On Solutions of Quadratic Integral Equations in Orlicz Spaces

AbstractIn this paper we study the quadratic integral equation of the form
$$x(t) = g(t) + \lambda . G(x)(t) . \int\nolimits_a^bK(t, s)f(s, x(s)){\rm d}s.$$x(t)=g(t)+λ.G(x)(t).∫abK(t,s)f(s,x(s))ds.We discuss the existence of solutions for the above equation in different function spaces. We stress on the case when f has non-polynomial growth which leads to solutions in Orlicz spaces. The detailed theory for a wide class of spaces is presented. Some existence theorems for a.e. monotonic solutions in Orlicz spaces are proved either for strongly nonlinear functions f or for rapidly growing kernel K. The presented method allows us to extend the current results as well as to unify the proofs for both quadratic and non-quadratic cases.


Introduction
The paper is devoted to study the following quadratic integral equation (1.1) Since the quadratic problems are related with the pointwise product of two operators, it is usually solved in the space of continuous functions or in a context of Banach algebras of continuous functions. This approach has some disadvantages. Firstly, for classical equations (G(x) = const.) discontinuous solutions are frequently considered and such solutions are applicable. Secondly, for some quadratic problems like the Chandrasekhar equation [4,5] discontinuous solutions are expected (see some comments in [13]), then continuous solutions seem to be inadequate for integral problems and lead to several restrictions on considered functions. Moreover, we allow to consider problems in which either the growth of the function f or the kernel K is not MJOM polynomial. An operator G is supposed to be continuous on a required space of solutions. The problem is modelled on some quadratic integral equations (for which G is identity operator or the Nemytskii superposition operator, we are not restricted only to this case). Thus we have one more motivation: our approach allows to include also classical integral equations. For this class of integral equations there are different types of considered equations. Usually this implies some restriction for the growth of f and K. Recall that similar investigations for quadratic integral equations relate mainly to continuous solutions. The key point is to ensure that an operator of pointwise multiplication is well defined and has some compactness properties. We prefer an approach to this problem allowing us to consider a wide class of integral equations with solutions in some spaces of discontinuous functions (growth conditions are relaxed). Let us briefly recall a historical background and some motivations. We prefer a method, which allows us to unify classical and quadratic integral equations and to consider the same classes of solutions. We started such an approach in [13], but it was done only in that case of Banach-Orlicz algebras or in case when an intermediate space (described later) is L ∞ ( [12]). Here we show a detailed theory and omit such restrictions.
The new starting point is to consider some integral problems with exponential nonlinarities (see [10]) or with rapidly growing kernels ( [11], for instance). In such a case the Nemytskii operator should be considered as acting on some Orlicz spaces ( [11]). In our opinion, it is very convenient and we will try to keep this class of solutions for quadratic equations. This suggest an operator-oriented approach. In a class of Orlicz spaces we consider spaces associated with a kind of assumed growth for G and f . For a moment denote by X an Orlicz space of solutions for our problem and by F the Nemytskii superposition operator generated by f . Thus, we have G : X → W 1 , F : X → U and finally the linear integral operator H with a kernel K is acting from U into W 2 . The space U is depending on some growth assumptions of f -not necessarily of polynomial type. In a typical case of quadratic problems the spaces W 1 and W 2 are supposed both to be the space of continuous functions and then some properties of this Banach algebra allow to solve the problem. Unfortunately, this is really restrictive assumption. We started to replace this assumption by considering X = L 1 (I) and W 2 = L ∞ (I) ( [12]). Then, we considered some Banach-Orlicz algebras ( [13], but such spaces are "small" so this still leads to some restrictions on H. Here we present a complete theory for such problems. In general, allowing U be an Orlicz space we consider the triple of Orlicz spaces (not necessarily Banach algebras) for which the pointwise multiplication takes a pair of functions from W 1 and W 2 into X. For a given U a correct choice of W 1 and W 2 allows us to control both acting and growth conditions for considered operators. As will be clarified later some additional assumptions on U allow us to prove their boundedness and continuity. To show a detailed theory we need to consider a few different cases (for different classes of Orlicz spaces).
Since for equations of this type an approach via the Schauder fixed point theorem is not useful and the Banach contraction principle is too restrictive Vol. 12 (2015) On Solutions of Quadratic Integral Equations 903 in many applications, we prefer to investigate the properties of operators with respect to the topology of convergence in measure. We investigate some properties of this topology on considered Orlicz spaces and then use the Darbo fixed-point theorem for proving main results.

