2,…,n, ϕ∈C([0,1]×R,R∖{0}), Υ∈C([0,1]×R,R), τ∈R and ζ∈(a,b). In [16], the consideration of Hilfer-type generalized proportional fractional derivative operators was initiated.
Coupled systems of fractional order are also significant, as such systems appear in the mathematical models in science and engineering, such as bio-engineering [20], fractional dynamics [21], financial economics [22], etc. Coupled systems of FDEs with diverse boundary conditions have been the focus of many researches. In [23], the authors studied existence and Ulam-Hyers stability results of a coupled system of ψ-Hilfer sequential fractional differential equations. Existence and uniqueness results are derived in [24] for a coupled system of Hilfer-Hadamard fractional differential equations with fractional integral boundary conditions. Recently, in [25] a coupled system of nonlinear fractional differential equations involving the (k,ψ)-Hilfer fractional derivative operators complemented with multi-point nonlocal boundary conditions were discussed. Moreover, Samadi et al. [26] have considered a coupled system of Hilfer-type generalized proportional fractional differential equations.
In this article, motivated by the above works, we study a coupled system of ψ-Hilfer sequential generalized proportional FDEs with boundary conditions generated by the problem (1.1). More precisely, we consider the following coupled system of nonlinear proportional ψ-Hilfer sequential fractional differential equations with multi-point nonlocal boundary conditions of the form
{HDν1,ϑ1;ς;ψ[HDν2,ϑ2;ς;ψp1(w)Φ1(w,p1(w),p2(w))−n∑i=1pIηi,ς,ψHi(w,p1(w),p2(w))]=Υ1(w,p1(w),p2(w)),w∈[t1,t2],HDν3,ϑ3;ς;ψ[HDν4,ϑ4;ς;ψp1(w)Φ2(w,p1(w),p2(w))−m∑j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))]=Υ2(w,p1(w),p2(w)),w∈[t1,t2],p1(t1)=HDν2,ϑ2;ς;ψp1(t1)=0,p1(t2)=θ1p2(ξ1),p2(t1)=HDν4,ϑ4;ς;ψp2(t1)=0,p2(t2)=θ2p1(ξ2), | (1.2) |
where HDν,ϑ1;ς;ψ denotes the ψ-Hilfer generalized proportional derivatives of order ν∈{ν1,ν2,ν3,ν4}, with parameters ϑl, 0≤ϑl≤1, l∈{1,2,3,4}, ψ is a continuous function on [t1,t2], with ψ′(w)>0, pIη,ς,ψ is the generalized proportional integral of order η>0, η∈{ηi,ηj}, θ1,θ2∈R, ξ1,ξ2∈[t1,t2], Φ1,Φ2∈C([t1,t2]×R×R,R∖{0}) and Hi,Gj,Υ1,Υ2∈C([t1,t2]×R×R,R), for i=1,2,…,n and j=1,2,…,m.
We emphasize that:
● We study a general system involving ψ-Hilfer proportional fractional derivatives.
● Our equations contain fractional derivatives of different orders as well as sums of fractional integrals of different orders.
● Our system contains nonlocal coupled boundary conditions.
● Our system covers many special cases by fixing the parameters involved in the problem. For example, taking ψ(w)=w, it will reduce to a coupled system of Hilfer sequential generalized proportional FDEs with boundary conditions, while if ς=1, it reduces to a coupled system of ψ-Hilfer sequential FDEs. Besides, by taking Φ1,Φ2=1 in the problem (1.2), then we obtain the following new coupled system of the form:
{HDν1,ϑ1;ς;ψ[HDν2,ϑ2;ς;ψp1(w)−n∑i=1pIηi,ς,ψHi(w,p1(w),p2(w))]=Υ1(w,p1(w),p2(w)),w∈[t1,t2],HDν3,ϑ3;ς;ψ[HDν4,ϑ4;ς;ψp1(w)−m∑j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))]=Υ2(w,p1(w),p2(w)),w∈[t1,t2],p1(t1)=HDν2,ϑ2;ς;ψp1(t1)=0,p1(t2)=θ1p2(ξ1),p2(t1)=HDν4,ϑ4;ς;ψp2(t1)=0,p2(t2)=θ2p1(ξ2). |
In obtaining the existence result of the problem (1.2), first the problem (1.2) is converted into a fixed-point problem and then a generalization of Krasnosel'ski˘i's fixed-point theorem due to Burton is applied.
The structure of this article has been organized as follows: In Section 2, some necessary concepts and basic results concerning our problem are presented. The main result for the problem (1.2) is proved in Section 3, while Section 4 contains an example illustrating the obtained result.
In this section, we summarize some known definitions and lemmas needed in our results.
Definition 2.1. [17,18] Let ς∈(0,1] and ν>0. The fractional proportional integral of order ν of the continuous function F is defined by
pIν,ς,ψF(w)=1ςνΓ(ν)∫wt1eς−1ς(ψ(w)−ψ(s))(ψ(w)−ψ(s))ν−1F(s)ψ′(s)ds,t1>w. |
Definition 2.2. [17,18] Let ς∈(0,1], ν>0, and ψ(w) is a continuous function on [t1,t2], ψ′(w)>0. The generalized proportional fractional derivative of order ν of the continuous function F is defined by
(pDν,ς,ψF)(w)=(pDn,ς,ψ)ςn−νΓ(n−ν)∫wt1eς−1ς(ψ(w)−ψ(s))(ψ(w)−ψ(s))n−ν−1F(s)ψ′(s)ds, |
where n=[ρ]+1 and [ν] denotes the integer part of the real number ν, where Dn,ς,ψ=Dς,ψ⋯Dς,ψ⏟n−times.
Now the generalized Hilfer proportional fractional derivative of order ν of function F with respect to another function ψ is introduced.
