In this paper, the Kaldor business cycle model with variable depreciation rate of the capital stock are investigate the existence, uniqueness and stability of the positive equilibrium point, and the existence of the periodic solution and Hopf bifurcation respectively. Finally, we analyze the dynamic behaviors of the specific system and perform numerical simulations.
1.
Introduction
Mathematical modelling provides a systematic formalism for the understanding of the corresponding real-world problem. Moreover, adequate mathematical tools for the analysis of the translated real-world problem are at our disposal. Fixed point theory (FPT), an important branch of nonlinear functional analysis, is prominent for modelling a variety of real-world problems. It is worth mentioning that the real-world phenomenon can be translated into well known existential as well as computational FPP.
The EP theory provides an other systematic formalism for modelling the real-world problems with possible applications in optimization theory, variational inequality theory and game theory [7,10,13,17,18,19,21,25,28,31,32]. In 1994, Blum and Oettli [13] proposed the (monotone-) EP in Hilbert spaces. Since then various classical iterative algorithms are employed to compute the optimal solution of the (monotone-) EP and the FPP. It is remarked that the convergence characteristic and the speed of convergence are the principal attributes of an iterative algorithm. All the classical iterative algorithms from FPT or EP theory have a common shortcoming that the convergence characteristic occurs with respect to the weak topology. In order to enforce the strong convergence characteristic, one has to assume stronger assumptions on the domain and/or constraints. Moreover, strong convergence characteristic of an iterative algorithm is often more desirable than weak convergence characteristic in an infinite dimensional framework.
The efficiency of an iterative algorithm can be improved by employing the inertial extrapolation technique [29]. This technique has successfully been combined with the different classical iterative algorithms; see e.g., [2,3,4,5,6,8,9,14,15,16,23,27]. On the other hand, the parallel architecture of the algorithm helps to reduce the computational cost.
In 2006, Tada and Takahashi [33] suggested a hybrid framework for the analysis of monotone EP and FPP in Hilbert spaces. On the other hand, the iterative algorithm proposed in [33] fails for the case of pseudomonotone EP. In order to address this issue, Anh [1] suggested a hybrid extragradient method, based on the seminal work of Korpelevich [24], to address the pseudomonotone EP together with the FPP. Inspired by the work of Anh [1], Hieu et al. [21] suggested a parallel hybrid extragradient framework to address the pseudomonotone EP together with the FPP associated with nonexpansive operators.
Inspired and motivated by the ongoing research, it is natural to study the pseudomonotone EP together with the FPP associated with the class of an η-demimetric operators. We therefore, suggest some variants of the classical Mann iterative algorithm [26] and the Halpern iterative algorithm [20] in Hilbert spaces. We formulate these variants endowed with the inertial extrapolation technique and parallel hybrid architecture for speedy strong convergence results in Hilbert spaces.
The rest of the paper is organized as follows. We present some relevant preliminary concepts and useful results regarding the pseudomonotone EP and FPP in Section 2. Section 3 comprises strong convergence results of the proposed variants of the parallel hybrid extragradient algorithm as well as Halpern iterative algorithm under suitable set of constraints. In Section 4, we provide detailed numerical results for the demonstration of the main results in Section 3 as well as the viability of the proposed variants with respect to various real-world applications.
2.
Preliminaries
Throughout this section, the triplet (H,<⋅,⋅>,‖⋅‖) denotes the real Hilbert space, the inner product and the induced norm, respectively. The symbolic representation of the weak and strong convergence characteristic are ⇀ and →, respectively. Recall that a Hilbert space satisfies the Opial's condition, i.e., for a sequence (pk)⊂H with pk⇀ν then the inequality lim infk→∞‖pk−ν‖<lim infk→∞‖pk−μ‖ holds for all μ∈H with ν≠μ. Moreover, H satisfies the the Kadec-Klee property, i.e., if pk⇀ν and ‖pk‖→‖ν‖ as k→∞, then ‖pk−ν‖→0 as k→∞.
