Iter | f(μ) | Iter | f(μ) |
μ=2 | 13.87 | μ=13 | 25.21 |
μ=3 | 164.6 | μ=14 | 23.21 |
μ=4 | 56.16 | μ=15 | 21.20 |
μ=5 | 63.72 | μ=16 | 20.20 |
μ=6 | 54.28 | μ=17 | 18.71 |
μ=7 | 44.23 | μ=18 | 17.64 |
μ=8 | 36.45 | μ=19 | 16.53 |
μ=9 | 33.78 | ... | ... |
μ=10 | 33.78 | μ=60 | 5.02 |
In this paper, we firstly formulate a new hyperchaotic Hamiltonian system and demonstrate the existence of multi-equilibrium points in the system. The characteristics of equilibrium points, Lyapunov exponents and Poincaré sections are studied. Secondly, we investigate the complex dynamical behaviors of the system under holonomic constraint and nonholonomic constraint, respectively. The results show that the hyperchaotic system can generated by introducing constraint. Additionally, the hyperchaos control of the system is achieved by applying linear feedback control. The numerical simulations are carried out in order to analyze the complex phenomena of the systems.
Citation: Junhong Li, Ning Cui. Dynamical behavior and control of a new hyperchaotic Hamiltonian system[J]. AIMS Mathematics, 2022, 7(4): 5117-5132. doi: 10.3934/math.2022285
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In this paper, we firstly formulate a new hyperchaotic Hamiltonian system and demonstrate the existence of multi-equilibrium points in the system. The characteristics of equilibrium points, Lyapunov exponents and Poincaré sections are studied. Secondly, we investigate the complex dynamical behaviors of the system under holonomic constraint and nonholonomic constraint, respectively. The results show that the hyperchaotic system can generated by introducing constraint. Additionally, the hyperchaos control of the system is achieved by applying linear feedback control. The numerical simulations are carried out in order to analyze the complex phenomena of the systems.
The most papular Banach contraction mapping principle (BCMP) [1] is the largest powerful fundamental fixed point result. This principle has a lot of applications in pure and applied mathematics (see [2,3,4]). In the past few decades, many authors extended and generalized the (BCMP) in several ways (see [5,6,7,8,9,10]). Ran and Reurings [11] obtained positive definite solutions of matrix equations using the aid of the Banach contraction principle in partially ordered sets. Nieto and Rodriguez-Lopez [12] also used partially ordered spaces and fixed point theorems to find solutions of some differential equations. Very recently, Wardowski [13] furnished the idea of an F-contraction, which is an extension of the (BCMP). Furthermore, common fixed point theorems for rational FR -contractive pairs of mappings with applications are announced in [14] as an extension of F-contractions in relation theoretic metric spaces. On the other hand, Matthews [15] introduced the notion of a partial metric space as a part of the study of semantics of dataflow network, and for more results in this direction see ([16,17,18,19,20]). One of the latest extensions of a metric space and a partial metric space is initiated through the concept of a m-metric space [21], and some researchers work in this direction (see more [22,23,24,25,26,27,28,29,30]). In our article, we utilize two last notions to give an interesting type of generalized FmR-contractions in the frame of relation theoretic m-metric spaces and to prove some fixed point results.
Generally saying that, we generalize and extend some recent results in [31]. We also extend the earlier mentioned results in the setting of relation theoretic m-metric spaces, that contain only the last two conditions imposed on the Wardowski function F in the first section. Furthermore, the consequences of our main results improve and generalize some corresponding theorems appearing in the literature.
Our article consists four sections. In the first section, we recall some fundamental definitions and theorems concerning m-metric spaces and different types of F-contractions. In the second section, we define the notion of generalized FmR-contractions of rational type and generalized FmR-contractions of cyclic type. In the third section, we use the whole Wardowski function in the setting of FmR-contractions of rational type as consequences of main results in section Ⅱ. Using these ideas, we prove some new fixed point results in the frame of relation theoretic m-metric spaces and we present some examples to show that our obtained results are meaningful. In section Ⅳ, we present an application and we ensure the existence of a solution of a class of nonlinear matrix equations.
Throughout this article, N indicates a set of all natural numbers, R indicates set of real numbers and R+ indicates set of positive real numbers, respectively. We also denote N0=N∪{0}. Henceforth, U will denote a non-empty set. Given a self mapping γ:U→U. A Picard sequence based on an arbitrary ζ0 in U is given by ζμ=γ(ζμ−1)=γμ(ζ0) for all μ in N, where γμ denotes the μth-iteration of γ.
In 2013, the notion of a m-metric space was introduced by Asadi et al. [21]. They also extended the well known Banach contraction fixed point theorem from partial metric spaces to m-metric spaces. We start recalling some definitions and properties:
Definition 1.1. [21] Let U≠∅. The function m:U×U→R+ is a m-metric on the set U if for all ζ,ℑ,ℵ∈U,
(i) ζ=ℑ⟺m(ζ,ζ)=m(ℑ,ℑ)=m(ζ,ℑ)(T0 -separation axiom);
(ii) mζℑ≤m(ζ,ℑ) (minimum self distance axiom);
(iii) m(ζ,ℑ)=m(ℑ,ζ) (symmetry);
(iv) m(ζ,ℑ)−mζℑ≤(m(ζ,ℵ)−mζℵ)+(m(ℵ,ℑ)−mℵℑ) (modified triangle inequality),
where
mζℑ=min{m(ζ,ζ),m(ℑ,ℑ)},Mζℑ=max{m(ζ,ζ),m(ℑ,ℑ)}. |
Here, the pair (U,m) is called a m-metric space.
On among the classical examples of m-metric spaces is the pair (ζ,m) where U={ζ,ℑ,ℵ} and m(ζ,ζ)=1, m(ℑ,ℑ)=9, m(ℵ,ℵ)=5. Other examples of m-metric spaces may be found, for instance in [21]. Clearly, each partial metric is a m -metric space, but the converse does not hold (see [32,33,34]).
