jj ii the convergence of solutions for nonlinear j...

Background and . . .

Existence and . . .

Existence of Extremal . . .

Convergence

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The Convergence of Solutions for Nonlinear

Singular Differential Systems

Peiguang Wang

Hebei University

[email protected]

http://www.amss.ac.cn/~guoxh


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• Background and History of Studies• Background and significance

• Review of Previous Studies

• Existence and Uniquenness of Solutions• IVP for nonlinear singular systems

• Existence of Extremal Solutions• BVP for nonlinear singular systems with delay

• IVP for nonlinear singular systems with "maxima"

• Convergence of Nonlinear Singular Systems• quadratic convergence of IVP for singular systems

• quadratic convergence of PBVP for singular systems

• rapid convergence for singular differential systems

• rapid convergence for singular systems with "maxima"



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1 Background and History of Studies

The background and history of studies on singular systems are summarized,

and the problems to be studied in the fields are briefly introduced.

1.1. Background and significance

In practical fields, many problems were found to be modeled by singular d-

ifferential systems, such as optimal control problems, constrained control prob-

lems, electrical circuits, some population growth models, singular perturbations

problems, and so on.

In 1974, Rosenbrock[1] gave the model when he analyzed electrical networks,

in the course of describing dynamical processes with constraints.

Since 1980, the theory of singular systems began to form and gradually de-

veloped into separate branch of modern control theory. The studies for singular

systems not only has the widespread practical significance, moreover its theory

value also has broad prospects for development.



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Singular systems also known as descriptor systems, semi-state space systems,

differential-algebraic systems, and generalized state space systems, etc.

The applications for singular systems in economics (the Leontieff model, see

[Luenberger and Arbel 1977]) and demography (the Leslie model, see [Camp-

bell 1980]) are well known.

Example 1.1[1]. The Leontief model of economic systems.

Bx(k + 1) = (I − A+B)x(k)− d(k), (1.1)

where A ∈ Rn×n is an input-output (or production) matrix, B ∈ Rn×n is the

capital coefficient matrix. Most of the elements in B are zero except for a few.

B is often singular. In this sense the system (1.1) is a typical discrete-time

singular system.



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Some models of singular systems.• The linear singular systems having constant coefficient matrices

EX = AX +BU

Y = CX

where E is a singular matrix, implies rankE = r < n. E, A ∈ Rn×n, B ∈Rn×n and C ∈ Rn×n are constant matrices.

• The linear time-varying singular systemsE(t)X = A(t)X +B(t)U

Y = C(t)X

where E(t) is a singular matrix for ∀t ∈ J , J = [t0,+∞), t0 ≥ 0.



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• The general form of singular systemF (t,X, X, U) = 0

G(t,X, Y , U) = 0

where ∂F∂X

is a singular matrix for ∀t ∈ J , X(t) is a semi-state vector function

with n× 1, U(t) is a control with m× 1, Y (t) is a output with r × 1, F and G

are vector valued functions, J = [t0,+∞), t0 ≥ 0.

• The form of differential-algebraic systemsX1 = F1(t,X1, X2, U)

0 = F2(t,X1, X2, U)

Y = G(t,X1, X2, U)

When ∂F2

∂X2is nonsingular, the system is singular system with 1-th Index. When

∂F2

∂X2is singular, the system is singular system with high index.



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1.2. Review of Previous Studies

The methods of study for singular system are mostly geometric approach,

frequency domain method and state-space techniques.

On the discussion of singular systems, researchers still had different views to

the some related questions, thus the research achievement of singular systems

appears extremely fragmentary. For examples, for stability of singular systems,

compared with nonsingular systems, there are three main difficulties:

i) it isn’t easy to satisfy the existence and uniqueness of solutions, since the

initial conditions may not be consistent;

ii) it is difficult to calculate the derivatives of Lyapunov functions;

iii) there often happen impulses and jumps in the solutions.



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•Outline of existence results

[1] H.H. Rosenbrock. Structural properties of linear dynamical systems. Int. J.

Control, 1974, 20(2): 191-202.

[2] S.L. Campbell. Singular systems of differential equations. Pitman Advanced

Publishing Program (I), London, 1980.

[3] S.L. Campbell. Singular systems of differential equations. Pitman Advanced

Publishing Program (II), London, 1982.

[4] H.S. Xi. On the state of continuous-time boundary value descriptor systems.

Control Theory Appl., 1993, 10(6): 692-697.

[5] J.Y. Lin and Z.H. Yang. Existence and uniqueness of solutions for non-linear

singular systems. Int. J. Systems Sci., 1988, 19(11): 2179-2184.



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[6] W. Wang and Y. Liu. Solvability of a class of nonlinear boundary-value

singular systems. J. South China Univ. Tech., 1997, 25(3): 72-76.

[7] M.S.N. Murty. Nonlinear three-point boundary value problems associated

with system of first order matrix differential equation. Bull. Inst. Math. Acad.

Sinica, 1987, 15(2): 243-249.

[8] F. Wang and Y. An. Existence and uniqueness of solutions to n-point bound-

ary value problems associated with a system of first order matrix differential

equations. J. Math. Phy. Sci., 1990, 24(3): 159-169.



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Convergence

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•Outline of convergence results

[9] J.A. Uvah and A.S. Vatsala. Monotone method for first order singular sys-

tems with boundary conditions. Internat. J. Stoch. Anal., 1989, 2(4): 217-224.

[10] W. Wang and Y.Q. Liu. Monotone iterative technique for boundary value

problems of singular integro-differential systems. J. South China Univ. Tech.,

1995, 23(6): 48-52.

[11] W. Wang and Y.Q. Liu. Monotone iterative technique for boundary value

problem of second-order singular differential system. J. Syst. Sci. Syst. Eng.,

1995, 4(4): 266-272.

[12] P.G. Wang and T.T. Kong. Quasilinearization for the boundary value prob-

lem of second-order singular differential system. Abstr. Appl. Anal., 2003,

Volume 2003, Article ID 308413, 7 pages.



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[13] A.R. Abd-Ellateef Kamar, G.M. Attia, K. Vajravelu, et al. Generlized

quasilinearization for singular system of differential equations. Appl. Math.

Comput., 2000, 114(1): 69-74.

[14] T. Jankowski. Minimal and maximal solutions to systems of differential

equations with a singular matrix. Anziam J., 2003, 45: 223-231.

[15] T. Jankowski. General quasilinearization method for systems of differential

equations with a singular matrix. Miskolc Math. Notes, 2006, 7(1): 13-26.

[16] P.G. Wang and J. Zhang. Monotone iterative technique for initial value

problems of nonlinear singular discrete systems. J. Comput. Appl. Math., 2008,

221: 158õ164.

[17] F. Wang and Y. An. A generalized quasilinearization method for telegraph

system. Nonlinear Anal. Real World Appl., 2010, 11: 407-413.



Existence and . . .


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2 Existence and Uniqueness of Solutions

2.1. IVP for nonlinear singular systems

In this section, nonlinear singular system and an expression for the set of

admissible initial conditions is given. The existence and uniqueness theorem of

solutions is proved.

