Positive solutions for generalized two-term fractional differential equations with integral boundary conditions

In this paper, we consider a class of boundary value problems for nonlinear two-term fractional differential equations with integral boundary conditions involving two ψ-Caputo fractional derivative. With the help of properties Green function, the fixed point theorems of Schauder and Banach, and the method of upper and lower solutions, we derive the existence and uniqueness of positive solution of proposed problem. Finally, an example is provided to illustrate the acquired results.

In the literature, nonlinear one-term FDEs of the form have been considered by many authors (see [27,28,29,30,31]). More generally, we can indicate to [32,33,34,35,36,37] on the equations of kind Recently, the authors in [38] investigated the positivity results of the Caputo-type problem by using the method of upper and lower solutions and some fixed point theorems. Very recently, Xu and Han in [39] studied the positivity results of the following nonlinear two-term FDEs in the Riemann-Liouville derivatives sense. Also, the positivity of solutions for the following nonlinear Hadamard-type FDEs is another great study by Ardjouni in [40]. Over time, due to the operator's reliance on the integration kernel, many types of new fractional derivatives and integrals emerge to obtain a distinct kernel and this makes the range of definitions wide-ranging, due to the evolution of these operators, we refer here to some recent results that dealt with the existence of solution and positive solution to various problems of FDEs [41,42,43,44,45,46].
To the best of our knowledge, no article has studied the existence of positive solutions for nonlinear FDEs with integral boundary conditions (1.1). This problem has two nonlinear terms and includes two generalized fractional derivatives. Compared to many two-term FDEs, the type of problem we considered is more general. To show the existence and uniqueness of the positive solution, we transform (1.1) into a fractional integral equation with the aid of the Green function, and then by the method of upper and lower solutions and use Schauder and Banach fixed point theorems we obtain our results.
The organization of this paper as follows: the representation of the problem with a brief survey for literature is presented in the introduction. In Section 2, we give the preliminary facts and some useful lemmas that will be used throughout the paper. In Section 3, we prove the existence and uniqueness of positive solutions to problem (1.1) via some fixed point theorems. An illustrative example is reported to justify our findings is presented in Section 4. Finally, the conclusions close the paper.

Preliminaries
Let Ω = [0, 1] be a compact interval subset R. By X = C (Ω, R) we indicate the Banach space of all continuous functions from Ω into R with the norm u = max t∈Ω |u(t)|. Define the following space By a positive solution u ∈ X, we mean a function u(t) > 0, for t ∈ Ω. Definition 2.1. Let a, b ∈ R + such that b > a. For any u ∈ [a, b], we define respectively the upper and lower contral functions as follows: Certainly, the functions U(t, u), L(t, u), U * (t, u)and L * (t, u) are monotonous nondecreasing with respect to u. Moreover, we have L(t, u) f(t, u) U(t, u), L * (t, u) g(t, u) U * (t, u).
We state some needful definitions and lemmas that will be used throughout this paper.
Theorem 2.8. Let K be a nonempty closed convex subset of a Banach space U and φ : K −→ K be a contraction operator. Then there is a unique u ∈ K with φu = u.
Theorem 2.9. Let K be a nonempty bounded, closed and convex subset of a Banach space U and φ : K −→ K be a completely continuous operator. Then φ has a fixed point in K.

Lemma 3.2. The function G ψ defined by (3.2) satisfies
Proof. The proof of part 1 was done, see [48]. To prove the part 2, we have N(t) = [ψ(t) − ψ(0)] and Υ(t) := N(t) N(1) . For 0 s t 1, we get and for 0 t s 1, we get Therefore, Now we are able to prove more results there on existence and uniqueness of positive solution to the problem (1.1).
To use the fixed point theorem, according to Lemma 3.1, we consider the operator φ : (3.6) We need the following assumptions to establish our reselts.
(H 1 ) Let u, u ∈ ε, such that a u u b and for any t ∈ Ω, where u and u are the upper and lower solutions for (1.1) respectively. Proof. Let P = {u ∈ X : u(t) u(t) u(t), t ∈ Ω} with the norm u = max 0 t 1 |u(t)| , then we have u b. Hence, P is a convex, bounded, and closed subset of the Banach space X. Moreover, the continuity of g and f implies the continuity of the operator φ defined by (3.6) on P. Now, if u ∈ P, there exist positive constants p f and p g such that max {f(t, u(t)) : t ∈ Ω, u(t) b} < p f , and max {g(t, u(t)) : t ∈ Ω, u(t) b} < p g .