Every m-metric m on U generates a T0 topology τm(say) on U which has a base of collection of m-open balls
{Bm(ζ,ϵ):ζ∈U,ϵ>0}, |
where
Bm(ζ,ϵ)={ℑ∈U:m(ζ,ℑ)<mζℑ+ϵ}for all ζ∈U,ε>0. |
If m is a m-metric space on U, then the functions mw, ms:U×U→R+ given by:
mw(ζ,ℑ)=m(ζ,ℑ)−2mζℑ+Mζℑ, |
ms={m(ζ,ℑ)−mζℑ, if ζ≠ℑ0, if ζ=ℑ. |
define ordinary metrics on U. It is easy to see that mw and ms are equivalent metrics on U.
Definition 1.2. According to [21],
(i) a sequence {ζμ} in a m-metric space (U,m) converges with respect to τm to ζ if and only if
limμ→∞(m(ζμ,ζ)−mζμζ)=0, |
(ii) a sequence {ζμ} in a m-metric space (U,m) is called m-Cauchy if limμ,ν→∞(m(ζμ,ζν)−mζμζν) and limμ,ν→∞(Mζμ,ζν−mζμζν) exist and are finite,
(iii) (U,m) is said to be complete if every m -Cauchy sequence {ζμ} in U is m-convergent to ζ with respect to τm in U such that
limμ→∞(m(ζμ,ζ)−mζμζ)=0, and limμ→∞(Mζμ,ζ−mζμζ)=0, |
(iv) {ζμ} is a Cauchy sequence in (U,m) if and only if it is a Cauchy sequence in the metric space (U,mw),
(v) (U,m) is complete if and only if (U,mw) is complete.
Consider a function F:(0,∞)→R so that:
(F1) F(ζ)<F(ℑ) for all ζ<ℑ,
(F2) for each sequence {ϖμ}⊆(0,∞), limμ→∞ϖμ=0 iff limμ→∞F(ϖμ)=−∞,
(F3) there exists p∈(0,1) such that limϖμ→0+ϖpF(ϖ)=0.
According to [13], denote by ∇(F) the collection of functions F:(0,∞)→R satisfying (F2) and (F3). Take also
Π(F)={F∈∇F:F,verifies(F1)}. |
Example 1.1. [13] The following below functions belong to Π(F):
(1) F(s)=lns,
(2) F(s)=s+lns,
(3) F(s)=ln(s2+s),
(4) F(s)=−1√s,
for all s>0.
Example 1.2. The following functions are not strictly increasing and belong to ∇(F):
(1) F(s)=100ln(s2+sins),
(2) F(s)=sins+lns,
(3) F(s)=sins−1√s,
for all s>0.
Let γ be a self-mapping on a mm-space U. The following are some valuable notations that are useful for the rest.
(i) (γ)Fix is the set of all fixed points of γ,
(ii) Θ(Ψ,S)={ζ∈U:(ζ,γ(ζ))∈R},
(iii) ϝ(ζ,ℑ,∇) is thefashion of all paths in ∇ from ζ to ℑ.
Altun et al. [35] gave two fixed point results for multivalued F -contractions on mm-spaces. We ensure the existence of fixed point results for generalized FmR-contractions by using the concept given in [35] to the metric space setup. The motivation of this study is to solve nonlinear matrix equations. First, inspired by Altun et al. [31] and Wardowski [13], we give the following concepts.
Theoretic relations have been used in many research articles, for examples see [36]. A non-empty subset R of U2 is said to be a relation on the m-metric space (U,m) if R={(ζ,ℑ)∈U2:ζ,ℑ∈U}. If (ζ,ℑ)∈R, then we say that ζ⪯ℑ (ζ precede ℑ) under R denoted by (ζ,ℑ)∈R, and the inverse of R is denoted as R−1={(ζ,ℑ)∈U2:(ℑ,ζ)∈R}. Set S=R∪R−1⊆U2. Consequently, we illustrate another relation on U denoted S∗ and is given as (ζ,ℑ)∈S∗⇔(ℑ,ζ)∈S and Ω≠ℑ.
Definition 1.3. [36] Let U≠∅ and R be a binary relation on U. Then R is transitive if (ζ,ξ)∈R and (ξ,ℑ)∈R ⇒ (ζ,ℑ)∈R, for all ζ,ℑ,ξ∈U.
Definition 1.4. [36] Let U≠∅. A sequence ζμ∈U is called R-preserving, if (ζμ,ζμ+1)∈R.
Definition 1.5. [36] Let U≠∅ and γ:U→U. A binary relation R on U is called γ-closed if for any ζ,ℑ in U, we deduce (ζ,ℑ)∈R⇒(γ(ζ),γ(ℑ))∈R.
We begin with the following definitions.
Definition 2.1. We say that (U,m,R) is regular if for each sequence {ζμ} in U,
(ζμ,ζμ+1)∈R for all μ∈Nlimμ→∞(m(ζμ,ζ)−mζμζ)=0, i.e., ζμtm→ζ∈R,}⇒(ζμ,ζ)∈R, for all μ∈N. |
Definition 2.2. A relation theoretic m-metric space (U,m,R) is said to be R-complete if for an R-preserving m-Cauchy sequence {ζμ} in U, there exists some ζ in U such that
limμ→∞m(ζμ,ζ)−mζμζ=0, and limμ→∞(Mζμ,ζ−mζμζ)=0. |
Definition 2.3. Let (U,m) be a m-metric space endowed with a binary relation R on U and γ be a self-mapping on U. Then, γ is said to be a FmR-contractions, if there exist FmR∈Π(∇) and ξ>0, such that
ξ+FmR(m(γ(ζ),γ(ℑ)))≤FmR(m(ζ,ℑ)) | (2.1) |
for all ζ,ℑ∈U with (ζ,ℑ)∈S∗.
Now, we introduce the concept of a generalized rational type FmR-contraction.