For a nonempty closed and convex subset K⊆H, the metric projection operator ΠHK:H→K is defined as ΠHK(μ)=argminν∈K‖μ−ν‖. If T:H→H is an operator then Fix(T)={ν∈H|ν=Tν} represents the set of fixed points of the operator T. Recall that the operator T is called η-demimetric (see [35]) where η∈(−∞,1), if Fix(T)≠∅ and
The above definition is equivalently represented as
Recall also that a bifunction g:K×K→R∪{+∞} is coined as (ⅰ) monotone if g(μ,ν)+g(ν,μ)≤0, for all μ,ν∈K; and (ⅱ) strongly pseudomonotone if g(μ,ν)≥0⇒g(ν,μ)≤−α‖μ−ν‖2,for all μ,ν∈K, where α>0. It is worth mentioning that the monotonicity of a bifunction implies the pseudo-monotonicity, but the converse is not true. Recall the EP associated with the bifunction g is to find μ∈K such that g(μ,ν)≥0 for all ν∈K. The set of solutions of the equilibrium problem is denoted by EP(g).
Assumption 2.1. [12,13] Let g:K×K→R∪{+∞} bea bifunction satisfying the following assumptions:
(A1) g is pseudomonotone, i.e., g(μ,ν)≥0⇒g(μ,ν)≤0, for allμ,ν∈K;
(A2) g is Lipschitz-type continuous, i.e., there exist two nonnegativeconstants d1,d2 such that
(A3) g is weakly continuous on K×K imply that, if μ,ν∈K and (pk), (qk) are two sequences in K such that pk⇀μ and qk⇀ν respectively, then f(pk,qk)→f(μ,ν);
(A4) For each fixed μ∈K, g(μ,⋅) is convex and subdifferentiable on K.
In view of the Assumption 2.1, EP(g) associated with the bifunction g is weakly closed and convex.
Let gi:K×K→R∪{+∞} be a finite family of bifunctions satisfying Assumption 2.1. Then for all i∈{1,2,⋯,M}, we can compute the same Lipschitz coefficients (d1,d2) for the family of bifunctions gi by employing the condition (A2) as
where d1=max1≤i≤M{d1,i} and d2=max1≤i≤M{d2,i}. Therefore, gi(μ,ν)+gi(ν,ξ)≥gi(μ,ξ)−d1‖μ−ν‖2−d2‖ν−ξ‖2. In addition, we assume Tj:H→H to be a finite family of η-demimetric operators such that Γ:=(⋂Mi=1EP(gi))∩(⋂Nj=1Fix(Tj))≠∅. Then we are interested in the following problem:
Lemma 2.2. [11] Let μ,ν∈H and β∈R then
(1) ‖μ+ν‖2≤‖μ‖2+2⟨ν,μ+ν⟩;
(2) ‖μ−ν‖2=‖μ‖2−‖ν‖2−2⟨μ−ν,ν⟩;
(3) ‖βμ+(1−β)ν‖2=β‖μ‖2+(1−β)‖ν‖2−β(1−β)‖μ−ν‖2.
Lemma 2.3. [35] Let T:K→H be an η-demimetric operator defined on a nonempty, closed and convex subset K of a Hilbert space H with η∈(−∞,1). Then Fix(T) is closed and convex.
Lemma 2.4. [36] Let T:K→H be an η-demimetric operator defined on a nonempty, closed and convex subset K of a Hilbert space H with η∈(−∞,1). Then the operator L=(1−γ)Id+γT is quasi-nonexpansive provided that Fix(T)≠∅ and 0<γ<1−η.
Lemma 2.5. [11] Let T:K→K be a nonexpansive operator defined on a nonempty closed convex subset K of a real Hilbert spaceH and let (pk) be a sequence in K. If pk⇀x and if (Id−T)pk→0, then x∈Fix(T).
Lemma 2.6. [37] Let h:K→R be aconvex and subdifferentiable function on nonempty closed and convex subset K of a real Hilbert space H. Then, p∗ solves the min{h(q):q∈K}, if and only if 0∈∂h(p∗)+NK(p∗), where ∂h(⋅) denotes the subdifferential of h and NK(ˉp) is the normal cone of K at ˉp.
3.
Algorithm and convergence analysis
Our main iterative algorithm of this section has the following architecture:
Theorem 3.1. Let the following conditions:
(C1) ∑∞k=1ξk‖pk−pk−1‖<∞;
(C2) 0<a∗≤γk≤min{1−η1,⋯,1−ηN},
hold. Then Algorithm 1 solves the problem 2.1.
The following result is crucial for the strong convergence result of the Algorithm 1.
Lemma 3.2. [1,30] Suppose that ν∗∈EP(gi), and pk, ek, ui,k, vi,k, i∈{1,2,⋯,M} are defined in Step 1 of the Algorithm 1. Then we have
Proof of Theorem 3.1.
Step 1. The Algorithm 1 is stable.
