On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions

Esta obra está bajo una Licencia Creative Commons Atribución 4.0 Internacional.

In this paper, we investigate some delay integral or integro-differential problems for both integer or fractional orders. For most of papers devoted to study such problems either on a finite interval or on a half-line, the expected solutions are continuous (on C([0,∞ )), cf. [29]) or continuous and bounded (on BC([0,∞ )), cf. [4, 9, 27, 28]). However, for integral problems, it is much more natural if solutions are not so regular and they are only integrable. This approach is not sufficiently investigated, and it requires to investigate operators acting on different spaces together with different qualitative indices in such spaces. We concentrate on delay integral equations.

Some particular problems are widely studied, but they are neither unified nor obtained for common and general assumptions. Delay integral or differential equations are quite frequent in mathematical biology, medicine, and physics. Let us recall some of our motivations. Many of the models of machining operations fall into the class of autonomous delay differential equations of the form

where the problem is studied in finite-dimensional spaces and h > 0 (cf. [24]).

In [15] the authors formulated a model to explain the observed periodic outbreaks of certain infectious diseases with periodic contact rate that varies seasonally, and it was also studied in [16, 32,33,34]. This model can also be interpreted as an equation describing the growth of a population when the birth rate varies seasonally. Then this model takes the form

and is usually studied in its integral form.

However, many physical and biological models have been successfully described by delayed differential or integral problems with discontinuous functions, such as electric, pneumatic, and hydraulic networks (see [2, 11, 20]). For instance, in [10] the authors focused on the discontinuity solutions of problem with delays and proved some discontinuity properties for delay differential equations

Note that we propose to study such problems with possibly general initial-value functions ϕ, not necessarily continuous. In particular, we will study the delay fractional integral problem

(1)

All the problems mentioned above are extensions of earlier results, so we will try to extend and unify many of them. To achieve our goal, let us consider first the delay integral problem of Volterra–Hammerstein type

(2)

still being a general form of many previous problems considered, for instance, in [5, 7, 16, 17, 25]. It should be stressed that we will investigate these problems on unbounded interval with integrable solutions as well as locally integrable ones. The above mentioned results are investigated either in the one of cases discussed here or even only on compact intervals. We will consider here the case of finite delay, and the the initial function describing the past will be defined on an interval [τ, 0].

More precisely, in this paper, we study the existence and the uniqueness of a.e. monotonic solutions of problem (2). We will also unify some known results being particular cases of (2), and we will extend some of them from bounded interval to unbounded one in the space L1(R+) or L1loc(R+). As applications, we discuss the solvability of the integro-differential problems of fractional order (1). Proofs are based on operators form for considered problems with operators acting on appropriate function spaces and with the use of some special measures of noncompactness on these spaces together with the Darbo fixed point theorem. In particular, we need to investigate different properties of operators and different properties of considered subsets of functions paces. Thus proofs will be different than that for continuous solutions and for results about integrable solutions based on the weak topology argument (cf. [5, 8]).

Let R be the field of real numbers, and let R+ be the interval [0,∞ ). Denote by L1 = L1(R+) the Banach space of all real functions defined and Lebesgue integrable on R+ endowed with the norm

and by L1loc(R+) the Fréchet space of all locally integrable spaces on R+ endowed with a family of seminorms ǁxǁ = ∫0 T |x(t)| dt (T > 0). A nonempty set A in L1loc(R+) is bounded if it is bounded in every seminorm, i.e., supT>0{ǁχ[0,T ]xǁT : x ∈ A} < ∞.

Definition 1. (See [1].) Assume that a function f (t, x) = f : R+ × R → R satisfies the Carathéodory conditions, i.e., it is measurable in t for any x ∈ R and continuous in x for almost all t ∈ R+. Then for every function x being measurable on R+, we may assign

Ff (x)(t) = f t, x(t) , t ∈ R+.

The operator Ff defined in such a way is called the superposition (Nemytskii) operator generated by the function f .