Definition 2.3. [27] Let ς∈(0,1], F,ψ∈Cm([t1,t2],R) in which ψ is positive and strictly increasing with ψ′(w)≠0 for all w∈[t1,t2]. The ψ-Hilfer generalized proportional fractional derivative of order ν and type ϑ for F with respect to another function ψ is defined by
(HDν,ϑ,ς,ψF)(w)=pIϑ(n−ν),ς,ψ[pDn,ς,ψ(pI(1−ϑ)(n−ν),ς,ψF)](w), |
where n−1<ν<n and 0≤ϑ≤1.
Lemma 2.4. [27] Let m−1<ν<m,n∈N, 0<ς≤1, 0≤ϑ≤1 and m−1<γ<m such that γ=ν+mϑ−νϑ. If F∈C([t1,t2],R) and pI(m−γ,ς,ψ)F∈Cm([t1,t2],R), then
(pIν,ς,ψHDν,ϑ,ς,ψF)(w)=F(w)−n∑j=1eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ−jςγ−jΓ(γ−j+1)(pIj−γ,ς,ψF)(t1). |
To prove the main result we need the following lemma, which concerns a linear variant of the ψ-Hilfer sequential proportional coupled system (1.2). This lemma plays a pivotal role in converting the nonlinear problem in system (1.2) into a fixed-point problem.
Lemma 2.5. Let 0<ν1,ν3≤1, 1<ν2,ν4≤2, 0≤ϑi≤1, γi=νi+ϑi(1−νi), i=1,3 and γj=νj+ϑj(2−νj), j=2,4, Θ=M1N2−M2N1≠0, ψ is a continuous function on [t1,t2], with ψ′(w)>0, and Q1,Q2∈C([t1,t2],R), Φ1,Φ2∈C([t1,t2]×R×R,R∖{0}) and Hi,Gj,Q1,Q2∈C([t1,t2]×R×R,R), for i=1,2,…,n and j=1,2,…,m, and pI(1−γi,ς,ψ)Qj∈Cm([t1,t2],R),i=1,2,3,4,j=1,2. Then the pair (p1,p2) is a solution of the system
{HDν1,ϑ1;ς;ψ[HDν2,ϑ2;ς;ψp1(w)Φ1(w,p1(w),p2(w))−n∑i=1pIηi,ς,ψHi(w,p1(w),p2(w))]=Q1(w),w∈[t1,t2],HDν3,ϑ3;ς;ψ[HDν4,ϑ4;ς;ψp1(w)Φ2(w,p1(w),p2(w))−m∑j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))]=Q2(w),w∈[t1,t2],p1(t1)=HDν2,ϑ2;ς;ψp1(t1)=0,p1(t2)=θ1p2(ξ1),p2(t1)=HDν4,ϑ4;ς;ψp2(t1)=0,p2(t2)=θ2p1(ξ2), |
if and only if
p1(w)=pIν2,ς,ψΦ1(w,p1(w),p2(w))(n∑i=1pIηi,ς,ψHi(w,p1(w),p2(w))+pIν1,ς,ψQ1(w))+eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ2−1Θςγ2−1Γ(γ2){N2[θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1)) |
×(m∑j=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψQ2(ξ1)))−pIν2,ς,ψΦ1(t2,p1(t2),p2(t2))×(n∑i=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψQ1(t2))]+M2[θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))×(n∑i=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψQ1(ξ2)))−pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))×(m∑j=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+pIν3,ς,ψQ2(t2))]} | (2.1) |
and
p2(w)=pIν4,ς,ψΦ2(w,p1(w),p2(w))(m∑j=1pIˉηj,ς,ψGj(w,p1(w),p2(w))+pIν3,ς,ψQ2(w)) |
+eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ4−1Θςγ4−1Γ(γ4){N1[θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))×(m∑j=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψQ2(ξ1)))−pIν2,ς,ψΦ2(t2,p1(t2),p2(t2))×(n∑i=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψQ1(t2))]+M1[θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))×(n∑i=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψQ1(ξ2)))−pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))×(m∑j=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+pIν3,ς,ψQ2(t2))]}, | (2.2) |
where
M1=eς−1ς(ψ(t2)−ψ(t1))(ψ(t2)−ψ(t1))γ2−1ςγ2−1Γ(γ2),M2=θ1eς−1ς(ψ(ξ1)−ψ(t1))(ψ(ξ1)−ψ(t1))γ4−1ςγ4−1Γ(γ4),N1=θ2eς−1ς(ψ(ξ2)−ψ(t1))(ψ(ξ2)−ψ(t1))γ2−1ςγ2−1Γ(γ2),N2=eς−1ς(ψ(t2)−ψ(t1))(ψ(t2)−ψ(t1))γ4−1ςγ4−1Γ(γ4). | (2.3) |
Proof. Due to Lemma 2.4 with m=1, we get
HDν2,ϑ2;ς;ψp1(w)Φ1(w,p1(w),p2(w))−n∑i=1pIηi,ς,ψHi(w,p1(w),p2(w))=pIν1;ς;ψQ1(w)+c0eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ1−1ςγ1−1Γ(γ1),HDν4,ϑ4;ς;ψp2(w)Φ2(w,p1(w),p2(w))−m∑j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))=pIν3;ς;ψQ2(w)+d0eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ3−1ςγ3−1Γ(γ3), | (2.4) |
where c0,d0∈R. Now applying the boundary conditions
HDν2,ϑ2;ς;ψp1(t1)=HDν4,ϑ4;ς;ψp1(t1)=0, |
we get c0=d0=0. Hence
HDν2,ϑ2;ς;ψp1(w)=Φ1(w,p1(w),p2(w))(n∑i=1pIηi,ς,ψHi(w,p1(w),p2(w))+pIν1;ς;ψQ1(w)),HDν4,ϑ4;ς;ψp2(w)=Φ2(w,p1(w),p2(w))(m∑j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))+pIν3;ς;ψQ2(w)). | (2.5) |
Now, by taking the operators pIν2,ς,ψ and pIν4,ς,ψ into both sides of (2.5) and using Lemma 2.4, we get
p1(w)=pIν2;ς;ψΦ1(w,p1(w),p2(w))(n∑i=1pIηi,ς,ψHi(w,p1(w),p2(w))+pIν1;ς;ψQ1(w))+c1eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ2−1ςγ2−1Γ(γ2)+c2eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ2−2ςγ2−2Γ(γ2−1),p2(w)=pIν4;ς;ψΦ2(w,p1(w),p2(w))(m∑j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))+pIν3;ς;ψQ2(w))+d1eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ4−1ςγ4−1Γ(γ4)+d2eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ4−2ςγ4−2Γ(γ4−1). | (2.6) |
