In recent years, the frequency adjustment of U.S. monetary policy has a dynamic and global impact on other countries' economy. Based on the financial conditions index (FCI), the paper employs the time-varying parameter vector autoregressive model with stochastic volatility (TVP-VAR-SV) and spillover index respectively to investigate the time-varying impact of U.S. financial conditions (UFCI) on China's inflation (CINF) and its impact mechanisms. Some results are achieved as follows: first, the impacts of UFCI on CINF vary greatly over time both in the dimension of action duration and time point. Second, the effects of UFCI on CINF directly relate to different types of major events, and they are heterogeneous in action duration, degree, direction as well as the trend and range of fluctuations. In addition, UFCI can work on CINF through trade flow and China's financial market, and the China's financial market plays a main conductive role, and its conductive effect changes over time.
Citation: Yanhong Feng, Shuanglian Chen, Wang Xuan, Tan Yong. Time-varying impact of U.S. financial conditions on China's inflation: a perspective of different types of events[J]. Quantitative Finance and Economics, 2021, 5(4): 604-622. doi: 10.3934/QFE.2021027
[1] | Ridha Dida, Hamid Boulares, Bahaaeldin Abdalla, Manar A. Alqudah, Thabet Abdeljawad . On positive solutions of fractional pantograph equations within function-dependent kernel Caputo derivatives. AIMS Mathematics, 2023, 8(10): 23032-23045. doi: 10.3934/math.20231172 |
[2] | Hamid Boulares, Manar A. Alqudah, Thabet Abdeljawad . Existence of solutions for a semipositone fractional boundary value pantograph problem. AIMS Mathematics, 2022, 7(10): 19510-19519. doi: 10.3934/math.20221070 |
[3] | Reny George, Fahad Al-shammari, Mehran Ghaderi, Shahram Rezapour . On the boundedness of the solution set for the ψ-Caputo fractional pantograph equation with a measure of non-compactness via simulation analysis. AIMS Mathematics, 2023, 8(9): 20125-20142. doi: 10.3934/math.20231025 |
[4] | Saeed M. Ali, Mohammed S. Abdo, Bhausaheb Sontakke, Kamal Shah, Thabet Abdeljawad . New results on a coupled system for second-order pantograph equations with ABC fractional derivatives. AIMS Mathematics, 2022, 7(10): 19520-19538. doi: 10.3934/math.20221071 |
[5] | Weerawat Sudsutad, Chatthai Thaiprayoon, Aphirak Aphithana, Jutarat Kongson, Weerapan Sae-dan . Qualitative results and numerical approximations of the (k,ψ)-Caputo proportional fractional differential equations and applications to blood alcohol levels model. AIMS Mathematics, 2024, 9(12): 34013-34041. doi: 10.3934/math.20241622 |
[6] | Choukri Derbazi, Zidane Baitiche, Mohammed S. Abdo, Thabet Abdeljawad . Qualitative analysis of fractional relaxation equation and coupled system with Ψ-Caputo fractional derivative in Banach spaces. AIMS Mathematics, 2021, 6(3): 2486-2509. doi: 10.3934/math.2021151 |
[7] | Abdelkader Moumen, Ramsha Shafqat, Zakia Hammouch, Azmat Ullah Khan Niazi, Mdi Begum Jeelani . Stability results for fractional integral pantograph differential equations involving two Caputo operators. AIMS Mathematics, 2023, 8(3): 6009-6025. doi: 10.3934/math.2023303 |
[8] | Iyad Suwan, Mohammed S. Abdo, Thabet Abdeljawad, Mohammed M. Matar, Abdellatif Boutiara, Mohammed A. Almalahi . Existence theorems for Ψ-fractional hybrid systems with periodic boundary conditions. AIMS Mathematics, 2022, 7(1): 171-186. doi: 10.3934/math.2022010 |
[9] | Mohamed Houas, Kirti Kaushik, Anoop Kumar, Aziz Khan, Thabet Abdeljawad . Existence and stability results of pantograph equation with three sequential fractional derivatives. AIMS Mathematics, 2023, 8(3): 5216-5232. doi: 10.3934/math.2023262 |
[10] | Karim Guida, Lahcen Ibnelazyz, Khalid Hilal, Said Melliani . Existence and uniqueness results for sequential ψ-Hilfer fractional pantograph differential equations with mixed nonlocal boundary conditions. AIMS Mathematics, 2021, 6(8): 8239-8255. doi: 10.3934/math.2021477 |
In recent years, the frequency adjustment of U.S. monetary policy has a dynamic and global impact on other countries' economy. Based on the financial conditions index (FCI), the paper employs the time-varying parameter vector autoregressive model with stochastic volatility (TVP-VAR-SV) and spillover index respectively to investigate the time-varying impact of U.S. financial conditions (UFCI) on China's inflation (CINF) and its impact mechanisms. Some results are achieved as follows: first, the impacts of UFCI on CINF vary greatly over time both in the dimension of action duration and time point. Second, the effects of UFCI on CINF directly relate to different types of major events, and they are heterogeneous in action duration, degree, direction as well as the trend and range of fluctuations. In addition, UFCI can work on CINF through trade flow and China's financial market, and the China's financial market plays a main conductive role, and its conductive effect changes over time.