Definition 2.4. Let (U,m) be a m-metric space endowed with a binary relation R on U. Let γ:U→U be a self-mapping on U. It is called a generalized rational type FmR-contraction if there are FmR∈∇(F) and ξ>0 such that
ξ+FmR(m(γ(ζ),γ(ℑ)))≤FmR(max{m(ζ,ℑ),m(ζ,γ(ζ)),m(ℑ,γ(ℑ)),m(ζ,γ(ζ))[1+m(ℑ,γ(ℑ))]1+m(ζ,ℑ)}), | (2.2) |
for all ζ,ℑ∈U with (ζ,ℑ)∈S∗.
Theorem 2.1. Let (U,m) be a complete m-metric space with a binary relation R on U and γ be a self-mapping on U such that:
(i) the class Θ(γ,R) is nonempty;
(ii) R is γ-closed;
(iii) the mapping γ is R-continuous;
(iv) γ is a generalized rational type FmR-contraction mapping.
Then γ possesses a fixed point in U.
Proof. Let ζ0∈Θ([γ,R]). We define a sequence {ζμ} by ζμ+1=γ(ζμ)=γμ(ζ0) for each μ∈N. If there is μ0 in N so that γ(ζμ0)=ζμ0, then γ has a fixed point ζμ0 and the proof is complete. Let ζμ+1≠ζμ for all μ in N, so m(ζμ+1,ζμ)>0. Since (γ(ζ0),ζ0)∈S∗, using γ-closedness of R, we get (γ(ζμ+1),ζμ)∈S∗. Then using the fact that γ is a generalized rational type FmR-contraction mapping, one writes
FmRm(ζμ+1,ζμ)=FmR(m(ζμ+1,ζμ)). |
≤FmR(max{m(ζμ,ζμ−1),m(ζμ,γ(ζμ)),m(ζμ−1,γ(μ−1)),m(ζμ,ζμ+1)[1+m(ζμ−1,ζμ)]1+m(ζμ−1,ζμ)})−ξ | (2.3) |
≤FmR(max{m(ζμ,ζμ−1),m(ζμ,ζμ+1),m(ζμ−1,ζμ),m(ζμ,ζμ+1)[1+m(ζμ−1,ζμ)]1+m(ζμ−1,ζμ)})−ξ≤FmR(max{m(ζμ,ζμ−1),m(ζμ,ζμ+1)})−ξ. |
If max{m(ζμ,ζμ−1),m(ζμ,ζμ+1)}=m(ζμ,ζμ+1), then from (2.3), we have
FmR(m(ζμ,ζμ+1))≤FmR(m(ζμ,ζμ+1))−ξ<FmR(m(ζμ,ζμ+1)), |
which is a contradiction. Thus, max{m(ζμ,ζμ−1),m(ζμ,ζμ+1)}=m(ζμ,ζμ−1) and so from (2.3), we have
FmR(m(ζμ,ζμ+1))≤FmR(m(ζμ−1,ζμ)) for all μ∈N. | (2.4) |
Denote δμ=m(ζμ,ζμ+1). We have δμ>0 for all μ∈N and using (2.4) we deduce that
FmR(δμ)≤FmR(δμ−1)−ξ≤FmR(δμ−1)−2ξ≤...FmR(δμ−1)−μξ. | (2.5) |
It implies that limμ→∞FmR(δμ)=−∞, then by (F2), we have limμ→∞δμ=0. Due to (F3), there exists k∈(0,1) such that limμ→∞δkμFmR(δμ)=0.
From (2.4) the following is true for all μ∈N,
δkμ(FmR(δμ)−FmR(δ0))≤−δkμμτ≤0. | (2.6) |
Letting μ→∞ in (2.6), we get
limμ→∞μδkμ=0. | (2.7) |
From (2.7), there exists μ1∈N so that μδkn≤1 for all μ≥μ1, then we deduce
δμ≤1μ1k for all μ≥μ1. |
We claim that {ζμ} is a m-Cauchy sequence in the m-metric space. Let ν,μ∈N such that ν>μ≥μ1. Using the triangle inequality of a m-metric space, one writes
m(ζμ,ζν)−mζμ,ζν≤m(ζμ,ζμ+1)−mζμ,ζμ+1+m(ζμ+1,ζμ+2)−mζμ+1,ζμ+2+...+m(ζν−1,ζν)−mζν−1,ζν≤m(ζμ,ζμ+1)+m(ζμ+1,ζμ+2)+⋯+m(ζν−1,ζν)≤δμ+δμ+1+⋯+δv−1=ν−1∑i=μδi≤∞∑i=μδi≤∞∑i=μ1i1k. |
The convergence of the series ∑∞i=μ1i1k yields that m(ζμ,ζν)−mζμ,ζν→0. Thus, {ζμ} is a M-Cauchy sequence in (U,m). Since (U,m,R) is R-complete, there exists ζ∈U such that {ζμ} converges to ζ with respect to tκ, that is, m(ζμ,ζ)−mζμ,ζ→0 as μ→∞. Now, the R-continuity of γ implies that
ζ=limμ→∞ζμ+1=limμ→∞γ(ζμ)=γ(ζ). |
Hence, ζ is a fixed point of γ.