Then
(φu) (t) 1 0 G ψ (t, s)ψ (s)f(s, u(s))ds Thus, Hence, φ(P) is uniformly bounded. Next, we prove the equicontinuity of φ(P). Let u ∈ P, then for any t 1 , t 2 ∈ Ω with t 1 < t 2 , we have As t 1 → t 2 the right-hand side of the previous inequality is independent of u and tends to zero. Therefore, (φu) is equicontinuous. The Arzela-Ascoli theorem shows that φ : X −→ X is compact. To apply Theorem 2.9 it remains to prove that φP ⊆ P. Let u ∈ P. Then by assumption (H 1 ) and Definition 2.1, we have Hence, u (φu) (t) u, t ∈ Ω, that is, φ(P) ⊆ P. According to Theorem 2.9, the operator φ has at least one fixed point u ∈ P. Therefore, the problem (1.1) has at least one positive solution u ∈ X and u u u, t ∈ Ω.
Next, we give further special cases of the preceding theorem.

8)
Then the problem (1.1) has at least one positive solution u ∈ P. Moreover, (3.10) Proof. Consider the following problems In view of Lemma 3.1, the problems (3.11) and (3.12) are equivalent to By the given assumption (3.8) and the definition of control function, we have where a, b are the minimum and maximum of y on Ω. It follows that Obviously, (3.13) and (3.14) are the upper and lower solutions of the problem (1.1). An application of Theorem 3.3 shows that (1.1) has at least one solution u ∈ P and satisfies z(t) u(t) y(t).

Corollary 3.5. Suppose that
. Then the problem (1.1) has at least a positive solution u ∈ X.
Proof. Consider the following problem (3.16) Problem (3.16) is equivalent to fractional integral equation Let be a positive real number such that Then, the set B = {u ∈ X : u } is convex, closed, and bounded subset of X. The operator F : is completely continuous in X as in the proof of Theorem 3.3. Moreover, If u ∈ B , then it follows from (3.15) and (3.17) that This shows that F : B → B is a compact operator. Hence, the Theorem 2.9 ensures that F has at least one fixed point in B , and then problem (3.16) has at least one positive solution u(t), where 0 < t < 1. Therefore, if t ∈ Ω one can asserts that By the Definition 2.1, we obtain u(t) and by the Definition 2.1, we get Thus, u is a lower positive solution of problem (1.1). By Theorem 3.3, the problem (1.1) has at least one positive solution u ∈ X, where u(t) u(t) u(t).
Our final result discusses the uniqueness of positive solution to (1.1) using Theorem 2.8. Theorem 3.6. Suppose that f, g : Ω × R + −→ R + are continuous functions, and there exist two constants M 1 , M 2 > 0 such that for t ∈ Ω and u, v ∈ R + . Then, if Proof. In view of Theorem 3.3, the problem (1.1) has at least one positive solution in P.
Hence, we just prove that the operator defined by (3.6) is a contraction on P. Obviously, if u ∈ P, then φu ∈ P. Indeed, for any t ∈ Ω and u, v ∈ R + we have As R < 1, the operator φ is a contraction mapping due to (3.18). So,Theorem 2.8 shows that the problem (1.1) has a unique positiv solution u ∈ P.

Conclusions
In this paper, we have considered a class of boundary value problems for nonlinear two-term fractional differential equations with integral boundary conditions involving two ψ-Caputo fractional derivative. The studied problem has two nonlinear terms and includes two generalized fractional derivatives. Compared to many two-term FDEs, the type of problem we considered is more general. With the aid of the properties Green function, known fixed point theorems, and the method of upper and lower solutions, we have established the existence and uniqueness of positive solutions for a proposed problem. Finally, the main results are well illustrated with the help of an example. Many results of problems that contain classical fractional operators are obtained as special cases of (1.1). The reported results in this paper are novel and an important contribution to the existing literature on the topic.