Boundary Value Problem of Nonlinear Hybrid Differential Equations with Linear and Nonlinear Perturbations

)e aim of this paper is to study a boundary value problem of the hybrid differential equation with linear and nonlinear perturbations. It generalizes the existing problem of second type. )e existence result is constructed using the Leray–Schauder alternative, and the uniqueness is guaranteed by Banach’s fixed-point theorem. Towards the end of this paper, an example is provided to illustrate the obtained results.


Introduction
e application of differential equations in different real domains has increased the importance of this theory which is still under development. Hybrid differential equations are a subfield of differential equations which also has enough importance. Recently, it has attracted the attention of several mathematicians [1][2][3][4][5]. In [6], the authors studied the following hybrid differential equation with linear perturbation: x(t)), where the existence of solutions to these problems has been ensured using Dhage's theorem. Motivated by the abovementioned problem, we consider the following boundary value problem for hybrid differential equation: where f ∈ C(I × R; R/ 0 { }) and g, h ∈ C(I × R; R) are given functions and α, β ∈ R such that α ≠− 1.
e proposed problem can be considered as generalization of problem (1) which becomes a special case if we take f � 1, and also, the novelty is at the level of the relationship between the boundary values. Using Banach's fixed-point theorem, we show the existence and the uniqueness of the solution of the proposed problem. e fixed-point theorems used for hybrid differential equations with perturbation of first or second type are those based on the composition of the solution as sum or product of two operators such as the Dhage case. For our case, we have a mixed problem which brings together the two types, where we thought of using Leray-Schauder's Fixed-Point eorem as a second existence result for which we will have a single operator.

Preliminaries
First, we recall some basic results used in this paper. We start by recalling Leray-Schauder alternative.
Let Π: Y ⟶ Y be a completely continuous operator and en, either the set P Π is unbounded or Π has at least one fixed point. Now, we recall the following lemmas on which we will base ourselves to build the solution of our problem.
Lemma 2 (see [6]). Suppose that x↦x − g(t, x) is increasing in R for each t ∈ I. en, for any h: if and only if x satisfies the following hybrid integral equation:

Existence Result
Before presenting the existence results, we pose the following hypotheses: (iii) ere exists positive constants λ f , λ g , and λ h such that for each t ∈ I and x, y ∈ R.
Denote C :� C(I, R), the space of all continuous mapping defined on I into R endowed with the norm ‖x‖ � sup t∈I ‖x(t)‖.

Lemma 3.
Let h ∈ C(I, R), then x is an integral solution of (2) if and only if it satisfies the following integral equation: Proof. Suppose that x is a solution for (2), then we obtain Hence, By using the second equation in (2), we obtain By replacing in (9), we obtain e other implication is trivial. Now, we can give the definition of an integral solution of problem (2).
International Journal of Differential Equations Definition 1. An integral solution of problem (2) is a function x ∈ C which satisfies the following: (1) e map t ⟼ xf(t, x) − g(t, x) is continuous for each x ∈ R, and (2) x satisfies the following integral equation: for each t ∈ I.
To reduce the form of mathematical expressions, consider the following notations: where r is a real number which will be defined later. Now, we can provide our first existence result.

Theorem 1.
Suppose that (A 0 ) − (A 3 ) are satisfied. In addition, assume that the following condition is verified: en, the problem (2) has a unique solution.
Proof. First, we define the following closed ball: where Also, we define the following operator Π on C by for each t ∈ I. e proof will be made in two steps: International Journal of Differential Equations (i) ΠB r ⊆ B r . Indeed, for x ∈ B r and t ∈ I, we have (20) (ii) Hence, according to (18), we obtain (iii) en, (iv) Π is a contraction: For x, y ∈ B r and t ∈ I, we have 4 International Journal of Differential Equations

h(s, y(s))ds
which shows Π is a contraction.
us, Π is a contraction. en, the existence and uniqueness of the solution is guaranteed by Banach's fixedpoint theorem. Now, we present the second existence result using Leray-Schauder alternative. □ Theorem 2. Suppose that (A 0 ) and (A 1 ) are satisfied. In addition, assume that there exist c 1 , c 2 > 0, such that |h(t, x)| ≤ c 1 + c 2 ‖x‖, for each (t, x) ∈ I × R. (24) Also, ] f > ac 2 . en, problem (2) has at least one solution.
International Journal of Differential Equations