Example 2.1. Let U=[0,∞) and m be defined by m(ζ,ℑ)=min{ζ,ℑ} for all ζ,ℑ∈U. (U,m) is a complete m-metric space. Consider the sequence {zμ}⊆U given by zμ=μ(μ+1)(2μ+1)6 for all μ≥2. Set an binary relation on U denoted by R given by R={(z1,z1),(zμ−1,zμ):μ=2,3,...100}. Now, give γ:U→U as
γ(ζ)={ζ, if 0≤ζ≤z1,z1, if z1≤ζ≤z2,zμ+(zμ−zμ−1zμ+1−zμ)(ζ−zμ), if zμ+1≤ζ≤zμforallμ=2,3,⋯,100. |
Obviously, R is γ-closed and γ is continuous. Choosing ζ=zμ and ℑ=zμ+1 (for μ=1,2,3,⋯,100), for first condition of F (which is (F1)), we have
FmR(m(γ(zμ),γ(zμ+1)))=FmR(m(zμ−1,zμ))=FmR(zμ−1)=100ln(zμ−12−sinzμ−1), |
and
FmR(max{m(zμ,zμ+1),m(zμ,γ(zμ)),m(zμ+1,γ(zμ+1)),m(zμ,γ(zμ))(1+m(zμ+1,γ(zμ+1)))1+m(zμ,zμ+1)})=FmR(max{m(zμ,zμ+1),m(zμ,zμ−1),m(zμ+1,zμ),m(zμ,zμ−1)(1+m(zμ+1,zμ))1+m(zμ,zμ+1)}).=FmR(zμ). |
Now, for μ=2,3,4,⋯,100, we have
ξ+100ln(zμ−12+sinzμ−1)≤100ln(zμ2+sinzμ), | (2.9) |
implies that
ξ≤100ln(zμ2+sinzμzμ−12+sinzμ−1). | (2.10) |
Let
f(μ)=100ln(zμ2+sinzμzμ−12+sinzμ−1). | (2.11) |
In view of Table 1 and Figure 1, since the function {f(μ)}μ≥2 is decreasing and discontinuous, the smallest value in (2.11) is 5.02. Therefore, the Eq (2.10) holds for 0<ξ<5. So
ξ+FmR(m(γ(ζ),γ(ℑ)))≤FmR(max{m(ζ,ℑ),m(ζ,γ(ζ)),m(ℑ,γ(ℑ)),m(ζ,γ(ζ))[1+m(ℑ,γ(ℑ))]1+m(ζ,ℑ)}), |
Iter | f(μ) | Iter | f(μ) |
μ=2 | 13.87 | μ=13 | 25.21 |
μ=3 | 164.6 | μ=14 | 23.21 |
μ=4 | 56.16 | μ=15 | 21.20 |
μ=5 | 63.72 | μ=16 | 20.20 |
μ=6 | 54.28 | μ=17 | 18.71 |
μ=7 | 44.23 | μ=18 | 17.64 |
μ=8 | 36.45 | μ=19 | 16.53 |
μ=9 | 33.78 | ... | ... |
μ=10 | 33.78 | μ=60 | 5.02 |
for all ζ,ℑ∈U such that (ζ,ℑ)∈S∗ with mm-space. Hence, γ is a generalized rational type FmR-contraction mapping with 0<ξ<5. Generally, we can say that γ has infinite (F.Ps).
Theorem 2.2. Theorem 2.1 remains true if the condition (ii) is replaced by the following:
(i) (ii)′,
(ii) (X,κ,∇) is regular.
Proof. It is a same argument as Theorem 2.1. Here, the sequence {ζμ} is m-Cauchy and converges to some Ω in U such that m(ζμ,ζ)−mζμ,ζ as limit μ→∞ which implies that
limμ→∞m(ζμ,ζ)=limμ→∞mζμ,ζ=limμ→∞min{m(ζμζμ),m(ζ,ζ)}=m(ζ,ζ)=limμ,ν→∞m(ζμ,ζν)=0 and limμ,ν→∞mζμ,ζν=0. |
As (ζμ,ζμ+1)∈R, then (ζμ,ζ)∈R for all μ∈N. Set μ={μ∈N:γ(ζμ)=γ(ζ)}. We will take two cases depending on μ
C-1. If μ is a finite set, then there exists μ0 in N, so that γ(ζμ)≠γ(ζ) for every μ≥μ0. In particular, (ζμ,ζ)∈S∗ and (γ(ζμ),γ(ζ))∈S∗, then for all μ≥μ0,
ξ+FmR(m(γ(ζμ),γ(ζ)))≤F(m(ζμ,ζ)). |
Since limμ→∞m(ζμ,ζ)=0 implies that limμ→∞FmR(m(ζμ,ζ))=−∞, one writes limμ→∞FmR(m(γ(ζμ),γ(ζ)))=−∞. Therefore, limμ→∞m(γ(ζμ),γ(ζ))=0, which yields that γ(ζ)=ζ, that is, ζ is a fixed point of γ.
C-2. If μ is an infinite set, then there exists a subsequence {ζμk} of {ζμ} so that ζμk+1=γ(ζμk)=γ(ζ) for k∈N, so γ(ζμk)→γ(ζ) with respect to tm as ζμ converges ζ, then γ(ζ)=ζ, i.e., γ has a fixed point. Hence, the proof is complete.
Now, we prove a result of uniqueness.
Theorem 2.3. Following Theorems 2.1 and 2.2 γ possesses a unique fixed point if ϝ(ζ,ℑ,∇)≠∅, for all ζ,ℑ∈(γ)Fix.
Proof. Let ζ,ℑ∈(γ)Fix such that ζ≠ℑ. Since ϝ(ζ,ℑ,∇)≠∅, there exists a path ({a0,a1,...aμ}) of some finite length μ in ∇ from Ω to ℑ (with as≠as+1 for all s∈[0,p−1]). Then a0=ζ, ak=ℑ, (as,as+1)∈S∗ for every s∈[0,p−1]. As as∈γ(U), γ(as)=as for all s∈[0,p−1] we deduce that
FmR(m(as,as+1))=FmR(m(γ(as),γ(as+1)))≤FmR{max{m(as,as+1),m(as,γ(as)),m(as+1,γ(as+1)),m(as,γ(as)),[1+m(as,γ(as+1))]1+m(as,as+1)}}−ξ=FmR{max{m(as,as+1),m(as,as),m(as+1,as+1),m(as,as)[1+m(as,as+1)]1+m(as,as+1)}}−ξ<FmR{(m(as,as+1))}. |
It is a contradiction. Hence, γ possesses a unique fixed point.
Now, we say that γ:U→U has the property P if
(γμ)Fix=(γ)Fix for each μ is member of N. |
In this theorem, we use above condition having property P.
