# Weak solutions of functional differential inequalities with first-order partial derivatives

- Zdzisław Kamont
^{1}Email author

**2011**:15

https://doi.org/10.1186/1029-242X-2011-15

© Kamont; licensee Springer. 2011

**Received: **7 December 2010

**Accepted: **22 June 2011

**Published: **22 June 2011

## Abstract

The article deals with functional differential inequalities generated by the Cauchy problem for nonlinear first-order partial functional differential equations. The unknown function is the functional variable in equation and inequalities, and the partial derivatives appear in a classical sense. Theorems on weak solutions to functional differential inequalities are presented. Moreover, a comparison theorem gives an estimate for functions of several variables by means of functions of one variable which are solutions of ordinary differential equations or inequalities. It is shown that there are solutions of initial problems defined on the Haar pyramid.

**Mathematics Subject Classification: 35R10, 35R45**.

## Keywords

## 1 Introduction

Two types of results on first-order partial differential or functional differential equations are taken into considerations in the literature. Theorems of the first type deal with initial problems which are local or global with respect to spatial variables, while the second one are concerned with initial boundary value problems. We are interested in results of the first type. More precisely, we consider initial problems which are local with respect to spatial variables. Then, the Haar pyramid is a natural domain on which solutions of differential or functional differential equations or inequalities are considered.

Hyperbolic differential inequalities corresponding to initial problems were first treated in the monographs [1]. (Chapter IX) and [2] (Chapters VII, IX). As is well known, they found applications in the theory of first-order partial differential equations, including questions such as estimates of solutions of initial problems, estimates of domains of solutions, estimates of the difference between solutions of two problems, criteria of uniqueness and continuous dependence of solution on given functions. The theory of monotone iterative methods developed in the monographs [3, 4] is based on differential inequalities.

Two different types of results on differential inequalities are taken into consideration in [1, 2]. The first type allows one to estimate a function of several variables by means of an another function of several variables, while the second type, the so-called comparison theorems give estimates for functions of several variables by means of functions of one variable, which are solutions of ordinary differential equations or inequalities.

There exist many generalizations of the above classical results. We list some of them below. Differential inequalities and the uniqueness of semi-classical solutions to the Cauchy problem for the weakly coupled systems were developed in [5] (Chapter VIII). Hyperbolic functional differential inequalities and suitable comparison results for initial problems are given in [6, 7] (Chapter I). Infinite systems of functional differential equations and comparison results are discussed in [8, 9]. Impulsive partial differential inequalities were investigated in [10]. A result on implicit functional differential inequalities can be found in [11]. Differential inequalities with unbounded delay are investigated in [12]. Functional differential inequalities with Kamke-type comparison problems can be found in [13]. Viscosity solutions of functional differential inequalities were studied in [14, 15].

The aim of this article is to add a new element to the above sequence of generalizations of classical theorems on differential inequalities.

*U*and

*V*, we denote by

*C*(

*U*,

*V*) the class of all continuous functions from

*U*into

*V*. We use vectorial inequalities with the understanding that the same inequalities hold between their corresponding components. Suppose that ,

*a*> 0, ℝ

_{+}= [0, + ∞), is nondecreasing and

*M*(0) = 0

_{[n]}where 0

_{[n]}= (0, ..., 0) ∈ ℝ

^{ n }. Let

*E*be the Haar pyramid:

*b*∈ ℝ

^{ n }and

*b*>

*M*(

*a*). Write

*E*

_{0}= [-

*b*

_{0}, 0] × [-

*b*,

*b*] where

*b*

_{0}∈ ℝ

_{+}. For (

*t*,

*x*) ∈

*E*define

Write *r*_{0} = -*b*_{0} - *a*, *r* = 2*b* and *B* = [-*r*_{0}, 0] × [-*r*, *r*]. Then, *D*[*t*, *x*] ⊂ *B* for (*t*, *x*) ∈ *E*. Given *z*: *E*_{0} ∪ *E* → ℝ and (*t*, *x*) ∈ *E*, define *z*_{(t, x)}: *D*[*t*, *x*] → ℝ by *z*_{(t, x)}(*τ*, *s*) = *z*(*t* + *τ*, *x* + *s*), (*τ*, s) ∈ *D*[*t*, *x*]. Then *z*_{(t, x)}is the restriction of *z* to the set (*E*_{0} ∪ *E*) ∩ ([-*b*_{0}, *t*] × ℝ^{
n
}) and this restriction is shifted to *D*[*t*, *x*].

