Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 120, pp. 1–10.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ROTHE’S METHOD FOR SOLVING SEMI-LINEAR

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

DARSHANA DEVI, DURANTA CHUTIA, RAJIB HALOI

Abstract. We consider a semi-linear differential equation of parabolic type

with deviating arguments in a Banach space with uniformly convex dual, and
apply Rothe’s method to establish the existence and uniqueness of a strong

solution. We also include an example as an application of the main result.

1. Introduction

In differential equations with deviating arguments the unknown function and its
derivative are evaluated at different values of their arguments. They are consid-
ered as one of the most important and frequently used differential equations and
hence the study of these equations has been rapidly increasing. They are widely
used in various branches of science and technology such as self-oscillating systems,
automatic control, problems related with combustion in rocket motion, long-term
planning in economics, biological problems, and many other areas of science and
technology [11, 13]. The very familiar hot shower problem is closely related to
these differential equations. For an extensive reading on differential equations with
deviating arguments, we refer the reader to [8, 9, 10, 12, 14, 17].

Rothe’s method was introduced by Rothe [25] in 1930 to solve a scalar parabolic
initial value problem of second order. Rothe used time discretization to develop
his method so the method is also known as the method of semidiscretization or the
method of lines. Later on many authors have used and developed this method, see
[1, 2, 5, 7, 16, 24]. Rothe’s method is effectively used to establish the existence
and uniqueness of solution of equations such as linear, nonlinear, parabolic and
hyperbolic equations with higher orders. The method is also used to study the
diffusion problems [6, 18, 20, 22, 23]. Recently Rothe’s method is also applied to
study variational-hemivariational inequalities with applications to contact mechan-
ics [3, 4, 19],. Thus the application of this method is not limited to mathematics but
also applicable to physics and biology. The method becomes a strong and efficient
tool to analyze the existence and uniqueness of solution to differential equations.

2010 Mathematics Subject Classification. 34G20, 34K30, 35D35, 35K58.
Key words and phrases. Strong solution; deviating argument;

semigroup of bounded linear operators; semidiscretization method.
c©2020 Texas State University.

Submitted October 23, 2020. Published December 10, 2020.

1



2 D. DEVI, D. CHUTIA, R. HALOI EJDE-2020/120

Raheem and Bahuguna [23] applied Rothe’s method to study the fractional in-
tegral diffusion equation in a Banach space X,

∂u(t)

∂t
+Au(t) =

1

Γ(α)

∫ t

0

u(s)

(t− s)1−α
ds+ f(t), t ∈ (0, T ],

u(0) = u0,

where 0 < α < 1, -A is the infinitesimal generator of a C0-semigroup of contractions,
f is a given map from [0, T ] to X, and the initial point u0 ∈ D(A) ⊂ X, the domain
of A.

Dubey [7] used the method for the nonlinear nonlocal functional differential
equation in a Banach space X,

u′(t) +Au(t) = f(t, u(t), ut), t ∈ (0, T ],

h(u0) = φ on [−τ, 0],

where 0 < T < ∞, φ ∈ C0 := C([−τ, 0];X), τ > 0, the nonlinear operator A
is single-valued and m-accretive defined from the domain D(A) ⊂ X to X, the
nonlinear map f is defined from [0, T ]×X×C0 to X, the map h is defined from C0

to C0. For u ∈ CT := C([−τ, T ];X), the map ut ∈ C0 is defined by ut(s) = u(t+ s)
for s ∈ [−τ, 0]. Here, Ct := C([−τ, t];X) for t ∈ [0, T ] is the Banach space of all
continuous functions from [−τ, t] into X endowed with the supremum norm,

‖φ‖t = sup
τ≤η≤t

‖φ(η)‖, φ ∈ Ct,

where ‖ · ‖ is the norm in X.
In this article, we consider the following semi-linear differential equation with

deviating arguments. Let (X, ‖·‖) be a Banach space with a uniformly convex dual
X∗.