Theorem 2.4. Let (U,m) be a complete m-metric space with a binary relation R on U and γ be a self-mapping such that:
(i) the class Θ(γ,R) is nonempty,
(ii) the binary relation R is γ-closed,
(iii) γ is R-continuous,
(iv) there are FmR∈∇(F) and ξ>0 so that
ξ+FmR(m(γ(ζ),γ2(ζ)))≤FmR(max{m(ζ,γ(ζ)),m(γ(ζ),γ2(ζ)),m(ζ,γ(ζ))[1+m(γ(ζ),γ2(ζ))]1+m(γ(ζ),γ2(ζ))}) |
for all ζ∈U, with (γ(ζ),γ2(ζ))∈S∗.
Then γ has a fixed point. Furthermore, if
(v) (iv)′;
(vi) ζ∈(γμ)Fix (forsomeμ∈N) which implies that (ζ,γ(ζ))∈R,
then γ has a property P.
Proof. Let ζ0∈Θ([γ,R]), i.e., (ζ0,γ(ζ0))∈R ,therefore using assumption (ii), we get (ζμ,ζμ+1)∈R for each μ∈N. Denote ζμ+1=γ(ζμ)= γμ+1(ζ0), for all μ∈N. If there exists μ0∈N so that γ(ζμ0)=ζμ0, then γ has a fixed point ζμ0 and it completes the proof. Otherwise, assume that ζμ+1≠ζμ for every μ∈N. Then (ζμ,ζμ+1)∈R (forallμ∈N). Continuing this process and using the assumption (iv), we deduce (forallμ∈N)
FmR(m(γ(ζμ−1),γ2(ζμ−1)))≤FmR(max{m(ζμ−1,γ(ζμ−1)),m(γ(ζμ−1),γ2(ζμ−1)),m(ζμ−1,γ(ζμ−1))[1+m(γ(ζμ−1),γ2(ζμ−1))]1+m(γ(ζμ−1),γ2(ζμ−1))})−ξ=FmR(max{m(ζμ−1,ζμ),m(ζμ,ζμ+1)m(ζμ−1,ζμ)[1+m(ζμ,ζμ+1)]1+m(ζμ,ζμ+1)})−ξ≤FmR(max{m(ζμ−1,ζμ),m(ζμ,ζμ+1)})−ξ. |
Assume that max{m(ζμ−1,ζμ),m(ζμ,ζμ+1)}=m(ζμ,ζμ+1), then we get
FmR(m(ζμ,ζμ+1))≤FmR(m(ζμ,ζμ+1))−ξ<FmR(m(ζμ,ζμ+1)). |
It is a contradiction. Hence, max{m(ζμ−1,ζμ),m(ζμ,ζμ+1)}=m(ζμ−1,ζμ), and so
FmR(m(ζμ,ζμ+1))≤FmR(m(ζμ−1,ζμ))−ξforallμ∈N. |
This yields that (forallμ∈N)
FmR(m(ζμ,ζμ+1))≤FmR(m(ζμ−1,ζμ))−ξ≤FmR(m(ζμ−2,ζμ−1))−2ξ≤...≤FmR(m(ζ0,ζ1))−μξ. | (2.1) |
By applying limit as μ goes to ∞ in above equation, we deduce limμ→∞FmR(m(ζμ,ζμ+1))=−∞. Since FmR∈∇(F), we deduce that limμ→∞m(ζμ,ζμ+1)=0. Using (F3), there is k∈(0,1) so that
limμ→∞(m(ζμ,ζμ+1))kFmR(m(ζμ,ζμ+1))=0. |
Now, from (2.9), we have
(m(ζμ,ζμ+1))kFmR(m(ζμ,ζμ+1))−(m(ζμ,ζμ+1))kFmR(m(ζ0,ζ1))≤−(m(ζμ,ζμ+1))kμ≤0. | (2.2) |
Letting μ→∞ in (2.10), we get limμ→∞(m(ζμ,ζμ+1))k=0. There is μ1 in N so that
μ(m(ζμ,ζμ+1))k≤1forallμ≥μ1. |
That is,
m(ζμ,ζμ+1)≤1μ1kforallμ≥μ1. |
Now, for ν>μ>μ1, we have
m(ζμ,ζν)−mζμ,ζν≤ν−1∑i=μm(ζμ,ζν)−mζμ,ζν≤ν−1∑i=μm(ζμ,ζν)≤ν−1∑i=μ1i1k. |
Since the series ∑ν−1i=μ1i1k is convergent, i.e., m(ζμ,ζν)−mζμ,ζν converges to 0, the sequence {ζμ} is a m-Cauchy sequence. Since (U,m,R) is R-complete and (ζμ,ζμ+1)∈R for all μ∈N, {ζμ} converges to ζ∈U. Now, using the R-continuity of γ, we deduce that
ζ=limμ→∞ζμ+1=limμ→∞γ(ζμ)=γ(ζ). |
Finally, we will prove that (γμ)Fix=(γ)Fix where μ∈N. Assume on contrary that ζ∈(γμ)Fix and ζ∉(γ)Fix for some μ∈N. Then m(ζ,γ(ζ))>0, (ζ,γ(ζ))∈R(fromcondition(iv)′). From assumption (ii) we obtain (γμ(ζ),γμ+1(ζ))∈R for all μ∈N. Assumption (iv) implies that
FmR(m(ζ,γ(ζ)))=FmR(m(γ(γμ−1(ζ)),γ2(γμ−1(ζ))))≤FmR(m(γμ−1(ζ)),γμ(ζ))−ξ≤FmR(m(γμ−2(ζ)),γμ−1(ζ))−2ξ...≤FmR(m(ζ,γ(ζ)))−μξ. |
Taking μ→∞ in above inequality, we obtain FmR(m(ζ,γ(ζ)))=−∞, a contradiction. So, (γμ)Fix=(γ)Fix for any μ∈N.
Corollary 2.1. Let (U,m) be a complete m-metric space with a binary relation R on U and γ be a self-mapping such that:
(i) the class Θ(γ,R) is nonempty;
(ii) the binary relation R is γ-closed;
(iii) γ is R-continuous;
(iv) γ is a FmR-contraction mapping.
Then γ possesses a fixed point in U.