Recently, fractional calculus methods became of great interest, because it is a powerful tool for calculating the derivation of multiples systems. These methods study real world phenomena in many areas of natural sciences including biomedical, radiography, biology, chemistry, and physics [1,2,3,4,5,6,7]. Abundant publications focus on the Caputo fractional derivative (CFD) and the Caputo-Hadamard derivative. Additionally, other generalization of the previous derivatives, such as Ψ-Caputo, study the existence of solutions to some FDEs (see [8,9,10,11,12,13,14]).
In general, an m-point fractional boundary problem involves a fractional differential equation with fractional boundary conditions that are specified at m different points on the boundary of a domain. The fractional derivative is defined using the Riemann-Liouville fractional derivative or the Caputo fractional derivative. Solving these types of problems can be challenging due to the non-local nature of fractional derivatives. However, there are various numerical and analytical methods available for solving such problems, including the spectral method, the finite difference method, the finite element method, and the homotopy analysis method. The applications of m-point fractional boundary problems can be found in various fields, including physics, engineering, finance, and biology. These problems are useful in modeling and analyzing phenomena that exhibit non-local behavior or involve memory effects (see [15,16,17,18]).
Pantograph equations are a set of differential equations that describe the motion of a pantograph, which is a mechanism used for copying and scaling drawings or diagrams. The equations are based on the assumption that the pantograph arms are rigid and do not deform during operation, we can simply say that see [19]. One important application of the pantograph equations is in the field of drafting and technical drawing. Before the advent of computer-aided design (CAD) software, pantographs were commonly used to produce scaled copies of drawings and diagrams. By adjusting the lengths of the arms and the position of the stylus, a pantograph can produce copies that are larger or smaller than the original [20], electrodynamics [21] and electrical pantograph of locomotive [22].
Many authors studied a huge number of positive solutions for nonlinear fractional BVP using fixed point theorems (FPTs) such as SFPT, Leggett-Williams and Guo-Krasnosel'skii (see [23,24]). Some studies addressed the sign-changing of solution of BVPs [25,26,27,28,29].
In this work, we use Schauder's fixed point theorem (SFPT) to solve the semipostone multipoint Ψ-Caputo fractional pantograph problem
Dν;ψrϰ(ς)+F(ς,ϰ(ς),ϰ(r+λς))=0, ς in (r,ℑ) | (1.1) |
ϰ(r)=ϑ1, ϰ(ℑ)=m−2∑i=1ζiϰ(ηi)+ϑ2, ϑi∈R, i∈{1,2}, | (1.2) |
where λ∈(0,ℑ−rℑ),Dν;ψr is Ψ-Caputo fractional derivative (Ψ-CFD) of order ν, 1<ν≤2, ζi∈R+(1≤i≤m−2) such that 0<Σm−2i=1ζi<1, ηi∈(r,ℑ), and F:[r,ℑ]×R×R→R.
The most important aspect of this research is to prove the existence of a positive solution of the above m-point FBVP. Note that in [30], the author considered a two-point BVP using Liouville-Caputo derivative.
The article is organized as follows. In the next section, we provide some basic definitions and arguments pertinent to fractional calculus (FC). Section 3 is devoted to proving the the main result and an illustrative example is given in Section 4.
In the sequel, Ψ denotes an increasing map Ψ:[r1,r2]→R via Ψ′(ς)≠0, ∀ ς, and [α] indicates the integer part of the real number α.
Definition 2.1. [4,5] Suppose the continuous function ϰ:(0,∞)→R. We define (RLFD) the Riemann-Liouville fractional derivative of order α>0,n=[α]+1 by
RLDα0+ϰ(ς)=1Γ(n−α)(ddς)n∫ς0(ς−τ)n−α−1ϰ(τ)dτ, |
where n−1<α<n.
