On existence and uniqueness of positive solutions to a class of fractional boundary value problems

Differential equations of fractional order occur more frequently on different research areas and engineering such as physics, chemistry, economics, etc. Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic, etc. [1–6].

For an extensive collection of results about this type of equations, we refer the reader to the monograph by Kilbas and Trujillo [7], Samko, Kilbas, and Marichev [8], Miller and Ross [9], and Podlubny [10].

On the other hand, some basic theory for the initial value problems of fractional differential equations involving the Riemann-Lioville differential operator has been discussed by Lakshmikantham et al. [11, 12], Bai et al. [13–16], Zhang [17], etc.

and they proved the existence of positive solutions by means of the Krasnosel'skii fixed-point theorem and Legget-Williams fixed-point theorem.

where 2 < α ≤ 3, λ is a positive parameter, and f : (0, ∞) → (0, ∞) is continuous.

where 2 < α ≤ 3 and f : [0, 1] × [0, ∞) → [0, ∞) is a continuous function.

Notice that in [18] the question of uniqueness of solutions is not treated.

In our study, the main tool is a fixed-point theorem in partially ordered sets, which gives us uniqueness of the solution. This result appears in [19].

For the convenience of the reader, we present some definitions, lemmas, and basic results that will be used later.

where n = [α] + 1 and [α] denotes the integer part of α and Γ(α) denotes the gamma function, provided that the right side is pointwise defined on (0, ∞).

provided that the right side is pointwise defined on (0, ∞).

The following two lemmas can be found in [20].

where c _i ∈ ℝ (i = 1, ... n) and n = [α] + 1 as unique solution.

for some c _i ∈ ℝ (i = 1, ..., n) and n = [α] + 1.

In [21], it is proved the following result by using Lemmas 1 and 2.

is the Green's function associated to the boundary value problem (3).

Remark 1. In [21], it is proved that G(t, s) ≥ 0, for t, s ∈ [0, 1].

Now, we present the fixed-point theorems that we will use later. These results appear in [19].

where ψ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function such that ψ is positive in (0, ∞), ψ(0) = 0, and lim_t→∞ψ(t) = ∞. If there exists x₀ ∈ X with x₀ ≤ Tx₀ then T has a fixed point.

we have the following result.

Theorem 2. Adding condition (5) to the hypotheses of Theorem 1, we obtain uniqueness of the fixed point.

Remark 2. In Theorems 1 and 2, the condition lim_t→∞ψ(t) = ∞ is superfluous.

In our considerations, we will work in the Banach space C[0, 1] = {x : [0, 1] → ℝ, continuous} with the standard distance given by d(x, y) = sup_0≤t≤1{|x(t) - y(t)|}.

In [22], it is proved that (C[0, 1], ≤) with the above-mentioned metric satisfies condition (4) of Theorem 1. Moreover, for x, y ∈ C[0, 1], as the function max(x, y) ∈ C[0, 1], (C[0, 1], ≤) satisfies condition (5).

By $\mathcal{F}$, we denote the class of functions ψ : [0, ∞) → [0, ∞) continuous, nondecreasing, positive in (0, ∞) and ψ(0) = 0, and by $\mathcal{J}$ the class of functions φ : [0, ∞) → [0, ∞) continuous, nondecreasing, and satisfying that $I-\phi \in \mathcal{F}$, where I denotes the identity mapping on [0, ∞).

Our starting point of this section is the following result about Green's function appearing in Section 2.

Lemma 4. ${max}_{t\in \left[0,1\right]}\underset{0}{\overset{1}{\int }}G\left(t,s\right)ds=\frac{1}{\Gamma \left(\alpha +1\right)}\left[{\left(\frac{\alpha -1}{\alpha }\right)}^{\alpha -1}-\left(\frac{\alpha -1}{\alpha }\right)\right]$

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In the sequel, we present the main result of this paper.

For convenience, we put $A=\frac{1}{\Gamma \left(\alpha +1\right)}\left[{\left(\frac{\alpha -1}{\alpha }\right)}^{\alpha -1}-{\left(\frac{\alpha -1}{\alpha }\right)}^{\alpha }\right]$.

Theorem 3. Our Problem (2) has a unique nonnegative solution u(t) if the following conditions are satisfied:

(H1) f : [0, 1] × [0, ∞) → [0, ∞) is continuous and nondecreasing respect to the second argument.

where$\phi \in \mathcal{J}$.

Obviously, (P, d) with d(x, y) = sup{|x(t) - y(t)|: t ∈ [0, 1]} is a complete metric space satisfying conditions (4) and (5).

where G(t, s) is the Green's function appearing in Section 2. Obviously, T applies P into itself since f(t, x) and G(t, s) are nonnegative continuous functions.

In what follows we check that assumptions in Theorem 2 are satisfied.

Firstly, the operator T is nondecreasing.

Now, we prove that T satisfies the contractive condition appearing in Theorem 1.

This proves that T satisfies the contractive condition of Theorem 1.

where 0 denotes the zero function.

Therefore, Theorem 2 says us that Problem (2) has a unique nonnegative solution.