*E*× ℝ ×

*C*(

*B*, ℝ) × ℝ

^{ n }and suppose that

*f*: Ω → ℝ is a given function of the variables (

*t*,

*x*,

*p*,

*w*,

*q*),

*x*= (

*x*

_{1}, ...,

*x*

_{ n }),

*q*= (

*q*

_{1}, ...,

*q*

_{ n }). Let us denote by

*z*an unknown function of the variables (

*t*,

*x*). Given

*ψ*:

*E*

_{0}→ ℝ, we consider the functional differential equation:

where
. We will say that *f* satisfies condition (*V* ), if for each (*t*, *x*, *p*, *q*) ∈ *E* × ℝ × ℝ^{
n
} and for *w*,
such that
for (*τ*, *s*) ∈ *D*[*t*, *x*] then we have
. It is clear that condition (*V*) means that the value of *f* at the point (*t*, *x*, *p*,*w*, *q*) ∈ Ω depends on (*t*, *x*, *p*, *q*) and on the restriction of *w* to the set *D*[*t*, *x*] only.

*c*≤

*a*, is a weak solution of (1), (2) provided

- (i)
- (ii)
- (iii)
for each

*x*∈ [-*b*,*b*], the function satisfies equation 1 for almost all*t*∈*I*[*x*] ∩ [0,*c*] and condition (2) holds.

This class of solutions for nonlinear equations was introduced and widely studied in nonfunctional setting by Cinquini and Cinquini Cibrario [16, 17].

The paper is organized as follows. In Sections 2 and 3 we present theorems on functional differential inequalities corresponding to (1), (2). They can be used for investigations of solutions to (1), (2). We show that the set of solutions is not empty. In Section 4 we prove that there is a weak solution to (1), (2) defined on *E*_{
c
} where *c* ∈ (0, *a*] is a sufficiently small constant.

## 2 Functional differential inequalities

Let
, [*τ*, *t*] ⊂ ℝ, be the class of all integrable functions Ψ: [*τ*, *t*] → ℝ^{
n
}. The maximum norm in the space *C*(*B*, ℝ) will be denoted by ||·||_{
B
}. We will need the following assumptions on given functions.

**Assumption**

*H*

_{0}. The function

*f*: Ω → ℝ satisfies the condition (

*V*) and

**Theorem 2.1**. *Suppose that Assumption H*_{0} *is satisfied and*

*is satisfied for almost all t* ∈ *I*[*x*],

*t*,

*x*) ∈

*E*and . Suppose that [

*t*

_{0},

*t*] is the interval on which the solution

*g*(·,

*t*,

*x*) is defined. Then,

*τ*,

*g*(

*τ*,

*t*,

*x*)) ∈

*E*for

*τ*∈ [

*t*

_{0},

*t*] and, consequently, the function

*g*(·,

*t*,

*x*) is defined on [0,

*t*]. It follows from (10) that

and consequently
which contradicts (9). Hence, A_{+} is empty and the statement (7) follows.

Now we prove that a weak initial inequality for
and
on *E*_{0} and weak functional differential inequalities on *E* imply weak inequality for
and
on *E*.

**Assumption**

*H*[

*σ*]. The function

*σ*: [0,

*a*] × ℝ

_{+}→ ℝ

_{+}satisfies the conditions:

- (1)
*σ*(*t*, ·): ℝ_{+}→ ℝ_{+}is continuous for almost all*t*∈ [0,*a*], - (2)

*σ*(

*t*,

*p*) ≤

*m*

_{ σ }(

*t*) for

*p*∈ ℝ

_{+}and for almost all

*t*∈ [0,

*a*],

- (3)

**Theorem 2.2**. *Suppose that Assumptions H*_{0} *and H*[*σ*] *are satisfied and*

*is satisfied for almost all t* ∈ *I*[*x*].

