Research article

A new spectral method with inertial technique for solving system of nonlinear monotone equations and applications

  • 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 aR, 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)1αt1κ=a(ΔκG)(κ)(1+λ)tκ1=B(α)12αt1κ=a(ΔκG)(κ)(1+λ)tκ,[tNa+1], (2.1)

    and

    (CFR      aΔαG)(t)=B(α)1αΔtt1κ=aG(κ)(1+λ)tκ1=B(α)12αΔtt1κ=aG(κ)(1+λ)tκ,[tNa+1], (2.2)

    respectively, where λ=α1α for α[0,1), and B(α) is a normalizing positive constant.

    Moreover, for the higher order case when q<α<q+1 with q0, we have

    (CFC     aΔαG)(t)=(CFC     aΔαqΔqG)(t),[tNa+1]. (2.3)

    Remark 2.1. It should be noted that

    0<1+λ=12α1α<1,

    if α(0,12), where (as above) λ=α1α.

    Definition 2.2 (see [29,32]). Let G be defined on Na and α[1,2]. Then G is αconvexiff(ΔG) is (α1)monotone increasing. That is,

    G(t+1)αG(t)+(α1)G(t1)0,[tNa+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:NaR 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 tNa+2:

    (CFC     aΔαΔG)(t)=B(α)12αt1κ=a(Δ2κf)(κ)(1+λ)tκ=B(α)12α[t1κ=a(ΔG)(κ+1)(1+λ)tκt1κ=a(ΔG)(κ)(1+λ)tκ]=B(α)12α[(1+λ)(ΔG)(t)+λt1κ=a(ΔG)(κ)(1+λ)tκ]=B(α)12α[(1+λ)(ΔG)(t)(1+λ)ta(ΔG)(a)+λt1κ=a+1(ΔG)(κ)(1+λ)tκ]. (3.1)

    Since B(α)12α>0,1+λ>0 and (CFC     aΔαΔG)(t)0 for all tNa+2, then (3.1) gives us

    (ΔG)(t)(1+λ)ta1(ΔG)(a)λ1+λt1κ=a+1(ΔG)(κ)(1+λ)tκ. (3.2)

    We will now show that (ΔG)(a+i+1)0 if we assume that (ΔG)(a+i)0 for some iN1. 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)0λ1+λa+iκ=a+1(ΔG)(κ)(1+λ)a+i+1κ000,

    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 tNa+1. Then G is monotone increasing, positive and (12α)convex on Na.

    Proof. From the definition with q=1 we have

    0(CFC     aΔαG)(t)=(CFC     aΔα1ΔG)(t),[tNa+1].

    Since (ΔG)(a)0 is given we have

    (ΔG)(t)0,[tNa+1],

    by Lemma 3.1. This implies that G is a monotone increasing function. Therefore,

    G(t)G(t1)G(a+1)G(a)0,[tNa+1],

    and hence G is positive.

    From the idea in Lemma 3.1 we have (here λ=α12α for α(1,32)),

    (ΔG)(t)(1+λ)ta1(ΔG)(a)λ1+λt1κ=a+1(ΔG)(κ)(1+λ)tκ=(1+λ)ta1(ΔG)(a)0λ(ΔG)(t1)λ1+λt2κ=a+1(ΔG)(κ)(1+λ)tκ0since(ΔG)(t)0λ(ΔG)(t1)=(α12α)(ΔG)(t1)=(12α1)(ΔG)(t1).

    Consequently we have that G is (12α)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 tNa+1. Then, Then (Δ2G)(t)0, for all tNa. Furthermore, one has G convex on the set Na.

    Proof. Let (CFC     aΔαG)(t):=F(t) for each tNa+1. Since α(2,52), we have:

    (CFC     aΔαG)(t)=(CFC     aΔα2Δ2G)(t)=(CFC     aΔα2ΔF)(t)0,

    for each tNa+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 tNa+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 tNa+1. Then (Δ2G)(t)0, for each tNa.

