# Optimal Controls for a Class of Impulsive Katugampola Fractional Differential Equations with Nonlocal Conditions

Authors: Xingru Chen; Haibo Gu; Yu Sun
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we investigate a class of impulsive Katugampola fractional differential equations with nonlocal conditions in a Banach space. First, by using a fixed-point theorem, we obtain the existence results for a class of impulsive Katugampola fractional differential equations. Secondly, we derive the sufficient conditions for optimal controls by building approximating minimizing sequences of functions twice.


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## Body

## 1. Introduction

In recent years, fractional calculus has received more and more attention in many fields because of the local limit definition of integral-order ordinary differential equation or partial differential equation which is not suitable to describe the history-dependent process and has received more and more attention in many fields. Leibnitz discovered fractional derivatives in 1695. Because fractional-order differential equation can describe objective laws and the essence of things more accurately than integral differential equation, many scholars have devoted themselves to the study of fractional calculus. Now, Riemann-Liouville calculus definition, Caputo differential definition, Grunwald-Letnikov differential definition, etc. are the most commonly used fractional calculus definition in basic mathematics research and engineering application research (see [1]). Fractional differential equations are widely used in practice and greatly enriched the content of mathematical theory and penetrated into many fields of natural science.In 2011, Katugampola presented a new fractional integral, which generalizes the Riemann-Liouville and Hadamard integral into a single form. When a parameter was fixed at different values, it produces the above integrals as special cases; whenρ⟶1, we can get the Riemann-Liouville operators; when ρ⟶0, we can get the Hadamard operators (see [2]). And he presented two representations of the generalized derivative called Katugampola derivative in [3]. In the same work, he obtained boundedness in an extended Lebesgue measurable space and gave illustrative examples of generalized fractional integral. In [4], he obtained existence and uniqueness results to the solution of initial value problem for a class of generalized fractional differential equations. Since then, many scholars have done a lot of research on Katugampola derivative (see [5–15]).Since the end of the last century, many scholars have devoted themselves to the study of approximate controllability (see [15–20]). Impulsive differential equation, which provides a natural description of observed evolution processes, is an important mathematical tool to solve some practical problems. The theory of impulsive differential equations of integer order has been widely used in practical mathematical modeling and has become an important area of research in recent years, which steadily receives attention of many authors (see [21–34]). Sun et al. [35] considered a class of impulsive fractional differential equations with Riemann-Liouville fractional derivative, the existence of solution was proved by using Darbo-Sadovskii's fixed-point theorem, and the optimal control results were obtained.On the basis of previous work, in this paper, we initiate to study the existence of the following impulsive Katugampola fractional differential equation with nonlocal condition:(1)ρDa+αxt=ft,xt+Btut,t∈J′=a,T,t≠tk,ρIa+1−αxα+gx=x0,u∈Uad,ρDa+βxtk+−ρDa+βxtk−=Jkxtk,k=1,⋯,m,where 0<β<α<1, ρ>0, and a∈R and Iρa+1−α and Dρa+α are the Katugampola fractional integral and derivative, which will be given in next section. x∈X, and X is a real Banach space; f is a nonlinear perturbation; gx is a given X-valued function; Jkxtk is a nonlinear mapping; a<t1<t2<⋯<tk≤T; and Uad is a control set, the control u∈Uad.The rest of the paper is organised as follows. In Section2, we give some preliminary facts that we need in the subsequent sections. In Section 3, we obtain the existence of solution of system (1) by using Darbo-Sadovskii’s fixed-point theorem. In Section 4, we establish the optimal controls for a Lagrange problem. In Section 5, an illustrative example is given to show the practical usefulness of the analytical results.


