Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations with mixed-type boundary value conditions ∗

Abstract. In this article, we study a class of nonlinear fractional differential equations with mixedtype boundary conditions. The fractional derivatives are involved in the nonlinear term and the boundary conditions. By using the properties of the Green function, the fixed point index theory and the Banach contraction mapping principle based on some available operators, we obtain the existence of positive solutions and a unique positive solution of the problem. Finally, two examples are given to demonstrate the validity of our main results.

In [36], Zhang et al. studied the following fractional differential equation: where A is a function of bounded variation and dA can be a signed measure, f : (0, 1) × (0, +∞) × (0, +∞) → R + is continuous, and f (t, x, y) may be singular at both t = 0, 1 and x = y = 0.By analyzing the spectral of the relevant linear operator, they obtained positive solutions of the singular fractional differential equation.
In [30], Zhang studied the following nonlinear fractional differential equation with infinite-point boundary value conditions: 0 + is the standard Riemann-Liouville derivative, q : (0, 1) → R + and f : (0, 1) × (0, +∞) → R + are continuous functions, and q(t) may be singular at t = 0, 1.By using height functions of the nonlinear term on some bounded sets, the author obtained the positive solutions of the problem.
In [25], Qarout et al. studied the following semi-linear Caputo fractional differential equation: where c D q 0+ denotes the Caputo fractional derivative of order q, f : [0, 1] × R + → R + is a continuous function, a and b are real constants and α i are positive real constants.They got the existence of solutions by using some standard tools of fixed point theory.
Motivated by the above mentioned work, the purpose of this article is to investigate the existence of solutions for the BVP (1).The main new features presented in this paper are as follows.Firstly, the boundary value problem has wider and more general boundary conditions; it includes many situations, which were investigated before as special cases.Secondly, the presence of the fractional derivatives in the nonlinear term f and the boundary conditions makes the study extremely difficult.By using some available operators, the BVP (1) is transformed into a class of relatively simple low-order fractional differential equations.Thirdly, our technique and tools are novel.Consequently, conditions for the positive solutions and a unique positive solution of the BVP (1) are obtained.
The rest of the paper is organized as follows.In Section 2, we present some preliminaries and lemmas that are used to prove our main results, and we also develop some properties of the Green function, and reduce the original equation to a class of relatively simple equations by using some available operators.In Section 3, we discuss the existence of positive solutions of the BVP (1) by the tool of the fixed point index theory, and give an example to demonstrate the application of our theoretical results.In Section 4, we create a appropriate operator and discuss a unique positive solution of the BVP (1), and give an example to emphasize our two theories.
where N is the smallest integer greater than or equal to α.Now, we consider the following modified boundary value problem (BVP): where δ = γ − β.
We have https://www.mii.vu.lt/NA which means that By x(t) = I β 0 + u(t) and ( 3), we obtain Hence, we claim that given by is a solution of the following boundary value problem: where Nonlinear Anal.Model.Control, 24(1):73-94 Proof.By means of Lemma 2, we can turn (4) to an equivalent integral equation Considering the fact that u(0) = 0, we get that c 2 = 0. Then On the other hand, by the condition we have https://www.mii.vu.lt/NABy ( 6) and ( 7), we have where Then, by (8), we have Multiplying both sides of (9) by p 1 (t) and integrating from 0 to 1, we have Nonlinear Anal.Model.Control, 24(1):73-94 Then, from (10) we obtain Substituting ( 11) into ( 9), we have Multiplying both sides of (12) by p 2 (t) and integrating from 0 to η, we have From ( 12) and ( 13) we have The proof is complete.
Proof.According to the property in convergence of sequence, there exists 0 ζ 0 1 such that lim i→∞ ζ i = ζ 0 .For s ∈ [0, 1], we may discuss in two aspects: https://www.mii.vu.lt/NA (i) If 0 s < ζ 0 , then there exists The assumption a 3 < 1 gives a guarantee that the above function is well defined.Moreover, Hence, from ( 1) and ( 2) we know that l(s) is nondecreasing on [0, 1], and l(s) l(0 Then the Green function G(t, s) defined by (5) satisfies: Proof.(i) For 0 s t 1, (ii) For 0 s t 1, The proof is complete.
Lemma 7. (See [6].)Let E be a real Banach space, P be a cone of E. Let Ω ⊂ E be a bounded open set, T : Ω ∩ P → P be a completely continuous.If there exists u 0 ∈ P \{θ} such that u−T u = µu 0 for all µ 0, u ∈ ∂Ω ∩P , then i(T, Ω ∩P, P ) = 0.
Lemma 8. (See [6].)Let E be a real Banach space, P be a cone of E. Let Ω ⊂ E be a bounded open set with θ ∈ Ω, and T : Ω ∩ P → P be a completely continuous.If µu = T u for all µ 1, u ∈ ∂Ω ∩ P , then i(T, Ω ∩ P, P ) = 1.
Lemma 9. (See [16].)Let E be a real Banach space, P be a cone of E. Suppose that L : E → E is a completely continuous linear operator, and L(P ) ⊂ P .If there exist ψ ∈ P − P , ψ / ∈ −P and a constant c > 0 such that cLψ ψ, then the spectral radius r(L) = 0, and L has a positive eigenfunction ϕ * corresponding to its first eigenvalue λ 1 = (r(L)) −1 such that λ 1 Lϕ * = ϕ * .Definition 3. (See [6,16].)Let E be a real Banach space, P be a cone of E. Let T : E → E be a linear operator, and T : P → P .If there exists u 0 ∈ P \ {θ} such that for any x ∈ P \ {θ}, there exist a natural number n and real numbers α, β > 0, satisfying αu 0 T n x βu 0 , then T is called a u 0 -bounded linear operator on E. Lemma 10. (See [6,16].)Let E be a real Banach space, P be a cone of E. Let T be a completely continuous u 0 -bounded operator, λ 1 be the first eigenvalue of T .Then T must have a positive eigenfunction ϕ * , which belongs to P \ {θ} such that λ 1 Lϕ * = ϕ * ; and λ 1 is the unique positive eigenvalue of T corresponding to the positive eigenfunction.
In what follows, two operators T, L 1 : E → E are defined respectively by Lemma 11.As is defined by ( 14), where Proof.For any u ∈ E, t ∈ [0, 1], by ( 14) we obtain It is easy to verify that T : P → P and L 1 : P → P are completely continuous operators.
Lemma 12.The spectral radius r(L 1 ) = 0, and L 1 has a positive eigenfunction ϕ * corresponding to the first eigenvalue Proof.It is easy to check that L 1 : P → P is a completely continuous operator.In fact, by Lemma 6, there exists According to the density of R, there exists a constant c > 0 such that c(L 1 ψ)(t) ψ(t), t ∈ [0, 1].In view of Lemma 9, the spectral radius r(L 1 ) = 0, and L 1 has a positive eigenfunction ϕ * corresponding to its first eigenvalue The proof is complete.

