A generalization of Ascoli–Arzelá theorem in Cⁿ with application in the existence of a solution for a class of higher-order boundary value problem Open Access

In this paper, we consider the following higher-order boundary value problem:

where n is a given positive integer, α, γ > 0, β, δ ≥ 0; f is continuous and satisfies $f(s,u0,u1,…,un−2)≤a(s)+∑k=0n−2bk|uk|$ such that a is continuous on I and $bk∈R+,k=0,…,n−2$⁠.

Equation (1.1) and its particular forms have been studied by many authors (see, for example, [1–4, 6, 7, 9–13] and the references therein).

Wong and Agarwal in [13] and Patricia et al. in [12] have studied the following boundary value problem:

under the following condition: there exists continuous functions f: (0, + ∞) → (0, + ∞) and $p1,p,q1,q:(0,1)→R$ such that

Agarwal and Wong [1] have studied the existence of a positive solution for the problem (1.1) under the following condition: there exists L ≥ 0 such that

and some other conditions, where the function g is defined in (3.2).

Chyan and Henderson [3] have studied the existence of a positive solution of the following problem:

such that f and q are continuous and non-negative functions.

The following analogical problem has been studied by Eloe and Ahmad in [5],

The following more general form has been studied by J. R. Graef and T. Moussaoui in [8],

where the derivatives x⁽ⁱ⁾, 0 ≤ i ≤ n − 2 do not appear in the non-linear terms.

Our main task in this paper consists of giving a generalization of Ascoli–Arzelá theorem in the space Cⁿ(X, E) (the space of functions from a compact subset of $R$ into a Banach space E with continuous nth derivative) in order to prove the compactness criteria and to use Schauder fixed point theorem in the space Cⁿ to prove the existences of a solution for the higher-order boundary value problem (1.1).

The rest of this paper is organized as follows. In Section 2, we give a generalization of Ascoli–Arzelá theorem in the space Cⁿ. The existences of a solution to higher-order boundary value problem (1.1) are presented in Section 3.

Before stating the main result in this section, we provide the following notations and definition:

Let E be a Banach space endowed with the norm $.1$⁠, and X be a compact subset of $R$⁠. We note by Cⁿ(X, E) the space of all functions with n continuous derivatives from X to E; this space is endowed with the norm $f=∑i=0n‖f(i)‖∞$ such that $f∞=supx∈X{‖f(x)‖1}$⁠.

For our purpose, we need the following definition in Cⁿ(X, E).

The family F ⊂ Cⁿ(X, E) is called equi-bounded if there is a constant M such that $f(i)(x)1≤M$ for all i = 1, …, n, for all f ∈ F and for all x ∈ X.

The following result gives the Ascoli–Arzelá theorem in the space Cⁿ(X, E)

To see that F is equi-continuous, let ɛ > 0, then there exists f₁, …, f_m ∈ Cⁿ(X, E) such that

Since $fj(i)$ are uniformly continuous, then there exists δ > 0 such that for all x, y ∈ X, if |x − y| < δ, then for all i = 0, …, n and for all j = 1, …, m

Let f ∈ F, then there exists j ∈ {1, …, m} such that $f∈Bε3(fj)$⁠.

Hence, for all i = 0, …, n

which implies that F is equi-continuous.

Conversely, assume that F is equi-continuous and equi-bounded. To show that F is relatively compact it suffices to show that F is totally bounded; indeed if F is totally bounded, then $F¯$ is also totally bounded, which implies that $F¯$ is compact.

Since F is equi-continuous, then for all x ∈ X and ɛ > 0, there exists δ_x > 0 such that if y ∈ X and |x − y| < δ_x, we have for all i = 0, …, n

The collection ${Bδx(x)}x∈X$ is an open cover of the compact subset X; hence there exists x₁, x₂, …, x_m ∈ X such that $X=⋃mj=1Bδxj$⁠.

which implies that, for all $x∈Bδxj$ and for all i = 0, …, n

Since F is equi-bounded, then the set

$F={(f(xj),f′(xj),…,f(n)(xj)),j=1,…,m;f∈F}$ is bounded.

Since a bounded set in $Rn+1$ is totally bounded, then there exists a subset

${(y1,i,y2,i,…,yn+1,i),i=1,…,k}⊂Rn+1$ such that

For any application φ: {1, …, m} → {1, …, k}, we define the set

It is clear that $F=⋃Fφ$⁠. Now, we show that the diameter of $Fφ$ is less than ɛ.