## 2. Preliminaries

In this section, we present some results which will be useful throughout the paper.LetX be a real Banach space; £X is a kind of linear operators in X and not necessarily bounded.Ca,T;X denotes the Banach space of all continuous functions u:a,T⟶X with the norm
(2)u=suput,t∈a,T.L1a,T;X denotes the Banach space of all X-valued Bochner integrable functions defined on a,T with the norm
(3)u1=∫aTutdt.The Banach spacePCδ is defined by x:a,T⟶X:x∈Ctk,tk+1, and with the norm
(4)xδ=supt∈a,Ttρ−aρρδxt,where k=1,⋯,m; α+δ−1≥0; xtk− and xtk+ exist; xtk−=xtk; and limt→a+tρ−aρ/ρδxt exists.Consider the spaceXcpa,bc∈R,1≤p≤∞ of those functions f on a,b for which fXcp<∞, with the norm
(5)fXcp=∫abtcftpdtt1/p<∞.Definition 1 (see [2]).
The Katugampola left-sided fractional integral of orderα>0, ρ>0 of f∈Xcpa,b for −∞<a<t<∞, is defined by
(6)ρIa+αft=ρ1−αΓα∫atτρ−1tρ−τρ1−αfτdτ.Definition 2 (see [3]).
The Katugampola fractional derivative, associated with the generalized fractional integrals (6), is defined, for 0≤a<t<∞, n=α+1, by
(7)ρDa+αft=t1−ρddtnρIa+n−αft=ρα−n+1Γn−αt1−ρddtn∫atτρ−1tρ−τρα−n+1fτdτ.Lemma 3 (see [3]).
Let0≤α<1, then for a>0 and ρ>0,
(8)Dρa+αIρa+αxt=xt.Lemma 4 (see [36]).
Letα>0, 0≤γ<1. If x∈X and Iρa+1−αx∈Cγ1a,b, then
(9)Iρa+αDρa+αxt=xt−ρIa+1−αxaΓαtρ−aρρα−1.Lemma 5.
Let0≤β<α<1 and x∈X, then
(10)Dρa+βIρa+αxt=ρIa+α−βxt.
Hausdorff measure of noncompactnessσ· defined on each bounded subset Ω of the Banach space X, and σΩ=infε>0 (Ω has a finite ε-net in X).Lemma 6 (see [37]).
The measure of noncompactnessσ· satisfies the following:
(i)
σB=0 if and only if B is relatively compact in X(ii)
Ifσx∪B=σB for every x∈X and every nonempty subset B⊆X(iii)
σλB≤∣λ∣σB for any λ∈ℝ(iv)
σB1+B2≤σB1+σB2, where B1+B2=x+y:x∈B1,y∈B2(v)
σB1∪B2≤maxσB1,σB2
For anyD⊂Ca,T,X, we define
(11)∫atDsds=∫atusds:u∈Dfort∈a,T,
whereDs=us∈X:u∈D.Lemma 7 (see [37]).
IfG⊂X is a convex bounded and closed set, the continuous mapping Λ:G⟶G is a σ-contraction and then has at least one fixed point in G.
The setN which has controls u is another separable reflexive Banach space. Let PfN⊂N be nonempty closed and convex. We assume that mapping w:a,T⟶PfN is multivalued and measurable, w·⊂O, O⊂N is bounded, the control set Uad is defined as Uad=Vwp=u∈LpO∣ut∈wt,a.e., p>1, and Uad≠∅.