Applying the conditions p1(t1)=p2(t1)=0 in (2.6), we get c2=d2=0 since γ2∈[ν2,2] and γ4∈[ν4,2]. Thus we have
p1(w)=pIν2;ς;ψ(Φ1(w,p1(w),p2(w))(n∑i=1pIηi,ς,ψHi(w,p1(w),p2(w))+pIν1;ς;ψQ1(w)))+c1eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ2−1ςγ2−1Γ(γ2),p2(w)=pIν4;ς;ψ(Φ2(w,p1(w),p2(w))(m∑j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))+pIν3;ς;ψQ2(w)))+d1eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ4−1ςγ4−1Γ(γ4). | (2.7) |
In view of (2.7) and the conditions p1(t2)=θ1p2(ξ1) and p2(t2)=θ2p1(ξ2), we get
pIν2,ς,ψΦ1(t2,p1(t2),p2(t2))(n∑i=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψQ1(t2))+c1eς−1ς(ψ(t2)−ψ(t1))(ψ(t2)−ψ(t1))γ2−1ςγ2−1Γ(γ2)=θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))(m∑j=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψQ2(ξ1)))+d1θ1eς−1ς(ψ(ξ1)−ψ(t1))(ψ(ξ1)−ψ(t1))γ4−1ςγ4−1Γ(γ4), | (2.8) |
and
pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))(m∑j=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+Iν3,ς,ψQ2(t2))+d1eς−1ς(ψ(t2)−ψ(t1))(ψ(t2)−ψ(t1))γ4−1ςγ2−1Γ(γ2)=θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))(n∑i=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψQ1(ξ2)))+c1θ2eς−1ς(ψ(ξ2)−ψ(t1))(ψ(ξ2)−ψ(t1))γ2−1ςγ2−1Γ(γ2). | (2.9) |
Due to (2.3), (2.8), and (2.9), we have
c1M1−d1M2=M,−c1N1+d1N2=N, | (2.10) |
where
M=θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))(m∑j=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψQ2(ξ1)))−pIν2,ς,ψΦ1(t2,p1(t2),p2(t2))(n∑i=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψQ1(t2)),N=θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))(n∑i=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψQ1(ξ2)))−pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))(m∑j=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+pIν3,ς,ψQ2(t2)). |
By solving the above system, we conclude that
c1=1Θ[N2M+M2N],d1=1Θ[M1N+N1M]. |
Replacing the values c1 and d1 in Eq (2.7), we obtain the solutions (2.1) and (2.2). The converse is obtained by direct computation. The proof is complete.
Let Y=C([t1,t2],R)={p:[t1,t2]⟶Ris continuous}. The space Y is a Banach space with the norm ‖p‖=supw∈[t1,t2]|p(w)|. Obviously, the space (Y×Y,‖(p1,p2)‖) is also a Banach space with the norm ‖(p1,p2)‖=‖p1‖+‖p2‖.
Due to Lemma 2.5, we define an operator V:Y×Y→Y×Y by
V(p1,p2)(w)=(V1(p1,p2)(w)V2(p1,p2)(w)), | (3.1) |
where
V1(p1,p2)(w)=pIν2,ς,ψΦ1(w,p1(w),p2(w))(n∑i=1pIηi,ς,ψHi(w,p1(w),p2(w))+pIν1,ς,ψΥ1(w,p1(w),p2(w)))+eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ2−1Θςγ2−1Γ(γ2){N2[θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))×(m∑j=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψΥ2(ξ1,p1(ξ1),p2(ξ1)))−pIν2,ς,ψΦ1(t2,p1(t2),p2(t2))(n∑i=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψΥ1(t2,p1(t2),p2(t2)))]+M2[θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))×(n∑i=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψΥ1(ξ2,p1(ξ1),p2(ξ2)))−pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))(m∑j=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+pIν3,ς,ψΥ2(t2,p1(t2),p2(t2)))]},w∈[t1,t2], |
and
V2(p1,p2)(w)=pIν4,ς,ψΦ2(w,p1(w),p2(w))(m∑j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))+pIν3,ς,ψΥ2(w,p1(w),p2(w)))+eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ4−1Θςγ4−1Γ(γ4){N1[θ1pIν4,ς,ψΦ2(ξ1,p1(ξ1),p2(ξ1))×(m∑j=1pI¯ηj,ς,ψGj(ξ1,p1(ξ1),p2(ξ1))+pIν3,ς,ψΥ2(ξ1,p1(ξ1),p2(ξ1))))−pIν2,ς,ψΦ1(t2,p1(t2),p2(t2))(n∑i=1pIηi,ς,ψHi(t2,p1(t2),p2(t2))+pIν1,ς,ψΥ1(t2,p1(t2),p2(t2)))]+M1[θ2pIν2,ς,ψΦ1(ξ2,p1(ξ2),p2(ξ2))×(n∑i=1pIηi,ς,ψHi(ξ2,p1(ξ2),p2(ξ2))+pIν1,ς,ψΥ1(ξ2,p1(ξ2),p2(ξ2))))−pIν4,ς,ψΦ2(t2,p1(t2),p2(t2))(m∑j=1pI¯ηj,ς,ψGj(t2,p1(t2),p2(t2))+pIν3,ς,ψΥ2(t2,p1(t2),p2(t2)))]},w∈[t1,t2]. |
To prove our main result we will use the following Burton's version of Krasnosel'ski˘i's fixed-point theorem.
Lemma 3.1. [28] Let S be a nonempty, convex, closed, and bounded set of a Banach space (X,‖⋅‖) and let A:X→X and B:S→X be two operators which satisfy the following:
(i) A is a contraction,
(ii) B is completely continuous, and
(iii) x=Ax+By,∀y∈S⇒x∈S.