∂u(t)

∂t
+Au(t) = f(t, u(t), u(h(u(t), t))), t ∈ (0, T ],

u(0) = u0, u0 ∈ X.
(1.1)

We assume that for each t ∈ (0, T ], −A is the infinitesimal generator of a C0-
semigroup of contractions, the non-linear continuous maps f : [0, T ]×X ×X → X
and h : X×[0, T ]→ [0, T ] satisfy suitable growth conditions in its arguments stated
in next section.

2. Preliminaries and main result

In this section we briefly state some definitions and results need for proving the
main result. At the end of this section, we state our main result.

Definition 2.1. Let X be a Banach space and X∗ be its dual. For every x ∈ X
we define the duality map P as

P (x) = {x∗ ∈ X∗ : (x∗, x) = ‖x‖2 = ‖x∗‖2},
where (x∗, x) denotes the value of x∗ at x.

Definition 2.2. A nonlinear operator A : D(A) ⊂ X → X is called m-accretive if

(i) (Ax−Ay, P (x− y)) ≥ 0, for all x, y ∈ D(A),
(ii) R(I +A) = X, where R(·) is the range of an operator.

Lemma 2.3 ([21, Theorem 1.4.3]). If -A is the infinitesimal generator of a C0

semigroup of contractions, then A is m-accretive, i.e.,



EJDE-2020/120 ROTHE’S METHOD WITH DEVIATING ARGUMENTS 3

(i) (Au−Av, P (u− v)) ≥ 0, for all u, v ∈ D(A), where P is the duality map.
(ii) For each λ > 0, we have R(I + λA) = X, where I is the identity operator

on X and R(·) denotes range of an operator.

Lemma 2.4 ([15, Lemma 2.5]). Let X be a Banach space and X∗ be its uniformly
convex dual. Let -A be the infinitesimal generator of a C0 semigroup of contractions.
Consider the sequence Xn ∈ D(A), n = 1, 2, 3, . . . such that Xn → u ∈ X and if
‖AXn‖ are bounded, then u ∈ D(A) and AXn ⇀ Au.

Lemma 2.5 ([23]). Let α1, α2, . . . , αj be non-negative numbers satisfying

(i) α1 ≤ B,

(ii) αi ≤ B + Cλ
∑i−1
k=1 αk, where B, C and λ are positive constants.

Then for each i = 1, 2, . . . , j we have αi ≤ BeC(i−1)λ.

We use the following assumptions for proving our main result.

(A1) Suppose there exists a constant Lf > 0 such that for each x, y, x′, y′ ∈ X
and t, s ∈ [0, T ], the function f : [0, T ]×X ×X → X satisfies

‖f(t, x, x′)− f(s, y, y′)‖ ≤ Lf (|t− s|+ ‖x− y‖+ ‖x′ − y′‖).
(A2) Let there exists a constant L1 > 0 such that for each x, y ∈ X and t, s ∈

[0, T ] the map h : X × [0, T ]→ [0, T ] satisfies

|h(x, t)− h(y, s)| ≤ L1(‖x− y‖+ |t− s|).

Remark 2.6. For each x, y ∈ X and t, s ∈ [0, T ], we have

‖x(h(x, t))− y(h(y, s))‖ ≤ L2(|h(x, t)− h(y, s)|)
≤ L2[L1(‖x− y‖+ |t− s|)] = Lh(‖x− y‖+ |t− s|),

for some constants L1, L2, Lh > 0.

Theorem 2.7. Let (A1) and (A2) be satisfied. Then the initial value problem
(1.1) has a unique strong solution u on the interval [0, T ]. More precisely we have,
u ∈ C([0, T ];X) such that u(t) ∈ D(A), u is differentiable a.e. on [0, T ] and u
satisfies (1.1).

3. Approximation

In this section, we use time discretization to approximate the system (1.1) by
corresponding parabolic problems and construct an approximate solution to the
original problem. Also we prove the convergence of this approximate solution to
the solution of (1.1) with the help of analogues results for the approximate equations
of the original system.