Existence of a positive solution
In this section, let λ 1 be the first eigenvalue of operator L 1 .We need the following conditions: (H1) lim inf (H2) lim sup Theorem 1. Assume that (H1) or (H2) holds, then the BVP (1) has at least one positive solution.
By (H1 2 ), there exist R 1 > r and 0 < κ < 1 such that Now we define a linear operator It is obvious that L 1 : P → P is a bounded linear operator, and the spectral radius of L 1 is r( L 1 ) = κ < 1.Let Z = {u ∈ P : µu = T u, µ 1}.
It follows from ( 21) and ( 26) that i(T, (B R \ B r ) ∩ P, P ) = i(T, B R ∩ P, P ) − i(T, B r ∩ P, P ) = 1.So, the operator T has at least one fixed point on (B R \ B r ) ∩ P .This implies that BVP (1) has at least one positive solution.
https://www.mii.vu.lt/NAIf (H2) holds, similar to the proof of above, there exist R > r > 0 such that i(T, B r ∩ P, P ) = 1, i(T, B R ∩ P, P ) = 0. Therefore i(T, B R \ B r ∩ P, P ) = i(T, B R ∩ P, P ) − i(T, B r ∩ P, P ) = −1.It implies that T has at least one fixed point on (B R \ B r ) ∩ P .This implies that the BVP (1) has at least one positive solution.The proof of Theorem 1 is completed.
Example 1.We consider the following fractional equations: where It is obvious that Thus, the assumptions of Theorem 1 are satisfied, therefore the BVP ( 27) has at least one positive solution.

Existence of the unique positive solution
In this section, we need the following condition: (H3) There exist nonnegative functions Now, for t ∈ [0, 1], we define an operator L 2 : P → P as follows: For convenience, we set Lemma 13.The operator L 2 defined by (28) is a linear operator with Then for any u ∈ E, t ∈ [0, 1], we have  For any u i , v i , w i ∈ R + (i = 1, 2), t ∈ [0, 1], we have This means and, consequently, the assumptions of Theorem 2 are satisfied.Thus, the BVP (1) has a unique positive solution.