Then there exists a solution of the operator equation x=Ax+Bx.
Theorem 3.2. Assume that:
(H1) The functions Φk:[t1,t2]×R2→R∖{0}, Υk:[t1,t2]×R2→R for k=1,2 and hi,gj:[t1,t2]×R2→R for i=1,2,…,n,j=1,2,…,m, are continuous and there exist positive continuous functions ϕk, ωk:[t1,t2]→R, k=1,2, hi:[t1,t2]→R, gj:[t1,t2]→R i=1,2,…,nj=1,2,…,m, with bounds ‖ϕk‖, ‖ωk‖, k=1,2, and ‖hi‖, i=1,2,…,m, ‖gj‖,j=1,2,…,m, respectively, such that
|Φ1(w,u1,u2)−Φ1(w,¯u1,¯u2)|≤ϕ1(w)(|u1−¯u1|+|u2−¯u2|),|Φ2(w,u1,u2)−Φ2(w,¯u1,¯u2)|≤ϕ2(w)(|u1−¯u1|+|u2−¯u2|),|Υ1(w,u1,u2)−Υ1(w,¯u1,¯u2|≤ω1(w)(|u1−¯u1|+|u2−¯u2|),|Υ2(w,u1,u2)−Υ2(w,¯u1,¯u2|≤ω2(w)(|u1−¯u1|+|u2−¯u2|),|Hi(w,u1,u2)−Hi(w,¯u1,¯u2)|≤hi(w)(|u1−¯u1|+|u2−¯u2|),|Gj(w,u1,u2)−Gj(w,¯u1,¯u2)|≤gj(w)(|u1−¯u1|+|u2−¯u2|), | (3.2) |
for all w∈[t1,t2] and ui,¯ui∈R, i=1,2.
(H2) There exist continuous functions Fk,Lk,k=1,2, λi,μj,i=1,2,…,n,j=1,2,…,m such that
|Φ1(w,u1,u2)|≤F1(w),|Φ2(w,u1,u2)|≤F2(w),|Hi(w,u1,u2)|≤λi(w),|Gj(w,u1,u2)|≤μj(w),|Υ1(w,u1,u2)|≤L1(w),|Υ2(w,u1,u2)|≤L2(w), | (3.3) |
for all w∈[t1,t2] and u1,u2∈R.
(H3) Assume that
K:={(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)[1+(N2+M2|θ2|)(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)]+(N1+M1|θ2|)(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)(ψ(t2)−ψ(t1))γ4−1Θςγ4−1Γ(γ4)}×[‖F1‖n∑i=1‖hi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)+n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)‖ϕ1‖]+{(N2|θ1|+M2)(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)+(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)[1+(N1|θ1|+M1)(ψ(t2)−ψ(t1))γ4−1Θςγ4−1Γ(γ4)]}×[‖F2‖m∑j=1‖gj‖(ψ(t2)−ψ(t1))¯ηjς¯ηjΓ(¯ηj+1)+m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηjΓ(¯ηj+1)‖ϕ2‖]<1, |
where ‖Fk‖=supt∈[t1,t2]|Fk(t)|, ‖Lk‖=supt∈[t1,t2],k=1,2, ‖λi‖=supt∈[t1,t2], i=1,2,…,n, and ‖μj‖=supt∈[t1,t2], j=1,2,…,m.
Then the ψ-Hilfer sequential proportional coupled system (1.2) has at least one solution on [t1,t2].
Proof. First, we consider a subset S of Y×Y defined by S={(p1,p2)∈Y×Y:‖(p1,p2)‖≤r}, where r is given by
r=R1+R2 | (3.4) |
where
R1=[1+(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)(N2+M2|θ2|)]‖F1‖(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)×(n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)+n∑i=1‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1))+[N2|θ1|+M2](ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)‖F2‖(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)×(m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηΓ(¯η+1)+m∑j=1‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1)) |
and
R2=[1+(ψ(t2)−ψ(t1))γ4−1Θςγ4−1Γ(γ4)(N1|θ1|+M1)]‖F2‖(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)×(m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηΓ(¯η+1)+m∑j=1‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1))+[N1+M1|θ2|](ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)‖F1‖(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)×(n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)+n∑i=1‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)). |
Let us define the operators:
Hi(p1,p2)(w)=n∑i=1pIηi,ς,ψHi(w,p1(w),p2(w)),w∈[t1,t2], |
Gj(p1,p2)(w)=m∑j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w)),w∈[t1,t2], |
Y1(p1,p2)(w)=pIν1,ς,ψΥ1(w,p1(w),p2(w)),w∈[t1,t2], |
Y2(p1,p2)(w)=pIν3,ς,ψΥ2(w,p1(w),p2(w)),w∈[t1,t2], |
and
F1(p1,p2)(w)=Φ1(w,p1(w),p2(w)),w∈[t1,t2], |
F2(p1,p2)(w)=Φ2(w,p1(w),p2(w)),w∈[t1,t2]. |