To apply the Rothe’s Method, we consider the interval [0, T ] and divide it into
the subintervals of length λn = T

n . We use the following approximate equations to
replace the system (1.1). For i = 1, we have

un1 − un0
λn

+Aun1 = f0,

un0 = u0,
(3.1)

and for 2 ≤ i ≤ n, we use the equations,

uni − uni−1
λn

+Auni = fni−1, (3.2)



4 D. DEVI, D. CHUTIA, R. HALOI EJDE-2020/120

where,

fni = f(tni , u
n
i , u
′n
i ), uni = u(tni ), u′

n
i = u(h(uni , t

n
i )),

and f0 = fn(0, u0, u(h(u0, 0))).
Next we successively establish the existence and uniqueness of solution of the

approximate equations

un1 − un0
λn

+Aun1 = f0, un0 = u0, (3.3)

uni − uni−1
λn

+Auni = fni−1, i = 2, 3, . . . , n. (3.4)

The existence of a unique solution uni ∈ D(A) to the system (3.2) is a consequence
of Lemma 2.3.

We now define the Rothe’s sequence as

Un(t) =

{
u0, if t = 0,

uni−1 + 1
λn

(t− tni−1)(uni − uni−1), if t ∈ (tni−1, t
n
i ].

Before proving the main result, we now present some results which are needed.

Lemma 3.1. For each n ∈ N and i = 1, 2, . . . , n, the estimate

‖uni ‖ ≤ C

holds for some constant C > 0 and the constant is independent of n, i and λn.

Proof. From (3.3), we have

un1 + λnAu
n
1 = un0 + λnf0.

Applying P (un1 ) on both sides and using the definition of accretivity of A, we obtain

‖un1‖ ≤ ‖un0‖+ λn‖f0‖ ≤ ‖u0‖+ T‖f0‖ = C1(say).

From (3.4) and for 2 ≤ i ≤ n, we have

uni + λnAu
n
i = uni−1 + λnf

n
i−1.

We apply P (uni ) on both sides and use the definition of accretivity of A to obtain

‖uni ‖ ≤ ‖uni−1‖+ λn‖fni−1‖.

By using the hypotheses (A1) and (A2) in the above equation, we obtain

‖uni ‖ ≤ ‖uni−1‖+ λn
[
Lf
{
|tni−1|+ ‖uni−1 − u0‖+ Lh

(
‖uni−1 − u0‖+ |tni−1|

)}
+ ‖f0‖

]
= ‖uni−1‖+ λn

[
Lf |tni−1|(1 + Lh) + ‖f0‖

]
+ λnLf‖uni−1 − u0‖(1 + Lh).

Repeating above process, we obtain

‖uni ‖ ≤ ‖u0‖+ iλn
[
Lf |tni−1|(1 + Lh) + ‖f0‖

]
+ iλnLf‖u0‖(1 + Lh)

+ λnLf (1 + Lh)

i−1∑
j=1

‖unj ‖

≤ ‖u0‖
{

1 + TLf (1 + Lh)
}

+ T
{
LfT (1 + Lh) + ‖f0‖

}
+ λnLf (1 + Lh)

i−1∑
j=1

‖unj ‖.



EJDE-2020/120 ROTHE’S METHOD WITH DEVIATING ARGUMENTS 5

Applying Lemma 2.5 on the above inequality, we obtain

‖uni ‖ ≤
[
‖u0‖

{
1 + TLf (1 + Lh)

}
+ T

{
LfT (1 + Lh) + ‖f0‖

}]
eLf (1+Lh)(i−1)λn

≤
[
‖u0‖

{
1 + TLf (1 + Lh)

}
+ T

{
LfT (1 + Lh) + ‖f0‖

}]
eLf (1+Lh)T

= C.

This completes the proof. �

Lemma 3.2. For each n ∈ N and i = 1, 2, . . . , n there exists a positive constant C
which is independent of n, i and λn such that

‖
uni − uni−1

λn
‖ ≤ C.

Proof. From (3.3), we obtain

un1 − u0
λn

+Aun1 −Au0 = f0 −Au0.