    Proof. For tNa+1, we have

    Δ(CFC     aΔαG)(t)=B(α)12αΔ[t1κ=a(ΔκG)(κ)(1+λ)tκ]=B(α)12α[tκ=a(ΔκG)(κ)(1+λ)t+1κt1κ=a(ΔκG)(κ)(1+λ)tκ]=B(α)12α[(1+λ)(ΔG)(t)+t1κ=a(ΔκG)(κ)(1+λ)t+1κt1κ=a(ΔκG)(κ)(1+λ)tκ]=B(α)12α[(1+λ)(ΔG)(t)+λt1κ=a(ΔκG)(κ)(1+λ)tκ], (3.3)

    where λ=α1α. It follows from (3.3) that,

    Δ2(CFC     aΔαG)(t)=B(α)12αΔ[(1+λ)(ΔG)(t)+λt1κ=a(ΔκG)(κ)(1+λ)tκ]=(1+λ)B(α)12α(Δ2G)(t)+λB(α)12α[tκ=a(ΔκG)(κ)(1+λ)t+1κt1κ=a(ΔκG)(κ)(1+λ)tκ]=(1+λ)B(α)12α(Δ2G)(t)+λB(α)12α[(1+λ)t+1a(ΔG)(a)+t1κ=a(ΔκG)(κ+1)(1+λ)tκt1κ=a(ΔκG)(κ)(1+λ)tκ]=(1+λ)B(α)12α(Δ2G)(t)+λB(α)12α[(1+λ)t+1a(ΔG)(a)+t1κ=a(Δ2κG)(κ)(1+λ)tκ]. (3.4)

    Due to the nonnegativity of (1+λ)B(α)12α, from (3.4) we deduce

    (Δ2G)(t)λ1+λ[(1+λ)t+1a(ΔG)(a)+t1κ=a(Δ2κG)(κ)(1+λ)tκ]. (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(1α)>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(1α)>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 tNa 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 tNa+1. Then G is convex on the set Na.

    Proof. Let (CFC     aΔμG)(t):=F(t) for each tNa+1. Then, by assumption we have

    (CFCa+1ΔνCFC     aΔμG)(t)=(CFCa+1ΔνF)(t)0,

    for each tNa+1. From the definition with q=2 we have

    F(a+1)=(CFC     aΔμG)(a+1)=(CFC     aΔμ2Δ2G)(a+1)=B(μ2)52μaκ=a(Δ3κG)(κ)(1+λμ)aκ=B(μ2)52μ>0(Δ3G)(a)0byassumption0,

    where λμ=μ23μ. Since (Δ2G)(a)0, we find that (Δ2G)(t)0 for each tNa. 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)13μ(ΔG2)(a+1)0,

    and

    (Δ2G)(a+1)(Δ2G)(a)0,

    for each tNa+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 tNa+1. Then we have

    F(a+1)=(CFC     aΔμ2Δ2G)(a+1)=B(μ2)52μaκ=a(Δ3G)(κ)(1+λμ)a+1κ=B(μ2)52μ(1+λμ)(Δ3G)(a)0, (3.6)

    and

    F(a+2)=(CFC     aΔμ2Δ2G)(a+2)=B(μ2)52μa+1κ=a(Δ3G)(κ)(1+λμ)a+2κ=B(μ2)52μ[(1+λμ)2(Δ3G)(a)+(1+λμ)(Δ3G)(a+1)]=(1+λμ)B(μ2)52μ[(1+λμ)[(Δ2G)(a+1)(Δ2G)(a)]+[(Δ2G)(a+2)(Δ2G)(a+1)]](1+λμ)B(μ2)52μ[13μ1]0, (3.7)

    where λμ=μ252μ. On the other hand, one has

    F(a+2)F(a+1)=(1+λμ)B(μ2)52μ[(1+λμ)(Δ3G)(a)+(Δ3G)(a+1)(Δ3G)(a)](1+λμ)B(μ2)52μ(λμ1+13μ)(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)11μ(ΔG)(a+1)0,

    and

    (ΔG)(a+1)(ΔG)(a)0,

    for each tNa+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 tNa+1. Then, we see that

    (Δ2F)(a+1)Δ2(CFC     aΔμG)(a+1)by=(3.4)(1+λμ)B(μ)12μ(Δ2G)(a+1)+λμB(μ)12μ[(1+λμ)2(ΔG)(a)+aκ=a(Δ2κG)(κ)(1+λμ)a+1κ]=(1+λμ)B(μ)12μ[(Δ2G)(a+1)+λμ(1+λμ)(ΔG)(a)+λμ(Δ2G)(a)]=(1+λμ)B(μ)12μ[(ΔG)(a+2)+(λμ1)(ΔG)(a+1)+λ2μ(ΔG)(a)](1+λμ)B(μ)12μ[11μ(ΔG)(a+1)+(λμ1)(ΔG)(a+1)+λ2μ(ΔG)(a)0](1+λμ)B(μ)12μ[11μ+λμ1](ΔG)(a+1)0,

    where λμ=μ1μ. It follows that,

    (Δ2F)(t)=Δ2(CFC     aΔμG)(t)0,

    for each tNa 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.

    Figure 1.  Three different regions concerning Theorems 3.1–3.3.

    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, αconvexity and αmonotonicity 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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