Acting and continuity conditions in the case of σ-finite measure spaces were proved by Appell and Zabreiko:

Theorem 1. (See [1, Thm. 3.1].) Suppose that f satisfies the Carathéodory conditions.

The superposition operator Ff maps continuously the space L1 into L1 if and only if

f (t, x) ≤ a(t) + b|x|

for all t ∈ R+ and x ∈ R, where a ∈ L1 and b ≥ 0. Moreover, this operator Ff : L1 → L1 is continuous.

Let S = S(I) denotes the set of measurable (in Lebesgue sense) functions on an interval I. Identifying the functions equal almost everywhere the set S furnished with the metric

we obtain a complete metric space. Moreover, the convergence in measure on I is equivalent to the convergence with respect to the metric d in [35, Prop. 2.14].

For σ-finite subsets of R, we say that the sequence xn is convergent in finite measure to x if it is convergent in measure on each set T of finite measure. The compactness in such a space is called “compactness in measure”. The following property will be useful in our investigation.

Theorem 2. (See [12, Thm. 2.3].) Let X be a bounded subset of L1 consisting of functions, which are a.e. nonincreasing (or a.e. nondecreasing) on the half-line R+. Then X is compact in measure in L1.

Now, let us recall the concept of measure of noncompactness. Assume that (E, . ) is an arbitrary Banach space with zero element θ, and the symbol Br stands for the closed ball with radius r and centered at θ. Denote by ME the family of all nonempty and bounded subsets of E and by NE its subfamily consisting of all relatively compact sets.

The symbols X, XW stand for the closure, and the weak closure of a set X, respectively, and the symbol conv X will denote the convex closed hull of a set X.

Definition 2. (See [6].) A mapping µ : ME [0, ) is called a regular measure of noncompactness in E if it satisfies the following conditions:

1. (i) µ(X) = 0 ⇔ X ∈ NE.

2. (ii) X ⊂ Y ⇒ µ(X) ≤ µ(Y ).

3. (iii) µ(X) = µ(conv X) ≤ µ(X).

4. (iv) µ(λX) = |λ|µ(X) for λ ∈ R.

5. (v) µ(X + Y ) ≤ µ(X) + µ(Y ).

6. (vi) µ(X S Y ) = max{µ(X), µ(Y )}

7. (vii) If Xn is a sequence of nonempty, bounded, closed subsets of E, Xn = X W

   n such that Xn+1 ⊂ Xn for n = 1, 2, . . . , and if limn→∞ µ(Xn) = 0, then the set X ∞ = ∩n=1∞ Xn is nonempty.

An important example of such mappings is the following.

Definition 3. (See [3].) Let X be a nonempty and bounded subset of E. The Hausdroff measure of noncompactness χ(X) is defined as χ(X) = inf{r > 0: there exists a finite subset Y of E such that x ⊂ Y + Br}.

We need to construct a measure of noncompactness in L1. For nonempty and bounded subset X of the space L1, let

(3)

(cf. [8]), where

(4)

is a cocalled measure of uniform integrability (cf. [5]), and

(5)

Immediately, we get the following.

Lemma 1. The measure µ is a measure of noncompactness if restricted to the family of subsets being compact in measure in L1.

Proof. In [8, Lemma 2], it is proved that µ is a measure of weak noncompactness in L1 (see also [8, Thm. 4]). But [18, Thm. 1] implies that for any bounded subset X L1 being additionally compact in measure, this quantity is equal to the Hausdorff measure of (strong) noncompactness (see [6]).

Moreover, measures χ(x) and µ(x) are equivalent:

Theorem 3. (See [5, Thm. 5].) Let X be a nonempty, bounded, and compact in measure subset of L1. Then

χ(x) ≤ µ(x) ≤ 2χ(x).

Let us recall some fixed point theorems. In the proof of the existence of solutions of considered problems in L1, the following Darbo fixed point theorem will be useful:

Theorem 4. (See [6, Thm. 3.1].) Let Q be a nonempty, bounded, closed, and convex subset of E, and let H : Q → Q be a continuous transformation, which is a contraction with respect to the measure of noncompactness µ, i.e., there exists k ∈ [0, 1) such that

µ H(X) ≤ kµ(X)

for any nonempty subset X of E. Then H has at least one fixed point in the set Q.