Consider the nonlinear singular system

Ax′(t) = f(t, x(t)), (2.1)

where A is a singular n× n matrix, x ∈ Rn, x and f are vector-valued differen-

tiable functions, and t is a real variable.

The following results can be found in [5].



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Lemma 2.1.1. Assume that

(H2.1) There exist non-singular matrices P and Q such that (2.1) is decomposed

equivalently into the following form

y′1(t)= F1y1(t) + g1(t, y1(t), y2(t)), (2.2)

F2y′2(t) = y2(t) + g2(t, y1(t), y2(t)), (2.3)

where [y1

y2

]= y = Q−1x,

F1 and F2 are square matrices, Fm2 and Fm−1

2 6= 0. Furthermore, if the function

g2(t, y1(t), y2(t)) is differentiable m− 1 times.

Then (2.3) can be written equivalently as

y2(t) = −m−1∑i=0

F i2g

(i)2 (t, y1, y2),



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where

g(i)2 =

dig2dti

, i = 2, · · · ,m− 1, g02 = g2, g(1)2 =

dg2dt.

Definition 2.1.2. The vector x0 is an admissible initial condition for (2.1) at t0if there is a differentiable solution to (2.1), defined on some interval [t0, t0 + r]

and r > 0 such that x(t0) = x0.

Corollary 2.1.3. Assume that

(H2.2) The function g2(t, y1(t), y2(t)) is differentiable m − 1 times when (2.1)

is written in the form of (2.2) and (2.3).

Then the set of admissible initial conditions for (2.1) at t0 is given by

I0 =x0 = Q

[y01

y02

] ∣∣∣y02 = −m−1∑i=0

F i2g

(i)2 (t0, y1, y2)

.



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Theorem 2.1.4. Assume that (2.1) can be written as (2.2) and (2.3), and satisfies

the admissible initial condition x0 at t0, and

(H2.3) The function g2(t, y1, y2) in (2.3) is differentiable m times with respect

to y1, y2 and t, in the domain G ⊃ D, where

D : ‖y − y0‖ ≤ K, t0 ≤ t ≤ t0 + r.

(H2.4) The function g1(t, y1, y2) and

m−1∑i=0

F i2g

(i+1)2 (t, y1, y2)

are continuous and satisfy the Lipschitz condition in D.

Then (2.1) has a unique continuous solution satisfying x(t0) = x0 and defined

on the interval [t0, t0 + α], where α = minr, (K/M) and

M = maxsup ‖F1y1 + g1(t, y1, y2)‖, sup ‖m−1∑i=0

F i2g

(i+1)2 (t, y1, y2)‖.



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3 Existence of Extremal Solutions

3.1. BVP for nonlinear singular systems with delay

The method of upper and lower solutions combined with monotone iterative

has been widely used to prove the existence of extremal solutions on nonlin-

ear problems. Previous studies have mainly focused on nonlinear differential

equations.

In this section, we discuss the boundary value problem for nonlinear singular

systems by utilizing the method of upper and lower solutions coupled with the

monotone iterative technique. The existence of extremal solutions is obtained

as limits of monotone sequences, each member of these sequence is a solution

of linear systems which can be explicitly computed.



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Consider the boundary value problem for nonlinear singular system with delay

(BVP) Ax′(t) = f(t, x(t), x(t− τ)), t ∈ J,D1x(t0 − τ) = a, D2x(t1) = b,

(3.1)

where A is a singular n × n matrix, Rank(A) = m < n, x(t) ∈ Rn, f ∈C[J × Rn × Rn, Rn], J = [t0; t1], K = [t0 − τ, t1], D1, D2 are nonsingular

matrices, a, b are constant vectors, and τ > 0 is a constant.

Assume that the conditions hold

(H3.1) LetA andM be matrices such that (λA+M)−1 exists and is nonnegative

for some λ ∈ R. Also, let T , T−1 exist and be nonnegative such that

T−1AT =

(C 0

0 0

),

where A = (λA + M)−1A, M , T and T−1 are real n × n matrices. C is a real

s× s diagonal nonsigular matrix with s < n and C−1 > 0, (I1 − λC) ≥ 0.



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(H3.2) There exist α0, β0 ∈ C1[K,Rn], with α0(t) ≤ β0(t), t ∈ J , such that

Aα′0 ≤ f(t, α0(t), α0(t− τ)), D1α0(t0 − τ) ≤ a, D2α0(t1) ≤ b,

Aβ′0(t) ≥ f(t, β0(t), β0(t− τ)), D1β0(t0 − τ) ≥ a, D2β0(t1) ≥ b,

(H3.3) There exists a matrix M ∈ Rn×n such that f(t, x1, u) − f(t, x2, u) ≤−M(x1 − x2), whenever α0 ≤ x1 ≤ x2 ≤ β0 and α0(t− τ) ≤ u ≤ β0(t− τ).

(H3.4) f(t, x, u) is increasing in u for fixed t and x, with f(t, x, u) = f(t, u, x).

Remark 3.1.1. In (H3.1), let (λA+M)−1, T , T−1 are nonnegative andC−1 > 0,

(I1 − λC) ≥ 0. Thus, we give the following definition.

Definition 3.1.2. Let A = (aij), B = (bij) ∈ Rm×n, if aij ≥ bij for any i, j, we

write A ≥ B, if aij > bij for any i, j, we write A > B. Especially, if A ≥ 0,

then A is a nonnegative matrix, if A > 0, then A is a positive matrix.



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In order to develop the monotone method of BVP (3.1), we need an existence

result of BVP for the linear singular system with delay. Thus, we discuss the

following linear singular system

Ax′ +Mx = g(t), D1x(t0 − τ) = a, D2x(t1) = b. (3.2)

The existence result of BVP (3.2) is given by the following lemmas.

Lemma 3.1.3. (See [2]) The following are equivalent.

(H3.5) The following boundary value problem has a unique solution

Ax′ +Mx = g(t), D1x(t0) = a, D2x(t1) = b. (3.3)



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(H3.6) The associated homogeneous boundary value problem

Ax′ +Mx = 0, D1x(t0) = 0, D2x(t1) = 0 (3.4)

has only zero solution.

(H3.7) Rank(Q) = Rank(ADA) = Rank(An), where A = (λA + M)−1A,

M = (λA+M)−1B, AD is the Drazin inverse of A,

Q =

(D1A

DA

D2e−ADM(t1−t0)ADA

), AD = T

(C−1 0

0 0

)T−1.

Lemma 3.1.4. (See [2]) For a given g, a and b, BVP (3.3) is consistent if and

only if

Qq =

(a−D1h(t0)

b−D2h(t1)−D2ADAe−A

DM(t1−t0)∫ t1t0eAMsg(s)ds

)

has a solution q, where h(t) = (I − AADA)x.



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(H3.8) The following associated homogeneous boundary value problem

Ax′ +Mx = 0, D1x(t0 − τ) = 0, D2x(t1) = 0 (3.5)

has only zero solution.

(H3.9) For a given g, a and b,

Qq =

(a−D1h(t0 − τ)

b−D2h(t1)−D2ADAe−A

DM(t1−t0+τ)∫ t1t0−τ e

AMsg(s)ds

)

has a solution q, where h(t) = (I − AADA)x.