Notation and Auxiliary Facts
Let R be the field of real numbers. In this paper by I we will denote a compact interval [a, b] ⊂ R. Assume that (E, · ) is an arbitrary Banach space with zero element θ. Denote by B r (x) the closed ball centered at x and with radius r. The symbol B r stands for the ball B(θ, r). When necessary we will also indicate the space using the notation B r (E). Let S = S(I) denotes the set of measurable (in Lebesgue sense) functions on I and let meas stands for the Lebesgue measure in R. Identifying the functions equals almost everywhere the set S becomes a complete metric space. Note that the topology of convergence in measure on I is metrizable (cf. Proposition 2.14 in [14]). The compactness in such spaces we will call a "compactness in measure" and such sets have important properties when considered as subsets of some Orlicz spaces.
To make the paper self-contained we need to recall some basic notions and facts in the theory of Orlicz spaces.
Let For Orlicz spaces we have different situations than for Lebesgue ones. The inclusion L M ⊂ L P holds if, and only if, there exist positive constants u 0 and a such that P (u) ≤ aM (u) for u ≥ u 0 .
An important property of E M spaces lies in the fact that this is a class of functions from L M having absolutely continuous norms. Moreover, we have An N -function M is said to satisfy Δ -condition if there exists K, t 0 ≥ 0 such that for t, s ≥ t 0 , we have M (ts) ≤ KM (t)M (s). If the N -function M satisfies the Δ -condition, then it also satisfies Δ 2 -condition.
The last important class of N -functions consists of functions which increase more rapidly than power functions. An N -function M is said to satisfy Δ 3 -condition if there exists K, t 0 ≥ 0 such that for t ≥ t 0 , we have tM (t) ≤ M (Kt).
Sometimes, we will use more general concept of function spaces, i.e. ideal spaces. A normed space (X, · ) of (classes of) measurable functions x : I → U (U is a normed space) is called pre-ideal if for each x ∈ X and MJOM each measurable y : I → U the relation |y(s)| ≤ |x(s)| (for almost all s ∈ I) implies y ∈ X and y ≤ x . If X is also complete, it is called an ideal space (see [15]). The class of Orlicz spaces stands for an important (but not unique) example of ideal spaces. If possible, we will describe our results in terms of ideal spaces. This will indicate the possible extensions for our results.

Nonlinear Operators
In our paper we propose to reduce the considered problem to the operator form. In particular, we will investigate many properties of operators acting on different function spaces.
One of the most important operator studied in nonlinear functional analysis is the so-called superposition (or : Nemytskii) operator [16]. Assume that a function f : I × R → R satisfies Carathéodory conditions, i.e. it is measurable in t for any x ∈ R 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 (x)(t) = f (t, x(t)), for t ∈ I. The operator F in such a way is called the superposition operator generated by the function f . We will be interested in the case when F acts between some Orlicz spaces.
A full discussion about necessary and sufficient conditions for continuity and boundedness of such a type of operators can be found in [16]. The following property will be used: Assume that a function f : I × R → R satisfies Carathéodory conditions. Then, the superposition operator F transforms measurable functions into measurable functions.
We will utilize the fact that Carathéodory mappings transforming measurable functions into the same space are sequentially continuous with respect to topology of convergence in (finite) measure. Lemma 17.5] in S and [18] in L M (I)). Assume that a function f : I × R → R satisfies Carathéodory conditions. The superposition operator F maps a sequence of functions convergent in measure into a sequences of functions convergent in measure.
In our proofs we have to control the domain and the range of considered operators. The following lemma seems to be useful for the superposition operator:  In Orlicz spaces there is no automatic continuity of superposition operators like in L p spaces, but we have a useful result (remember, that the Orlicz space L M is ideal and if M satisfies Δ 2 condition it is also regular cf. [ Let us note, that in the case of functions of the form f (t, x) = g(t)h(x), the superposition operator F is continuous from the space of continuous functions C(I) into L M (I) even when M does not satisfy Δ 2 condition ( [19]). Since E M (I) is a regular part of an Orlicz space L M (I) (cf. [14, p.72]), in the context of Orlicz spaces, we will use the following (see also Lemma 3.3): The problem of boundedness of such a type of operators in different classes of Orlicz spaces will be described in the proofs of our main results (cf. also [11] or [16]).
Two more operators will play an important role it this paper, namely the linear integral operator H(x) = λ b a K(t, s)x(s)ds and the pointwise multiplication operator. The first one is well known and all necessary results concerning the properties of such a kind of operators in Orlicz spaces can be found in [11], so here we omit the details and important results will be pointed out in the proofs of our main results. Now, we need to describe the second one. By U (x)(t) we will denote the operator of the form: Generally speaking, the product of two functions x, y ∈ L M (I) is not in L M (I). However, if x and y belong to some particular Orlicz spaces, then the product x · y belongs to a third Orlicz space.
We need to consider the converse direction: if we are looking or solutions in a given Orlicz space L ϕ , then we need to indicate the spaces L ϕ1 and L ϕ2 such that the product of functions from that spaces belong to L ϕ . We are interested in finding that spaces as big as possible.
Let us note, that one can find two functions belonging to Orlicz spaces: u ∈ L U (I) and v ∈ L V (I) such that the product uv does not belong to any Orlicz space (this product is not integrable). Nevertheless, we have: 1. For every functions u ∈ L ϕ1 (I) and w ∈ L ϕ2 , u · w ∈ L ϕ (I).
2. There exists a constant k > 0 such that for all measurable u, w on I we