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In the sequel, we present a sufficient condition for the existence and uniqueness of positive solutions for Problem (2) (positive solution means x(t) > 0 for t ∈ (0, 1)). The proof of this condition is similar to the proof of Theorem 2.3 of [23]. We present this proof for completeness.

Theorem 4. Under assumptions of Theorem 3 and suppose that f(t₀, 0) ≠ 0 for certain t₀ ∈ [0, 1]. Then, Problem (2) has a unique positive solution.

Proof. Consider the nonnegative solution x(t) for Problem (2) whose existence is guaranteed by Theorem 3.

In the sequel, we will prove that x(t) is a positive solution.

This gives us $x\left(t^{*}\right)={\int }_0^{1}G\left(t^{*},s\right)f\left(s,0\right)\mathsf{\text{d}}s=0$.

On the other hand, as f(t₀, 0) ≠ 0 for certain t₀ ∈ [0, 1], the nonnegative character of f(t, y) gives us f(t₀, 0) > 0. As f(t, y) is a continuous function, we can find a set A ⊂ [0, 1] with t₀ ∈ A, μ(A) > 0, where μ is the Lebesgue measure and f(t, 0) > 0 for any t ∈ A. This contradicts (6).

Therefore, x(t) > 0 for t ∈ (0, 1). This finishes the proof. □

In fact, if f(t, 0) = 0 for each t ∈ [0, 1], it is easily seen that the zero function satisfies Problem (2) and the uniqueness of the solution gives us x(t) = 0. The reverse implication is obvious.

has a unique positive solution.

Now, we present an example that illustrates our results.

In this case, $\alpha =\frac{5}{2}$ and f(t, u) = c + λ · arctg u. It is easily seen that f(t, u) satisfies (H1) of Theorem 3.

In the sequel, we prove that f(t, u) satisfies (H2) of Theorem 3.

In fact, put ϕ(u) = arctag u = α and ϕ(v) = arctg v = β (notice that, as u ≥ v and ϕ is nondecreasing, α ≥ β).

This proof our previous claim.

Now, we prove that ϕ(u) = arctg u belongs to $\mathcal{J}$. Obviously, ϕ : [0, ∞) → [0, ∞) is a continuous and nondecreasing function. Moreover, ψ(u) = u - ϕ(u) = u - arctg u is also continuous and nondecreasing and satisfies ψ(u) > 0 for u > 0 and ψ(0) = 0. Consequently, $\varphi \in \mathcal{J}$.

where 2 < α ≤ 3, λ is a positive parameter and f : (0, ∞) → (0, ∞) is continuous. The main tool used by the authors in this paper is Guo-Kranosel'skii fixed-point theorem on cones. In [18], the question about the uniqueness of solutions is not treated.

One of the results of [18] is the following theorem.

Theorem 5. [[18], Theorem 3.2] If there exists l ∈ (0, 1) such that q(l)c₂f₀ > F_∞c₁ holds then, for each λ ∈ ((q(l)c₂f₀)^-1, (F_∞c₁)^-1), the boundary value problem (8) has at least one positive solution.

Here, we consider (q(l)c₂f₀)^-1 = 0 if f₀ = ∞ and (F_∞c₁)^-1 = ∞ if F_∞ = 0, where $f_0={lim}_{u\to 0^{+}}inf\frac{f\left(u\right)}{u}$, $F_{\infty }={lim}_{u\to +\infty }sup\frac{f\left(u\right)}{u}$, q(t) = t^α-1(1 - t), k(s) = s(1 - s)^α-1, $c_1=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^1\left(\alpha -1\right)k\left(s\right)\mathsf{\text{d}}s$, and $c_2=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^1\frac{1}{\alpha -1}q\left(s\right)k\left(s\right)\mathsf{\text{d}}s$.

Now, we present the following example.

In this case, $\alpha =\frac{5}{2}$ and f(u) = c + arctg u. Then, we have F_∞ = 0 and f₀ = ∞. Moreover, c₁ = 0.129, c₂ = 0.0077, and $q\left(1/2\right)=\frac{\sqrt{2}}{8}=0.1768$ [[18], Example 5.1]. Thus, q(1/2)c₂f₀ > F_∞c₁ holds. Theorem 5 gives us the existence of a positive solution for Problem (9) for each λ ∈ (0, ∞). The question of uniqueness cannot be treated by the results of [18].

On the other hand, following a similar reasoning that in Example 1, Theorem 4 gives us the existence of a unique positive solution for Problem (9) when $0<\lambda \le {\left(\frac{1}{\Gamma \left(5/2+1\right)}\left[{\left(\frac{3}{5}\right)}^{3/2}-{\left(\frac{3}{5}\right)}^{5/2}\right]\right)}^{-1}\approx 17.8682$.

Our main contribution is the uniqueness of positive solution for Problem (9) when 0 < λ ≤ 17.8682.

Now, we present an example that cannot be studied by the results of [18], and it can be treated by the ones obtained in this paper.

In this case, the boundary value problem is nonautonomous, and thus, this problem cannot be studied by the results of [18].

On the other hand, using a similar argument that in example 1, and using Theorem 4, we obtain the existence of a unique positive solution for Problem (10) when 0 < λ ≤ 17.868.