Here, we use the definition of F-contractions with the standard conditions (i−iii).
Definition 2.5. Given a mm-space (U,m) and a binary relation R on U. Suppose that
ϖ={(ζ,ℑ)∈S∗:κ(ζ,ℑ)>0}. |
We say that a self-mapping γ:U→U is a rational type FmR-contraction if there exists FmR∈Π(∇) such that
ξ+FmR(m(γ(ζ),γ(ℑ)))≤FmR(max{m(ζ,ℑ),m(ζ,γ(ζ)),m(ℑ,γ(ℑ)),m(ζ,γ(ζ))[1+m(ℑ,γ(ℑ))]1+m(ζ,ℑ)}) | (3.1) |
for all (ζ,ℑ)∈Ξ.
Theorem 2.5. Let (U,m) be a complete m-metric space, R be a binary relation on U and γ be a self-mapping on U. Assume that:
(i) the class Θ(γ,R) is non-empty;
(ii) the binary relation R is γ-closed;
(iii) γ is R-continuous;
(iv) γ is a rational type FmR -contraction mapping.
Then γ possesses a fixed point in U.
Proof. Let ζ0∈Θ([γ,R]), i.e., ([ζ0,γ(ζ0)])∈R . We define a sequence {ζμ+1} given as ζμ+1=γ(ζμ)= γμ+1(ζ0). We have (ζμ,ζμ+1)∈R for all μ in N. If there exists μ0 in N such that γ(ζμ0)=ζμ0, then ζμ0 is a fixed point of γ and the proof is finished. Now, assume that ζμ+1≠ζμ for all μ∈N. Then (ζμ,ζμ+1)∈R(for all μ∈N). Using the condition (iv), we deduce (for all μ∈N)
F(m(γ(ζμ−1),γ2(ζμ−1)))≤FmR(max{m(ζμ−1,γ(ζμ−1)),m(γ(ζμ−1),γ2(ζμ−1)),m(ζμ−1,γ(ζμ−1))[1+m(γ(ζμ−1),γ2(ζμ−1))]1+m(γ(ζμ−1),γ2(ζμ−1))})−ξ=FmR(max{m(ζμ−1,ζμ),m(ζμ,ζμ+1),m(ζμ−1,ζμ)[1+m(ζμ,ζμ+1)]1+m(ζμ,ζμ+1)})−ξ≤FmR(max{m(ζμ−1,ζμ),m(ζμ,ζμ+1)})−ξ. | (2.3) |
By (F1), we have max{m(ζμ−1,ζμ),m(ζμ,ζμ+1)}=m(ζμ,ζμ+1), then we get a contradiction. Thus, max{m(ζμ,ζμ−1),m(ζμ,ζμ+1)}=m(ζμ,ζμ−1) and so from (2.3) we have
FmR(m(ζμ,ζμ+1))≤FmR(m(ζμ,ζμ−1))−ξ for all μ∈N. | (3.3) |
The proof of Theorem 2.1 is complete.
Corollary 2.2. Let (U,m) be a complete m-metric space, R be a binary relation on U and γ be a self-mapping on U. Assume that:
(i) the class Θ(γ,R) is non-empty;
(ii) R is γ-closed;
(iii) γ is R-continuous;
(iv) γ is a FmR-contraction mapping.
Then γ possesses a fixed point in U.
Example 2.2. Let U=[0,1] and m be a relation theoretic m-metric defined by
m(ζ,ℑ)=ζ+ℑ2 for all ζ,ℑ∈U. |
We define the binary relation
(ζ,ℑ)∈S∗⇔m(ζ,ζ)=m(ζ,ℑ)⇔ζ+ℑ2. |
(U,m) is a complete m-metric space with a binary relation. Define a mapping γ:U→U by
γ(ζ)={ζ5 if ζ∈[0,1)0 if ζ=1. |
Obviously, ℜ is γ-closed, also and γ is ℜ -continuous. Define Fmℜ:(0,∞)→R by
Fmℜ(a)=ln(a+a2) forall ξ∈(0,∞). |
Assume that (ζ,ℑ)∈Ξ={(ζ,ℑ)∈S∗:m(γ(ζ),γ(ℑ))>0}. Therefore, for all ζ,ℑ∈U, with 0<ζ<1,ℑ=1, we have
Fmℜ(m(γ(ζ),γ(ℑ)))=Fmℜ(m(ζ5,0))=Fmℜ(ζ10)=ln(ζ1002+ζ10). |
Now, consider ZA=max{m(ζ,1),m(ζ,γ(ζ)),m(ℑ,γ(ℑ)),m(ζ,γ(ζ))[1+m(ℑ,γ(ℑ))]1+m(Ω,1)}
FmR(max{m(ζ,1),m(ζ,γ(ζ)),m(ℑ,γ(ℑ)),m(ζ,γ(ζ))[1+m(ℑ,γ(ℑ))]1+m(ζ,1)})=FmR(max{ζ+12,ζ+ζ52,1+02,ζ+ζ52[1+1+02]1+ζ+12})=FmR(ζ+12)=FmR(ZA)=ln((ζ+12)2+ζ+12). |
From Table Table.2, γ is a rational type FmR -contraction mapping with ξ=2. Moreover, there is ζ0=0.1 in U so that(ξ0,γ(ξ0))∈S∗ and the class Θ(γ,R) is nonempty. Hence, all conditions of Theorem 2.5 hold, and therefore γ has a fixed point.
In [37], Kirk et al. gave the concept of a cyclic contraction, which is the extension of the Banach contraction. It is utilized in the following theorem.
Theorem 3.1. Suppose that (U,m) is a compete m-metric space, G, H are two nonempty closed subsets of U and γ:U→U verifies the following conditions:
(i) γ(B)⊆D and γ(D)⊆B;
(ii) there exists a constant k∈(0,1) such that
m(γ(ζ),γ(ℑ))≤km(ζ,ℑ)forallζ∈B,ℑ∈D. |
Then B∩D is nonempty and there is ζ∈B∩D a fixed point of γ.