*ε*

_{0}> 0 such that, for every 0 <

*ε*<

*ε*

_{0}, the solution

*ω*(·,

*ε*) is defined on [0,

*a*] and

which completes the proof of (17). It follows from Theorem 2.1 that
on *E*. From this inequality, we obtain in the limit, letting *ε* tend to zero, inequality (16). This completes the proof.

The results presented in Theorems 2.1 and 2.2 have the following properties. In both the theorems, we have assumed that
on *E*_{0}. It follows from Theorem 2.1 that the strong inequality (6) and the strong functional differential inequality (5) for almost all *t* ∈ *I*[*x*] imply the strong inequality (7). Theorem 2.2 shows that the weak initial inequality
on *E* and the weak functional differential inequality (15) for almost all *t* ∈ *I*[*x*] imply the weak inequality (16).

In the next two lemmas, we assume that
on *E*_{0} and we prove that the strong initial inequality (6) and the weak functional inequality (15) imply the strong inequality (7).

We prove also that the weak initial inequality
on E_{0} and the strong functional differential inequality (5) imply the inequality
for (*t*, *x*) ∈ *E*, 0 < *t* ≤ *a*.

**Lemma 2.3**. *Suppose that Assumptions H*_{0} *and H*[*σ*] *are satisfied and*

*(1) the estimate (14) holds on* Ω *for*
,

*(2)*
*for* (*t*, *x*) ∈ *E*_{0} *and for each x* ∈ [-*b*, *b*] *the functional differential inequality (5) is satisfied for almost all t* ∈ *I*[*x*].

*Under these assumption, we have*
for (*t*, *x*) ∈ *E*, 0 < *t* ≤ *a*.

*Proof*It follows from Theorem 2.2 that for (

*t*,

*x*) ∈

*E*. Suppose that there is , such that . By repeating the argument used in the proof of Theorem 2.1, we obtain

where *g*(·, *t*, *x*) is the solution to (12). Then,
, which completes the proof of the lemma.

**Lemma 2.4**. *Suppose that Assumption H*_{0} *and H*[*σ*] *are satisfied and*

*(1) the estimate (14) holds on* Ω *for*
,

*(2)*
*for* (*t*, *x*) ∈ *E*_{0} *and*
*for x*∈ [-*b*, *b*],

*(3) for each x* ∈ [-*b*, *b*] *the functional differential inequality (15) is satisfied for almost all t* ∈ *I*[*x*].

*E*

_{0}and for

*x*∈ [-

*b*,

*b*]. Suppose that

*z*

^{⋆}:

*E*

_{0}∪

*E*→ ℝ is defined by

*E*

_{0}and for

*x*∈ [-

*b*,

*b*]. We prove that for each

*x*∈ [-

*b*,

*b*], the functional differential inequality

which completes the proof of (22). We get from Theorem 2.1 that (21 holds. Inequalities (20), (21), imply (18), which completes the proof of the lemma.

**Remark 2.5**.

*The results presented in Section 2 can be extended on functional differential inequalities corresponding to the system:*

*where z* = (*z*_{1}, ..., *z*_{
k
}) *and f* = (*f*_{1}, ..., *f*_{
k
}): *E* × ℝ^{
k
} × *C*(*B*, ℝ^{
k
}) × ℝ^{
n
} → ℝ^{
n
} *is a given function of the variables* (*t*, *x*, *p*,*w*, *q*), *p* = (*p*_{1}, ..., *p*_{
k
}), *w* = (*w*_{1}, ..., *w*_{
k
}), *Some quasi-monotone assumptions on the function f with respect to p are needed in this case*.