Then we have
|Hi(¯p1,¯p2)(w)−Hi(p1,p2)(w)|≤n∑i=1pIηi,ς,ψ|Hi(w,¯p1(w),¯p2(w))−Hi(w,p1(w),p2(w))|≤n∑i=1‖hi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)(‖¯p1−p1‖+‖¯p2−p2‖) |
and
|Hi(p1,p2)(w)|≤n∑i=1pIηi,ς,ψ|Hi(w,p1(w),p2(w))|≤n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1). |
Also, we obtain
|Gj(¯p1,¯p2)(w)−Gj(p1,p2)(w)|≤m∑j=1pI¯ηj,ς,ψ|Gj(w,¯p1(w),¯p2(w))−Gj(w,p1(w),p2(w))|≤m∑j=1‖gj‖(ψ(t2)−ψ(t1))¯ηjςηiΓ(¯ηj+1)(‖¯p1−p1‖+‖¯p2−p2‖) |
and
|Gj(p1,p2)(w)|≤m∑j=1pI¯ηi,ς,ψ|Hi(w,p1(w),p2(w))|≤m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηiς¯ηiΓ(¯ηi+1). |
Moreover, we have
|Y1(¯p1,¯p2)(w)−Y1(p1,p2)(w)|≤pIν1,ς,ψ|Υ1(w,¯p1(w),¯p2(w))−Υ1(w,p1(w),p2(w))|≤‖ω1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)(‖¯p1−p1‖+‖¯p2−p2‖), |
|Y1(p1,p2)(w)|≤pIν1,ς,ψ|Υ1(w,p1(w),p2(w))|≤‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1), |
and
|Y2(¯p1,¯p2)(w)−Y2(p1,p2)(w)|≤pIν3,ς,ψ|Υ2(w,¯p1(w),¯p2(w))−Υ2(w,p1(w),p2(w))|≤‖ω2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1)(‖¯p1−p1‖+‖¯p2−p2‖), |
|Y2(p1,p2)(w)|≤pIν1,ς,ψ|Υ2(w,p1(w),p2(w))|≤‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1). |
Finally, we get
|F1(¯p1,¯p2)(w)−F1(p1,p2)(w)|≤|Φ1(w,¯p1(w),¯p2(w))−Φ1(w,p1(w),p2(w))|≤‖ϕ1‖(‖¯p1−p1‖+‖¯p2−p2‖), |
|F1(p1,p2)(w)|≤|Φ1(w,p1(w),p2(w))|≤‖F1‖, |
and
|F2(¯p1,¯p2)(w)−F2(p1,p2)(w)|≤|Φ2(w,¯p1(w),¯p2(w))−Φ2(w,p1(w),p2(w))|≤‖ϕ2‖(‖¯p1−p1‖+‖¯p2−p2‖), |
|F2(p1,p2)(w)|≤|Φ2(w,p1(w),p2(w))|≤‖F2‖. |
Now we split the operator V as
V1(p1,p2)(w)=V1,1(p1,p2)(w)+V1,2(p1,p2)(w),V2(p1,p2)(w)=V2,1(p1,p2)(w)+V2,2(p1,p2)(w), |
with
V1,1(p1,p2)(w)=pIν2,ς,ψF1(p1,p2)(w)Hi(p1,p2)(w)+eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)×{N2[θ1pIν4,ς,ψF2(p1,p2)(w)Gj(p1,p2)(w)−pIν2,ς,ψF1(p1,p2)(w)Hi(p1,p2)(w)]+M2[θ2pIν2,ς,ψF1(p1,p2)(w)Hi(p1,p2)(w)−pIν4,ς,ψF2(p1,p2)(w)Gj(p1,p2)(w)]},V1,2(p1,p2)(w)=pIν2,ς,ψF1(p1,p2)(w)Y1(p1,p2)(w)+eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)×{N2[θ1pIν4,ς,ψF2(p1,p2)(w)Y2(p1,p2)(w)−pIν2,ς,ψF1(p1,p2)(w)Y1(p1,p2)(w)]+M2[θ2pIν2,ς,ψF1(p1,p2)(w)Y1(p1,p2)(w)−pIν4,ς,ψF2(p1,p2)(w)Y2(p1,p2)(w)]},V2,1(p1,p2)(w)=pIν4,ς,ψF2(p1,p2)(w)Gj(p1,p2)(w)+eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ4−1Θςγ4−1Γ(γ4)×{N1[θ1pIν4,ς,ψF2(p1,p2)(w)Gj(p1,p2)(w)−pIν2,ς,ψF1(p1,p2)(w)Hi(p1,p2)(w)]+M1[θ2pIν2,ς,ψF1(p1,p2)(w)Hi(p1,p2)(w)−pIν4,ς,ψF2(p1,p2)(w)Gj(p1,p2)(w)]}, |
and
V2,2(p1,p2)(w)=pIν4,ς,ψF2(p1,p2)(w)Y2(p1,p2)(w)+eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ4−1Θςγ4−1Γ(γ4)×{N1[θ1pIν4,ς,ψF2(p1,p2)(w)Y2(p1,p2)(w)−pIν2,ς,ψF1(p1,p2)(w)Y1(p1,p2)(w)]+M1[θ2pIν2,ς,ψF1(p1,p2)(w)Y1(p1,p2)(w)−pIν4,ς,ψF2(p1,p2)(w)Y2(p1,p2)(w)]}. |
In the following, we will show that the operators V1 and V2 fulfill the assumptions of Lemma 3.1. We divide the proof into three steps:
Step 1. The operators V1,1 and V2,1 are contraction mappings. For all (p1,p2),(¯p1,¯p2)∈Y×Y we have
|V1,1(¯p1,¯p2)(w)−V1,1(p1,p2)(w)|≤(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)|F1(¯p1,¯p2)(w)Hi(¯p1,¯p2)(w)−F1(p1,p2)(w)Hi(p1,p2)(w)| |
+(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2){N1|θ1|(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)|F2(¯p1,¯p2)(w)Gj(¯p1,¯p2)(w) |
−F2(p1,p2)(w)Gj(p1,p2)(w)|+(N2+M2|θ2|)(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)|F1(¯p1,¯p2)(w)Hi(¯p1,¯p2)(w)−F1(p1,p2)(w)Hi(p1,p2)(w)| |