Applying P (un1 − u0) on the above equation and using the definition of accretivity
of A, we obtain

‖u
n
1 − u0
λn

‖ ≤ ‖f0‖+ ‖Au0‖ = C1(say). (3.5)

We rewrite the equation (3.4) for the index i− 1 and subtract it from (3.4), we
obtain

uni − uni−1
λn

+Auni −Auni−1 =
uni−1 − uni−2

λn
+ fni−1 − fni−2.

Again we apply P (uni − uni−1) on both sides to deduce the following estimates∥∥uni − uni−1
λn

∥∥
≤
∥∥uni−1 − uni−2

λn

∥∥+ ‖fni−1 − fni−2‖

≤
∥∥uni−1 − uni−2

λn

∥∥+ Lf
{
|tni−1 − tni−2|+ ‖uni−1 − uni−2‖

+ Lh

(
‖uni−1 − uni−2‖+ |tni−1 − tni−2|

)}
=
∥∥uni−1 − uni−2

λn

∥∥+ Lf (1 + Lh)|tni−1 − tni−2|+ Lf (1 + Lh)λn
∥∥uni−1 − uni−2

λn

∥∥
= Lf (1 + Lh)λn +

{
1 + Lf (1 + Lh)λn

}∥∥uni−1 − uni−2
λn

∥∥ .
We put C2 = Lf (1 + Lh) and repeating the above inequality, we obtain∥∥uni − uni−1

λn

∥∥ ≤ K + (1 + C2λn)i−1
∥∥un1 − u0

λn

∥∥, (3.6)

where,

K = C2λn
{

1 + (1 + C2λn) + (1 + C2λn)2 + · · ·+ (1 + C2λn)i−2
}

= (1 + C2λn)i−1 − 1.

Now, we have

(1 + C2λn)i−1 ≤ eC2λn(i−1) ≤ eC2T .



6 D. DEVI, D. CHUTIA, R. HALOI EJDE-2020/120

Hence K is a constant independent of n, i and λn. Thus from the estimates (3.5)
and (3.6), we obtain ∥∥uni − uni−1

λn

∥∥ ≤ K + eC2TC1 = C.

This completes the proof. �

Next we define a sequence of step functions

Y n(t) =

{
u0 if t = 0,

uni if t ∈ (tni−1, t
n
i ].

Remark 3.3. From Lemma 3.2, we can conclude that Un(t) is uniformly Lipschitz
continuous and Un(t)− Y n(t)→ 0 as n→∞.

We define fn(t) = f(tni , u
n
i , u
′n
i ). Then equation (3.3) and (3.4) can be rewritten

as
d

dt
Un(t) +AY n(t) = fn(t), t ∈ (0, T ]. (3.7)

where d
dt denotes the left derivative in the interval (0, T ]. For t ∈ (0, T ], we have∫ t

0

AY n(s)ds = u0 − Un(t) +

∫ t

0

fn(s)ds. (3.8)

Lemma 3.4. There exists u ∈ C([0, T ];X) such that Un → u in C([0, T ];X) as
n→∞. Moreover, u is Lipschitz continuous on [0, T ].

Proof. From (3.7), we see that

d

dt
Un(t)− d

dt
Um(t) +AY n(t)−AY m(t) = fn(t)− fm(t).

Applying P (Y n(t)− Y m(t)), using the definition of accretivity of A, we obtain

(
d

dt
Un(t)− d

dt
Um(t), P (Y n(t)− Y m(t)) ≤ (fn(t)− fm(t), P (Y n(t)− Y m(t)).

