On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces

Abstract The existence of a.e. monotonic solutions for functional quadratic Hammerstein integral equations with the perturbation term is discussed in Orlicz spaces. We utilize the strategy of measure of noncompactness related to the Darbo fixed point principle. As an application, we discuss the presence of solution of the initial value problem with nonlocal conditions.


Introduction
This article is dedicated to examine the presence of solutions for the functional quadratic integral equation: It is helpful to find solutions in Orlicz spaces, when we deal with some problems involving strong nonlinearities in which either the growth of the functions f i or the kernel K is not polynomial (of exponential growth, for instance), then discontinuous solutions are expected. This is motivated by some mathematical models in physics and statistical physics [1][2][3]. The considered thermodynamic problem leads to the integral equation with exponential nonlinearities of the form In [9,10], the authors extend the results in [6] to arbitrary Orlicz spaces which are not necessary Banach algebras using different sets of assumptions controlling the intermediate spaces. The authors in [11] discussed the quadratic functional-integral equations in Orlicz spaces L φ for φ satisfying Δ 2 -condition. The solutions for the Volterra integral equation in generalized Orlicz spaces (Musielak-Orlicz spaces) were studied in [12], see also [13].
The presented class of Orlicz spaces permits us to include the case of Lebesgue spaces L p for ≥ p 1 as a special case. The key point is to dominate optimally the acting, continuity, and monotonicity conditions for the considered operators between the target spaces, which was not sufficiently utilized in previous studies such as [14].
As an application of our results, we will discuss the solvability of the initial value problem (IVP) with nonlocal condition The IVP for ordinary differential equations has applications in various regions, for example, in physics and different areas of applied mathematics such as theory of elasticity (see [15][16][17]) and has the preferable effect with nonlocal conditions than the initial or Darboux conditions. The results displayed in this article are motivated by unifying some known results for particular cases of equation (1) in one proof and will extend them to general functional quadratic Hammerstein integral equations with linear perturbation of second kind in Orlicz spaces on the bounded interval. We use the strategy of measure of noncompactness with the Darbo fixed point principle to prove the existence of a.e. monotonic solution of the considered problems. The solution of IVP (2) with nonlocal condition (3) is also examined.

Preliminaries
Let be the field of real numbers and I be an interval [ ] ⊂ a b , . Assume that ( ∥⋅∥) E, is an arbitrary Banach space with zero element θ. The symbol B r stands for the closed ball centered at θ and with radius r and we will recall the space by the notation ( ) B E r . If X is a subset of E, then X and X conv denote the closure and convex closure of X, respectively.
Next, we give some lemmas and theorems in the Orlicz spaces theory (cf. also [2,18] d .
It is well-known that the Hausdorff measure of noncompactness [19] is defined by: where χ D denotes the characteristic function of a measurable subset ⊂ D I . It forms a regular measure of noncompactness if restricted to the family of subsets being compact in measure in a class of regular ideal (or Orlicz) spaces (cf. [21]). be a continuous transformation which is a contraction with respect to the measure of noncompactness μ, i.e., there exists ∈ [ ) k 0, 1 such that for any nonempty subset X of E. Then, V has at least one fixed point in the set Q and the set of all fixed points for V is compact in E.

Definition 2.2. [22]
Assume that a function × → f I : satisfies Carathéodory conditions, i.e., it is measurable in t for any ∈ x and continuous in x for almost all ∈ t I . Then, to every function ( ) x t being measurable on I we may assign the function , . f The operator F f is said to be the superposition (Nemytskii) operator generated by the function f.
be a complete metric space. Moreover, the convergence in measure on I is equivalent to the convergence with respect to the metric d (cf. Proposition 2.14 in [20]). The compactness in such a space is called a "compactness in measure". , and for every ∈

Main results
Rewrite equation (1) as and assume that φ is an N-function which satisfies Δ 2 -condition and that: (iv) Assume that function K is measurable in ( ) t s , and assume that the linear integral operator K 0 with kernel ( ) for ∈ t t I , 1 2 with < t t 1 2 ; (vi) Let ≤ r 1 be a positive solution of the equation By assumption (iv), the operator K 0 is continuous and the norm of ( ) K x 0 is estimated by (cf. [10]) Now, we are ready to proclaim our main results.    consisting of all functions which are a.e. nondecreasing on I. Similarly, as claimed in [10] this set is nonempty, bounded, closed and convex in ( ) L I φ . The set Q r is compact in measure in view of Lemma 2.4. IV. Now, we will show that B preserves the monotonicity of functions. Take ∈ x Q r , then x is a.e. nondecreasing on I and consequently F g and = F i , 1, 2, 3 fi , are also of the same type in virtue of assumption (ii). Furthermore, ( ) K x 0 is a.e. nondecreasing on I (thanks for assumption (v)). Since the pointwise product of a.e. monotone functions is still of the same type, the operator A is a.e. nondecreasing on I. Then by assumption (i), we deduce that is also a.e. nondecreasing on I. This gives us that → B Q Q : r r and is continuous. V. We will prove that B is a contraction with respect to a measure of strong noncompactness.
Assume that ⊂ X Q r is a nonempty and let the fixed constant > ε 0 be arbitrary. Then, for an arbitrary ∈ x X and for an arbitrary measurable set Hence, taking into account that ∈ ( ) h a a a E I , , , Thus, by definition of ( ) c x , we get ( ) Since ⊂ X Q r is a nonempty, bounded and compact in measure subset of an ideal regular space ( ) E I φ , we can use Lemma 2.1 and get ( ) Next, we present L p -solutions for equation (1), which is still a more general result than the earlier ones.  (iv) Assume that the function K is measurable in ( ) t s , and the linear integral operator K 0 with kernel (⋅ ⋅) K , maps ( ) L I q into ( ) ∞ L I and is continuous, such that for ∈ t t I , 1 2 with < t t 1 2 . (vi) Assume that there exists a positive number ≤ r 1 such that

IVP
Next, we will discuss the existence of some special class of solutions for IVP (2) with nonlocal condition (3). As a consequence of our main result solutions are not absolutely continuous, but they are in a narrower space. Since ∈ x E φ (thanks to Theorem 3.1), then we deduce that y is a solution for IVP (2) with nonlocal condition (3), which completes the proof. □

Remarks
Remark 5.1. The quadratic equations have many applications in astrophysics, neutron transport, radiative transfer theory and in the kinetic theory of gases [4,5,23]. . This is the standard non-quadratic case which is reduced to the classical integral equation (cf. [12,24,25]). Remark 5.3. Shragin [26] proved that the Nemytskii operators are bounded on "small" balls and in [27] the authors apply these results for Hammerstein integral equations in Orlicz spaces.
Remark 5.4. The continuity of the linear integral operator of the form K 0 is depending on the kernel K (cf. assumption (iv)). For example, the fractional integral operator