Boundary value problems for a second-order differential equation with involution in the second derivative and their solvability

: We consider the two-point boundary value problems for a nonlinear one-dimensional second-order differential equation with involution in the second derivative and in lower terms. The questions of existence and uniqueness of the classical solution of two-point boundary value problems are studied. The definition of the Green’s function is generalized for the case of boundary value problems for the second-order linear differential equation with involution, indicating the points of discontinuities and the magnitude of discontinuities of the first derivative. Uniform estimates for the Green’s function of the linear part of boundary value problems are established. Using the contraction mapping principle and the Schauder fixed point theorem, theorems on the existence and uniqueness of solutions to the boundary value problems are proved. The results obtained in this paper cover the boundary value problems for one-dimensional differential equations with and without involution in the lower terms.


Introduction
Functional-differential equations are widely used in mathematical models.Among them, a special place is occupied by differential equations with deviating arguments and, in particular, differential equations with involution.
Recently, the boundary value problems for one-dimensional nonlinear differential equations with deviating arguments have been considered in [1,2] (see also references therein).If a simple model of a stationary temperature distribution in a straight wire with a length l is written as an equation [2] ( ) ( ) ( ) with some boundary conditions, then modeling the stationary temperature distribution of the bent wire leads to an equation with deviating arguments [2].
It should be noted that interest in nonlinear one-dimensional differential equations is closely related to specific mathematical models.For example, the motion of a point under the action of a restoring force in a medium with hydraulic resistance is described by a second-order nonlinear equation [23,24] ( ) ( ) ( ) ( ) 0 Nonlinear equations of the second order of the type arise in radio engineering [25].The electrical engineering equation has the form [26] ( ) ( ) ( ) The equations of a more general form were also studied The forced oscillations of the pendulum are described by the equation [27] ( ) 2 sin sin y x b y аx  += .
The Thomas-Fermi equation ( ) ( ) arises in studying the distribution of electrons in an atom [28].Later, generalizations of the type and more general equations were considered.
The impetus to studying Eq (1.1) (or (1.3)) was triggered by the fact that one of the generalizations of such equations is one-dimensional differential equations with involution.The bibliography on the theory of differential equations with involution can be found in monographs [29][30][31].
Another reason is that the properties of solutions to differential equations with involution may differ significantly from the properties of solutions to equations without involution (see, for example, [21][22][23][24][25]).Therefore, it is important to study the properties of differential equations with involution.
As far as nonlinear one-dimensional differential equations without involution are concerned, it should be noted that the fourth-order equations have been actively studied recently (see [32][33][34][35]).
We are devoted to studying the existence of a solution to the differential equation with boundary conditions where   .The other subject of studies is the existence of a solution to the equation = is called an involution [29].Equations (1.1) and (1.3) contain an involution of the form ( ) . More detailed information on differential equations with involution can be found in monographs [29][30][31].
In the papers cited above, various aspects of the theory of boundary value problems for linear and nonlinear differential equations with involution are studied.The major results of most works on the theory of nonlinear differential equations are based on fixed point theorems.It is well known that fixed point theorems are used to prove the existence of solutions to boundary value problems for one-dimensional differential equations.Very often, the specific form of the Green's function of the problems associated with the studied boundary value problems significantly affects the results.
In this paper, we also use fixed point theorems.As far as we know, equations of the type (1.1), (1.2) have not been considered before.We generalize the definition of the Green's function to the case of boundary value problems for the second-order linear differential equation with involution.An important contribution of this paper is the study of the expression and properties of the Green's function of the problem described by Eq (1.1), and the proof of the theorem on the existence (uniqueness) of solutions to the studied problems.
The results obtained for 0  = imply the corresponding results for differential equations with or without involution in lower terms.
The paper consists of three sections.In Section 2, the expression and properties of the Green's function of the boundary value problem related to problems (1.1), (1.2) and (1.3), (1.2) are considered.In Section 3, using theorems on fixed points, we prove the solvability of the studied problems.

Preliminary results
Let us introduce the definition of the Green's function of a boundary value problem for a second-order linear differential equation with involution.Consider the equation ( ) ( ) ( ) with boundary conditions where ( ) ) ( ) has a unique solution of the type , where ( ) , G x t is the Green's function of the homogeneous boundary value problem (2.1), (2.2).It is well known that the Green's function is one of the best tools for studying boundary value problems.The specific form of the Green's function is important.Lemma 2.1.[36] The Green's function of the boundary value problem is written as , .
For 0  = from (2.4), we obtain the Green's function of the boundary value problem for a second-order ordinary linear differential equation (see, for example, [37]) To obtain the main results, we will use the following estimates for the Green's function.
Lemma 2.2.If 11  −   , then the Green's function (2.4) has the following estimates: ( ) ( ) Proof.Let us prove the first assertion of the Lemma.It can be assumed that 0 x  .For 1 tx −   − , the Green's function (2.4) takes the form . This implies the first assertion of the Lemma ( ) x − , then the Green's function can be written as For tx  the first assertion of the Lemma is proved in a similar way Let us prove the second inequality of the Lemma.After calculation of ( ) ( ) we get the estimate ( ) ( ) The third assertion of the Lemma is proved by direct calculation.As The Lemma is proved.Lemma 2.3.If 1   , then the following estimates hold for the Green function (2.4): Proof.Let us prove the first assertion of the Lemma.The case 1 tx −   − .Let 1 . By simple analysis, we get The estimate for the Green's function has the following form The first assertion of the Lemma is proved.The other assertions of the Lemma are proved in a similar way.The Lemma is proved.
From the proved lemmas, we have the following corollary.Corollary 2.1.If 1   , then the following estimates hold for the Green's function (

Results
Consider the boundary value problem (1.1), (1.2).We can consider Eq (1.1) as an inhomogeneous form of Eq (2.3).According to Theorem 2.3, the boundary value problem (1.1), (1.2) is equivalent to the integral equation x t is Green's function of the form (2.4).Here, the first two terms are the solution to the homogeneous Eq (1.1), and the third term is the solution to the inhomogeneous Eq (1.1).Let   ( ) with the norm ( ) Let us use the equality

Ay x y y y y x G x t F t y t y t dt
to define the operator : A X X → .Thus, the problem of the existence of a solution to the boundary value problem (1.1), (1.2) is reduced to the problem of the existence of a fixed point of the operator A defined by formula (3.1).Proof.The theorem will be proved (by the contraction mapping principle) if we show that the operator A defined by equality (3.1) is contractive.Consider the difference   , , , l l l l , such that   Proof.To prove the theorem, it suffices to show that the operator B defined by equality (3.
→ .As above, the problem of the existence of a solution to the boundary value problem (1.3), (1.2) is reduced to the problem of the existence of a fixed point of the operator B defined by formula(3.7 the boundary value problem (1.3), (1.2) has a unique solution.
Ay x the estimate Volume 8, Issue 11, 26275-26289.G x t is the Green's function of the form (2.4).Let , ).The following theorem is valid.