Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line

: In this work, we consider a generalized coupled system of integral equations of Hammerstein-type with, eventually, discontinuous nonlinearities. The main existence tool is Schauder’s ﬁxed point theorem in the space of bounded and continuous functions with bounded and continuous derivatives on R , combined with the equiconvergence at ± ∞ to recover the compactness of the correspondent operators. To the best of our knowledge, it is the ﬁrst time where coupled Hammerstein-type integral equations in real line are considered with nonlinearities depending on several derivatives of both variables and, moreover, the derivatives can be of different order on each variable and each equation. On the other hand, we emphasize that the kernel functions can change sign and their derivatives in order to the ﬁrst variable may be discontinuous. The last section contains an application to a model to study the deﬂection of a coupled system of inﬁnite beams.


Introduction
Integral equations are of many types and Hammerstein-type is a particular case of them. These equations appear naturally in inverse problems, fluid dynamics, potential theory, spread of interdependent epidemics, elasticity, . . . (see References [1][2][3]). Hammerstein-type integral equations usually arise from the reformulation of boundary value problem associated of partial or ordinary differential equations.
Hammerstein-type integral equations in real line play an important role in physical problems and are often used to reformulate or rewrite mathematical problems. For example, the propagation of mono-frequency acoustic or electromagnetic waves over flat nonhomogeneous terrain modeled by the Helmholtz equation ϕ + k 2 ϕ = 0, in the upper half plane D = {(x 1 , x 2 ) ⊂ R 2 : x 2 > 0} with a Robin or impedance condition ∂ ∂x 2 + ikβϕ = ϕ 0 on the boundary line ∂D, where k is a wave number, β ∈ L ∞ (∂D) is the surface admittance describing the local properties of the ground surface ∂D and ϕ 0 ∈ L ∞ (∂D) the inhomogeneous term, can be reformulated as Hammerstein-type integral equations (see Reference [4]). In fact, the authors have shown that the above problem is equivalent to Hammerstein-type integral equations in the real line where ψ is given,k ∈ L 1 ∩ C (R \ {0}), z ∈ L ∞ is closely connected with the surface admittance β (z = i(1 − β)) and u ∈ BC is to be determined. In Reference [5] new variants of some nonlinear alternatives of Leray-Schauder and Krasnosel'skij type were introduced, involving the weak topology of Banach spaces. Along with the proof of theorems on the existence of solutions, profound constructive solvability theorems were proposed with analysis of branching solutions of nonlinear Hammerstein integral equation presented in Reference [6]. Interested readers can find explicit and implicit parameterizations in the construction of branching solutions by iterative methods in Reference [7]. However, due to the lack of compactness, there are only a few studies in the literature on Hammerstein integral equations in the real line compared to works in bounded domains.
By reviewing existence results for various types of functional, differential, and integral equations, in Reference [8], Banaś and Sadarangani use arguments associated with the measure of non-compactness and illustrate applications in proving existence for some integro-functional equations in the set of continuous functions.
Information on the utility, and some mathematical tools used to address Hammerstein-type integral equations in real line or half-line can be found in References [9][10][11][12][13][14].
On the other hand, Cabada et al. [23] deal with Hammerstein-type integral equations in unbounded domains via spectral theory. More concretely they study the existence of fixed points of the integral operator where f : R 2 → [0, +∞) satisfies a sort of L ∞ −Carathéodory conditions, k : R 2 → R is the kernel function and η(t) ≥ 0 for a.e. t ∈ R.
IIhan and Ozdemir [24], study the nonlinear perturbed integral equation where the functions u(t, s, x) and the operators T i x, (i = 1, 2) are given, while x = x(t) is an unknown function. Using the technique of a suitable measure of noncompactness, they prove an existence theorem for the mentioned system. Motivated by the works above, we consider the following generalized coupled systems of integral equations of Hammerstein-type The main existence tool is Schauder's fixed point theorem in the space of bounded and continuous functions with bounded and continuous derivatives on R, combined with the equiconvergence at ±∞ to recover the compactness of the correspondent operators. To the best of our knowledge, it is the first time where coupled Hammerstein-type integral equations in real line are considered with nonlinearities depending on several derivatives of both variables and, moreover, the derivatives can be of different order on each variable and each equation. On the other hand, we emphasize that the kernel functions can change sign and their derivatives in order to the first variable may be discontinuous.
The outline of the present paper is as follows-Section 2 contains auxiliary results and assumptions of this paper. In Section 3, we present an existence result. Lastly, an application to a real phenomenon is shown-a model to study the deflection of a coupled system of infinite beams.