## 3. Existence of Solution

In this section, we obtain the existence of solution of system (1) by using Darbo-Sadovskii’s fixed-point theorem.Lemma 8.
A solutionx∈PCδa,T;X of problem (1) can be expressed as
(12)xt=x0−gxΓαtρ−aρρα−1+Γα−βΓα∑a<tk<ttkρ−aρρ1+β−αJkxtktρ−aρρα−1+ρ1−αΓα∫atτρ−1tρ−τρ1−αfτ,xτ+Bτuτdτ.Proof.
Suppose thatx∈PCδa,T;X is a solution of problem (1). By (9), we can get
(13)xt=Iρa+αDρa+αxt+Iρa+1−αx0Γαtρ−aρρα−1=x0−gxΓαtρ−aρρα−1+ρIa+αf·,x·+B·u·t,t∈a,t1.
By using formula (10), we can get
(14)Dρa+βxt=x0−gxΓα−βtρ−aρρα−β−1+ρIa+α−βf·,x·+B·u·t,t∈a,t1,where the second term in the right side is a continuous function on a,T. In particular,
(15)Dρa+βxt1−=x0−gxΓα−βt1ρ−aρρα−β−1+ρIa+α−βf·,x·+B·u·t1.
Obviously, the impulsive conditionDρa+βxtk+−Dρa+βxtk−=Jkxtk is a discontinuity condition, which means we can get
(16)xt=C1tρ−aρρα−1+x0−gxΓαtρ−aρρα−1+ρIa+αf·,x·+B·u·t,t∈t1,t2.
Since this function satisfies problem (1) in view of (8), for t∈t1,t2, we obtain
(17)Dρa+βxt=C1ΓαΓα−βtρ−aρρα−β−1+x0−gxΓα−βtρ−aρρα−β−1+ρIa+α−βf·,x·+B·u·t,so that
(18)Dρa+βxt1+=C1ΓαΓα−βt1ρ−aρρα−β−1+x0−gxΓα−βt1ρ−aρρα−β−1+ρIa+α−βf·,x·+B·u·t1.
Combining (15) and (18) and applying problem (1) yield
(19)C1=Γα−βΓαt1ρ−aρρ1+β−αJ1xt1,xt=x0−gxΓαtρ−aρρα−1+Γα−βΓαt1ρ−aρρ1+β−αJ1xt1tρ−aρρα−1+ρ1−αΓα∫atτρ−1tρ−τρ1−αfτ,xτ+Bτuτdτ,t∈t1,t2.
Repeating the above process, the solutionxt for tk−1,tk can be written as
(20)xt=x0−gxΓαtρ−aρρα−1+Γα−βΓα∑a<tk<ttkρ−aρρ1+β−αJkxtktρ−aρρα−1+ρ1−αΓα∫atτρ−1tρ−τρ1−αfτ,xτ+Bτuτdτ.
Conversely, by a simple calculation of (12), we can easily get (1). This completes the proof.LetBr≔x∈X:xδ≤r and Wr≔y∈PCδa,T;X:yt∈Br,∀t∈a,T, where r is a finite positive constant.We introduce the following hypotheses:(H1).
f∈Ca,Tt1,t2,⋯,tm×X and ftk±,x exist, ∀x∈X,k=1,⋯,m.(H2).
There exists a functionφ such that
(21)ft,x≤φrt,where r is a finite positive constant and r>0. For a.e., t∈a,T and x∈Br, Iρaαφrt exists and is bounded and continuous.(H3).
For each bounded setQ⊂X,
(22)σft,Q≤qσQ,where q is a constant, q>0, and σ is the Hausdorff measure of noncompactness.(H4).
B:a,T⟶£N,X is an essentially bounded and compact mapping, i.e., B∈L∞a,T,£N,X.(H5).
g:PCδa,T;X⟶X is a compact and continuous mapping.(H6).
For∀x1,x2∈PCδa,T;X, Jk:X⟶X satisfies Lipschitz condition:
(23)Jkx1−Jkx2≤hkx1−x2,k=1,2,⋯,m,where hk is the Lipschitz constant and hk>0.Remark 9.
(i)
From the definition ofUad and hypothesis (H4), we can easily obtain that Bu∈Lpa,T;X with p>1 for any u∈Uad. Therefore, Bu∈L1a,T;X and BuL1<+∞(ii)