From the above inequality and using that

(
d

dt
Un(t)− d

dt
Um(t), P (Un(t)− Um(t)) =

1

2

d

dt

∥∥Un(t)− Um(t)
∥∥2,

we obtain

(
d

dt
Un(t)− d

dt
Um(t), P (Un(t)− Um(t))

≤ (fn(t)− fm(t), P (Y n(t)− Y m(t)) + (
d

dt
Un(t)− d

dt
Um(t), P (Un(t)− Um(t))

− (
d

dt
Un(t)− d

dt
Um(t), P (Y n(t)− Y m(t))

=
∥∥ d
dt

(Un(t)− Um(t)
∥∥(‖Un(t)− Um(t)‖ − ‖Y n(t)− Y m(t)‖

)
+ ‖fn(t)− fm(t)‖‖Y n(t)− Y m(t)‖

≤
∥∥ d
dt

(Un(t)− Um(t)
∥∥(‖Un(t)− Um(t)− Y n(t) + Y m(t)‖

)
+ ‖fn(t)− fm(t)‖‖Y n(t)− Y m(t)‖.



EJDE-2020/120 ROTHE’S METHOD WITH DEVIATING ARGUMENTS 7

which implies

1

2

d

dt

∥∥Un(t)− Um(t)
∥∥2

≤
∥∥ d
dt

(Un(t)− Um(t)
∥∥(‖Un(t)− Y n(t)‖+ ‖Um(t)− Y m(t)‖

)
+ Lf

{
|tni − tmi |+ ‖uni − umi ‖+ Lh(‖uni − umi ‖+ |tni − tmi |)

}
‖Y n(t)− Y m(t)‖

which implies
d

dt

∥∥Un(t)− Um(t)
∥∥2 ≤ σ1

nm(t) (say),

and σ1
nm(t)→ 0 as n,m→∞. This implies

‖Un(t)− Um(t)‖2 ≤ σ2
nm(t),

where

σ2
nm(t) =

∫ t

0

σ1
nm(s)ds

and σ2
nm(t)→ 0 as n,m→∞. Taking the supremum, we obtain

sup
tε(0,T ]

‖Un(t)− Um(t)‖2 ≤ σ2
nm(t).

Using the above inequality, we conclude that Un → u in C([0, T ], X). Since each
Un is uniformly Lipschitz continuous, it follows that u is Lipschitz continuous. �

Remark 3.5. As the sequence Un(t) − Y n(t) → 0 as n → ∞, Y n(t) → u(t).
Furthermore it is clear that Y n(t) ∈ D(A) for each n ∈ N. Also ‖AY n‖ are
bounded so by Lemma 2.4, we can conclude that AY n ⇀ Au.

Proof of Theorem 2.7. For every x∗ ∈ X∗ and t ∈ (0, T ], we have∫ t

0

(AY n(s), x∗)ds = (u0, x
∗)− (Un(t), x∗) +

∫ t

0

(fn(s), x∗)ds. (3.9)

From the Lemma 3.4, Remark 3.5 and the bounded convergence theorem, from
(3.9) after considering the limit as n→∞, we obtain∫ t

0

(Au(s), x∗)ds = (u0, x
∗)−(u(t), x∗)+

∫ t

0

(f(s, u(s), u(h(u(s), s))), x∗)ds. (3.10)

Since Au(t) is Bochner integrable on [0, T ], from equation (3.10), we obtain

d

dt
u(t) +Au(t) = f(t, u(t), u(h(u(t), t))) a.e. t ∈ (0, T ]. (3.11)

Now it is clear that u ∈ C([0, T ];X) and differentiable on (0, T ] with u(t) ∈ D(A);
u(0) = u0 and satisfies the problem (3.11). Therefore it will be a strong solution of
the problem (1.1) on [0, T ].

Next we prove the uniqueness of the solution. For this, if possible we assume
that u1 and u2 are two strong solutions of (1.1). We put u = u1 − u2, from (3.11),
we have

(
du(t)

dt
, P (u(t))) + (Au(t), P (u(t)))

= (f(t, u1(t), u(h(u1(t), t))− f(t, u2(t), u(h(u2(t), t)), P (u(t))).



8 D. DEVI, D. CHUTIA, R. HALOI EJDE-2020/120

By using the definition of accretivity of A, we obtain(du(t)

dt
, P (u(t))

)
≤ (f(t, u1(t), u1(h(u1(t), t))− f(t, u2(t), u2(h(u2(t), t)), P (u(t))).