+M2(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)|F2(¯p1,¯p2)(w)Gj(¯p1,¯p2)(w)−F2(p1,p2)(w)Gj(p1,p2)(w)|}≤(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)[1+(N2+M2|θ2|)(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)] |
×|F1(¯p1,¯p2)(w)Hi(¯p1,¯p2)(w)−F1(p1,p2)(w)Hi(p1,p2)(w)|+(N2|θ1|+M2)(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)(ψ(t2)−ψ(t1))γ2−1Θςγ2−1×|F2(¯p1,¯p2)(w)Gj(¯p1,¯p2)(w)−F2(p1,p2)(w)Gj(p1,p2)(w)| |
≤(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)[1+(N2+M2|θ2|)(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)]×[|F1(¯p1,¯p2)(w)||Hi(¯p1,¯p2)(w)−Hi(p1,p2)(w)|+|Hi(p1,p2)(w)||F1(¯p1,¯p2)(w)−F1(p1,p2)(w)]+(N2|θ1|+M2)(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)(ψ(t2)−ψ(t1))γ2−1Θςγ2−1×[|F2(¯p1,¯p2)(w)||Gj(¯p1,¯p2)(w)−Gj(p1,p2)(w)| |
+|Gj(p1,p2)(w)||F2(¯p1,¯p2)(w)−F2(p1,p2)(w)|]≤(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)[1+(N2+M2|θ2|)(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)]×[‖F1‖n∑i=1‖hi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)(‖¯p1−p1‖+‖¯p2−p2‖)+n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)‖ϕ1‖(‖¯p1−p1‖+‖¯p2−p2‖)] |
+(N2|θ1|+M2)(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)(ψ(t2)−ψ(t1))γ2−1Θςγ2−1×[‖F2‖m∑j=1‖gj‖(ψ(t2)−ψ(t1))¯ηjςηiΓ(¯ηj+1)(‖¯p1−p1‖+‖¯p2−p2‖)+m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηjΓ(¯ηj+1)‖ϕ2‖(‖¯p1−p1‖+‖¯p2−p2‖)] |
={(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)[1+(N2+M2|θ2|)(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)]×[‖F1‖n∑i=1‖hi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)+n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)‖ϕ1‖]+(N2|θ1|+M2)(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)×[‖F2‖m∑j=1‖gj‖(ψ(t2)−ψ(t1))¯ηjς¯ηiΓ(¯ηj+1)+m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηjΓ(¯ηj+1)‖ϕ2‖]}×(‖¯p1−p1‖+‖¯p2−p2‖). |
Similarly we can find
|V2,1(¯p1,¯p2)(w)−V2,1(p1,p2)(w)|≤{(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)[1+(N1|θ1|+M1)(ψ(t2)−ψ(t1))γ4−1Θςγ4−1Γ(γ4)]×[‖F2‖m∑j=1‖gj‖(ψ(t2)−ψ(t1))¯ηjςηiΓ(¯ηj+1)+n∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηjΓ(¯ηj+1)‖ϕ2‖]+(N1+M1|θ2|)(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)(ψ(t2)−ψ(t1))γ4−1Θςγ4−1Γ(γ4)×[‖F1‖n∑i=1‖hi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)+n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)‖ϕ1‖]}×(‖¯p1−p1‖+‖¯p2−p2‖). |
Consequently, we get
‖(V1,1,V2,1)(¯p1,¯p2)−(V1,1,V2,1)(p1,p2)‖≤K(‖¯p1−p1‖+‖¯p2−p2‖), |
which means that (V1,1,V2,1) is a contraction.
Step 2. The operator V2=(V1,2,V2,2) is completely continuous on S. For continuity of V1,2, take any sequence of points (pn,qn) in S converging to a point (p,q)∈S. Then, by the Lebesgue dominated convergence theorem, we have
limn→∞V1,2(pn,qn)(w)=pIν2,ς,ψlimn→∞F1(pn,qn)(w)limn→∞Y1(pn,qn)(w)+eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)×{N2[θ1pIν4,ς,ψlimn→∞F2(pn,qn)(w)limn→∞Y2(pn,qn)(w)−pIν2,ς,ψlimn→∞F1(pn,qn)(w)limn→∞Y1(pn,qn)(w)]+M2[θ2pIν2,ς,ψlimn→∞F1(pn,qn)(w)limn→∞Y1(pn,qn)(w)−pIν4,ς,ψlimn→∞F2(pn,qn)(w)limn→∞Y2(pn,qn)(w)]}=pIν2,ς,ψF1(p,q)(w)Y1(p,q)(w)+eς−1ς(ψ(w)−ψ(t1))(ψ(w)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)×{N2[θ1pIν4,ς,ψF2(p,q)(w)Y2(p,q)(w)−pIν2,ς,ψF1(p,q)(w)Y1(p,q)(w)]+M2[θ2pIν2,ς,ψF1(p,q)(w)Y1(p,q)(w)−pIν4,ς,ψF2(p,q)(w)Y2(p,q)(w)]}=V1,2(p,q)(w), |
for all w∈[t1,t2]. Similarly, we prove limn→∞V2,2(pn,qn)(w)=V2,2(p,q)(w) for all w∈[t1,t2]. Thus V2(pn,qn)=(V1,2(pn,qn),V2,2(pn,qn)) converges to V2(p,q) on [t1,t2], which shows that V2 is continuous.
