
Many problems arising from science and engineering are in the form of a system of nonlinear equations. In this work, a new derivative-free inertial-based spectral algorithm for solving the system is proposed. The search direction of the proposed algorithm is defined based on the convex combination of the modified long and short Barzilai and Borwein spectral parameters. Also, an inertial step is introduced into the search direction to enhance its efficiency. The global convergence of the proposed algorithm is described based on the assumption that the mapping under consideration is Lipschitz continuous and monotone. Numerical experiments are performed on some test problems to depict the efficiency of the proposed algorithm in comparison with some existing ones. Subsequently, the proposed algorithm is used on problems arising from robotic motion control.
Citation: Sani Aji, Aliyu Muhammed Awwal, Ahmadu Bappah Muhammadu, Chainarong Khunpanuk, Nuttapol Pakkaranang, Bancha Panyanak. A new spectral method with inertial technique for solving system of nonlinear monotone equations and applications[J]. AIMS Mathematics, 2023, 8(2): 4442-4466. doi: 10.3934/math.2023221
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Many problems arising from science and engineering are in the form of a system of nonlinear equations. In this work, a new derivative-free inertial-based spectral algorithm for solving the system is proposed. The search direction of the proposed algorithm is defined based on the convex combination of the modified long and short Barzilai and Borwein spectral parameters. Also, an inertial step is introduced into the search direction to enhance its efficiency. The global convergence of the proposed algorithm is described based on the assumption that the mapping under consideration is Lipschitz continuous and monotone. Numerical experiments are performed on some test problems to depict the efficiency of the proposed algorithm in comparison with some existing ones. Subsequently, the proposed algorithm is used on problems arising from robotic motion control.
Recent years have witnessed the development of distributed discrete fractional operators based on singular and nonsingular kernels with the aim of solving a large variety of discrete problems arising in different application fields such as biology, physics, robotics, economic sciences and engineering (see for example [1,2,3,4,5,6,7,8,9]). These operators depend on their corresponding kernels overcoming some limits of the order of discrete operators, for example the most popular operators are Riemann-Liouville and Caputo with standard kernels, Caputo-Fabrizio with exponential kernels, Attangana-Baleanu with Mittag-Leffler kernels (see for example [10,11,12,13]). We also refer the reader to [14,15,16,17,18] for discrete fractional operators. Modeling and positivity simulations have been developed or adapted for discrete fractional operators, ranging from continuous fractional models to discrete fractional frameworks; see for example [1,19,20]). For other results on positivity and monotonicity we refer the reader to [21,22,23,24,25] and for discrete fractional models with monotonicity and positivity which is important in the context of discrete fractional calculus we refer the reader to [26,27,28,29].
In this work, we are interested in finding positivity and monotonicity results for the following single and composition of delta fractional difference equations:
(CFC aΔνG)(t) |
and
(CFCa+1ΔνCFC aΔμG)(t), |
where we will assume that G is defined on Na:={a,a+1,…}, and ν and μ are two different positive orders.
The paper is structured as follows. The mathematical backgrounds and preliminaries needed are given in Section 2. Section 3 presents the problem statement and the main results. Conclusions are provided in Section 4.
Let us start this section by recalling the notions of discrete delta Caputo-Fabrizio fractional operators that we will need.
Definition 2.1 (see [30,31]). Let (ΔG)(t)=G(t+1)−G(t) be the forward difference operator. Then for any function G defined on Na with a∈R, the discrete delta Caputo-Fabrizio fractional difference in the Caputo sense and Caputo-Fabrizio fractional difference in the Riemann sense are defined by
(CFC aΔαG)(t)=B(α)1−αt−1∑κ=a(ΔκG)(κ)(1+λ)t−κ−1=B(α)1−2αt−1∑κ=a(ΔκG)(κ)(1+λ)t−κ,[∀t∈Na+1], | (2.1) |
and
(CFR aΔαG)(t)=B(α)1−αΔtt−1∑κ=aG(κ)(1+λ)t−κ−1=B(α)1−2αΔtt−1∑κ=aG(κ)(1+λ)t−κ,[∀t∈Na+1], | (2.2) |
respectively, where λ=−α1−α for α∈[0,1), and B(α) is a normalizing positive constant.
Moreover, for the higher order case when q<α<q+1 with q⩾0, we have
(CFC aΔαG)(t)=(CFC aΔα−qΔqG)(t),[∀t∈Na+1]. | (2.3) |
Remark 2.1. It should be noted that
0<1+λ=1−2α1−α<1, |
if α∈(0,12), where (as above) λ=−α1−α.
Definition 2.2 (see [29,32]). Let G be defined on Na and α∈[1,2]. Then G is α−convexiff(ΔG) is (α−1)−monotone increasing. That is,
G(t+1)−αG(t)+(α−1)G(t−1)⩾0,[∀t∈Na+1]. |
This section deals with convexity and positivity of the Caputo-Fabrizio operator in the Riemann sense (2.2). We first present some necessary lemmas.