Definition 2.2. [4,5] The Ψ-Riemann-Liouville fractional integral (Ψ-RLFI) of order α>0 of a continuous function ϰ:[r,ℑ]→R is defined by
Iα;Ψrϰ(ς)=∫ςr(Ψ(ς)−Ψ(τ))α−1Γ(α)Ψ′(τ)ϰ(τ)dτ. |
Definition 2.3. [4,5] The CFD of order α>0 of a function ϰ:[0,+∞)→R is defined by
Dαϰ(ς)=1Γ(n−α)∫ς0(ς−τ)n−α−1ϰ(n)(τ)dτ, α∈(n−1,n),n∈N. |
Definition 2.4. [4,5] We define the Ψ-CFD of order α>0 of a continuous function ϰ:[r,ℑ]→R by
Dα;Ψrϰ(ς)=∫ςr(Ψ(ς)−Ψ(τ))n−α−1Γ(n−α)Ψ′(τ)∂nΨϰ(τ)dτ, ς>r, α∈(n−1,n), |
where ∂nΨ=(1Ψ′(ς)ddς)n,n∈N.
Lemma 2.1. [4,5] Suppose q,ℓ>0, and ϰinC([r,ℑ],R). Then ∀ς∈[r,ℑ] and by assuming Fr(ς)=Ψ(ς)−Ψ(r), we have
1) Iq;ΨrIℓ;Ψrϰ(ς)=Iq+ℓ;Ψrϰ(ς),
2) Dq;ΨrIq;Ψrϰ(ς)=ϰ(ς),
3) Iq;Ψr(Fr(ς))ℓ−1=Γ(ℓ)Γ(ℓ+q)(Fr(ς))ℓ+q−1,
4) Dq;Ψr(Fr(ς))ℓ−1=Γ(ℓ)Γ(ℓ−q)(Fr(ς))ℓ−q−1,
5) Dq;Ψr(Fr(ς))k=0, k=0,…,n−1, n∈N, qin(n−1,n].
Lemma 2.2. [4,5] Let n−1<α1≤n,α2>0, r>0, ϰ∈L(r,ℑ), Dα1;Ψrϰ∈L(r,ℑ). Then the differential equation
Dα1;Ψrϰ=0 |
has the unique solution
ϰ(ς)=W0+W1(Ψ(ς)−Ψ(r))+W2(Ψ(ς)−Ψ(r))2+⋯+Wn−1(Ψ(ς)−Ψ(r))n−1, |
and
Iα1;ΨrDα1;Ψrϰ(ς)=ϰ(ς)+W0+W1(Ψ(ς)−Ψ(r))+W2(Ψ(ς)−Ψ(r))2+⋯+Wn−1(Ψ(ς)−Ψ(r))n−1, |
with Wℓ∈R, ℓ∈{0,1,…,n−1}.
Furthermore,
Dα1;ΨrIα1;Ψrϰ(ς)=ϰ(ς), |
and
Iα1;ΨrIα2;Ψrϰ(ς)=Iα2;ΨrIα1;Ψrϰ(ς)=Iα1+α2;Ψrϰ(ς). |
Here we will deal with the FDE solution of (1.1) and (1.2), by considering the solution of
−Dν;ψrϰ(ς)=h(ς), | (2.1) |
bounded by the condition (1.2). We set
Δ:=Ψ(ℑ)−Ψ(r)−Σm−2i=1ζi(Ψ(ηi)−Ψ(r)). |
Lemma 2.3. Let ν∈(1,2] and ς∈[r,ℑ]. Then, the FBVP (2.1) and (1.2) have a solution ϰ of the form
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)h(τ)Ψ′(τ)dτ, |
where
ϖ(ς,τ)=1Γ(ν){[(Ψ(ℑ)−Ψ(r))ν−1−Σm−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1]Ψ(ς)−Ψ(r)Δ−(Ψ(ς)−Ψ(τ))ν−1,τ≤ς,ηi−1<τ≤ηi,[(Ψ(ℑ)−Ψ(τ))ν−1−Σm−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1]Ψ(ℑ)−Ψ(r)Δ,ς≤τ,ηi−1<τ≤ηi, | (2.2) |
i=1,2,...,m−2.
Proof. According to the Lemma 2.2 the solution of Dν;ψrϰ(ς)=−h(ς) is given by
ϰ(ς)=−1Γ(ν)∫ςr(Ψ(ς)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+c0+c1(Ψ(ς)−Ψ(r)), | (2.3) |
where c0,c1∈R. Since ϰ(r)=ϑ1 and ϰ(ℑ)=∑m−2i=1ζiϰ(ηi)+ϑ2, we get c0=ϑ1 and
c1=1Δ(−1Γ(ν)m−2∑i=1ζi∫ηjr(Ψ(ηi)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+1Γ(ν)∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+ϑ1[m−2∑i=1ζi−1]+ϑ2). |
By substituting c0,c1 into Eq (2.3) we find,
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2−1Γ(ν)(∫ςr(Ψ(ς)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ+(Ψ(ς)−Ψ(r))Δm−2∑i=1ζi∫ηjr(Ψ(ηi)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ−Ψ(ς)−Ψ(r)Δ∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1h(τ)Ψ′(τ)dτ)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2+∫ℑrϖ(ς,τ)h(τ)Ψ′(τ)dτ, |
where ϖ(ς,τ) is given by (2.2). Hence the required result.