## 3 Comparison theorem

**Assumption**

*H*

_{⋆}. The functions Δ:

*E*×

*C*(

*B*, ℝ) → ℝ

^{ n }, Δ = (Δ

_{1}, ..., Δ

_{ n }), and

*ϱ*: [0,

*a*] × ℝ

_{+}→ ℝ

_{+}satisfy the conditions:

- (3)
ϱ(·,

*p*): [0,*a*] → ℝ_{+}is measurable for*p*∈ ℝ_{+}and ϱ(*t*, ·): ℝ_{+}→ ℝ_{+}is continuous and nondecreasing for almost all*t*∈ [0,*a*], and there is such that ϱ(*t*,*p*) ≤*m*ϱ(*t*) for*p*∈ ℝ_{+}and for almost all*t*∈ [0,*a*], - (4)
*z*^{⋆}:*E*_{0}∪*E*→ ℝ is continuous and - (i)
- (ii)
for each

*x*∈ [-*b*,*b*] the function*z*^{⋆}(·,*x*):*I*[*x*] → ℝ is absolutely continuous.

**Theorem 3.1**. *Suppose that Assumption H*_{
⋆
} *is satisfied and*

*is satisfied for almost all t* ∈ *I*[*x*],

*(2) the number η* ∈ ℝ_{+} *is defined by the relation:* |*z*^{⋆}(*t*, *x*)| *≤ η for (t*, *x)* ∈ *E*_{0}.

*t*,

*x*) ∈

*E*. It follows from condition 1) of Assumption

*H*

_{⋆}that

*g*[

*z*

^{⋆}](·,

*t*,

*x*) is defined on [0,

*t*]. We conclude from (23) that for each

*x*∈ [-

*b*,

*b*], the differential inequality

The function *ω*(·, *η*) satisfies the integral equation corresponding to the above inequality. From condition 3) of Assumption *H*_{⋆} we obtain (24), which completes the proof.

We give an estimate of the difference between two solutions of equation 1.

**Theorem 3.2**. *Suppose that the function f* : Ω → ℝ *satisfies condition (V) and*

*(1) conditions (1)*-*(3) of Assumption H*_{0} *hold*,

*(2) there is ϱ*: [0,

*a*] × ℝ

_{+}→ ℝ

_{+}

*such that condition (3) of Assumption H*

_{ ⋆ }

*is satisfied and*

*(3) the functions*
,
*are weak solutions to (1) and η* ∈ ℝ_{+} *is defined by the relation:*
*for* (*t*, *x*) ∈ *E*_{0}.

*where ω*(·, *η*) *is the maximal solution to (25)*.

is satisfied for almost all ∈ *I*[*x*]. From Theorem 3.1 we obtain (27), which completes the proof.

The next lemma on the uniqueness of weak solutions is a consequence of Theorem 3.2.

**Lemma 3.3**. *Suppose that the function f* : Ω → ℝ *satisfies condition* (*V* ) *and*

*(1) assumptions (1), (2) of Theorem 3.2 hold*,

*(2) the function*
*for t* ∈ [0, *a*] *is the maximal solution to (25) with η* = 0.

*Then, problem (1), (2) admits one weak solution at the most*.

*Proof* From (27) we deduce that for *η* = 0 we have
on *E* and the lemma follows.

## 4 Existence of solutions of initial problems

*E*×

*C*(

*B*, ℝ) × ℝ

^{ n }and suppose that

*F*: Ξ → ℝ is a given function of the variables (

*t, x, w, q*). Given

*ψ*:

*E*

_{0}→ ℝ, we consider the functional differential equation:

We assume that *F* satisfies condition (*V*) and we consider weak solutions to (28), (29).