Lemma 3.1. Let G:Na→R be a function satisfying
(CFC aΔαΔG)(t)⩾0 |
and
(ΔG)(a)⩾0, |
for α∈(0,12) and t in Na+2. Then (ΔG)(t)⩾0, for every t in Na+1.
Proof. From Definition 2.1, we have for each t∈Na+2:
(CFC aΔαΔG)(t)=B(α)1−2αt−1∑κ=a(Δ2κf)(κ)(1+λ)t−κ=B(α)1−2α[t−1∑κ=a(ΔG)(κ+1)(1+λ)t−κ−t−1∑κ=a(ΔG)(κ)(1+λ)t−κ]=B(α)1−2α[(1+λ)(ΔG)(t)+λt−1∑κ=a(ΔG)(κ)(1+λ)t−κ]=B(α)1−2α[(1+λ)(ΔG)(t)−(1+λ)t−a(ΔG)(a)+λt−1∑κ=a+1(ΔG)(κ)(1+λ)t−κ]. | (3.1) |
Since B(α)1−2α>0,1+λ>0 and (CFC aΔαΔG)(t)⩾0 for all t∈Na+2, then (3.1) gives us
(ΔG)(t)⩾(1+λ)t−a−1(ΔG)(a)−λ1+λt−1∑κ=a+1(ΔG)(κ)(1+λ)t−κ. | (3.2) |
We will now show that (ΔG)(a+i+1)⩾0 if we assume that (ΔG)(a+i)⩾0 for some i∈N1. Note from our assumption we have that (ΔG)(a)⩾0. But then from the lower bound for (ΔG)(a+i+1) in (3.2) and our assumption we have
(ΔG)(a+i+1)⩾(1+λ)i(ΔG)(a)⏟⩾0−λ1+λa+i∑κ=a+1(ΔG)(κ)(1+λ)a+i+1−κ⏟⩾0⏟⩾0⩾0, |
where we used λ1+λ<0. Thus, the result follows by induction.
Lemma 3.2. Let G be defined on Na and
(CFC aΔαG)(t)⩾0withtheinitialvaluesG(a+1)⩾G(a)⩾0, |
for α∈(1,32) and t∈Na+1. Then G is monotone increasing, positive and (12−α)−convex on Na.
Proof. From the definition with q=1 we have
0⩽(CFC aΔαG)(t)=(CFC aΔα−1ΔG)(t),[∀t∈Na+1]. |
Since (ΔG)(a)⩾0 is given we have
(ΔG)(t)⩾0,[∀t∈Na+1], |
by Lemma 3.1. This implies that G is a monotone increasing function. Therefore,
G(t)⩾G(t−1)⩾⋯⩾G(a+1)⩾G(a)⩾0,[∀t∈Na+1], |
and hence G is positive.
From the idea in Lemma 3.1 we have (here λ=−α−12−α for α∈(1,32)),
(ΔG)(t)⩾(1+λ)t−a−1(ΔG)(a)−λ1+λt−1∑κ=a+1(ΔG)(κ)(1+λ)t−κ=(1+λ)t−a−1(ΔG)(a)⏟⩾0−λ(ΔG)(t−1)−λ1+λt−2∑κ=a+1(ΔG)(κ)(1+λ)t−κ⏟⩾0since(ΔG)(t)⩾0⩾−λ(ΔG)(t−1)=(α−12−α)(ΔG)(t−1)=(12−α−1)(ΔG)(t−1). |
Consequently we have that G is (12−α)−convex on the set Na.
Lemma 3.3. Let G be defined on Na and
(CFC aΔαG)(t)⩾0with(Δ2G)(a)⩾0, |
for α∈(2,52) and t∈Na+1. Then, Then (Δ2G)(t)⩾0, for all t∈Na. Furthermore, one has G convex on the set Na.
Proof. Let (CFC aΔαG)(t):=F(t) for each t∈Na+1. Since α∈(2,52), we have:
(CFC aΔαG)(t)=(CFC aΔα−2Δ2G)(t)=(CFC aΔα−2ΔF)(t)⩾0, |
for each t∈Na+1, and by assumption we have
(ΔF)(a)=(Δ2G)(a)⩾0. |
Then, using Lemma 3.2, we get
(ΔF)(t)=(Δ2G)(t)⩾0 |
for each t∈Na+1. Hence, G is convex on Na.
Lemma 3.4. Let G be defined on Na and
Δ2(CFC aΔαG)(t)⩾0 |
and
(ΔG)(a+1)⩾(ΔG)(a)⩾0, |
for α∈(0,12) and t∈Na+1. Then (Δ2G)(t)⩾0, for each t∈Na.