Then BVP (3.2) has a unique solution, and the solution of BVP (3.2) is depen-

dent on λ.

Theorem 3.1.6. Assume that the conditions (H3.1), (H3.2)-(H3.4), (H3.8) and

(H3.9) hold.

Then there exist monotone sequences αn, βn such that αn → ρ, βn → γ

and ρ, γ are minimal and maximal solutions of BVP (3.1) respectively.



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3.2. IVP for nonlinear singular systems with "maxima"

Differential equations with "maxima" first appeared as an object of inves-

tigation about thirty years ago in connection with modeling of some applied

problems. For example, in the theory of automatic control of various techni-

cal systems, it often occurs that the law of regulation depends on the maximum

values of some regulated state parameters over certain time intervals.

E.P. Popov considered the system for regulating the voltage of a generator of

constant current. The equation describing the work of the regulator involves the

maximum of the unknown function and it has the form (See [Popov 1966])

T0u′(t) + u(t) + q max

s∈[t−h,t]u(s) = f(t),

where T0 and q are constants characterizing the object, u(t) is the regulated

voltage and f(t) is the perturbed effect.



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Generally, differential equations with "maxima" are characterized by two

main parts:

I) differential equations;

II) maximum of the unknown function over a past time interval.

The first part, differential equations, could be ordinary differential equations

of any order, linear or nonlinear, partial differential equations, etc.

The second part makes the set of differential equations with "maxima" too

wide since the maximum of the unknown function x(t) could be given

• on an interval with fixed length, i.e., maxs∈[t−r,t]

x(s), r = const > 0;

• on a retarded interval with variable length, i.e., maxs∈[σ(t),τ(t)]

x(s), where σ(t) ≤

τ(t) ≤ t;

• on several different intervals with fixed lengths or variable lengths; etc.



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In this section, we consider the following initial value problem for nonlinear

singular differential system with "maxima"(IVP) Ax′(t) = f(t, x(t), maxs∈[t−h,t]

x(s)), t ∈ [0, T ],

x(t) = ϕ(t), t ∈ [−h, 0],(3.6)

where A is a singular n × n matrix, x ∈ Rn, f ∈ C([0, T ] × Rn × Rn, Rn),

ϕ ∈ C([−h, 0], Rn), h and T are fixed positive constants.

We give the following sets for convenience.

Ω(α0, β0) = (t, x, y) ∈ [0, T ] × Rn × Rn | α0(t) ≤ x(t) ≤β0(t), max

s∈[t−h,t]α0(s) ≤ y(t) ≤ max

s∈[t−h,t]β0(s).



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Assume that the following conditions hold.

(H3.10) There exists a constant λ such that, L(t) = [λA + M(t)]−1 ≥ 0 exists

and A = L(t)A is a real matrix. Also, there exists a nonsingular matric T such

that T−1, (LT )−1 exist and T−1, (LT ), (LT )−1 ≥ 0, satisfying

T−1AT =

(C 0

0 0

), T−1[I − λA]T =

(I1 − λC 0

0 I2

),

where C is a diagonal matrix with C−1 ≥ 0.

(H3.11) There exist α0, β0 ∈ C1[K,Rn], with α0(t) ≤ β0(t), t ∈ J , such that

Aα′0 ≤ f(t, α0(t), maxs∈[t−h,t]

α0(s)), α0(t) ≤ ϕ(t),

Aβ′0(t) ≥ f(t, β0(t), maxs∈[t−h,t]

α0(s)), β0(t) ≥ ϕ(t),



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(H3.12) There exists a function f ∈ C0,1,1(Ω(α0, β0), Rn) such that

f(t, x(t), x(t))− f(t, y(t), y(t)) ≥ −L1(x(t)− y(t))− L2(x(t)− y(t)),

where α0(t) ≤ y(t) ≤ x(t) ≤ β0, maxs∈[t−h,t]

α0(s) ≤ y(t) ≤ x(t) ≤ maxs∈[t−h,t]

β0(s),

L1 = M(t0), L2 = N(t0) ≤ 0, t0 ∈ J .

(H3.13) The Fréchet derivatives fx, fy of the function f ∈ C0,1,1(Ω(α0, β0), Rn)

exist and are continuous.

(H3.14) The matrix [I − M ]−1 exists and nonnegative, N(t) ≤ 0, t ∈ [0, T ],

where

M = maxs∈[0,T ]

− [λA+M(s)]−1

[e−A

DMs

∫ s

0

eADMσADN(σ)dσ

+(I − AAD)MDN(s)].



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In order to develop the monotone iterative method of IVP (3.6), we need an

exists result of IVP for the linear singular system

Ax′(t) +M(t)x(t) = g(t), x(0) = x0. (3.7)

We have the following result.

Lemma 3.2.1. (See [2]) Assume that the condition (H3.10) hold, index(A)=1,

and

(H3.15) y0 satisfies (I − AAD)(y0 − w(0)) = 0, where w(t) = MDg(t), M =

M(t)L(t).

Then the unique solution y(t) of

Ay′(t) + My(t) = g(t), y(0) = y0 (3.8)

is given by

y(t) = e−ADMtAADy0 + e−A

DMt

∫ t

0

eADMσADg(σ)dσ + (I − AAD)MDg(t),



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where AD, MD mean the Drazin inverse of the matrices A, M , respectively.

Lemma 3.2.2. (See [13]) Assume that Ap′+Mp ≤ 0 such that A and B satisfy

assumption (H3.10). Then p(0) ≤ 0 implies p(t) ≤ 0 on [0, T ].

Consider the singular differential inequalities Ax′(t) +M(t)x(t) +N(t) maxs∈[t−h,t]

x(s) ≤ 0, t ∈ [0, T ],

x(t) ≤ 0, t ∈ [−h, 0],(3.9)

where A is a singular n× n matrix, M(t), N(t) are continuous n× n matrices

on [0, T ].

Lemma 3.2.3. Assume that the conditions (H3.10), (H3.14) and (H3.15) hold.

Then x(t) ≤ 0, t ∈ [−h, T ].

Lemma 3.2.4. Assume that the conditions (H3.10), (H3.11), (H3.13)-(H3.15) hold.

Then the solution x(t) of IVP (3.6) satisfies α0(t) ≤ x(t) ≤ β0(t), t ∈ [−h, T ].



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Theorem 3.2.5. Assume that conditions (H3.10)-(H3.15) hold. Then there exist

monotone sequences αn, βn such that αn → ρ, βn → γ and ρ, γ are

minimal and maximal solutions of IVP (3.6) respectively.

Proof. Letting L0 = mins∈[−h,0]

(ϕ(s) − α0(s)) ≥ 0. Choose numbers k0i ∈ [0, 1)

such that k0i ≤ L0i, (i = 1, 2, · · · , n).