By Theorems 2.1 and 3.1, we obtain successive fixed point results for cyclic rational type FmR- generalized contraction mappings.
Theorem 3.2. Let (U,m) be a complete m-metric space, G and H be two nonempty closed subsets of U and γ:U→U be an operator. Assume that the successive axioms hold:
(i) γ(G)⊆H and γ(H)⊆G;
(ii) there exist FmR∈∇(F) and ξ>0 such that
ξ+FmR(m(γ(ζ),γ(ℑ)))≤FmR(max{m(ζ,ℑ),m(ζ,γ(ζ)),m(ℑ,γ(ℑ)),m(ζ,γ(ζ))[1+m(ℑ,γ(ℑ))]1+m(ζ,ℑ)}), | (2.10) |
for all ζ in G and ℑ in H.
Then there is ζ∈G∩H a fixed point of γ.
Proof. Z=G∪H is closed, so Z is a closed subspace of U. Therefore, (U,m) is a complete m-metric space. Set a binary relation on Z denoted by R given as
R=G×H. |
It means that
(ζ,ℑ)∈R⇔(ζ,ℑ)∈B×D for all ζ,ℑ∈Z. |
Set S=R∪R−1 an asymmetric relation. Directly, (U,m,S) is regular. Assume {ζμ}∈Z is any sequence and ζ∈Z so that
(ζμ,ζμ+1)∈S for all μ∈N, |
and
limμ→∞m(ζμ,ζ)=limμ→∞min{m(ζμ,ζμ),m(ζ,ζ)}=m(ζ,ζ). |
Using the definition of S, we obtain
(ζμ,ζμ+1)∈(B×D)∪(D×B) for all μ∈N. | (3.2) |
Immediately, the product fashion Z×Z involves a mm-space m given as
m((ζ1,ℑ1),(ζ2,ℑ2))=m(ζ1,ℑ1)+m(ζ2,ℑ2)2. |
Since (U,m) is a complete m-metric space, we obtain (Z×Z,m) is complete. Furthermore, G×H and H×G are closed in (Z×Z,m), because G and H are closed in (U,m). Letting μ→∞ in (2.11), we have (ζ,ζ)∈(B×D)∪(D×B). This implies that ζ∈B∩D. Furthermore, from Eq (2.11), we have ζμ∈B∪D. Thus, we get (ζμ,ζ)∈S(for all μ∈N). Therefore, our assertions hold. Furthermore, since γ is a self-mapping and from condition (i), we obtain for all ζ,ℑ∈U,
(ζ,ℑ) in G×H which implies (γ(ζ),γ(ℑ))∈H×G,(ζ,ℑ) in H×G which implies (γ(ζ),γ(ℑ))∈G×H. |
The binary relation R is γ-closed. As B≠∅, there exists ζ0∈B, such that γ(ζ0)∈D that is (ζ0,γ(ζ0))∈R. Therefore, all the hypotheses of Theorem 2.2 are satisfied. Hence, (γ)Fix ≠∅ and also(γ)Fix⊆B∩D. Finally, as (ζ,ℑ)∈R for all ζ,ℑ∈G∩H, G∩H is ∇-directed. Hence, the main conditions of Theorem 2.2 are satisfied, so γ has a unique fixed point. It finishes the proof.
In this section, we illustrate how to guarantee existence of a solution of a matrix type equation. We shall use the following notations. Let A(μ) be the set of all μ×μ complex matrices, let H(μ)⊆A(μ) be the family of all μ×μ Hermitian matrices, let G(μ)⊆A(μ) be the set of all μ×μ positive definite matrices, H+(μ)⊆F(μ) be the set of all μ×μ positive semidefinite matrices. For Λ in G(μ), we will also denote Λ≻0. Furthermore, Λ⪰0 means that Λ in H+(μ) . As a different notation for Λ−Δ⪰0 and Λ−Δ≻0, we will denote Λ⪰Δ andΛ≻Δ, respectively. Also, for each Λ,Δ in A(μ) there is a greatest lower bound and least upper bound, see [38]. In addition, take
‖.‖= denote the spectral norm of matrix Q i.e ‖Q‖=(λ+(Q∗Q))12, |
such that
λ+(Q∗Q)= is the largest eigenvalue of Q∗Q, where Q∗ is the conjugate transport of Q. |
We use the m-metric induced by the trace norm ‖.‖tr given as ‖Q‖tr=∑μi=1Ξi(Q), where Ξi(Q), i=1,2,...,μ are the singular values of Q in A(μ). The set H(μ) endowed with this norm is a complete m-metric space. Moreover, we see that
H(μ)=is a partial ordered set with partial order⪯, where Λ⪯Δ⇔Λ⪰Δ. |
Consider the following nonlinear matrix equation
Λ=S+μ∑i=1Q∗iΞ(Λ)Qi, | (4.1) |
where ϑ is a positive definite matrix, Q1,Q2,⋯,Qm are μ×μ matrices and Ξ is an order persevering continuous map from H(μ) to G(μ). Then, FmR∈∇(F) and (A(μ),m) is a complete mm-space, where
m(Λ,Δ)=‖Λ+Δ2‖tr=12(tr(Λ+Δ)). | (4.2) |
In this section, we prove the existence of the positive definite solution to the nonlinear matrix Eq (4.1).
Theorem 4.1. Assume that there are positive real numbers C and ξ such that:
(i) for each Λ,Δ in H(μ) such that (Λ,Δ) in ⪯ with μ∑i=1Q∗iΞ(Λ)Qi≠μ∑i=1Q∗iΞ(Δ)Qi,
|tr(Ξ(Λ)+Ξ(Δ))2|≤|tr(Λ+Δ)2|C(1+ξ√tr(Λ+Δ)2)2, |
(ii) there exists a positive number N for which μ∑i=1QiQ∗i<CIμ and μ∑i=1Q∗iΞ(Λ)Qi>0.