Lemma 2.4. If 0<∑m−2i=1ζi<1, then
i) Δ>0,
ii) (Ψ(ℑ)−Ψ(τ))ν−1−∑m−2j=iζj(Ψ(ηj)−Ψ(τ))ν−1>0.
Proof. i) Since ηi<ℑ, we have
ζi(Ψ(ηi)−Ψ(r))<ζi(Ψ(ℑ)−Ψ(r)), |
−m−2∑i=1ζi(Ψ(ηi)−Ψ(r))>−m−2∑i=1ζi(Ψ(ℑ)−Ψ(r)), |
Ψ(ℑ)−Ψ(r)−m−2∑i=1ζi(Ψ(ηi)−Ψ(r))>Ψ(ℑ)−Ψ(r)−m−2∑i=1ζi(Ψ(ℑ)−Ψ(r))=(Ψ(ℑ)−Ψ(r))[1−m−2∑i=1ζi]. |
If 1−Σm−2i=1ζi>0, then (Ψ(ℑ)−Ψ(r))−Σm−2i=1ζi(Ψ(ηi)−Ψ(r))>0. So we have Δ>0.
ii) Since 0<ν−1≤1, we have (Ψ(ηi)−Ψ(τ))ν−1<(Ψ(ℑ)−Ψ(τ))ν−1. Then we obtain
m−2∑j=iζj(Ψ(ηj)−Ψ(τ))ν−1<m−2∑j=iζj(Ψ(ℑ)−Ψ(τ))ν−1≤(Ψ(ℑ)−Ψ(τ))ν−1m−2∑i=1ζi<(Ψ(ℑ)−Ψ(τ))ν−1, |
and so
(Ψ(ℑ)−Ψ(τ))ν−1−m−2∑j=iζj(Ψ(ηj)−Ψ(τ))ν−1>0. |
Remark 2.1. Note that ∫ℑrϖ(ς,τ)Ψ′(τ)dτ is bounded ∀ς∈[r,ℑ]. Indeed
∫ℑr|ϖ(ς,τ)|Ψ′(τ)dτ≤1Γ(ν)∫ςr(Ψ(ς)−Ψ(τ))ν−1Ψ′(τ)dτ+Ψ(ς)−Ψ(r)Γ(ν)Δm−2∑i=1ζi∫ηir(Ψ(ηj)−Ψ(τ))ν−1Ψ′(τ)dτ+Ψ(ς)−Ψ(r)ΔΓ(ν)∫ℑr(Ψ(ℑ)−Ψ(τ))ν−1Ψ′(τ)dτ=(Ψ(ς)−Ψ(r))νΓ(ν+1)+Ψ(ς)−Ψ(r)ΔΓ(ν+1)m−2∑i=1ζi(Ψ(ηi)−Ψ(r))ν+Ψ(ς)−Ψ(r)ΔΓ(ν+1)(Ψ(ℑ)−Ψ(r))ν≤(Ψ(ℑ)−Ψ(r))νΓ(ν+1)+Ψ(ℑ)−Ψ(r)ΔΓ(ν+1)m−2∑i=1ζi(Ψ(ηi)−Ψ(r))ν+(Ψ(ℑ)−Ψ(r))ν+1ΔΓ(ν+1)=M. | (2.4) |
Remark 2.2. Suppose Υ(ς)∈L1[r,ℑ], and w(ς) verify
{Dν;ψrw(ς)+Υ(ς)=0,w(r)=0, w(ℑ)=Σm−2i=1ζiw(ηi), | (2.5) |
then w(ς)=∫ℑrϖ(ς,τ)Υ(τ)Ψ′(τ)dτ.
Next we recall the Schauder fixed point theorem.
Theorem 2.1. [23] [SFPT] Consider the Banach space Ω. Assume ℵ bounded, convex, closed subset in Ω. If ϝ:ℵ→ℵ is compact, then it has a fixed point in ℵ.
We start this section by listing two conditions which will be used in the sequel.
● (Σ1) There exists a nonnegative function Υ∈L1[r,ℑ] such that ∫ℑrΥ(ς)dς>0 and F(ς,ϰ,v)≥−Υ(ς) for all (ς,ϰ,v)∈[r,ℑ]×R×R.
● (Σ2) G(ς,ϰ,v)≠0, for (ς,ϰ,v)∈[r,ℑ]×R×R.