Consider the following singular differential system with "maxima"

Ax′(t) = f(t, α0(t), maxs∈[t−h,t]

α0(s))− L1(x(t)− α0(t))

− L2( maxs∈[t−h,t]

x(s)− maxs∈[t−h,t]

α0(s))

≡ F0(t, x(t), maxs∈[t−h,t]

x(s)), t ∈ [0, T ],

x(t) = ϕ(t)− k0L0, t ∈ [−h, 0].

(3.10)

We shall now show that α0(t) and β0(t) are lower and upper solutions of IVP

(3.10) respectively. Consequently, by Lemma 3.2.4, the solution α1(t) of IVP

(3.10) satisfies α0(t) ≤ α1(t) ≤ β0(t) on [−h, T ].



Existence and . . .


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Similarly, letting C0 = mins∈[−h,0]

(β0(s)− ϕ(s)) ≥ 0. Choose numbers p0i ∈ [0, 1)

such that p0i ≤ C0i, (i = 1, 2, · · · , n).

We consider the singular differential system with "maxima"

Ax′(t) = f(t, β0(t), maxs∈[t−h,t]

β0(s))− L1(x(t)− β0(t))

− L2( maxs∈[t−h,t]


β0(s))

≡ G0(t, x(t), maxs∈[t−h,t]

x(s)), t ∈ [0, T ],

x(t) = ϕ(t) + p0C0, t ∈ [−h, 0].

(3.11)

We can prove that α1(t) and β0(t) are lower and upper solutions of IVP (3.13)

respectively. Thus, it follows by Lemma 3.2.4 that the solution β1(t) of IVP

(3.11) satisfies α1(t) ≤ β1(t) ≤ β0(t) on [−h, T ].



Existence and . . .


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We can prove by induction that

α0(t) ≤ α1(t) ≤ · · · ≤ αn(t) ≤ βn(t) ≤ · · · ≤ β1(t) ≤ β0(t), t ∈ [−h, T ].

Using the Ascoli-Arzela theorem, thus both sequences αn(t) and βn(t) are

uniformly convergent on [−h, T ].

Denote

limn→∞

αn(t) = ρ(t), limn→∞

βn(t) = r(t).

From the uniform convergence and the definition of the functions αn(t) and

βn(t), it follows the validity of the inequalities

α0(t) ≤ ρ(t) ≤ r(t) ≤ β0(t), t ∈ [−h, T ].

where the functions ρ(t) and r(t) are solutions of IVP (3.6).

Furthermore, we can show that ρ(t) and r(t) are minimal and maximal solu-

tions of IVP (3.6).



Existence and . . .


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Example 3.2.6. Consider the following singular system with "maxima"x′1(t) =

1

1− x1(t)− 2 max

s∈[t−h,t]x1(s)− 1,

0 = x22(t)− maxs∈[t−h,t]

x2(s), t ∈ [0, 1],

x1(t) = x2(t) = 0, t ∈ [−1, 0].

(3.12)

It is easy to check that IVP (3.12) has a zero solution. α0(t) = (−14,−

14)T is

a lower solution and β0(t) = (14,14)T is an upper solution of IVP (3.12). We will

construct sequences of functions that converge uniformly to zero solution.

Now, we can construct an increasing sequence, which converge to zero solu-

tion. It is easy to see that the matrices L1 and L2 can be chosen as(−2 0

0 −1

),

(8 0

0 294

),

respectively.



Existence and . . .


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Choosing k01 = 45 , k02 = 4

5 , and consider the following singular differential

system

x′1(t) =1

1 + 14

+ 2× 1

4− 1 + 2(x1(t) +

1

4)− 8( max

s∈[t−h,t]x1(s) +

1

4)

= 2x1(t)− 8 maxs∈[t−h,t]

x1(s)−6

5,

0 = (−1

4)2 − (−1

4) + (x2(t) +

1

4)− 29

4( maxs∈[t−h,t]

x2(s) +1

4)

= x2(t)−29

4max

s∈[t−h,t]x2(s)−

5

4, t ∈ [0, 1],

x1(t) = −k014, x2(t) = −k02

4, t ∈ [−1, 0].

(3.13)

Then IVP (3.13) has an exact solution α1(t) = (−15,−

15)T .



Existence and . . .


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Choosing k11 = 2936, k12 = 101

125 and consider the following singular differential

system

x′1(t) =1

1 + 15

+ 2× 1

5− 1 + 2(x1(t) +

1

5)− 8( max

s∈[t−h,t]x1(s) +

1

5)

= 2x1(t)− 8 maxs∈[t−h,t]

x1(s)−29

30,

0 = (−1

5)2 − (−1

5) + (x2(t) +

1

5)− 29

4( maxs∈[t−h,t]

x2(s) +1

5)

= x2(t)−29

4max


101

100, t ∈ [0, 1],

x1(t) = −k115, x2(t) = −k12

5, t ∈ [−1, 0].

(3.14)

Then IVP (3.14) has an exact solution α2(t) = (− 29180,−

101625)

T .



Existence and . . .


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Analogously, we will construct a decreasing sequence that uniformly converge

to zero solution.

Choosing p01 = 89 , p02 = 44


system

x′1(t) =1

1− 14

− 2× 1

4− 1 + 2(x1(t)−

1

4− 8( max


1

4)

= 2x1(t)− 8 maxs∈[t−h,t]

x1(s) +4

3,

0 = (1

4)2 − 1

4+ (x2(t)−

1

4)− 29

4( maxs∈[t−h,t]

x2(s)−1

4)

= x2(t)−29

4max

s∈[t−h,t]x2(s) +

11

8, t ∈ [0, 1],

x1(t) =p014, x2(t) =

p024, t ∈ [−1, 0].

(3.15)

Then IVP (3.15) has an exact solution β1(t) = (29,1150)

T .



Existence and . . .


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Choosing p11 = 3742, p12 = 6017


system

x′1(t) =1

1− 29

− 2× 2

9− 1 + 2(x1(t)−

2

9)− 8( max


2

9)

= 2x1(t)− 8 maxs∈[t−h,t]

x1(s) +74

63,

0 = (11

50)2 − 11

50+ (x2(t)−

11

50)− 29

4( maxs∈[t−h,t]

x2(s)−11

50)

= x2(t)−29

4max

s∈[t−h,t]x2(s) +

6017

5000, t ∈ [0, 1],

x1(t) =2

9p11, x2(t) =

11

50p12, t ∈ [−1, 0].

(3.16)

Then IVP (3.16) has an exact solution β2(t) = ( 37189,

601731250)

T .

We can see that α0(t) < α1(t) < α2(t) < x(t) = 0 < β2(t) < β1(t) < β0(t)

on [−1, 1]. Example 3.2.6 illustrates that Theorem 3.2.5 is feasible.



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4 Convergence

4.1. Quadratic convergence of IVP for singular systems

In this section, By using the method of quasilinearization, we study the

quadratic convergence of approximate solutions of the initial value problem for

nonlinear singular differential systems with "maxima" via assuming the concav-

ity or convexity on the right hand side function.

Consider the following initial value problem for singular differential system

with "maxima"(IVP) Ax′(t) = f(t, x(t), maxs∈[t−h,t]

x(s)), t ∈ [0, T ],

x(t) = ϕ(t), t ∈ [−h, 0],(4.1)

where A is a singular n × n matrix, x ∈ Rn, f ∈ C([0, T ] × Rn × Rn, Rn),




Existence and . . .