Observe the following representation of the set Ck+1:
This infers that Ck+1 is closed and convex for all k≥1. It is well-known that EP(gi) and Fix(Tj) (from the Assumption 2.1 and Lemma 2.3, respectively) are closed and convex. Hence Γ is nonempty, closed and convex. For any p∗∈Γ, it follows from Algorithm 1 that
From (3.1) and recalling Lemma 2.4, we obtain
Now recalling Lemma 3.2, the above estimate implies that
The above estimate (3.2) infers that Γ⊂Ck+1. It is now clear from these facts that the Algorithm 1 is well-defined.
Step 2. The limit limk→∞‖pk−p1‖ exists.
From pk+1=ΠHCk+1p1, we have ⟨pk+1−p1,pk+1−ν⟩≤0 for each ν∈Ck+1. In particular, we have ⟨pk+1−p1,pk+1−p∗⟩≤0 for each p∗∈Γ. This proves that the sequence (‖pk−p1‖) is bounded. However, from pk=ΠH1Ckp1 and pk+1=ΠH1Ck+1p1∈Ck+1, we have that
This infers that (‖pk−p1‖) is nondecreasing and hence
Step 3. ~p∗∈Γ.
Compute
Utilizing (3.3), the above estimate infers that
Recalling the definition of (ek) and the condition (C1), we have
Recalling (3.4) and (3.5), the following relation
infers that
Note that pk+1∈Ck+1, therefore the following relation
infers, on employing (3.4) and the condition (C1), that
Again, recalling (3.4) and (3.7), the following relation
infers that
In view of the condition (C2), observe the variant of (3.2)
Recalling (3.8) and condition (C1), we get
The above estimate (3.9) implies that
Reasoning as above, recalling (3.5), (3.8) and (3.10), we have
● ‖ˉvk−ek‖≤‖ˉvk−uik,k‖+‖uik,k−ek‖→0;
● ‖ˉvk−pk‖≤‖ˉvk−ek‖+‖ek−pk‖→0;
● ‖wk−ek‖≤‖wk−pk‖+‖pk−ek‖→0;
● ‖wk−ˉvk‖≤‖wk−ek‖+‖ek−ˉvk‖→0.
In view of the estimate limk→∞‖wk−ˉvk‖=0, we have
Next, we show that ~p∗∈⋂Mi=1EP(gi).
Observe that
Recalling Lemma 2.6, we get
This implies the existence of ˜x∈∂2gi(ek,ui,k) and ~x∗∈NK(ui,k) such that
Since ~x∗∈NK(ui,k) and ⟨~x∗,ν−ui,k⟩≤0 for all ν∈K. Therefore recalling (3.12), we have
Since ˜x∈∂2gi(ek,ui,k),
Therefore recalling (3.13) and (3.14), we obtain
Observe from the fact that (pk) is bounded then pkt⇀~p∗∈H as t→∞ for a subsequence (pkt) of (pk). This also infers that ˉwkt⇀~p∗, ˉvkt⇀~p∗ and bkt⇀~p∗ as t→∞. Since ek⇀~p∗ and ‖ek−ui,k‖→0 as k→∞, this implies ui,k⇀~p∗. Recalling the assumption (A3) and (3.15), we deduce that gi(~p∗,ν)≥0 for all ν∈K and i∈{1,2,⋯,M}. Therefore, ~p∗∈⋂Mi=1EP(gi). Moreover, recall that ˉvkt⇀~p∗ as t→∞ and (3.11) we have ~p∗∈⋂Nj=1Fix(Tj). Hence ~p∗∈Γ.
Step 4. pk→p∗=ΠHΓp1.
Since p∗=ΠHΓp1 and ~p∗∈Γ, therefore we have pk+1=ΠHCk+1p1 and p∗∈Γ⊂Ck+1. This implies that
By recalling the weak lower semicontinuity of the norm, we have
Recalling the uniqueness of the metric projection operator yields that ~p∗=p∗=ΠHΓp1. Also limt→∞‖pkt−p1‖=‖p∗−p1‖=‖~p∗−p1‖. Moreover, recalling the Kadec-Klee property of H with the fact that pkt−p1⇀~p∗−p1, we have pkt−p1→~p∗−p1 and hence pkt→~p∗. This completes the proof.
Corollary 3.3. Let K⊆H be a nonempty closed and convex subset of a real Hilbert space H. For all i∈{1,2,⋯,M}, let gi:K×K→R∪{+∞} be a finite family of bifunctions satisfying Assumption 2.1. Assume that Γ:=⋂Mi=1EP(gi)≠∅, such that
Assume that the condition (C1) holds, then the sequence (pk) generated by (3.16) strongly converges to a point in Γ.