Let ℵ=C([r,ℑ],R) the Banach space of CFs (continuous functions) with the following norm
‖ϰ‖=sup{|ϰ(ς)|:ς∈[r,ℑ]}. |
First of all, it seems that the FDE below is valid
Dν;ψrϰ(ς)+G(ς,ϰ∗(ς),ϰ∗(r+λς))=0, ς∈[r,ℑ]. | (3.1) |
Here the existence of solution satisfying the condition (1.2), such that G:[r,ℑ]×R×R→R
G(ς,z1,z2)={F(ς,z1,z2)+Υ(ς), z1,z2≥0,F(ς,0,0)+Υ(ς), z1≤0 or z2≤0, | (3.2) |
and ϰ∗(ς)=max{(ϰ−w)(ς),0}, hence the problem (2.5) has w as unique solution. The mapping Q:ℵ→ℵ accompanied with the (3.1) and (1.2) defined as
(Qϰ)(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)G(ς,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ, | (3.3) |
where the relation (2.2) define ϖ(ς,τ). The existence of solution of the problems (3.1) and (1.2) give the existence of a fixed point for Q.
Theorem 3.1. Suppose the conditions (Σ1) and (Σ2) hold. If there exists ρ>0 such that
[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≤ρ, |
where L≥max{|G(ς,ϰ,v)|:ς∈[r,ℑ], |ϰ|,|v|≤ρ} and M is defined in (2.4), then, the problems (3.1) and (3.2) have a solution ϰ(ς).
Proof. Since P:={ϰ∈ℵ:‖ϰ‖≤ρ} is a convex, closed and bounded subset of B described in the Eq (3.3), the SFPT is applicable to P. Define Q:P→ℵ by (3.3). Clearly Q is continuous mapping. We claim that range of Q is subset of P. Suppose ϰ∈P and let ϰ∗(ς)≤ϰ(ς)≤ρ, ∀ς∈[r,ℑ]. So
|Qϰ(ς)|=|[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+∫ℑrϖ(ς,τ)G(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ|≤[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≤ρ, |
for all ς∈[r,ℑ]. This indicates that ‖Qϰ‖≤ρ, which proves our claim. Thus, by using the Arzela-Ascoli theorem, Q:ℵ→ℵ is compact. As a result of SFPT, Q has a fixed point ϰ in P. Hence, the problems (3.1) and (1.2) has ϰ as solution.
Lemma 3.1. ϰ∗(ς) is a solution of the FBVP (1.1), (1.2) and ϰ(ς)>w(ς) for every ς∈[r,ℑ] iff the positive solution of FBVP (3.1) and (1.2) is ϰ=ϰ∗+w.
Proof. Let ϰ(ς) be a solution of FBVP (3.1) and (1.2). Then
ϰ(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+(Ψ(ς)−Ψ(r))Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)G(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)(F(τ,ϰ∗(τ),ϰ∗(r+λτ))+p(τ))Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ+1Γ(ν)∫ℑrϖ(ς,τ)p(τ)Ψ′(τ)dτ=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)G(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ+w(ς). |
So,
ϰ(ς)−w(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,(ϰ−w)(τ),(ϰ−w)(r+λτ))Ψ′(τ)dτ. |
Then we get the existence of the solution with the condition
ϰ∗(ς)=[1+Σm−2i=1ζi−1Δ(Ψ(ς)−Ψ(r))]ϑ1+Ψ(ς)−Ψ(r)Δϑ2+1Γ(ν)∫ℑrϖ(ς,τ)F(τ,ϰ∗(τ),ϰ∗(r+λτ))Ψ′(τ)dτ. |
For the converse, if ϰ∗ is a solution of the FBVP (1.1) and (1.2), we get
Dν;ψr(ϰ∗(ς)+w(ς))=Dν;ψrϰ∗(ς)+Dν;ψrw(ς)=−F(ς,ϰ∗(ς),ϰ∗(r+λς))−p(ς)=−[F(ς,ϰ∗(ς),ϰ∗(r+λς))+p(ς)]=−G(ς,ϰ∗(ς),ϰ∗(r+λς)), |
which leads to
Dν;ψrϰ(ς)=−G(ς,ϰ∗(ς),ϰ∗(r+λς)). |
We easily see that
ϰ∗(r)=ϰ(r)−w(r)=ϰ(r)−0=ϑ1, |
i.e., ϰ(r)=ϑ1 and
ϰ∗(ℑ)=m−2∑i=1ζiϰ∗(ηi)+ϑ2, |
ϰ(ℑ)−w(ℑ)=m−2∑i=1ζiϰ(ηi)−m−2∑i=1ζjw(ηi)+ϑ2=m−2∑i=1ζi(ϰ(ηi)−w(ηi))+ϑ2. |
So,
ϰ(ℑ)=m−2∑i=1ζiϰ(ηi)+ϑ2. |
Thus ϰ(ς) is solution of the problem FBVP (3.1) and (3.2).