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Definition 4.1.1. The function α0 ∈ C0([−h, T ], Rn) ∪ C1([0, T ], Rn) is called

a lower solution of IVP (4.1), if the following inequalities are satisfied: Aα′0(t) ≤ f(t, α0(t), maxs∈[t−h,t]

α0(s)), t ∈ [0, T ],

α0(t) ≤ ϕ(t), t ∈ [−h, 0].(4.2)

Analogously, the function α0 ∈ C0([−h, T ], Rn)∪C1([0, T ], Rn) is called an

upper solution of IVP (4.1), if the equalities hold in an opposite direction.

Consider the singular differential inequalities

Ax′ +M(t)x ≤ 0, x(0) ≤ 0, t ∈ [0, T ], (4.3)

where A is a singular n×n matrix, M(t) is a continuous n×n matrix on [0, T ].



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Lemma 4.1.2. (See [13]) Assume that

(H4.1) There exists a constant λ such that L(t) = [λA + M(t)]−1 exists and

A = AL(t) is a constant matrix.

(H4.2) There exists a nonsingular matrix Q such that Q−1, (LQ)−1 exist and

Q−1, (LQ), (LQ)−1 ≥ 0, satisfying

Q−1AQ =

(C 0

0 0

), Q−1[I − λA]Q =

(I1 − λC 0

0 I2

),

where C is a diagonal matrix with C−1 ≥ 0. Then x(t) ≤ 0, t ∈ [0, T ].


x(s) ≤ 0, t ∈ [0, T ],

x(t) ≤ 0, t ∈ [−h, 0],(4.4)

where M(t) and N(t) are continuous n× n matrices on [0, T ].



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Lemma 4.1.3. Assume that the conditions (H4.1)-(H4.2) hold, and

(H4.3) The matrix [I −M ]−1 exists and is nonnegative, N(t) ≤ 0, t ∈ [0, T ],

where

M = maxs∈[0,T ]

− [λA+M(s)]−1

[e−A

DMs

∫ s

0

eADMσADN(σ)dσ

+(I − AAD)MDN(s)].

Then x(t) ≤ 0, t ∈ [−h, T ].


(H4.4) The functions α0, β0 ∈ C0([−h, T ], Rn) ∪ C1([0, T ], Rn) are lower and

upper solutions of IVP (5.1) respectively, and α0(t) ≤ β0(t), t ∈ [−h, T ].



Then the solution x(t) of IVP (4.1) satisfies α0(t) ≤ x(t) ≤ β0(t), t ∈ [−h, T ].



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Theorem 4.1.5. Assume that the conditions (H4.1)-(H4.5) for M(t) =

−fx(t, x, y), N(t) = −fy(t, x, y), (t, x, y) ∈ Ω(α0, β0) hold, and

(H4.6) The Fréchet derivatives fx, fy, fxx, fxy, fyy of the function f ∈C0,2,2(Ω(α0, β0), R

n) exist and fy(t, x, y) ≥ 0, fxx(t, x, y) ≥ 0, fxy(t, x, y) ≥ 0,

fyy(t, x, y) ≥ 0.

Then there exist two monotone sequences αn(t), βn(t), which converge

uniformly to the solution of IVP (4.1) and the convergence is quadratic.

Proof. Consider the following singular differential systems with "maxima"

Ax′(t) = f(t, αn(t), maxs∈[t−h,t]

αn(s)) + fx(t, αn(t), maxs∈[t−h,t]

αn(s))(x(t)− αn(t))

+ fy(t, αn(t), maxs∈[t−h,t]

αn(s))( maxs∈[t−h,t]


αn(s))


x(s)), t ∈ [0, T ],

x(t) = ϕ(t)− k0L0, t ∈ [−h, 0],(4.5)



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and

Ay′(t) = f(t, βn(t), maxs∈[t−h,t]

βn(s)) + fx(t, αn(t), maxs∈[t−h,t]

αn(s))(y(t)− βn(t))

+ fy(t, αn(t), maxs∈[t−h,t]

αn(s))( maxs∈[t−h,t]

y(s)− maxs∈[t−h,t]

βn(s))

≡ G0(t, y(t), maxs∈[t−h,t]

y(s)), t ∈ [0, T ],

y(t) = ϕ(t) + p0C0, t ∈ [−h, 0].(4.6)

According to the IVPs (4.5), (4.6) and Lemma 4.1.4, we obtain the sequences

αn(t), βn(t) satisfying

α0(t) ≤ α1(t) ≤ · · · ≤ αn(t) ≤ βn(t) ≤ · · · ≤ β1(t) ≤ β0(t), t ∈ [−h, T ].

Finally, we shall show that the convergence of the sequences αn(t) and

βn(t) to the solution x(t) of IVP (4.1) is quadratic. For this purpose, define

an+1(t) = x(t)− αn+1(t) ≥ 0, bn+1(t) = βn+1(t)− x(t) ≥ 0, t ∈ [−h, T ].



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Consider the following two casesµ

Case I. If t ∈ [−h, 0], in view of x(t) − αn+1(t) = knLn and kni ≤ Lni, we

have

an+1(t) ≤ maxs∈[−h,0]

|an(s)|2. (4.7)

Case II. If t ∈ [0, T ], in view of the assumption fy ≥ 0, we have

Aa′n+1(t) ≤ −M(t)an+1(t)−N(t) maxs∈[−h,T ]

an+1(s) +M1 maxs∈[−h,T ]

|an(s)|2,

By Lemma 4.1.2, we get an+1(t) ≤ u(t), t ∈ [0, T ], where u(t) is the solution

of

Au′(t) +M(t)u(t) = −N(t) maxs∈[−h,T ]

an+1(s) +M1 maxs∈[−h,T ]

|an(s)|2, t ∈ [0, T ],

u(t) = knLn, t ∈ [−h, 0].



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By using the expression of x(t) and taking suitable estimates, we conclude that

maxs∈[−h,T ]

|an+1(s)| ≤ K1 maxs∈[−h,T ]

|an(s)|2,

where K1 is a positive n× n matrix.

Analogously, we can prove that

maxs∈[−h,T ]

|bn+1(s)| ≤ K2 maxs∈[−h,T ]

|bn(s)|2 +K3 maxs∈[−h,T ]

|an(s)|2,

where K2, K3 are positive n× n matrices. The proof is complete.




n) exist and fy(t, x, y) ≥ 0, fxx(t, x, y) ≤ 0, fxy(t, x, y) ≤ 0,

fyy(t, x, y) ≤ 0.

Then there exist two monotone sequences αn(t), βn(t), which converge

uniformly to the solution of IVP (4.1) and the convergence is quadratic.



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4.2. Quadratic convergence of PBVP for singular systems

In this section, we devote to discuss the quadratic convergence of approximate

solutions of the periodic boundary value problem for nonlinear singular differ-

ential systems with "maxima" by assuming suitable conditions on the right hand

side function.