Let E1 be a Fréchet space, and let its topology is defined by a family of seminorms ( n). By µn (n N) denote the family of measures of noncompactness related to this family of seminorms. For instance, it could be done like in the Hausdorff measure of noncompactness, i.e.,

µn(X) = {inf r > 0: there exists a finite subset Y of E such that x ⊂ Y + Bn} ,

where Bn = {x ∈ E1: ǁxǁn < r}. A detailed study of the case Lloc(R+) can be found, for instance, in [30].

Because the space 1LlocR+ is Fréchet, but not a Banach space, we need the following version of the Darbo fixed point theorem, which is sufficient for our investigation.

Theorem 5. (See [21].) Let Q be a nonempty, bounded, closed, and convex subset of a Fréchet space E1, and let H : Q → Q be a continuous transformation such that for the family of measures of noncompactness, (µn)n∈N, i.e., there exist constants kn ∈ [0, 1) (n ∈ N) such that

for any nonempty bounded subset X of E1 and n ϵ N. Then H has at least one fixed point in the set Q.

Next, we give short notes about fractional operators. We will restrict our attention to the case of L1 because for Lloc, they will result directly.

Definition 4. (See [26].) Let f ϵ L1 and α ϵ R+. The Riemann–Liouville (RL) fractional integral of the function f of order α is defined as

where Γ(α) is the Euler gamma function.

Lemma 2. (See [25, 31].) If f ∈ L1 and α ∈ (0, 1), then

(a) The operator Iaα maps L1 into itself continuously.

(b) The operator Iaα maps the monotonic nondecreasing function into functions of the same type.

First, let us consider problem (2). We will rewrite it in an operator form. The key difference between the case of integrable solutions and continuous ones is the action of operators on appropriate function spaces. It will not be surprising to assume that the expected solution observed in the past should have the same properties as in the future, so we will assume that ϕ ∈ L1([−τ, 0]) is a.e. nonincreasing and positive.

3.1 Existence of integrable solution

Rewrite problem (2) in the operator form

(6)

where xτ = x(t − τ ), τ < t, Kx(t) = ∫0t k(t, s)x(s) ds, and Ff , Fg be the superposition operators generated by f and g, respectively. Note that if ϕ is integrable, then for any integrable function x, a function xτ is integrable too.

First, we will prove the existence of solutions in L1. Consider the following assumptions.

1. (i) Let m, h : R+ → R+ be a.e. nonincreasing functions, where m is a bounded function with supt R+ m(t) ≤ M and h ϵ L1. Moreover, let ϕ L1([ - τ, 0]) be a.e. nonincreasing and positive.

2. (ii) Assume that the functions f, g : R+ R R satisfy Carathéodory conditions. Moreover, f (t, x) ≥ 0, g(t, x) ≥ 0 for x ≥ 0, and f , g are a.e. nonincreasing with respect to t and nondecreasing with respect to x.

3. (iii) There are positive integrable functions ai ∈ L1 and constants bi ≥ 0 (i = 1, 2) such that |f (t, x)| ≤ a1(t) + b1|x|, |g(t, x)| ≤ a2(t) + b2|x| for all t ∈ R+ and x ∈ R.

4. (iv) k : R+ → R+ → R satisfies Carathéodory conditions such that the linear operator K : L1 L1 is continuous and maps the set of a.e. nonincreasing and positive functions into functions of the same type.

5. (v) b2M + b1ǁKǁL1 < 1/2, where ǁKǁL1 is an operator norm ǁKǁL1 →L1 .

Let us recall that some sufficient conditions for the acting and continuity conditions in (iv) can be found in [22] (a full description is unknown). Some conditions guaranteeing preservation of monotonicity of functions by K exactly on R+ can be found in [23, Sect. 4]. As this paper is not easily accessible, let us recall that criterion:

Proposition 1. (See [23].) The operator K with the kernel k(t, s) being locally integrable with respect to s on R+ for each fixed t preserves the monotonicity of functions from Lloc(R+) if and only if

(7)

for t1 < t2, t1, t2 ∈ [0, T ] and for any b ≥ 0.