We propose the given FBVP as follows
D75ϰ(ς)+F(ς,ϰ(ς),ϰ(1+0.5ς))=0, ς∈(1,e), | (4.1) |
ϰ(1)=1, ϰ(e)=17ϰ(52)+15ϰ(74)+19ϰ(115)−1. | (4.2) |
Let Ψ(ς)=logς, where F(ς,ϰ(ς),ϰ(1+12ς))=ς1+ςarctan(ϰ(ς)+ϰ(1+12ς)).
Taking Υ(ς)=ς we get ∫e1ςdς=e2−12>0, then the hypotheses (Σ1) and (Σ2) hold. Evaluate Δ≅0.366, M≅3.25 we also get |G(ς,ϰ,v)|<π+e=L such that |ϰ|≤ρ, ρ=17, we could just confirm that
[1+Σm−2i=1ζi−1Δ(Ψ(ℑ)−Ψ(r))]ϑ1+Ψ(ℑ)−Ψ(r)Δϑ2+LM≅16.35≤17. | (4.3) |
By applying the Theorem 3.1 there exit a solution ϰ(ς) of the problem (4.1) and (4.2).
In this paper, we have provided the proof of BVP solutions to a nonlinear Ψ-Caputo fractional pantograph problem or for a semi-positone multi-point of (1.1) and(1.2). What's new here is that even using the generalized Ψ-Caputo fractional derivative, we were able to explicitly prove that there is one solution to this problem, and that in our findings, we utilize the SFPT. The results obtained in our work are significantly generalized and the exclusive result concern the semi-positone multi-point Ψ-Caputo fractional differential pantograph problem (1.1) and (1.2).
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Small Groups (RGP.1/350/43).
The authors declare no conflict of interest.
[1] |
Abbate A, Eickmeier S, Lemke W, et al. (2016) The Changing International Transmission of Financial Shocks: Evidence from a Classical Time-Varying FAVAR. J Money Credit Bank 48: 573-601. doi: 10.1111/jmcb.12311
![]() |
[2] |
Alessandri P, Mumtaz H (2017) Financial conditions and density forecasts for US output and inflation. Rev Econ Dyn 24: 66-78. doi: 10.1016/j.red.2017.01.003
![]() |
[3] |
Anaya P, Hachula M, Offermanns CJ (2017) Spillovers of US unconventional monetary policy to emerging markets: The role of capital flows. J Int Money Financ 73: 275-295. doi: 10.1016/j.jimonfin.2017.02.008
![]() |
[4] |
Angelopoulou E, Balfoussia H, Gibson HD (2013) Building a financial conditions index for the euro area and selected euro area countries what does it tell us about the crisis. Econ Model 38: 392-403. doi: 10.1016/j.econmod.2014.01.013
![]() |
[5] |
Antonakakis N, Gabauer D, Gupta R, et al. (2018) Dynamic connectedness of uncertainty across developed economies: A time-varying approach. Econ Lett 166: 63-75. doi: 10.1016/j.econlet.2018.02.011
![]() |
[6] | Anzuini A, Lombardi MJ, Pagano P (2012) The impact of monetary policy shocks on commodity prices. Available from: http://dx.doi.org/10.2139/ssrn.2030797. |
[7] | Bernanke B, Gertler M (1989) Agency Costs, Net Worth, and Business Fluctuations. Am Econ Rev 79: 14-31. |
[8] | Bernanke BS, Mihov I (1995) Measuring Monetary Policy. National Bureau of Economic Research Working Paper Series. |
[9] | Buthe T, Milner HV (2008) The politics of foreign direct investment into developing countries: Increasing FDI through international trade agreements? Am J Polit Sci 52: 741-762. |
[10] | Bowles P, Wang BT (2013) Renminbi Internationalization: A Journey to Where? Dev Change 44: 1363-1385. |
[11] |
Bowman D, Londono JM, Sapriza H (2015) US unconventional monetary policy and transmission to emerging market economies. J Int Money Financ 55: 27-59. doi: 10.1016/j.jimonfin.2015.02.016
![]() |
[12] |
Von Borstel J, Eickmeier S, Krippner L (2016) The interest rate pass-through in the euro area during the sovereign debt crisis. J Int Money Financ 68: 386-402. doi: 10.1016/j.jimonfin.2016.02.014
![]() |
[13] |
Bi Y, Anwar S (2017) US monetary policy shocks and the Chinese economy: a GVAR approach. Appl Econ Lett 24: 553-558. doi: 10.1080/13504851.2016.1210761
![]() |
[14] | Bräuning F, Ivashina V (2019) U.S. monetary policy and emerging market credit cycles. J Monetary Econ 112: 57-76. |
[15] | Canova F, Marrinan J (1998) Sources and propagation of international cycles: Common shocks or transmission? J Int Econ 46: 133-166. |
[16] |