Consider the following nonlinear singular system with "maxima"(PBVP) Ax′(t) = f(t, x(t), maxs∈[t−h,t]

x(s)), t ∈ [0, T ],

x(0) = x(T ), x(t) = x(0), t ∈ [−h, 0],(4.8)

where A is a singular n × n matrix, x ∈ Rn, f ∈ C([0, T ] × Rn × Rn, Rn), h,

T are positive constants.



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Definition 4.2.1. The function α0 ∈ C([−h, T ], Rn)∪C1([0, T ], Rn) is called a

lower solution of PBVP (4.8), if the following inequalities are satisfied: Aα′0(t) ≤ f(t, α0(t), maxs∈[t−h,t]

α0(s)), t ∈ [0, T ],

α0(0) ≤ α0(T ), α0(t) = α0(0), t ∈ [−h, 0].(4.9)

Analogously, the function α0 ∈ C([−h, T ], Rn) ∪ C1([0, T ], Rn) is called an

upper solution of PBVP (4.8), if the inequalities hold in an opposite direction.

Consider the singular differential inequalitiesAx′(t) +M(t)x(t) ≤ 0, t ∈ [0, T ],

x(0) ≤ x(T ),(4.10)

where A is a singular n×n matrix, M(t) is a continuous n×n matrix on [0, T ].



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(H4.8) There exists a constant λ such that L(t) = [λA + M(t)]−1 exists, A =

AL(t) is a constant matrix.

(H4.9) There exists a nonsingular matrix Q such that Q−1 exists and Q−1,

(LQ) ≥ 0, satisfying

Q−1AQ =

(C 0

0 0

), Q−1MQ =

(I1 − λC 0

0 I2

),

where C is a C−1 ≥ 0, (I1 − λC) ≤ 0.

(H4.10) The matrix D−1 = (I − e−∫ T0 C−1(I1−λC)ds)−1 exists and is positive.

Then x(t) ≤ 0, t ∈ [0, T ].

For the boundary value problem

Ax′ +M(t)x = g(t), EL−1(0)x(0)− FL−1(T )x(T ) = η. (4.11)

We have the following known result.



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Lemma 4.2.3. (See [4]) Assume that the condition (H4.8) hold, index(A)=1, and

(H4.11) J = E − F exp−ADMT is invertible.

Then the unique solution y(t) of

Ay′ + My = g(t), Ey(0)− Fy(T ) = η. (4.12)

is given by

y(t) = e−ADMtAADJ−1(η−ξ1)+AAD

∫ T

0

G(t, σ)g(σ)dσ+(I−AAD)MDg(t),

where ξ1 = E(I − AAD)MDg(0)− F (I − AAD)MDg(T ),

G(t, s) =

e−A

DMtJ−1EeADMσAD, t > σ,

e−ADMtJ−1Fe−A

DM(T−σ)AD, t < σ.

Here AD, MD mean the Drazin inverse of the matrices A, M . Note that once

we have y(t), we get x(t) = L(t)y(t), where x(t) is the solution of (4.12).



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x(s) ≤ 0, t ∈ [0, T ],

x(0) ≤ x(T ), x(t) = x(0), t ∈ [−h, 0],(4.13)

where M(t) and N(t) are continuous n× n matrices on [0, T ].


(H4.12) There exists a matrix N such that N ≤ N(t) ≤ 0, t ∈ [0, T ], and the

matrix [I −M ]−1 exists and is nonnegative, where

M = maxs∈[0,T ]

− [λA+M(s)]−1

[AAD

∫ T

0

G(t, σ)Ndσ + (I − AAD)MDN].

Then x(t) ≤ 0, t ∈ [−h, T ].



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(H4.13) The function α0, β0 ∈ C([−h, T ], Rn) ∪ C1([0, T ], Rn) are lower and

upper solutions of PBVP (4.9), and α0(t) ≤ β0(t), t ∈ [−h, T ].



Then the solution x(t) of PBVP (4.9) satisfies α0(t) ≤ x(t) ≤ β0(t).




n) exist and fy ≥ 0, H(f) ≥ 0, where

H(f) =

∫ 1

0

[(x(t)− y(t))

∂

∂x+ ( max

s∈[t−h,t]x(s)− max

s∈[t−h,t]y(s))

∂

∂y

]2×f(t, σx(t) + (1− σ)y(t), σ max

s∈[t−h,t]x(s) + (1− σ) max

s∈[t−h,t]y(s))dσ,

for α0(t) ≤ x, y ≤ β0(t), t ∈ [−h, T ].



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Then there exist two sequences αn(t), βn(t), which converge uniformly to

the solution x(t) of PBVP (4.9) and the convergence is quadratic.



(H4.16) The Fréchet derivatives fx, fy, fxx, fxy, fyy, φx, φy, φxx, φxy, φyy of

the function f , φ ∈ C0,2,2(Ω(α0, β0), Rn) exist and fy ≥ 0, H(f + φ) ≥ 0,

H(φ) ≥ 0.

Then there exist two sequences αn(t), βn(t), which converge uniformly to

the solution x(t) of PBVP (4.9) and the convergence is quadratic.



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4.3. Rapid convergence for singular differential systems

In this section, by weakening the restrictions of concavity or convexity on

the right hand side function, we investigate the higher order of convergence of

approximate solutions of the initial value problem for nonlinear singular differ-

ential systems under less restrictive conditions.

Consider the following initial value problem for nonlinear singular differential

system(IVP) Ax′ = f(t, x), t ∈ J,

x(0) = x0,(4.14)

where A is n × n singular matrix, x ∈ Rn, f ∈ C(J × Rn, Rn), J = [0, T ],

T > 0 is a fixed constant.



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Lemma 4.3.1. Assume that the functions α0, β0 ∈ C1(J,Rn) are lower and

upper solutions of IVP (4.14), and the following conditions hold for M(t) =

−fx(t, x), (t, x) ∈ Ω(α0, β0).

(H4.17) There exists a constant λ such that L(t) = [λA + M(t)]−1 exists and

A = AL(t) is a constant matrix.

(H4.18) There exists a nonsingular matrix Q such that Q−1, (LQ)−1 exist and

Q−1, (LQ), (LQ)−1 ≥ 0, satisfying

Q−1AQ =

(C 0

0 0

), Q−1[I − λA]Q =

(I1 − λC 0

0 I2

),

where C is a diagonal matrix with C−1 ≥ 0.

(H4.19) y0 satisfies (I − AAD)(y0 − w(0)) = 0, where w(t) = MDg(t), M =

M(t)L(t).

(H4.20) The Fréchet derivative fx of the function f ∈ C(Ω(α0, β0), Rn) exists

and is continuous.

Then the solution x(t) of IVP (4.14) satisfies α0(t) ≤ x(t) ≤ β0(t), t ∈ J .



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−fx(t, x), (t, x) ∈ Ω(α0, β0) hold, and

(H4.21) The Fréchet derivatives ∂if(t,x)∂xi (i = 0, 1, 2, · · · ,m) of the function f ∈

C(Ω(α0, β0), Rn) exist and are continuous, satisfying f(t, x)+Mxm is (m−1)-

hyperconvex, f(t, x)−Nxm is (m− 1)-hyperconcave, that is, ∂m(f(t,x)+Mxm)

∂xm ≥0, ∂

m(f(t,x)−Nxm)∂xm ≤ 0, where M , N are positive n× n matrix.