*M*

_{n × n}the class of all

*n*×

*n*matrices with real elements. For

*x*∈ ℝ

^{ n },

*W*∈

*M*

_{n × n}, where

*x*= (

*x*

_{1}, ...,

*x*

_{ n }),

*W*= [

*w*

_{ ij }]

_{i,j = 1,...,n}, we put

*W*∈

*M*

_{n × n}, then

*W*

^{ T }denotes the transpose matrix. Suppose that

*v ∈ C*(

*E*

_{0}∪

*R*, ℝ

^{ n }),

*U*∈

*C*(

*E*

_{0}∪

*R, M*

_{n × n}). The following seminorms will be needed in our considerations:

where *t ∈* [0, *a*]. The scalar product in ℝ^{
n
} will be denoted by "∘". We will use the symbol *CL*(*B*, ℝ) to denote the class of all linear and continuous operators defined on *C*(*B*, ℝ) and taking values in ℝ. The norm in the space *CL*(*B*, ℝ) generated by the maximum norm in *C*(*B*, ℝ) will be denoted by *||*·*||*_{⋆}. The maximum norms in *C*(*E*_{0}, ℝ) and *C*(*E*_{0}, ℝ^{
n
}) will be denoted by
and
, respectively.

**Assumption**

*H*

_{0}[

*F*]. The function

*F*: Ξ → ℝ satisfies the condition (

*V*) and

- (1)
- (2)

*θ*∈

*C*(

*B*, ℝ) is given by

*θ*(

*τ, s*) = 0 on

*B*,

- (3)

where (*t, x, w, q*) ∈ Ξ, and
is given by (3).

*C*(

*E*

_{0}, ℝ) such that

- (i)
- (ii)

are satisfied on *E*_{0}.

Let
be given and 0 < *c* ≤ *a*. We denote by *C*_{ψ.c}the class of all *z* ∈ *C*(*E*_{
c
}, R) such that *z*(*t*, *x*) = *ψ* (*t*, *x*) on *E*_{0}. For the above *ψ* and *c* we denote by C_{∂ψ.c}the class of all *v* ∈ *C*(*Ec*, ℝ^{
n
}) such that *v*(*t*, *x*) = ∂_{
x
} *ψ*(*t*, *x*) on *E*_{0}.

*H*

_{0}[

*F*] is satisfied and ,

*z*∈

*C*

_{ψ.c},

*u*∈ C

_{∂ψ.c}where 0 <

*c*≤

*a*. We consider the Cauchy problem

where (*t*, *x*) ∈ *E* and 0 ≤ *t* ≤ *c*. Let us denote by *g*[*z*, *u*](·, *t*, *x*) the solution of (30). The function *g*[*z*, *u*](·, *t*, *x*) is the bicharacteristic of (28) corresponding to (*z*, *u*).

*u*= ∂

_{ x }

*z*in (28). Then, we consider the linearization of (28) with respect to the last variable, and we obtain the equation

*u*= ∂

_{ x }

*z*in (37). If we consider (36) and (37) along the bicharacteristic

*g*[

*z*,

*u*](·,

*t*,

*x*), then we obtain

By integrating of (38) and (39) on [0, *t*] with respect to *τ*, we get (35).

We prove that there is a solution
to (35) defined on *E*_{
c
} where *c* ∈ (0, *a*] is sufficiently a small constant, and
and
are weak solutions to (28), (29). We first give estimates of solutions to (35).

**Lemma 4.1**. *Suppose that Assumption H*_{0}[*F*] *is satisfied and*

*(2) the functions*
,
*are continuous and they satisfy (35)*.

where *t* ∈ [0, *c*]. The functions
satisfy integral equations corresponding to the above inequalities. This proves the lemma.

where *t* ∈ [0, *c*]. It is clear that
satisfy the above conditions.

*d*,

*h*∈ ℝ

_{+},

*d*≥

*c*

_{1},

*h*≥

*c*

_{2}and 0 <

*c*≤

*a*. Suppose that . We denote by

*C*

_{ψ.c}[ζ,

*d*] the class of all

*z*∈

*C*

_{ψ.c}such that

**Assumption**

*H*[

*F*]. The function

*F*: Ξ → ℝ satisfies Assumption

*H*

_{0}[

*F*], and there is such that the terms

are bounded from above on Ξ[*A*,*C*] by

**Remark 4.2**. *It is important that we have assumed the Lipschitz condition for* ∂_{
x
}*F*, ∂_{
w
}*F*, ∂_{
q
}*F for*
*satisfying the condition:* ||*w*||_{
B
},
.