Canova F, De Nicolo G (1999) On the Sources of Business Cycles in the G-7. J Int Econ 59: 77-100. doi: 10.1016/S0022-1996(02)00085-5
![]() |
[17] |
Canova F, De Nicolo G (2000) Stock Returns, Term Structure, Inflation and Real Activity: An International Perspective. Macroecon Dyn 4: 343-372. doi: 10.1017/S1365100500016047
![]() |
[18] |
Chor D, Manova K (2012) Off the cliff and back? Credit conditions and international trade during the global financial crisis. J Int Econ 87: 117-133. doi: 10.1016/j.jinteco.2011.04.001
![]() |
[19] |
Chen Q, Filardo A, He D, et al. (2016) Financial crisis, US unconventional monetary policy and international spillovers. J Int Money Financ 10: 62-81. doi: 10.1016/j.jimonfin.2015.06.011
![]() |
[20] |
Caputo R, Herrera LO (2016) Following the leader? the relevance of the FED funds rate for inflation targeting countries. J Int Money Financ 71: 25-52. doi: 10.1016/j.jimonfin.2016.10.006
![]() |
[21] |
Chadwick MG (2019) Dependence of the "Fragile Five" and "Troubled Ten" emerging market financial systems on US monetary policy and monetary policy uncertainty. Res Int Bus Financ 49: 251-268. doi: 10.1016/j.ribaf.2019.04.002
![]() |
[22] |
Chang CY, Shie FS, Yang SL (2019) The relationship between herding behavior and firm size before and after the elimination of short-sale price restrictions. Quant Financ Econ 3: 526-549. doi: 10.3934/QFE.2019.3.526
![]() |
[23] | Dooley M, Hutchison M (2009) Transmission of the U.S. subprime crisis to emerging markets: Evidence on the decoupling-recoupling hypothesis. J Int Money Financ 28: 1331-1349. |
[24] |
Diebold FX, Yilmaz K (2014) On the network topology of variance decompositions: Measuring the connectedness of financial firms. J Econom 182: 119-134. doi: 10.1016/j.jeconom.2014.04.012
![]() |
[25] | Dahlhaus T, Vasishtha G (2014) The Impact of U.S. Monetary Policy Normalization on Capital Flows to Emerging-Market Economies. Bank of Canada Working Paper, 2014-53. |
[26] | Dedola L, Rivolta G, Stracca L (2017) If the Fed sneezes, who catches a cold? J Int Econ 108: 23-41. |
[27] |
Dong H, Liu Y, Chang JQ (2019) The heterogeneous linkage of economic policy uncertainty and oil return risks. Green Financ 1: 46-66. doi: 10.3934/GF.2019.1.46
![]() |
[28] | Eichengreen B, Rose AK, Wyplosz C (1996) Contagious Currency Crises. NBER Working Paper, No.5681. |
[29] | Edwards S (2015) Monetary Policy Independence under Flexible Exchange Rates: An Illusion? World Econ 38: 773-787. |
[30] |
Evgenidis A, Philippas D, Siriopoulos C (2019) Heterogeneous effects in the international transmission of the US monetary policy: a factor-augmented VAR perspective. Empir Econ 56: 1549-1579. doi: 10.1007/s00181-018-1448-1
![]() |
[31] | Frank N, Hesses H (2009) Financial Spillovers to Emerging Markets During the Global Financial Crisis. IMF Working Papers, 507-521. |
[32] |
Ferrario A, Guidolin M, Pedio M (2018) Comparing in- and out-of-sample approaches to variance decomposition-based estimates of network connectedness an application to the Italian banking system. Quant Financ Econ 2: 661-701. doi: 10.3934/QFE.2018.3.661
![]() |
[33] | Goodhart C, Hofmann B (2000) DO ASSET PRICES HELP TO PREDICT CONSUMER PRICE INFLATION? Manch Sch 68: 122-140. |
[34] | Goodhart C, Hofmann B (2001) Asset Prices, Financial Conditions, and the Transmission ofMonetary Policy. Conference on Asset Prices, Exchange Rates, and Monetary Policy, Stanford University. |
[35] | Grant JH, Lambert DM (2008) Do regional trade agreements increase members' agricultural trade? Am J Agric Econ 90: 765-782. |
[36] | Gelos G, Ustyugova Y (2017) Inflation responses to commodity price shocks - How and why do countries differ? J Int Money Financ 72: 28-47. |
[37] |
Gabauer D, Gupta R (2018) On the transmission mechanism of country-specific and international economic uncertainty spillovers: Evidence from a TVP-VAR connectedness decomposition approach. Econ Lett 171: 63-71. doi: 10.1016/j.econlet.2018.07.007