Then there exist monotone sequences αn, βn, which converge uniformly to

the solution of IVP (4.14) and the convergence is of order m, that is, there exist

positive n× n matrices K1, K2 such that

maxt∈J|x(t)− αn+1(t)| ≤ K1 max

t∈J|x(t)− αn(t)|m,

maxt∈J|βn+1(t)− x(t)| ≤ K2 max

t∈J|βn(t)− x(t)|m,

where maxt∈J|u(t)| = (max

t∈J|u1(t)|,max

t∈J|u2(t)|, · · · ,max

t∈J|un(t)|)T , |u|m =

(|u1|m, |u2|m, · · · , |un|m)T , u ∈ C(J,Rn).



Existence and . . .


Convergence

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Proof. Consider the singular differential systemsAx′ =

m−1∑i=0

∂if(t, αn)

∂xi(x− αn)i

i!−M(x− αn)m

≡ F (t, x, αn), t ∈ J,

x(0) = x0,

(4.15)

and

Ay′ =

∑m−1i=0

∂if(t,βn)∂xi

(y−βn)ii! −M(y − βn)m, m = 2k + 1,∑m−1

i=0∂if(t,βn)∂xi

(y−βn)ii! +N(y − βn)m, m = 2k,

≡ G(t, y, βn), t ∈ J,y(0)= x0.

(4.16)

According to the IVPs (4.15), (4.16) and Lemma 4.3.1, we see that the se-

quences αn, βn satisfying

α0 ≤ α1 ≤ · · · ≤ αn ≤ βn ≤ · · · ≤ β1 ≤ β0, t ∈ J.



Existence and . . .


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Finally, we shall show that the convergence of the sequences αn(t) and

βn(t) to the solution x(t) of IVP (4.14) is of order 2k + 1. For this purpose,

define

an+1 = x− αn+1, bn+1 = βn+1 − x, t ∈ J,

such that an+1(0) = 0, bn+1(0) = 0.

Using the mean value theorem, we have

Aa′n+1 = f(t, x)−m−1∑i=0

∂if(t, αn)

∂xi(αn+1 − αn)i

i!+M(αn+1 − αn)m

= f(t, x)−[f(t, αn+1)

−(∫ 1

0

(1− σ)m−1∂mf(t, σαn+1 + (1− σ)αn)

∂xmdσ)(αn+1 − αn)m

(m− 1)!

]+M(αn+1 − αn)m

≤(∫ 1

0

fx(t, σx+ (1− σ)αn+1)dσ)

(x− αn+1) + (M +N)amn

= −M(t)an+1 + camn ,

where c = N +M .



Existence and . . .


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Furthermore, we have an+1 ≤ u, t ∈ J , where u is the solution of

Au′ +M(t)u = camn , x(0) = 0.

Thus, using the expression of x and taking suitable estimates, we have

maxt∈J|an+1(t)| ≤ K1 max

t∈J|an(t)|m,

where K1is positive n× n matrix.

Similarly, we can show that

maxt∈J|bn+1(t)| ≤ K2 max

t∈J|bn(t)|m.

where K2 is positive n× n matrix. The proof is complete.

Remark 4.3.3. If the function f(t, x) is (m − 1)-hyperconvex or (m − 1)-

hyperconcave in x, by the method of quasilinearization, we also can obtain the

convergence of the monotone sequences αn, βn is of order m.



Existence and . . .


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4.4. Rapid convergence for singular systems with "maxima"

In this section, we obtain the higher order of convergence of approximate

solutions of initial value problem for nonlinear singular differential systems with

"maxima" under the assumptions of hyperconcavity or hyperconvexity on the

right hand side function, an example is given to illustrate the main results.

Consider the following initial value problem for singular differential system

with “maxima"(IVP) Ax′(t) = f(t, x(t), maxs∈[t−h,t]

x(s)), t ∈ [0, T ],

x(t) = ϕ(t), t ∈ [−h, 0],(4.17)

where A is n × n singular matrix, x ∈ Rn, f ∈ C([0, T ] × Rn × Rn, Rn),




Existence and . . .


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Theorem 4.4.1. Assume that the conditions of Lemma 4.3.1 for M(t) =


(H4.22) The Fréchet derivatives ∂mf(t,x,y)∂xiym−i , m = 1, 2, · · · , 2k + 1, i =

0, 1, 2, · · · ,m of the function f ∈ C0,2k+1,2k+1(Ω(α0, β0), Rn) exist and

fy(t, x, y) ≥ 0, ∂2k+1f(t,x,y)∂xiy2k+1−i ≥ 0, k ≥ 1, i = 0, 1, 2, · · · , 2k + 1.

Then there exist monotone sequences αn(t), βn(t), which converge uni-

formly to the solution x(t) of IVP (4.17) and the convergence is of order 2k+ 1,

that is, there exist positive n× n matrices K1, K2 such that

|x− αn+1|0 ≤ K1|x− αn|2k+10 ,

|βn+1 − x|0 ≤ K2|βn − x|2k+10 ,

where |u|0 = maxs∈[−h,T ]

|u(s)| = ( maxs∈[−h,T ]

|u1(s)|, maxs∈[−h,T ]

|u2(s)|, · · · , maxs∈[−h,T ]

|un(s)|)T .



Existence and . . .


Convergence

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Proof. Consider the singular differential systems with "maxima"

Ax′(t) = f(t, αn(t), maxs∈[t−h,t]

αn(s)) +2k∑i=1

1

i!

[(x(t)− αn(t))

∂

∂x

+ ( maxs∈[t−h,t]


αn(s))∂

∂y

]if(t, αn(t), max

s∈[t−h,t]αn(s))


x(s)), t ∈ [0, T ],

x(t) = ϕ(t)− k2k0 L0, t ∈ [−h, 0],(4.18)

and

Ay′(t) = f(t, βn(t), maxs∈[t−h,t]

βn(s)) +2k∑i=1

1

i!

[(y(t)− βn(t))

∂

∂x


y(s)− maxs∈[t−h,t]

βn(s))∂

∂y

]if(t, βn(t), max

s∈[t−h,t]βn(s))

≡ G0(t, y(t), maxs∈[t−h,t]

y(s)), t ∈ [0, T ],

y(t) = ϕ(t) + p2k0 C0, t ∈ [−h, 0].(4.19)



Existence and . . .


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(H4.23) The Fréchet derivatives ∂mf(t,x,y)∂xiym−i , m = 1, 2, · · · , 2k, i = 0, 1, 2, · · · ,m

of the function f ∈ C0,2k,2k(Ω(α0, β0), Rn) exist and fy(t, x, y) ≥ 0, ∂

2kf(t,x,y)∂xiy2k−i ≥

0, k ≥ 1, i = 0, 1, 2, · · · , 2k.


formly to the solution x(t) of IVP (4.17) and the convergence is of order 2k,

that is, there exists positive n× n matrices K3, K4 such that

|x− αn+1|0 ≤ K3|x− αn|2k0 ,

|βn+1 − x|0 ≤ K4|βn − x|2k−10 (|βn − x|0 + |x− αn|0).