Under hypothesis (H5),supx∈Wrgx<+∞Theorem 10.
Suppose that hypotheses (H1)–(H6) be satisfied and there is a constantr>0 such that
(24)Tρ−aρρα+δ−1supx∈wrgx+Γα−β∑1≤k≤mtkρ−aρρ1+β−αJk0+hkr+Tρ−aρρδsupt∈a,TIρaαφrt+Tρ−aρρα+δBuL1α+x0≤r,then problem (1) has at least one solution.Proof.
Letx0∈X be fixed. We consider the operator V on PCδa,T;X such that
(25)Vxt=x0−gx+Γα−β∑a<tk<ttkρ−aρρ1+β−αJkxtktρ−aρρα−1+ρ1−α∫atτρ−1tρ−τρ1−αfτ,xτ+Bτuτdτ.t∈a,T.
The proof is given in following several steps.
Step 1.V maps Wr into Wr.
For anyx∈Wr, by (H1)–(H6) and (12), we can obtain
(26)tρ−aρρδVxt≤tρ−aρρδx0−gx+Γα−β∑a<tk<ttkρ−aρρ1+β−αJkxtk×tρ−aρρα−1+tρ−aρρδρ1−α∫atτρ−1tρ−τρ1−αfτ,xτ+Bτuτdτ≤x0+gx+Γα−β∑1<k<mtkρ−aρρ1+β−αJk0+hkrtρ−aρρα+δ−1+tρ−aρρδρ1−α∫atτρ−1tρ−τρ1−αfτ,xτ+Bτuτdτ≤x0+gx+Γα−β∑1<k<mtkρ−aρρ1+β−αJk0+hkrTρ−aρρα+δ−1+tρ−aρρδρ1−α∫atτρ−1tρ−τρ1−αφrτdτ+tρ−aρρδρ1−αBuL1∫atτρ−1tρ−τρ1−αdτ≤x0+gx+Γα−β∑1<k<mtkρ−aρρ1+β−αJk0+hkrTρ−aρρα+δ−1+tρ−aρρδIρaα∣φr∣+tρ−aρρα+δBuL1α≤Tρ−aρρα+δ−1supx∈wrgx+Γα−β∑1≤k≤mtkρ−aρρ1+β−αJk0+hkr+Tρ−aρρδsupt∈a,TIρaαφrt+Tρ−aρρα+δBuL1α+x0≤r.
Step 2.V is continuous in Wr.
Letx1,x2∈Wr; we can get
(27)tρ−aρρδVx1t−tρ−aρρδVx2t≤tρ−aρρδgx1−gx2+Γα−β∑a<tk<ttkρ−aρρ1+β−αJkx1tk−Jkx2tk×tρ−aρρα−1+tρ−aρρδρ1−α∫atτρ−1tρ−τρ1−αfτ,x1τ−fτ,x2τdτ≤gx1−gx2+Γα−β∑a<tk<ttkρ−aρρ1+β−αJkx1tk−Jkx2tkTρ−aρρα+δ−1+Tρ−aρρδρ1−α∫atτρ−1tρ−τρ1−αfτ,x1τ−fτ,x2τdτ.
Becauseg, Jk, and f are continuous, we can obtain V is continuous.
Step 3.V has at least one fixed point.
Let(28)I1xt=x0−gxtρ−aρρα−1,I2xt=Γα−β∑a<tk<ttkρ−aρρ1+β−αJkxtktρ−aρρα−1,I3xt=ρ1−α∫atτρ−1tρ−τρ1−αfτ,xτ+Bτuτdτ,for each x∈Wr. By condition (H5), we can obtain that the operator I1 is compact in Wr, and σI1Wr=0. From (H1) and (H3), we can get that ft,xt is a compact mapping in Wr; we omit the proof of this step as it is similar to that of Theorem 4.34 in [38]. Combining with (H4), we know the operator I3 is compact in Wr, and σI3Wr=0.
Then, by condition (H6), we can know thatJkk=1m are Lipschitz continuous. For x1,x2∈Wr, we have
(29)tρ−aρρδI2x1t−tρ−aρρδI2x2t≤tρ−aρρδΓα−β∑a<tk<ttkρ−aρρ1+β−αJkx1tk−Jkx2tktρ−aρρα−1≤Γα−β∑a<tk<ttkρ−aρρ1+β−αJkx1tk−Jkx2tkTρ−aρρα+δ−1≤Tρ−aρρα+δ−1Γα−β∑a<tk<ttkρ−aρρ1+β−αhkx1−x2.
Therefore,(30)σI2Wr≤Tρ−aρρα+δ−1Γα−β∑a<tk<ttkρ−aρρ1+β−αhkσWr,σVWr≤σI1Wr+σI2Wr+σI3Wr≤Tρ−aρρα+δ−1Γα−β∑a<tk<ttkρ−aρρ1+β−αhkσWr.
By (12), we can obtain
(31)Tρ−aρρα+δ−1Γα−β∑a<tk<ttkρ−aρρ1+β−αhk<1.
By Lemma3, we can easily obtain that the operator V has a fixed point in x∈Wr; i.e., (1) has at least one solution. This completes the proof.