Our main result is the following.

Theorem 6. Let assumptions (i)–(v) be satisfied. Then problem (2) has at least one solution x ∈ L1, which is additionally a.e. nonincreasing function on R+.

Proof. We need to investigate acting, continuity, and contraction conditions for all operators describing equation (6).

By assumptions (ii), (iii) and due to Theorem 1, we can conclude that Ff , Fg map L1 into itself continuously. By assumption (iv) the operator K : L1 → L1 is continuous, and then KFf : L1 → L1 and is continuous too. For a given x ϵ L1, by assumption (i) we can deduce that H(x) belongs to L1 and H is continuous. Then

Put t − τ = u, so du = dt and then

From the above estimate we deduce that the function H(x) is bounded on R+, thus H :

L1 → L1. Moreover, we get

Where

The inequality obtained above infer that the operator H maps the ball Br into itself, i.e., H : Br → Br and is continuous.

Further, let Qr denote the subset of Br consisting of all function being a.e. nonincreasing and positive on R+. The set Qr is nonempty, bounded, closed, convex, and compact in measure in view of Theorem 2 (cf. [13], for instance).

Now, we will show that H preserves the monotonicity and positivity of functions from Qr. Take an arbitrary x Qr, then x(t) is a.e. nonincreasing and positive on R+, and consequently, f , g are also of the same type virtue of assumption (ii). By assumption (i) m, h are a.e. nonincreasing and positive functions on R+, and from assumption (iv) the operator K maps a.e. nonincreasing and positive functions into functions of the same type.

Thus, we can deduce that (H(x)) is also a.e. nonincreasing and positive on R+. This fact gives that H : Qr → Qr and is continuous.

Next, to prove that H is a contraction, we will assume that x ϵ = X C Qr, and let a constant ε > 0 be arbitrary, but fixed.

Then for an arbitrary x ϵ X C Qr and for any measurable set D C R+, meas D ≤ ε, we obtain

where the symbol K L1 (D) denotes the norm of the operator K acting from the space L1(D) into itself. Recall that D is arbitrary, so the set D ∩ [0, τ ] need not be empty. Then the above estimation require the use of assumption (i) as for such points we get t - τ ϵ [ - τ, 0], on this interval, we have x(s) = ϕ(s). Hence we keep the expected properties of a.e. monotonicity and positivity of functions.

It suffice to prove that H is a contraction with respect to some regular measure of noncompactness. Since h, ai ∈ L1, i = 1, 2, then we have the equality

As ǁKǁL1 (D) ≤ ǁKǁL1 , from definition 4 it follows that

(8)

For any T > 0, we have the following estimate:

where the symbol ǁ·ǁL1 (T ) denotes the operator norm acting from the space L1[T, ∞) into itself. Because T → ∞, by the definition 5 we get

(9)

By combining (8) and (9) and by applying definition 3 we have

Since X ⊂ Qr and we know that Qr is compact in measure, by Theorem 3 we have

Using all properties of Qr, by assumption (iv) 2(b2M + b1 K L1 ) < 1, then we can apply Theorem 4, which completes the proof.

Corollary 1. This theorem is also true for solutions in Lp(R+) (p > 1) with a suitable set of modified assumptions assuring acting and continuity conditions for considered operators (cf. [14]).

We are interested in studying delay integral problem, but in view of our proof, we can also formally generalize our main theorem by considering functional integral problem:

(10)

Corollary 2. Under the assumption of Theorem 6, there exists solution x L1, which is additionally a.e. nonincreasing function on R+ of the functional integral problem (10), provided the functional delay ψ : R+ → R+ is increasing, absolutely continuous, and there is a constant B > 0 such that ψ′(t) ≥ B for a.e. t R+ and when assumption (v) is modified to the form

3.3 Uniqueness of solutions

We are able to discusses the uniqueness of solution of problem (2) in the case of integrable solutions (the case of locally integrable solutions can be studied in a similar manner).