Proof. For t∈Na+1, we have
Δ(CFC aΔαG)(t)=B(α)1−2αΔ[t−1∑κ=a(ΔκG)(κ)(1+λ)t−κ]=B(α)1−2α[t∑κ=a(ΔκG)(κ)(1+λ)t+1−κ−t−1∑κ=a(ΔκG)(κ)(1+λ)t−κ]=B(α)1−2α[(1+λ)(ΔG)(t)+t−1∑κ=a(ΔκG)(κ)(1+λ)t+1−κ−t−1∑κ=a(ΔκG)(κ)(1+λ)t−κ]=B(α)1−2α[(1+λ)(ΔG)(t)+λt−1∑κ=a(ΔκG)(κ)(1+λ)t−κ], | (3.3) |
where λ=−α1−α. It follows from (3.3) that,
Δ2(CFC aΔαG)(t)=B(α)1−2αΔ[(1+λ)(ΔG)(t)+λt−1∑κ=a(ΔκG)(κ)(1+λ)t−κ]=(1+λ)B(α)1−2α(Δ2G)(t)+λB(α)1−2α[t∑κ=a(ΔκG)(κ)(1+λ)t+1−κ−t−1∑κ=a(ΔκG)(κ)(1+λ)t−κ]=(1+λ)B(α)1−2α(Δ2G)(t)+λB(α)1−2α[(1+λ)t+1−a(ΔG)(a)+t−1∑κ=a(ΔκG)(κ+1)(1+λ)t−κ−t−1∑κ=a(ΔκG)(κ)(1+λ)t−κ]=(1+λ)B(α)1−2α(Δ2G)(t)+λB(α)1−2α[(1+λ)t+1−a(ΔG)(a)+t−1∑κ=a(Δ2κG)(κ)(1+λ)t−κ]. | (3.4) |
Due to the nonnegativity of (1+λ)B(α)1−2α, from (3.4) we deduce
(Δ2G)(t)⩾−λ1+λ[(1+λ)t+1−a(ΔG)(a)+t−1∑κ=a(Δ2κG)(κ)(1+λ)t−κ]. | (3.5) |
By substituting t=a+1 into (3.5), we get
(Δ2G)(a+1)⩾−λ1+λ[(1+λ)2(ΔG)(a)+(Δ2G)(a)(1+λ)]=α(1−α)⏟>0[(1+λ)(ΔG)(a)⏟⩾0+(Δ2G)(a)⏟⩾0]⩾0. |
Also, if we substitute t=a+2 into (3.5), we obtain
(Δ2G)(a+2)⩾−λ1+λ[(1+λ)3(ΔG)(a)+(1+λ)2(Δ2G)(a)+(1+λ)(Δ2G)(a+1)]=α(1−α)⏟>0[(1+λ)2(ΔG)(a)⏟⩾0+(1+λ)(Δ2G)(a)⏟⩾0+(Δ2G)(a+1)⏟⩾0]⩾0. |
By continuing this process, we obtain that (Δ2G)(t)⩾0 for each t∈Na as desired.
Now, we are in a position to state the first result on convexity. Furthermore, three representative results associated to different subregions in the space of (μ,ν)-parameter will be provided.
Theorem 3.1. Let G be defined on Na with ν∈(0,12) and μ∈(2,52), and
(CFCa+1ΔνCFC aΔμG)(t)⩾0 |
and
(Δ2G)(a+1)⩾(Δ2G)(a)⩾0, |
for each t∈Na+1. Then G is convex on the set Na.
Proof. Let (CFC aΔμG)(t):=F(t) for each t∈Na+1. Then, by assumption we have
(CFCa+1ΔνCFC aΔμG)(t)=(CFCa+1ΔνF)(t)⩾0, |
for each t∈Na+1. From the definition with q=2 we have
F(a+1)=(CFC aΔμG)(a+1)=(CFC aΔμ−2Δ2G)(a+1)=B(μ−2)5−2μa∑κ=a(Δ3κG)(κ)(1+λμ)a−κ=B(μ−2)5−2μ⏟>0(Δ3G)(a)⏟⩾0byassumption⩾0, |
where λμ=−μ−23−μ. Since (Δ2G)(a)⩾0, we find that (Δ2G)(t)⩾0 for each t∈Na. Furthermore, we see that G is convex on Na from Lemma 3.3.
Theorem 3.2. Let G be defined on Na with ν∈(1,32) and μ∈(2,52), and
(CFCa+1ΔνCFC aΔμG)(t)⩾0,(Δ2G)(a+2)⩾13−μ(ΔG2)(a+1)⩾0, |
and
(Δ2G)(a+1)⩾(Δ2G)(a)⩾0, |
for each t∈Na+1. Then G is convex on Na.