Let $f,g∈Fφ$ and x ∈ X, then there exists j ∈ {1, …, m} such that $x∈Bδxj$⁠.

Hence, for all i = 1, …, n

which implies that the diameter of $Fφ$ is less than ɛ. Therefore, F can be covered by finitely many sets of diameter less than ɛ.

Thus F is totally bounded, and the proof is completed. □

In this section, we study the existence of a solution for the problem (1.1).

It is easy to check, (see [1]), that u is a solution of (1.1) in $Cn(I,R)$ if and only if u is a solution of the following integro-differential equation:

in $Cn−2(I,R)$⁠, such that $g(t,s)=∂n−2G(t,s)∂tn−2$ is the Green's function of the second-order boundary value problem

Moreover,

Before stating our main result, we recall the following Schauder fixed point theorem.

Equation (3.1) will be studied under the following assumptions:

• $[(i)]f∈C(I×Rn−1,R)$⁠.

• [(ii)] There exists a function $a∈C(I,R+)$ and constants $bk∈R+(k=0,…,n−2)$ such that

Under the assumptions (i) and (ii), we will make use of Schauder fixed point theorem to prove the following main result:

Then, the integro-differential Equation (3.1) has a solution in $Cn−2(I,R)$⁠.

It is clear that the operator A is well defined from E into itself.

Moreover for all x ∈ E, t ∈ I and i = 0, …, n − 2, we have

The proof is split into three steps.

• Step I. There exists α > 0 such that A transforms C = {x ∈ E, ‖x‖ ≤ α} into itself. It is clear that C is non-empty, bounded, closed and convex subset of E.

Moreover, for all x ∈ C, t ∈ I and i = 0, … n − 2, we have

Hence, for r = Max{b₀, …, b_n−2}, we obtain

We deduce that, A transforms C into itself if

which implies, under the condition of Theorem (3.2), that

Then, A transforms C into itself for

• Step 2: The operator A is continuous.

Let (x_m) ∈ C be a convergence sequence to x ∈ C, which implies that $(xm(i))$ converges to x⁽ⁱ⁾ in the space C(I, [ − α, α]) for all i = 0, …, n − 2.

Since f is uniformly continuous on the compact set $I×[−α,α]×⋯×[−α,α]︸n−1times$⁠, then the sequence $(f(s,xm,xm′,…,xm(n−2)))$ converges to f(s, x, x′, …, x⁽ⁿ⁻²⁾) in $C(I,R)$⁠.

It follows that

which implies that (Ax_m) converges to Ax , and the operator A is continuous.

• Step 3: A(C) is relatively compact; it is clear that A(C) is equi-bounded.

Now, to show that A(C) is equi-continuous, take t₁ and t₂ in I.

Then, for all i = 0, … n − 3, there exists ξ_i between t₁ and t₂ such that

Hence, for all i = 0, … n − 3,

Now, let ɛ > 0. We note $λ=max0≤i≤n−3∫01|∂1(i+1)G(t,s)|ds∞$⁠.

Then from (3.4), if $|t2−t1|≤δ1=ε1+‖a‖∞+rαλ$⁠, we have for all i = 0, …, n − 3,

On the other hand, since the function g(t, s) is uniformly continuous on I × I,

there exists δ₂ > 0 such that if |t₂ − t₁| ≤ δ₂, then for all s ∈ I

which implies, for i = n − 2, that

Hence, the third step is completed by setting δ = min (δ₁, δ₂). Therefore, the set A(C) is equi-continuous.

The proof of Theorem 3.2 then follows from Schauder fixed point theorem. □

On the other hand, we have

which implies that $∫01|G(t,s)|ds=14(2t+t2)$ and $‖∫01|G(t,s)|ds‖=34$⁠.

It is easy to see that |f(s, u₀, u₁)| ≤ λ ln(2) + λ|u₀| + λ|u₁|.

Hence, the conditions (i) and (ii) are fulfilled with a(s) = λ ln(2), b₀ = b₁ = λ.

Therefore, the inequality in Theorem 3.2 takes the form

Then by Theorem 3.2, we conclude that the third-order boundary value problem (3.5) has a solution $u∈C3(I,R)$ if $λ<811$⁠.

The authors are very grateful to the anonymous referees for their valuable comments and suggestions.