Theorem 8. Let assumptions of Theorem 6 be satisfied, but instead of assumption (iii), consider the following condition holds:

(vi) There exist constants bi ≥ 0 and positive functions ai ∈ L1, i = 1, 2, such that |f (t, x) − f (t, y)| ≤ b1|x − y|, |g(t, x) − g(t, y)| ≤ b2|x − y|, x, y ∈ Qr, and |f (t, 0)| ≤ a1(t), |g(t, 0)| ≤ a2(t), where Qr is defined in Theorem 6.

Then problem (2) has a unique solution in Qr.

Proof. From assumption (vi) we have

Similarly, g(t, x) ≤ a2(t) + b2 x . Thus, all assumptions of Theorem 6 be satisfied, then problem (2) has at least one solution x L1.

To prove the uniqueness of a solution of problem (2), suppose that x, y be any two different solutions of problem (2), and then we have

As x(t) = y(t) = ϕ(t) on [−τ, 0), letting u = t − τ , we have

Therefore,

which implies that ǁx − yǁL1 = 0 ⇒ x = y a.e., which completes the proof.

As fractional integral equations are special forms of a general results of Hammerstein– Volterra equations with a convolutions kernel K, then the continuity property is dependent on it. It is worthwhile to note that in a particular case of Riemann–Liouville fractional integral operators, i.e., with the kernel

for α = 1, it is not continuous from L1([0, T ]) into L∞([0, T ]) ( [19, Remark 4.1.2]) and discontinuous as an operator from L1([0, T ]) into itself ( [19, Remark 4.1.1]). But in a considered case 0 < α < 1, it is continuous (see Lemma 2).

Despite that the lemma presented below seems to be known, we are unable to find its proof, so let us prove it (it is also claimed in Lemma 2):

Lemma 3. For any 0 < α < 1, the kernel k of the Riemann–Liouville fractional integral operator satisfies condition (7), so our result applies also for the fractional problems of any order α.

Proof. Let t1 < t2, t1, t2 [0, T ]. Recall that in the kernel, we apply the characteristic function, so the limits of integration depend on the choice b. As

then, in view of arbitrariness of b ≥ 0, we need to consider three cases.

1. Let b > t2. Then

2. Let 0 < t1 < b < t2. Then

But t2 − b < t2 − t1 and then ∆ > (tα − tα − (t2 − t1)α)/α.

We need to use the fact that for any 0 < α < 1, a function g(t) = xα is concave.

For arbitrary points x1, x2 ∈ [0, T ], we have g((x1 + x2)/2) ≥ (g(x1) + g(x2))/2. Put x1 = t1, x2 = t2 − t1. Then (x1 + x2)/2 = t2/2 and g((x1 + x2)/2) = 2−αtα.

Consequently, 2 · g((x1 + x2)/2) = 21−α · tα > tα.

Therefore tα > tα + (t2 − t1)α and ∆ > 0.

3. Let 0 < b < t1 < t2. Then

As in this case tα − tα > 0 and (t2 − b)α − (t1 − b)α > 0, we get the thesis.

Consequently, we get an existence result for the fractional problem (1):

Theorem 9. Let assumptions of Theorem 7 be satisfied. Then for any 0 < α < 1, the fractional integral problem (1) has at least one locally integrable solution on R+.

The uniqueness results can be also obtained in the same manner.

We give an example to illustrate the applicability of our assumptions.

Consider the following delay integral problem for t ∈ R+:

(11)

It is clear that problem (11) is a particular case of problem (2), where ϕ(t) = x0 = const, m(t) = e−t, g(t, x) = (1 + x(t − τ ))/(t + 19)2, k(t, s) = 1/(t2 + s2), f (t, x) = 1/(t + 2)3 + (x(t − τ ))/20, and

One can easily check that

Thus, all assumptions of Theorem 6 are satisfied. Then problem (11) has at least one integrable solution a.e. nonincreasing on R+. Clearly, this problem cannot have continuous solutions.