![]() |
[38] |
Hosen M, Broni MY, Uddin MN (2020) What bank specific and macroeconomic elements influence non-performing loans in Bangladesh? Evidence from conventional and Islamic banks. Green Financ 2: 212-226. doi: 10.3934/GF.2020012
![]() |
[39] | Hung HF (2013) China: Saviour or Challenger of the Dollar Hegemony? Dev Change 44: 1341-1361. |
[40] |
Keep G, Pesaran MH, Potter SM (1996) Impulse response analysis in nonlinear multivariate models. J Econom 74: 119-147. doi: 10.1016/0304-4076(95)01753-4
![]() |
[41] | Kim S (2001) International transmission of U.S. monetary policy shocks: Evidence from VAR's. J Monetary Econ 48: 339-372. |
[42] | Kazi IA, Wagan H, Akbar F (2013) The changing international transmission of U.S. monetary policy shocks: Is there evidence of contagion effect on OECD countries. Econ Model 30: 90-116. |
[43] |
Krokida SI, Makrychoriti P, Spyrou S (2020) Monetary policy and herd behavior: International evidence. J Econ Behav Organ 170: 386-417. doi: 10.1016/j.jebo.2019.12.018
![]() |
[44] | Lin S, Ye H (2018) The international credit channel of U.S. monetary policy transmission to developing countries: Evidence from trade data. J Dev Econ 133: 33-41. |
[45] | Modigliani F (1971) Consumer Spending and Monetary Policy: The Linkages. Proceedings of a MONETARY CONFERENCE NANTUCKET ISLAND, ASSACHUSETTS, 9-84. |
[46] | Maćkowiak B (2007) External shocks, U.S. monetary policy and macroeconomic fluctuations in emerging markets. J Monetary Econ 54: 2512-2520. |
[47] |
Magud NE, Reinhart CM, Vesperoni ER (2014) Capital Inflows, Exchange Rate Flexibility and Credit Booms. Rev Dev Econ 18: 415-430. doi: 10.1111/rode.12093
![]() |
[48] |
Nakajima J, Kasuya M, Watanabe T (2009) Bayesian Analysis of Time-Varying Parameter Vector Autoregressive Model for the Japanese Economy and Monetary Policy. J Jpn Int Econ 25: 225-245. doi: 10.1016/j.jjie.2011.07.004
![]() |
[49] | Nakajima J (2011) Time-Varying Parameter VAR Model with Stochastic Volatility: An Overview of Methodology and Empirical Applications. IMES Discussion Paper, Series 2011-E-9. Available from: https://www.imes.boj.or.jp/research/papers/english/11-E-09.pdf. |
[50] |
Primiceri GE (2005) Time varing structure Vector Autoregressions and monetary policy. Rev Econ Stud 72: 821-852. doi: 10.1111/j.1467-937X.2005.00353.x
![]() |
[51] | Rosa C (2014) The high-frequency response of energy prices to U.S. monetary policy: Understanding the empirical evidence. Energy Econ 45: 295-303. |
[52] | Rafig S (2014) What Do Energy Prices Tell Us About UK Inflation? Economica 81: 293-310. |
[53] |
Rey H (2016) International Channels of Transmission of Monetary Policy and the Mundellian Trilemma. IMF Econ Rev 64: 6-35. doi: 10.1057/imfer.2016.4
![]() |
[54] |
Tobin J (1969) A General Equilibrium Approach To Monetary Theory. J Money Credit Bank 1: 15-29. doi: 10.2307/1991374
![]() |
[55] |
Tillmann P (2016) Unconventional monetary policy and the spillovers to emerging markets. J Int Money Financ 66: 136-156. doi: 10.1016/j.jimonfin.2015.12.010
![]() |
[56] |
Wang YS, Chueh YL (2013) Dynamic transmission effects between the interest rate, the US dollar, and gold and crude oil prices. Econ Model 30: 792-798. doi: 10.1016/j.econmod.2012.09.052
![]() |
[57] |
Wen F, Zhao Y, Zhang M, et al. (2019) Forecasting realized volatility of crude oil futures with equity market uncertainty. Appl Econ Lett 51: 6411-6427. doi: 10.1080/00036846.2019.1619023
![]() |
[58] | World Bank (1985) World development report 1985. |
[59] |
Zhong JH, Wang MD, Drakeford BM, et al. (2019) Spillover effects between oil and natural gas prices: Evidence from emerging and developed markets. Green Financ 1: 30-45. doi: 10.3934/GF.2019.1.30
![]() |
![]() |
![]() |