Next, we show that the operator (V1,2,V2,2) is uniformly bounded on S. For any (p1,p2)∈S we have
|V1,2(p1,p2)(w)|≤pIν2,ς,ψ|F1(p1,p2)(w)Y1(p1,p2)(w)|+(ψ(w)−ψ(t1))γ2−1Θςγ2−1Γ(γ2){N2[|θ1|pIν4,ς,ψ|F2(p1,p2)(w)Y2(p1,p2)(w)|+pIν2,ς,ψ|F1(p1,p2)(w)Y1(p1,p2)(w)|]+M2[|θ2|pIν2,ς,ψ|F1(p1,p2)(w)Y1(p1,p2)(w)|+pIν4,ς,ψ|F2(p1,p2)(w)Y2(p1,p2)(w)|]}≤(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)‖F1‖‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)+(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)×{N2|θ1|(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1)+N2(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)‖F1‖‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)+M2|θ2|‖F1‖‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)+M2(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1)}:=Λ1. |
Similarly we can prove that
|V2,2(p1,p2)(w)|≤(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1)+(ψ(t2)−ψ(t1))γ4−1Θςγ2−1Γ(γ2)×{N1|θ1|(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1)+N1(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)‖F1‖‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)+M1|θ2|‖F1‖n∑i=1‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)+M1(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1)}:=Λ2. |
Therefore ‖V1,2‖+‖V2,2‖≤Λ1+Λ2,(p1,p2)∈S, which shows that the operator (V1,2,V2,2) is uniformly bounded on S. Finally we show that the operator (V1,2,V2,2) is equicontinuous. Let τ1<τ2 and (p1,p2)∈S. Then, we have
|V1,2(p1,p2)(τ2)−V1,2(p1,p2)(τ1)|≤|1ςν2Γ(ν2)∫τ1t1ψ′(s)[(ψ(τ2)−ψ(s))ν2−1−(ψ(τ1)−ψ(s))ν2−1]×|F1(p1,p2)(s)Y1(p1,p2)(s)|ds+1ςν2Γ(ν2)∫τ2τ1ψ′(s)(ψ(τ2)−ψ(s))ν2−1|F1(p1,p2)(s)Y1(p1,p2)(s)|ds|+|(ψ(τ2)−ψ(t1))γ2−1−(ψ(τ1)−ψ(t1))γ2−1|Θςγ2−1Γ(γ2)W≤1ςν2Γ(ν2+1)n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)[|(ψ(τ2)−ψ(t1))ν2−(ψ(τ1)−ψ(t1))ν2|+2(ψ(τ2)−ψ(τ1))ν2]+|(ψ(τ2)−ψ(t1))γ2−1−(ψ(τ1)−ψ(t1))γ2−1|Θςγ2−1Γ(γ2)W, |
where
W=N2|θ1|(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1)+N2(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)‖F1‖‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)+M2|θ2|‖F1‖‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)+M2(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1). |
As τ2−τ1→0, the right-hand side of the above inequality tends to zero, independently of (p1,p2). Similarly we have |V2,2(p1,p2)(τ2)−V2,2(p1,p2)(τ1)|→0 as τ2−τ1→0. Thus (V1,2,V2,2) is equicontinuous. Therefore, it follows by the Arzelá-Ascoli theorem that (V1,2,V2,2) is a completely continuous operator on S.
Step 3. We show that the third condition (iii) of Lemma 3.1 is fulfilled. Let (p1,p2)∈Y×Y be such that, for all (¯p1,¯p2)∈S
(p1,p2)=(V1,1(p1,p2),V2,1(p1,p2))+(V1,2(¯p1,¯p2,V2,2(¯p1,¯p2)). |
Then, we have
|p1(w)|≤|V1,1(p1,p2)(w)|+|V1,2(¯p1,¯p2)(w)|≤pIν2,ς,ψ|F1(p1,p2)(w)Hi(p1,p2)(w)|+(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2){N2[|θ1|pIν4,ς,ψ|F2(p1,p2)(w)Gj(p1,p2)(w)|+pIν2,ς,ψ|F1(p1,p2)(w)Hi(p1,p2)(w)|]+M2[|θ2|pIν2,ς,ψ|F1(p1,p2)(w)Hi(p1,p2)(w)|+pIν4,ς,ψ|F2(p1,p2)(w)Gj(p1,p2)(w)|]}+pIν2,ς,ψ|F1(¯p1,¯p2)(w)Y1(¯p1,¯p2)(w)|+(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2){N2[|θ1|pIν4,ς,ψ|F2(¯p1,¯p2)(w)Y2(¯p1,¯p2)(w)|+pIν2,ς,ψ|F1(¯p1,¯p2)(w)Y1(¯p1,¯p2)(w)|]+M2[|θ2|pIν2,ς,ψ|F1(¯p1,¯p2)(w)Y1(¯p1,¯p2)(w)|+pIν4,ς,ψ|F2(¯p1,¯p2)(w)Y2(¯p1,¯p2)(w)|]}≤(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)‖F1‖n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)+(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)×{N2|θ1|(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηΓ(¯η+1)‖F2‖+N2(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)‖F1‖n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)+M2|θ2|‖F1‖n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)+M2(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηΓ(¯η+1)}+(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)‖F1‖n∑i=1‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)+(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)×{N2|θ1|(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖m∑j=1‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1)+N2(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)‖F1‖n∑i=1‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)+M2|θ2|‖F1‖n∑i=1‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1)+M2(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖m∑j=1‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1)}=[1+(ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)(N2+M2|θ2|)]‖F1‖(ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)×(n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)+n∑i=1‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1))+[N2|θ1|+M2](ψ(t2)−ψ(t1))γ2−1Θςγ2−1Γ(γ2)‖F2‖(ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)×(m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηΓ(¯η+1)+m∑j=1‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1))=R1. |
In a similar way, we find
|p2(w)|≤|V2,1(p1,p2)(w)|+|V2,2(¯p1,¯p2)(w)|≤[1+(ψ(t2)−ψ(t1))γ4−1Θςγ4−1Γ(γ4)(N1|θ1|+M1)](ψ(t2)−ψ(t1))ν4ςν4Γ(ν4+1)‖F2‖×(m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηΓ(¯η+1)+m∑j=1‖L2‖(ψ(t2)−ψ(t1))ν3ςν3Γ(ν3+1))+[N1+M1|θ2|](ψ(t2)−ψ(t1))ν2ςν2Γ(ν2+1)‖F1‖(ψ(t2)−ψ(t1))γ4−1ςγ4−1Γ(γ4)×(n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)+n∑i=1‖L1‖(ψ(t2)−ψ(t1))ν1ςν1Γ(ν1+1))=R2. |
Adding the previous inequalities, we obtain
‖p1‖+‖p2‖≤R1+R2=r. |
As ‖(p1,p2)‖=‖p1‖+‖p2‖, we have that ‖(p1,p2)‖≤r and so condition (iii) of Lemma 3.1 holds.
By Lemma 3.1, the ψ-Hilfer sequential proportional coupled system (1.2) has at least one solution on [t1,t2]. The proof is finished.