We now propose an other variant of the hybrid iterative algorithm embedded with the Halpern iterative algorithm [20].
Remark 3.4. Note for the Algorithm 2 that the claim pk is a common solution of the EP and FPP provided that pk+1=pk, in general is not true. So intrinsically a stopping criterion is implemented for k>kmax for some chosen sufficiently large number kmax.
Theorem 3.5. Let Γ≠∅ and the following conditions:
(C1) ∑∞k=1ξk‖pk−pk−1‖<∞;
(C2) 0<a∗≤γk≤min{1−η1,⋯,1−ηN} and limk→∞βk=0,
hold. Then the Algorithm 2 solves the problem 2.1.
Proof. Observe that the set Ck+1 can be expressed in the following form:
Recalling the proof of Theorem 3.1, we deduce that the sets Γand Ck+1 are closed and convex satisfying Γ⊂Ck+1 for all k≥0. Further, (pk) is bounded and
Since pk+1=ΠHCk+1(q)∈Ck+1, we have
Recalling the estimate (3.17) and the conditions (C1) and (C2), we obtain
Reasoning as above, we get
The rest of the proof of Theorem 3.5 follows from the proof of Theorem 3.1 and is therefore omitted.
The following remark elaborate how to align condition (C1) in a computer-assisted iterative algorithm.
Remark 3.6. We remark here that the condition (C1) can easily be aligned in a computer-assisted iterative algorithm since the value of ‖pk−pk−1‖ is quantified before choosing ξk such that 0≤ξk≤^ξk with
Here {σk} denotes a sequence of positives ∑∞k=1σk<∞ and ξ∈[0,1).
As a direct application of Theorem 3.1, we have the following variant of the problem 2.1, namely the generalized split variational inequality problem associated with a finite family of single-valued monotone and hemicontinuous operators Aj:K→H defined on a nonempty closed convex subset K of a real Hilbert space H for each j∈{1,2,⋯,N}. The set VI(K,A) represents all the solutions of the following variational inequality problem ⟨Aμ,ν−μ⟩≥0∀ν∈C.
Theorem 3.7. Assume that Γ=⋂Mi=1VI(C,Ai)∩⋂Nj=1Fix(Tj)≠∅ and the conditions (C1)–(C4) hold. Then the sequence (pk)
generated by (3.18) solves the problem 2.1.
Proof. Observe that, if we set gi(ˉμ,ˉν)=⟨Ai(ˉμ),ˉν−ˉμ⟩ for all ˉμ,ˉν∈K, then each Ai being L-Lipschitz continuous infers that gi is Lipschitz-type continuous with d1=d2=L2. Moreover, the pseudo-monotonicity of Ai ensures the pseudo-monotonicity of gi. Recalling the assumptions (A3)–(A4) and the Algorithm 1, note that
can be transformed into
Hence recalling gi(ˉμ,ˉν)=⟨Ai(ˉμ),ˉν−ˉμ⟩ for all ˉμ,ˉν∈K and for all i∈{1,2,⋯,M} in Theorem 3.1, we have the desired result.
4.
Numerical experiment and results
This section provides the effective viability of the algorithm via a suitable numerical experiment.
Example 4.1. Let H=R be the set of all real numbers with the inner product defined by ⟨p,q⟩=pq, for all p,q∈R and the induced usual norm |⋅|. For each i={1,2,⋯,M}, let the family of pseudomonotone bifunctions gi(p,q):K×K→R on K=[0,1]⊂H, is defined by gi(p,q)=Si(p)(q−p), where
where 0<λ1<λ2<...<λM<1. Note that EP(gi)=[0,λi] if and only if 0≤p≤λi and q∈[0,1]. Consequently, ⋂Mi=1EP(gi)=[0,λ1]. For each j∈{1,2,⋯,N}, let the family of operators Tj:R→R be defined by
Clearly, Tj defines a finite family of η-demimetric operators with ⋂Nj=1Fix(Tj)={0}. Hence Γ=(⋂Mi=1EP(gi))∩(⋂Nj=1Fix(Tj))=0. In order to compute the numerical values of the Algorithm 1, we choose ξ=0.5, αk=1100k+1, μ=17, λi=i(M+1), M=2×105 and N=3×105. Since
Observe that the expression
in the Algorithm 1 is equivalent to the following relation ui,k=ek−μSi(ek),for alli∈{1,2,⋯,M}. Similarly vi,k=ek−μSi(ui,k),for alli∈{1,2,⋯,M}. Hence, we can compute the intermediate approximation ˉvk which is farthest from ek among vi,k, for all i∈{1,2,⋯,M}. Generally, at the kth step if Ek=‖pk−pk−1‖=0 then pk∈Γ implies that pk is the required solution of the problem. The terminating criteria is set as Ek<10−6. The values of the Algorithm 1 and its variant are listed in the following table (see Table 1):
The values of the non-inertial and non-parallel variant of the Algorithm 1 referred as Alg.1∗ are listed in the following table (see Table 2):
The error plotting Ek against the Algorithm 1 and its variants for each choices in Tables 1 and 2 are illustrated in Figure 1.