Existence and . . .


Convergence

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(H4.24) The Fréchet derivatives ∂mf(t,x,y)∂xiym−i , m = 1, 2, · · · , 2k + 1, i =

0, 1, 2, · · · ,m of the function f ∈ C0,2k+1,2k+1(Ω(α0, β0), Rn) exist and

fy(t, x, y) ≥ 0, ∂2k+1f(t,x,y)∂xiy2k+1−i ≤ 0, k ≥ 1, i = 0, 1, 2, · · · , 2k + 1.


formly to the solutionx(t) of IVP (4.17) and the convergence is of order 2k+ 1,


|x− αn+1|0 ≤ K5|x− αn|2k0 (|x− αn|0 + |βn − x|0),

|βn+1 − x|0 ≤ K6|βn − x|2k0 (|βn − x|0 + |x− αn|0).



Existence and . . .


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(H4.25) The Fréchet derivatives ∂mf(t,x,y)∂xiym−i , m = 1, 2, · · · , 2k, i = 0, 1, 2, · · · ,m

of the function f ∈ C0,2k,2k(Ω(α0, β0), Rn) exist and fy(t, x, y) ≥ 0, ∂

2kf(t,x,y)∂xiy2k−i ≤

0, k ≥ 1, i = 0, 1, 2, · · · , 2k.


formly to the solution x(t) of IVP (4.17) and the convergence is of order 2k,


|x− αn+1|0 ≤ K7|x− αn|2k−10 (|x− αn|0 + |βn − x|0),

|βn+1 − x|0 ≤ K8|βn − x|2k0 .



Existence and . . .


Convergence

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Now, we give an example to illustrate the application of the established in the

previous section.

Example 4.4.5. Consider the following singular system with "maxima"x′1(t) = max

s∈[t−h,t]x1(s)− x1(t)−

1

1− x1(t)+ 1,

0 =1

x2(t)− 1+ 1, t ∈ [0, 1],

x1(t) = x2(t) = 0, t ∈ [−1, 0],

(4.20)

where

A =

(1 0

0 0

), f =

maxs∈[t−h,t]

x1(s)− x1(t)− 11−x1(t) + 1

1x2(t)−1 + 1

.

We can see easily that IVP (4.20) has a zero solution. Taking α0(t) = (0, 0)T ,

β0(t) = (12,12)T , it is easy to verify that α0(t), β0(t) are lower and upper solutions

of IVP (4.20) respectively, and the conditions of Theorem 4.4.4 are satisfied.



Existence and . . .


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Thus, we can construct two monotone sequences of functions which quadrat-

ically converge to zero solution.

The successive approximations αn+1(t) and βn+1(t) are solutions of the sin-

gular differential systems

x′1(t) = maxs∈[t−h,t]

αn1(s)− αn1(t)−1

1− αn1(t)+ 1


x1(s)− maxs∈[t−h,t]

αn1(s))− (x1(t)− αn1(t))

− 1

(1− βn1(t))2(x1(t)− αn1(t)),

0 =1

αn2(t)− 1+ 1− 1

(βn2(t)− 1)2(x2(t)− αn2(t)), t ∈ [0, 1],

x1(t) = x2(t) = −kn1Ln1 = −kn2Ln2, t ∈ [−1, 0],(4.21)

and



Existence and . . .


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βn1(s)− βn1(t)−1

1− βn1(t)+ 1


x1(s)− maxs∈[t−h,t]

βn1(s))− (x1(t)− βn1(t))

− 1

(1− βn1(t))2(x1(t)− βn1(t)),

0 =1

βn2(t)− 1+ 1− 1

(βn2(t)− 1)2(x2(t)− βn2(t)), t ∈ [0, 1],

x1(t) = x2(t) = pn1Cn1 = pn2Cn2, t ∈ [−1, 0],(4.22)

where Ln = mins∈[−1,0]

(−αn(s)) ≥ 0, Cn = mins∈[−1,0]

(βn(s)) ≥ 0, and the numbers

kni, pni ∈ [0, 1) are such that kni ≤ Lni and pni ≤ Cni, i = 1, 2, · · · , n.



Existence and . . .


Convergence

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First, we will construct an increasing sequence of lower solutions which

converges quadratically to zero solution. In view of the fact that α0(t) = (0, 0)T ,

we have L0 = (0, 0)T .

Taking k0 = (0, 0)T , the linear singular differential system (4.23) reduces tox′1(t) = max

s∈[t−h,t]x1(s)− 5x1(t),

0 = −4x2(t), t ∈ [0, 1],

x1(t) = x2(t) = 0, t ∈ [−1, 0].

(4.23)

Then, the IVP (4.23) has a zero solution, that is, α1(t) = (0, 0)T .

Proceeding as before, we can find that all successive approximations αn(t)

are equal to zero vector.

Next, we shall construct a decreasing sequence of upper solutions which will

converge quadratically to the exact solution.



Existence and . . .


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Since β0(t) = (12,12)T , we get C0 = (12,

12)T . Let p0 = (12,

12)T , then the linear

singular differential system (4.22) can be written asx′1(t) = max

s∈[t−h,t]x1(s)− 5x1(t) + 1,

0 = −4x2(t) + 1, t ∈ [0, 1],

x1(t) = x2(t) =1

22, t ∈ [−1, 0].

(4.24)

The function β1(t) = (14,14)T is a solution of IVP (4.24).

Choose p1 = (14,14)T , then the linear singular differential system (4.22) reduces

to the following x′1(t) = max


25

9x1(t) +

1

9,

0 = −16

9x2(t) +

1

9, t ∈ [0, 1],

x1(t) = x2(t) =1

42, t ∈ [−1, 0].

(4.25)

The IVP (4.27) has an exact solution β2(t) = ( 116,

116)

T .



Existence and . . .


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Now, choose pn−1 = ( 1

22(n−1) ,

1

22(n−1) )

T , then the linear singular differential

system (4.22) reduces to


x1(s)− (1 +22

n

(22(n−1) − 1)2)x1(t) +

1

(22(n−1) − 1)2,

0 = − 22n

(22(n−1) − 1)2x2(t) +

1

(22(n−1) − 1)2, t ∈ [0, 1],

x1(t) = x2(t) =1

22n, t ∈ [−1, 0].

(4.26)

Then, the IVP (4.26) has an exact solution βn(t) = ( 122n, 122n

)T .

By induction, we obtain β1(t) = (14,14)T , β2(t) = ( 1

16,116)

T , · · · ,βn(t) =

( 122n, 122n

)T ,· · · . It is easy to see that(1414

)≤

(1 1

1 1

)(122

122

),

(116116

)≤

(1 1

1 1

)(142

142

), · · · ,

(122n

122n

)≤

(1 1

1 1

) 1

(22(n−1)

)2

1

(22(n−1)

)2

, · · · .

Therefore, the convergence of the sequence βn(t) is quadratic.



Existence and . . .


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