We used that (du(t)

dt
, P (u(t))

)
=

1

2

d

dt
‖u(t)‖2.

From an easy calculation we obtain

d

dt
‖u(t)‖2 ≤ C‖u(t)‖2 a.e. t ∈ (0, T ],

where C = 2Lf (1 + Lh). Integrating over the interval (0, t), we obtain

‖u(t)‖2 ≤ C
∫ t

0

‖u(s)‖2ds.

Applying Gronwall’s inequality, we obtain u = 0 on [0, T ]. This shows the unique-
ness of the strong solution and hence it completes the proof. �

4. Application

We consider the equation with deviating argument,

∂u(t, x)

∂t
+
∂2u(t, x)

∂x2
= F (x, u(t, x)) +G(t, x, u(t, x)),

u(t, 0) = u(t, 1), 0 < t ≤ T,
u(0, x) = u0(x), x ∈ Ω.

(4.1)

where Ω is a bounded domain in Rn. Here

F (x, u(t, x)) =

∫ x

0

ξ(x, y)u(f(t)|u(t, y)|, y)dy ∀(t, x) ∈ (0, T ]× Ω.

We assume that f : [0, T ]→ R+ is locally Hölder continuous in t with f(0) = 0;
ξ ∈ C1(Ω × Ω;R); the function G : [0, T ] × Ω × R → R is measurable in x, locally
Hölder continuous in t, locally Lipschitz continuous in u and uniformly continuous
in x.

Let X = L2(Ω;R). We define X1 = D(A) = H2(Ω) ∩ H1
0 (Ω) and Au = ∂2u

∂x2 .

Then X1/2 = D((A)1/2) = H1
0 (Ω).

For x ∈ Ω, we define the map f : [0, T ]×H2(Ω)× L2(Ω)→ H1
0 (Ω) by

g(t, φ, ψ) = F (x, ψ) +G(t, x, φ),

where F (x, ψ(x, t)) =
∫ x
0
ξ(x, y)ψ(y, t)dy. We also assume that the map G : [0, T ]×

Ω×H2(Ω)→ H0
1 (Ω) that a C > 0,

‖G(t, x, u)−G(r, x, w)‖ ≤ C(|t− r|+ ‖u− v‖).
Thus, the map g satisfies assumption (A1) (see [9]) and h : H2(Ω) × [0, T ] → R+

defined by h(φ(x, t), t) = f(t)|φ(x, t)| satisfies assumption (A2) (see [9]).
Then problem (4.1) reduces to the system

∂u(t)

∂t
+Au(t) = f(t, u(t), u(h(u(t), t))), t ∈ (0, T ],

u(0) = u0, u0 ∈ X,



EJDE-2020/120 ROTHE’S METHOD WITH DEVIATING ARGUMENTS 9

which is the same as in equation (1.1) and satisfies all the assumptions. By applying
Theorem 2.7 we obtain a unique strong solution of (4.1).

Acknowledgements. D. Chutia wants to thank the DST INSPIRE for their grant
DST/INSPIRE Fellowship/2017/IF170509. R. Haloi wants to acknowledge funding
from NBHM (Grant No. 02011/9/2019 NBHM(R.P.)/R, D II/1324), and the DST
MATRICS (Grant No. SERB/F/12082/2018-2019).

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Darshana Devi

Department of Mathematical Sciences, Tezpur University, Sonitpur, Assam, Pin 784028,

India
Email address: darsana.mou@gmail.com

Duranta Chutia

Department of Mathematical Sciences, Tezpur University, Sonitpur, Assam, Pin 784028,
India

Email address: durantachutia123@gmail.com

Rajib Haloi

Department of Mathematical Sciences, Tezpur University, Sonitpur, Assam, Pin 784028,

India
Email address: rajib.haloi@gmail.com, Phone +913712-275511, Fax +913712-267006


	1. Introduction
	2. Preliminaries and main result
	3. Approximation
	4. Application
	Acknowledgements

	References