Then the matrix Eq (4.1) has a solution. Furthermore, the iteration
Λμ=S+μ∑i=1Q∗iΞ(Λμ−1)Qi, | (4.2) |
where Λ0 in F(μ) satisfies Λ0⪯ϑ+μ∑i=1Q∗iΞ(Sμ−1)Qi, converges in the sense of trace norm ‖.‖tr to the solution of the matrix Eq (4.1).
Proof. We define the mapping γ:H(μ)→H(μ) and FmR:R+→R by
γ(Λ)=S+μ∑i=1Q∗iΞ(Λ)Qi, for all Λ∈F(μ), |
and set
H+(μ)(γ,⪯)={Q∈F(μ):Q⪯γ(Q) or γ(Q)−Q ⪰0}. |
Then, γ is well defined and ⪯ is a relation under R, and ⪯ on F(μ) is γ-closed. FmR(a)=−1√a for all a∈R+. Furthermore, a fixed point of γ is a positive solution of (4.1). Now, we want to prove that γ is a FmR-contraction mapping with ξ Let (Λ,Δ)∈ϖ={((Λ,Δ)∈R:Ξ(Λ)≠Ξ(Δ))} which implies that Λ≺Δ. Since Ξ is an order preserving mapping, we deduce that Ξ(Λ)≺Ξ(Δ). We have
‖γ(Λ)+γ((Δ))2‖tr=12(tr(γ(Λ)+γ(Δ)))=m∑i=112(tr(QiQ∗i(γ(Λ)+γ(Δ))))=12tr((m∑i=1QiQ∗i)γ(Λ)+γ(Δ))≤‖m∑i=1EiE∗i‖12‖γ(Λ)+γ(Δ)‖1≤‖m∑i=1EiE∗i‖C(‖Λ+Δ2‖(1+ξ√‖Λ+Δ2‖)2)<(‖Λ+Δ2‖(1+ξ√‖Λ+Δ2‖)2), |
and so
(1+ξ√‖Λ+Δ2‖tr)2‖Λ+Δ2‖tr≤1‖γ(Λ)+γ(Λ)2‖tr. |
This implies that
(ξ+1√‖Λ+Δ2‖tr)2≤1‖γ(Λ)+γ(Δ)2‖tr, |
and then
ξ+1√‖Λ+Δ2‖tr≤1√‖γ(Λ)+γ(Δ)2‖tr. |
Consequently,
ξ−1√‖γ(Λ)+γ(Δ)2‖tr≤−1√‖Λ+Δ2‖tr. |
Now, we get
ξ+FmR(‖γ(Λ)+γ(Δ)2‖tr)≤FmR(‖Λ+Δ2‖tr). |
This shows that γ is a FmR-contraction. Using μ∑i=1Q∗iΞ(ϑ)Qi≻0, we deduce that ϑ⪯γ(ϑ). This means that ϑ in H+(γ,⪯). From Corollary 2.2, there exists Λ0∈H(μ) such that γ(Λ0)= Λ0. Hence, the matrix Eq (4.1) has a solution.
Example 4.1. Now, consider the matrix equation
Λ=S+2∑i=1Q∗iΞ(Λ)Qi, |
where
ϑ=(0.10.010.010.010.10.010.010.010.1), |
Q1=(0.20.010.010.010.40.010.010.010.4), |
Q2=(0.60.010.010.010.60.010.010.010.6), |
and Define FmR:R+→R by
FmR(a)=−1√a, |
for all a∈R+, and Ξ:H(μ)→H(μ) is given by Ξ(Λ)=Λ3. Then, all conditions of Corollary 2.2 are satisfied for N=610 by using the iterative sequence
Λμ+1=S+2∑i=1Q∗iΞ(Λ)Qi, |
Λ0=(000000000). |
After some iterations, we get the approximation solution
ϑ15=(0.02330.01020.09120.01020.04660.02140.09120.05320.0326). |
Hence, all the conditions of Theorem 4.1 are satisfied.
In this paper, a relation theoretic M-metric fixed point algorithm under rational type FmR-contractions (respectively, rational type generalized FmR-contractions) is proposed to solve the nonlinear matrix equation Λ=S+μ∑i=1Q∗iΞ(Λ)Qi. Some numerical comparison experiments with existing algorithms are presented within given tables and figures. Analogously, this proposed work can be extended to generalized distance spaces, such as symmetric spaces, mbm -spaces rmm-spaces, rmbm-spaces, pm-spaces, pbm-spaces, etc. Some problems of fixed point results could be studied in near future.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work grant code: 23UQU4331214DSR001.
The authors declare that they have no conflict of interest.
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1. | Muhammad Tariq, Muhammad Arshad, Eskandar Ameer, Ahmad Aloqaily, Suhad Subhi Aiadi, Nabil Mlaiki, On Relational Weak Fℜm,η-Contractive Mappings and Their Applications, 2023, 15, 2073-8994, 922, 10.3390/sym15040922 | |
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Iter | f(μ) | Iter | f(μ) |
μ=2 | 13.87 | μ=13 | 25.21 |
μ=3 | 164.6 | μ=14 | 23.21 |
μ=4 | 56.16 | μ=15 | 21.20 |
μ=5 | 63.72 | μ=16 | 20.20 |
μ=6 | 54.28 | μ=17 | 18.71 |
μ=7 | 44.23 | μ=18 | 17.64 |
μ=8 | 36.45 | μ=19 | 16.53 |
μ=9 | 33.78 | ... | ... |
μ=10 | 33.78 | μ=60 | 5.02 |
Iter | f(μ) | Iter | f(μ) |
μ=2 | 13.87 | μ=13 | 25.21 |
μ=3 | 164.6 | μ=14 | 23.21 |
μ=4 | 56.16 | μ=15 | 21.20 |
μ=5 | 63.72 | μ=16 | 20.20 |
μ=6 | 54.28 | μ=17 | 18.71 |
μ=7 | 44.23 | μ=18 | 17.64 |
μ=8 | 36.45 | μ=19 | 16.53 |
μ=9 | 33.78 | ... | ... |
μ=10 | 33.78 | μ=60 | 5.02 |