Auxiliary Results and Assumptions
For ι = 1, 2, let r ι = max {m ι , n ι } and consider the Banach spaces defined by E ι := BC r ι (R) (space of bounded and continuous functions with bounded and continuous derivatives on R, till order r ι ).
The spaces E ι defined above are equipped with the norms · Eι , where is also a Banach space.
and a.e. t ∈ R.
Next lemma and theorem give, respectively, a criterion to guarantee the compacity on R and the existence of solution via Schauder's fixed point (see References [26,27]). (iii) all functions from M are equiconvergent at ±∞, that is, for any given > 0, there exists t > 0 such that . . , r ι and x ∈ M.
Theorem 1. Let Y be a nonempty, closed, bounded and convex subset of a Banach space X, and suppose that P : Y → Y is a compact operator. Then P as at least one fixed point in Y.

Main Theorem
This section is dedicated to the main result of this article, that is, its statement and its proof and provides the existence of solution of problem (1).
Consider that assumptions (A1), (A2) hold, and, moreover, assume that there is R > 0, such that, for j = 0, 1, . . . , r ι , Proof. Consider the operators T 1 : E → E 1 and T 2 : 2 (s)) ds, Next, we will show that the operator T : E → E defined by T = (T 1 , T 2 ) has a fixed point on E.
The proof will follow several steps, presented in detail for operator T 1 (u, v). The technique for operator T 2 (u, v) is similar.

Step 1: T is well defined and uniformly bounded in E.
Consider a bounded set D ⊆ E and (u, v) ∈ D. Therefore, there is ρ 1 > 0 such that By the Lebesgue Dominated Theorem, (A1), (A2) and because f 1 is L ∞ −Carathéodory function, follow that, for i = 0, 1, . . . , r 1 , Taking into account these arguments, T 2 verifies similar bounds and T(u, v) E < +∞, that is TD ⊆ E is uniformly bounded.
Step 2: T is equicontinuous in E.
Consider t 1 , t 2 ∈ R and suppose without loss of generality, that t 1 ≤ t 2 . So, by (A1), for i = 0, 1, . . . , r 1 , Therefore, T 1 D is equicontinuous in E 1 . In the same way it can be shown that T 2 D is equicontinuous on E 2 . Thus, TD is equicontinuous on E.
then the solution of (1) is a homoclinic solution, otherwise it is a heteroclinic solution.

Application to Fourth Order Coupled Systems of Infinite Beams Deflection Model
Recently, in Reference [28], the authors studied two-beam coupled structure as two infinite beams and considering the coupling between the bending wave and the torsion, the conversion of wave types at the coupled interface, as well as others details on the coupling of beams.
Motivated by the concept of very large floating structures and ice plates in waves, in Reference [29], Jang et al. consider the inverse loading distribution from measured deflection of an infinite beam on elastic foundation. They express the relationship between the loading distribution and vertical deflection of the beam in the form of an integral equation of the first kind.
An efficient method for the static deflection analysis of an infinite beam on a nonlinear elastic foundation is developed in Reference [30], where the authors combine the quasilinear method and Green's functions to obtain the approximate solution.
Motivated by the works above and, specifically, in Reference [31], where the authors analyze of moderately large deflections of infinite nonlinear beams resting on elastic foundations under localized external loads, we consider an arbitrary family of nonlinear coupled system of Bernoulli-Euler-v. Karman problem composed by two fourth order differential equation (5) and the boundary conditions where w(±∞) := lim x→±∞ w(x).
We also emphasize that: • E i , I i , i = 1, 2 are the Young's modulus (the elastic modulus of the material) and the mass moment of inertia of the cross section of beam 1 and beam 2, respectively; • η 1 u(x), η 2 v(x) are the spring force upward of first and second beams, respectively; • A 1 , A 2 are the cross-sectional area of first and second beams, respectively; • ω 1 (x), ω 2 (x) are the positive localized applied loading downward on the corresponding beams.
So, by Theorem 2, there is at least a nontrivial solution (u, v) ∈ E 1 × E 2 of problems (5) and (6). In addition, by Remark 3 the solution is a nontrivial homoclinic solution.