Proof. Let F(t):=(CFC aΔμG)(t). Note that:
(CFCa+1ΔνCFC aΔμG)(t)=(CFCa+1ΔνF)(t)⩾0, |
for t∈Na+1. Then we have
F(a+1)=(CFC aΔμ−2Δ2G)(a+1)=B(μ−2)5−2μa∑κ=a(Δ3G)(κ)(1+λμ)a+1−κ=B(μ−2)5−2μ(1+λμ)(Δ3G)(a)⩾0, | (3.6) |
and
F(a+2)=(CFC aΔμ−2Δ2G)(a+2)=B(μ−2)5−2μa+1∑κ=a(Δ3G)(κ)(1+λμ)a+2−κ=B(μ−2)5−2μ[(1+λμ)2(Δ3G)(a)+(1+λμ)(Δ3G)(a+1)]=(1+λμ)B(μ−2)5−2μ[(1+λμ)[(Δ2G)(a+1)−(Δ2G)(a)]+[(Δ2G)(a+2)−(Δ2G)(a+1)]]⩾(1+λμ)B(μ−2)5−2μ[13−μ−1]⩾0, | (3.7) |
where λμ=−μ−25−2μ. On the other hand, one has
F(a+2)−F(a+1)=(1+λμ)B(μ−2)5−2μ[(1+λμ)(Δ3G)(a)+(Δ3G)(a+1)−(Δ3G)(a)]⩾(1+λμ)B(μ−2)5−2μ(λμ−1+13−μ)(Δ2G)(a+1)⩾0. | (3.8) |
Then, from Eqs (3.6)–(3.8), we see that F(a+2)⩾F(a+1)⩾0. Therefore, Lemma 3.2 gives
F(t)=(CFCa+1ΔνG)(t)⩾0 |
for all t in Na+1. Moreover, by considering (Δ2G)(a)⩾0 in Lemma 3.3, we can deduce that G is convex on the set Na.
Theorem 3.3. Let G be defined on Na with ν∈(2,52) and μ∈(0,12), and
(CFCa+1ΔνCFC aΔμG)(t)⩾0,(ΔG)(a+2)⩾11−μ(ΔG)(a+1)⩾0, |
and
(ΔG)(a+1)⩾(ΔG)(a)⩾0, |
for each t∈Na+1. Then we have that G is convex on Na.
Proof. Again, we write F(t):=(CFC aΔμG)(t), and therefore, (CFCa+1ΔνF)(t)⩾0 by assumption, for each t∈Na+1. Then, we see that
(Δ2F)(a+1)≡Δ2(CFC aΔμG)(a+1)by=(3.4)(1+λμ)B(μ)1−2μ(Δ2G)(a+1)+λμB(μ)1−2μ[(1+λμ)2(ΔG)(a)+a∑κ=a(Δ2κG)(κ)(1+λμ)a+1−κ]=(1+λμ)B(μ)1−2μ[(Δ2G)(a+1)+λμ(1+λμ)(ΔG)(a)+λμ(Δ2G)(a)]=(1+λμ)B(μ)1−2μ[(ΔG)(a+2)+(λμ−1)(ΔG)(a+1)+λ2μ(ΔG)(a)]⩾(1+λμ)B(μ)1−2μ[11−μ(ΔG)(a+1)+(λμ−1)(ΔG)(a+1)+λ2μ(ΔG)(a)⏟⩾0]⩾(1+λμ)B(μ)1−2μ[11−μ+λμ−1](ΔG)(a+1)⩾0, |
where λμ=−μ1−μ. It follows that,
(Δ2F)(t)=Δ2(CFC aΔμG)(t)⩾0, |
for each t∈Na by Lemma 3.3. Considering, (Δ2G)(a)⩾0, we can deduce that G is convex on Na by Lemma 3.4.
In Figure 1, we demonstrate the regions of the (μ,ν)-parameter space in which the above three Theorems 3.1–3.3 are applied.
In this study, we present some new positivity results for discrete fractional operators with exponential kernels in the sense of Caputo. In particular new positivity, α−convexity and α−monotonicity were presented. We now refer the reader to observations for discrete generalized fractional operators in [33] which combined with this paper may motivate future work.
Conceptualization, P.O.M., D.O., A.B.B. and D.B.; methodology, P.O.M., D.O.; software, D.O., D.B., K.M.A., A.B.B.; validation, P.O.M., D.O., D.B. and A.B.B.; formal analysis, K.M.A.; investigation, P.O.M., D.O., K.M.A.; resources, A.B.B.; writing-original draft preparation, P.O.M., D.O., D.B., K.M.A., A.B.B.; writing-review and editing, D.O., D.B. and A.B.B.; funding acquisition, D.B. and K.M.A. All authors read and approved the final manuscript.
This Research was supported by Taif University Researchers Supporting Project Number (TURSP-2020/217), Taif University, Taif, Saudi Arabia.
The authors declare that they have no conflicts of interest.
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