*There are differential integral equations and differential equations with deviated variables such that Assumption H*[*F*] *is satisfied and the functions* ∂_{
x
}*F*, ∂_{
w
}*F*, *and* ∂_{
q
}*F do not satisfy the Lipschitz condition with respect to the functional variable on* Ξ.

*It is clear that there are functional differential equations which satisfy Assumptions H*[*F*] *and they do not satisfy the assumptions of the existence theorem presented in* [18].

*where* 0 < *c* ≤ *a*.

*Then the bicharacteristics g*[*z*, *u*](·, *t*, *x*) and
*exist on intervals* [0, *δ*[*z*, *u*](*t*, *x*)] *and*
*such that for τ* = *δ*[*z*, *u*](*t*, *x*),
, *we have* (*τ*, *g*[*z*, *u*](*τ*, *t*, *x*)) ∈ ∂*E*_{
c
},
, *where* ∂*E*_{
c
} *is the boundary* of *E*_{
c
}.

*Proof*The existence and uniqueness of the solution to (30) follows from classical theorems on Carathéodory solutions of ordinary differential equations. We conclude from Assumption

*H*[

*F*] that the integral inequalities

are satisfied. Then, we obtain (40) and (41) from the Gronwall inequality.

**Assumption** *H*[*c*]. The constants *c* ∈ (0, *a*], *d*, *h* > 0 satisfy the relations: Λ(*c*) ≤ *d*, Γ(*c*) ≤ *h*.

*then there is c* ∈ (0, *a*] *such that* Λ(*c*) ≤ *d and* Γ(*c*) ≤ *h*.

**Theorem 4.5**. *Suppose that Assumptions H*[*F*], *H*[*c*] *are satisfied and*
. *Then there is a solution*
*of (28), (29)*.

*Proof* The proof will be divided into four steps

*z*

^{(m)}:

*E*

_{ c }→ ℝ,

*u*

^{(m)}:

*E*

_{ c }→ ℝ

^{ n }are already defined then

*u*

^{(m+1)}is a solution of the equation

We prove that

*I*

_{ m }), (

*II*

_{ m }) by induction. It is easily seen that conditions (

*I*

_{0}), (

*II*

_{0}) are satisfied. Suppose that (

*I*

_{ m }) and (

*II*

_{ m }) hold for a given

*m*≥ 0. We first prove that there is

*u*

^{(m+1)}:

*E*

_{ c }→ ℝ

^{ n }, and

*u*

^{(m+1)}∈

*C*

_{∂ψ.c}[χ,

*h*]. We claim that

From the above estimates and from (44), we deduce (47).

If follows from the Banach fixed point theorem that there is *u*^{(m+1)}∈ *C*_{∂ψ.c}[χ, *h*] and it is unique.

We conclude from the above estimates that *z*^{(m+1)}∈ *C*_{∂ψ.c}[ζ, *d*] which completes the proof of (*II*_{m+1}).

This proves (*II*_{m+1}).

**II**. We prove that the sequences {*z*^{(m)}} and {*u*^{(m)}} are uniformly convergent on *E*_{
c
}.

**III**. We prove that is a solution to (28), (29). We conclude from (46) that the functions , satisfy the relations

The relations
and
are equivalent. By differentiating (54) with respect to *τ* and by putting again
, we find that
is a weak solution to (28). Since
, it follows that initial condition (29) is satisfied.

**IV**. Now we prove (42). It follows from (31) - (35) and from Assumption

*H*[

*F*] that there are such that

This completes the proof of the theorem.

**Remark 4.6**. *It is easy to see that differential integral equations and equations with deviated variables are particular cases of (28)*.

*Then, equation*
*1*
*is equivalent to (28). It follows that existence results for (1), (2) can be obtained from Theorem 4.5*.

## Declarations

## Authors’ Affiliations

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