Example 4.2. Let H=Rn with the induced norm ‖p‖=√∑ni=1|pi|2 and the inner product ⟨p,q⟩=∑ni=1piqi, for all p=(p1,p2,⋯,pn)∈Rn and q=(q1,q2,⋯,qn)∈Rn. The set K is given by K={p∈Rn+:|pk|≤1}, where k={1,2,⋯,n}. Consider the following problem:
where gi:K×K→R is defined by:
where Si,k∈(0,1) is randomly generated for all i={1,2,⋯,M} and k={1,2,⋯,n}. For each j∈{1,2,⋯,N}, let the family of operators Tj:H→H be defined by
for all p∈H. It is easy to observe that Γ=⋂Mi=1EP(gi)∩⋂Nj=1Fix(Tj)=0. The values of the Algorithm 1 and its non-inertial variant are listed in the following table (see Table 3):
The values of the non-inertial and non-parallel variant of the Algorithm 1 referred as Alg.1∗ are listed in the following table (see Table 4):
The error plotting Ek≤10−6 against the Algorithm 1 and its variants for each choices in Tables 3 and 4 are illustrated in Figure 2.
Example 4.3. Let L2([0,1])=H with induced norm ‖p‖=(∫10|p(s)|2ds)12 and the inner product ⟨p,q⟩=∫10p(s)q(s)ds, for all p,q∈L2([0,1]) and s∈[0,1]. The feasible set K is given by: K={p∈L2([0,1]):‖p‖≤1}. Consider the following problem:
where gi(p,q) is defined as ⟨Sip,q−p⟩ with the operator Si:L2([0,1])→L2([0,1]) given by
Since each gi is monotone and hence pseudomonotone on C. For each j∈{1,2,⋯,N}, let the family of operators Tj:H→H be defined by
Then Tj is a finite family of η-demimetric operators. It is easy to observe that Γ=⋂Mi=1EP(gi)∩⋂Nj=1Fix(Tj)=0. Choose M=50 and N=100. The values of the Algorithm 1 and its non-inertial variant have been computed for different choices of p0 and p1 in the following table (see Table 5):
The values of the non-inertial and non-parallel variant of the Algorithm 1 referred as Alg.1∗ have been computed for different choices of p0 and p1 in the following table (see Table 6):
The error plotting Ek=<10−4 against the Algorithm 1 and its variants for each choices in Tables 5 and 6 are illustrated in Figure 3.
We can see from Tables 1–6 and Figures 1–3 that the Algorithm 1 out performs its variants with respect to the reduction in the error, time consumption and the number of iterations required for the convergence towards the common solution.
5.
Conclusions
In this paper, we have constructed some variants of the classical extragradient algorithm that are embedded with the inertial extrapolation and hybrid projection techniques. We have shown that the algorithm strongly converges towards the common solution of the problem 2.1. A useful instance of the main result, that is, Theorem 3.1, as well as an appropriate example for the viability of the algorithm, have also been incorporated. It is worth mentioning that the problem 2.1 is a natural mathematical model for various real-world problems. As a consequence, our theoretical framework constitutes an important topic of future research.
Conflict of interest
The authors declare that they have no competing interests.
Acknowledgments
The authors wish to thank the anonymous referees for their comments and suggestions.
The author Yasir Arfat acknowledge the support via the Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi, Thailand (Grant No.16/2562).
The authors Y. Arfat, P. Kumam, W. Kumam and K. Sitthithakerngkiet acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Moreover, this reserch was funded by King Mongkut's University of Technology North Bangkok, Contract No. KMUTNB-65-KNOW-28.