Let us consider the following coupled system of nonlinear sequential proportional Hilfer fractional differential equations with multi-point boundary conditions:
{HD13,15;37;logw[HD54,25;37;logwp1(w)Φ1(w,p1(w),p2(w))−2∑i=1pIηi,ς,ψHi(w,p1(w),p2(w))]=Υ1(w,p1(w),p2(w)),w∈[12,72],HD23,35;37;logw[HD74,45;37;logwp1(w)Φ2(w,p1(w),p2(w))−2∑j=1pI¯ηj,ς,ψGj(w,p1(w),p2(w))]=Υ2(w,p1(w),p2(w)),w∈[12,72],p1(12)=HD54,25;37;logwp1(12)=0,p1(72)=25p2(32),p2(12)=HD74,45;37;logwp2(12)=0,p2(72)=23p1(52), | (4.1) |
where
2∑i=1pIηi,ς,ψHi(w,p1,p2)=2∑i=1pI2(i+1)5,37,logw(|p1|(w+i2)(i+|p1|)+|p2|(w+i3)(i+|p2|)),2∑j=1pI¯ηj,ς,ψGj(w,p1,p2)=2∑j=1pI2(j+1)7,37,logw(|p1|(w2+j2)(j+|p1|)+|p2|(w2+j3)(j+|p2|)),Φ1(w,p1,p2)=1100(10w+255)(|p1|1+|p1|+|p2|1+|p2|+12),Φ2(w,p1,p2)=25(2w+99)2(|p1|1+|p1|+|p2|1+|p2|+14),Υ1(w,p1,p2)=1√w+2(|p1|3+|p1|)+12(√w+1)sin|p2|+13,Υ2(w,p1,p2)=1w2+4(12tan−1|p1|+|p2|2+|p2|)+15. |
Next, we can choose ν1=1/3, ν2=5/4, ν3=2/3, ν4=7/4, ϑ1=1/5, ϑ2=2/5, ϑ3=3/5, ϑ4=4/5, ς=3/7, ψ(w):=logw=logew, t1=1/2, t2=7/2, θ1=2/5, and θ2=2/3. Then, we have γ1=7/15, γ2=31/20, γ3=13/15, γ4=39/20, M1≈0.1930945138, M2≈0.2307306625, N1≈0.1816223751, N2≈0.3208292984, and Θ≈0.02004452646. Now, we analyse the nonlinear functions in the fractional integral terms. We have
|Hi(w,p1,p2)−Hi(w,¯p1,¯p2)|≤1i(w+i2)(|p1−¯p1|+|p2−¯p2|) |
and
|Gj(w,p1,p2)−Gj(w,¯p1,¯p2)|≤1j(w2+j2)(|p1−¯p1|+|p2−¯p2|), |
from which hi(w)=1/(i(w+i2)) and gj(w)=1/(j(w2+j2)), respectively. Both of them are bounded as
|Hi(w,p1,p2)|≤2w+i2and|Gj(w,p1,p2)|≤2w2+j2. |
Therefore λi(w)=2/(w+i2) and μj=2/(w2+j2). Moreover, we have
n∑i=1‖hi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)≈3.021061781, |
n∑i=1‖λi‖(ψ(t2)−ψ(t1))ηiςηiΓ(ηi+1)≈7.281499952, |
m∑j=1‖gj‖(ψ(t2)−ψ(t1))¯ηjς¯ηjΓ(¯ηi+1)≈2.776491121 |
and
m∑j=1‖μj‖(ψ(t2)−ψ(t1))¯ηjς¯ηjΓ(¯ηi+1)≈7.220966978. |
For the two non-zero functions Φ1 and Φ2 we have
|Φ1(w,p1,p2)−Φ1(w,¯p1,¯p2)|≤1100(10w+255)(|p1−¯p1|+|p2−¯p2|),|Φ2(w,p1,p2)−Φ2(w,¯p1,¯p2)|≤25(2w+99)2(|p1−¯p1|+|p2−¯p2|), |
|Φ1(w,p1,p2)|≤140(10w+255),and|Φ2(w,p1,p2)|≤910(2w+99)2, |
from which we get ‖ϕ1‖=1/26000, ‖ϕ2‖=1/25000, ‖F1‖=1/10400, ‖F2‖=9/100000, by setting ϕ1(w)=1/(100(10w+255)), ϕ2(w)=2/(5(2w+99)2), F1(w)=1/(40(10w+255)), and F2(w)=9/(10(2w+99)2), respectively.
Finally, for the nonlinear functions of the right sides in problem (4.1) we have
|Υ1(w,p1,p2)−Υ1(w,¯p1,¯p2)|≤12(√w+1)(|p1−¯p1|+|p2−¯p2|),|Υ2(w,p1,p2)−Υ2(w,¯p1,¯p2)|≤12(w2+4)(|p1−¯p1|+|p2−¯p2|), |
which give ω1(w)=1/(2(√w+1)), ω2(w)=1/(2(w2+4)) and
|Υ1(w,p1,p2)|≤1√w+2+12(√w+1)+13:=L1(w), |
and
|Υ2(w,p1,p2)|≤1w2+4(π4+1)+15:=L2(w). |
Therefore, using all of the information to compute a constant K in assumption (H3) of Theorem 3.2, we obtain
K≈0.9229566975<1. |
Hence, the given coupled system of nonlinear proportional Hilfer-type fractional differential equations with multi-point boundary conditions (4.1), satisfies all assumptions in Theorem 3.2. Then, by its conclusion, there exists at least one solution (p1,p2)(w) to the problem (4.1) where w∈[1/2,7/2].
In this paper, we have presented the existence result for a new class of coupled systems of ψ-Hilfer proportional sequential fractional differential equations with multi-point boundary conditions. The proof of the existence result was based on a generalization of Krasnosel'ski˘i's fixed-point theorem due to Burton. An example was presented to illustrate our main result. Some special cases were also discussed. In future work, we can implement these techniques on different boundary value problems equipped with complicated integral multi-point boundary conditions.
The authors declare that they have not used artificial intelligence (AI) tools in the creation of this article.
This research was funded by the National Science, Research and Innovation Fund (NSRF) and King Mongkut's University of Technology North Bangkok with contract no. KMUTNB-FF-66-11.
Professor Sotiris K. Ntouyas is an editorial board member for AIMS Mathematics and was not involved in the editorial review or the decision to publish this article. The authors declare no conflicts of interest.
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