## 4. Optimal Controls for Problem

In this section, we establish the optimal controls for a Lagrange problem.Letxu∈Wr denote the solution of problem (1) associated with the control u∈Uad.We research the following limited Lagrange problemP:
(32)Jxu,u=∫aTψt,xut,utdt.For allu∈Uad, there exist a x0∈Wr⊆PCδa,T;X and u0∈Uad such that
(33)Jx0,u0≤Jxu,u,where x0∈Wr denotes the solution of problem (1) associated with the control u0∈Uad.We introduce the following hypotheses:The functionψ:a,T×X×N satisfies the following:(Ψ1).
The functionψ:a,T×X×N⟶R∪∞ is Borel measurable.(Ψ2).
ψ∈X×N is sequentially lower continuous for a.e. t∈a,T.(Ψ3).
For allx∈X, ψ∈N is convex a.e. t∈a,T.(Ψ4).
There exist two constantsc≥0, d>0 such that
(34)ψt,xut,ut≥ϕt+cx+duNp,where ϕ∈L1a,T:R.Remark 11.
If the pairx·,u· satisfies system (1), ∀x·∈Wr, then the pair x·,u· is feasible. If the pair xu,u is feasible, then xu⊂Wr.Theorem 12.
Suppose that hypotheses (Ψ1)–(Ψ4) are satisfied, if the problem P satisfies the condition of Theorem 10., then it has at least one optimal feasible pair.Proof.
Step 1. For eachu∈Uad, let
(35)Ju=infxu∈SoluJxu,u.
IfSolu has infinite elements, we can easily to prove in the case of
(36)Ju=infxu∈SoluJxu,u=+∞.
IfSolu has finite elements, Jx∧u,u=infxu∈SoluJxu,u=Ju, if x∧u∈Solu.
We suppose thatJu=infxu∈SoluJxu,u<+∞. By conditions Ψ1–Ψ1, we can get Ju>−∞.
By the definition of the infimum, we can obtain a sequencexnun=1∞⊆Solu, which satisfies Jxnu,u⟶Ju for n⟶∞.
Sincexnu,un=1∞ is a sequence of feasible pairs, we obtain
(37)xnut=x0−gxnu+Γα−β∑a<tk<ttkρ−aρρ1+β−αJkxnutktρ−aρρα−1+ρ1−α∫atτρ−1tρ−τρ1−αfτ,xnuτ+Bτuτdτ,t∈a,T.
Step 2. There existsx∧u∈Solu such Jx∧u,u=infxu∈SoluJxu,u=Ju.Claim 1.
For∀u∈Uad, xnun=1∞ is precompact in PCδa,T;X.Let(38)xnu=I1xnu+I2xnu+I3xnu,we obtain that I1xnun=1∞⊂PCδa,T;X and I3xnun=1∞⊂PCδa,T;X are precompact (we omit the proof of this step as it is similar to that of Theorem 4.34 in [36]). Furthermore, by assumption (H6), the operator I2 is Lipschitz continuous in PCδa,T;X, where Lipschitz constant T+1α+δ−1Γα−β∑a<tk<ttk1+β−αhk. Hence, from the given condition and the Hausdorff measure of noncompactness, we can get
(39)σxnun=1∞≤σI1xnun=1∞+σI2xnun=1∞+σI3xnun=1∞≤Tρ−aρρα+δ−1Γα−β∑a<tk<ttkρ−aρρ1+β−αhkσxnun=1∞.Therefore, from the above inequality, we can obtainσxnun=1∞=0; then, the set xnun=1∞ is precompact in PCδa,T;X for u∈Uad.Claim 2.
x∧u∈Solu.In general, for anyu∈Uad, we assume that when n⟶∞, xnu⟶x∧u on PCδa,T;X. From the dominated convergence theorem and the continuity of g, Ji, and f, taking n⟶∞ on both sides for (37), we have
(40)x^ut=x0−gx∧u+Γα−β∑a<tk<ttkρ−aρρ1+β−αJkx^nutktρ−aρρα−1+ρ1−α∫atτρ−1tρ−τρ1−αfτ,x^nuτ+Bτuτdτ,a≤t≤T.Then,x∧u∈Solu.Step 3.
Jx∧u,u=infxu∈SoluJxu,u=Ju, ∀u∈Uad.By using the definition of a feasible pair, from the Balder theorem and hypotheses (Ψ1)–(Ψ4), we have
(41)Ju=limn→∞∫aTψt,xnut,utdt≥∫aTψt,x∧ut,utdt=Jx∧u,u≥Ju.Hence,Jx∧u,u≥Ju. This implies that for ∀u∈Uad, Ju at x∧u∈PCδa,T;X to the minimum.Step 4.
There existsu0∈Uad such that Ju0≤Ju for all u∈Uad.Ifinfu∈UadJu<+∞, we get that infu∈UadJu>−∞, and there is a sequence unn=1∞⊆Uad satisfying Jun⟶infu∈UadJu as n⟶∞ (we omit the proof of this step as it is similar to that of Step 1). Since Lpa,T;N is a reflexive Banach space and unn=1∞ is bounded in Lpa,T;N, for some uo∈Lpa,T;N, we can assume that unn=1∞ converges weakly, as n⟶∞.SinceUad is convex and closed, by Mazur's lemma, we can deduce that u0∈Uad. For n≥1, x∧un is the solution for (1) associated with un, where Jun attains its minimum. Hence, the pair x∧un,un is feasible, such that
(42)x^unt=x0−gx^un+Γα−β∑a<tk<ttkρ−aρρ1+β−αJkx^untktρ−aρρα−1+ρ1−α∫atτρ−1tρ−τρ1−αfτ,x^unτ+Bτuτdτ.where a≤t≤T.Let(43)I3′x∧unt=ρ1−α∫atτρ−1tρ−τρ1−αfτ,x∧unτdτ,Zunt=ρ1−α∫atτρ−1tρ−τρ1−αBτuτdτ.We have(44)x∧unt=I1x∧unt+I2x∧unt+I′3x∧unt+Zunt,a≤t≤T.SinceI1x∧unn=1∞, Zunt are both precompact in PCδa,T;X and I2 satisfies the Lipschitz continuity in PCδa,T;X with the Lipschitz constant, then
(45)Tρ−aρρα+δ−1Γα−β∑a<tk<ttkρ−aρρ1+β−αhk<1.Moreover, by given assumptions, we can obtain thatI′3x∧unn=1∞⊆PCδa,T;X is precompact.We can obtain thatρx∧unn=1∞=0; i.e., x∧unn=1∞⊆PCδa,T;X is precompact (we omit the proof of this step as it is similar to that of Theorem 4.34 in [36]). Thus, we can find a subsequence which is connected with x∧u0∈PCδa,T;X and x∧unn=1∞ satisfying that x∧un⟶x∧u0, as n⟶∞ in PCδa,T;X.Taking the limitn⟶∞ on both sides for (42), we obtain
(46)x∧u0t=x0−gx∧u0+Γα−β∑a<tk<ttkρ−aρρ1+β−αJkx∧u0tktρ−aρρα−1+ρ1−α∫atτρ−1tρ−τρ1−αfτ,x∧u0τ+Bτuτdτ,a≤t≤T.Therefore, the pairx∧u,u0 is feasible.By the Balder theorem and assumptions (Ψ1)–(Ψ4), we have
(47)infu∈UadJu=limn→∞∫aTψt,x∧unt,untdt≥∫aTψt,x∧u0t,u0tdt=Jx∧u0,u0≥infu∈UadJu.Then,(48)Jx∧u0,u0=Ju0=infxu0∈Solu0Jxu0,u0,which means J at u0∈Uad to the minimum.


## 5. An Example

Consider the following impulsive Katugampola fractional differential equations:(49)ρDa+αxt=ft,xt+Btut,t∈J′=a,1,a>0,t≠t1,ρIa+1−αxa=gx,ρDa+βxt1+−ρDa+βxt1−=J1xt1.Takingf:Ca,1;X⟶X and ft,x=2e−tx/1+et1+x,g:PCδa,1;X⟶X and gx is a continuous and compact map.Supposing thatB is a bounded and compact map,J1:X⟶X and J1x=3x/3+x.Letx1,x2∈Br and t∈a,1; we have
(50)ft,x1−ft,x2=2e−t1+etx11+x1−x21+x2=2e−tx1−x21+et1+x11+x2≤2e−t1+etx1−x2≤e−tx1−x2.For eachx1,x2∈Br and t∈a,1, we can easily get
(51)ft,x=2e−t1+etx1+x≤2e−t1+et≤e−t,J1x1−J1x2≤x1−x2.So hypotheses (H1)–(H6) are satisfied.We define the cost functionJxu,u=∫a1xutX2+utN2dt, where xu,u is a feasible pair. If the inequality
(52)1ρ−aρρα+δ−1supx∈wrgx+rΓα−βt11+β−α+1ρ−aρρδsupt∈a,1Iρaαe−t+1ρ−aρρα+δBuL1α+x0≤rholds, then it satisfied all the conditions in Theorem 10; our results can be applied to problem (49).



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