lim sup t→∞
Let us recall the following simple sufficient condition for the above statements hold true. MJOM are satisfied for large values of the argument and for certain constant α, then for every functions u ∈ L ϕ1 (I) and w ∈ L ϕ2 , u · w ∈ L ϕ (I). If moreover ϕ satisfies the Δ 2 -condition, then it is sufficient that the inequalities hold.
An interesting discussion about necessary and sufficient conditions for product operators can be found in [11,20]. We will consider the triple of spaces (ϕ, ϕ 1 , ϕ 2 ) satisfying the above property.

Monotone Functions
We are interested in finding of (almost everywhere) monotonic solutions for our problem. For the case of discontinuous functions we should describe this class of functions in considered solution spaces.
Let us recall, in metric spaces the set U 0 is compact if and only if each sequence from U 0 has a subsequence that converges in U 0 (i.e. sequentially compact). In particular, we need to use this simple fact in the space S.
Recall some consideration from [11] by adding some additional comments for Orlicz spaces. Let X be a bounded subset of measurable functions. Assume that there is a family of subsets (Ω c ) 0≤c≤b−a of the interval I such that measΩ c = c for every c ∈ [0, b − a], and for every x ∈ X, where Z is a set with measure zero, this family contains nonincreasing functions (possibly except for a set Z). We will call the functions from this family "a.e. nonincreasing" functions. This is the case, when we choose a measurable and nonincreasing function y and all functions equal a.e. to y satisfy the above condition. This means that such a notion can be also considered in the space S. Thus, we can write that elements from L M (I) belong to this class of functions. Further, let Q r stands for the subset of the ball B r consisting of all functions which are a.e. nonincreasing on I. Functions a.e. nondecreasing are defined by a similar way.
It is known that such a family constitutes a set which is compact in measure in S (cf. [17, section 19.8]). It is a little bit surprising that the proof of this property is not published anywhere. We are also interested, if the set is still compact in measure as a subset of some subspaces of S. In general, it is not true, but for the case of Orlicz spaces, we have the following: Proof. Since the convergence in measure is a metric convergence, we need only to check sequential compactness. Without loss of generality, let us restrict to the case of a.e. increasing functions. We will follow some ideas from the Helly theorem. Take an arbitrary sequence (x n ) ⊂ U . Denote by W the set such that the functions x n are increasing outside W (a countable union of countable sets), i.e. on J = I\W . Let Z be countable dense subset of J. Since {x n (t)} is bounded on Z (except perhaps the point t = b), by a diagonal procedure we can subtract a subsequence (x n k ) of (x n ) which is pointwise convergent (on Z). Then, a limit x = lim k→∞ x n k is an increasing function on Z. It is known that this function can be extended to an increasing function y defined on J in such a way that y is a limit of (x n k ) on Z.
Let t 0 be an arbitrary internal point in J. Since Z is dense in J we are able to find two sequences (s n ) and (τ n ) of points in Z tending to t 0 such that s n < t 0 < τ n . For any fixed k ∈ N, (x n k ) is increasing on J and we have and passing to the limit with k → ∞ we obtain By passing to the limit with n → ∞ we get For any point of continuity of y we have y(t) = lim k→∞ x n k (t). As the set D of all points of discontinuity of this function is at most countable, y is a.e. increasing.
Since the measure of I is finite and L M ⊂ S, the sequence (x n ) contains an a.e. convergent subsequence. As its limit is a.e. finite, by the Lebesgue theorem the subsequence is also convergent in measure. Summarising: arbitrary sequence in U contains a subsequence which is convergent in measure to some y ∈ U and then this set is compact in measure.
We have also an important nondecreasing on a finite interval I for each x ∈ R and the function x → f (t, x) is a.e. nondecreasing on R for any t ∈ I. Then, the superposition operator F generated by f transforms functions being a.e. nondecreasing on I into functions having the same property.
We will use the fact that the superposition operator takes the bounded sets compact in measure into the sets with the same property. Namely, we have Proof. Let V be a bounded and compact in measure subset of L M (I). By our assumption F (V ) ⊂ E M (I). As a subset of S the set F (V ) is compact in measure (cf. [22]). Since the topology of convergence in measure is metrizable, the compactness of the set is equivalent with the sequential compactness. By taking an arbitrary sequence (y n ) ⊂ F (V ) we get a sequence (x n ) in V such that y n = F (x n ). Since (x n ) ⊂ V , as follows from Lemma 3.2 F transforms this sequence into the sequence convergent in measure. Thus, (y n ) is compact in measure, so is F (V ).
Let us recall, that if M satisfies Δ 2 -condition, then the assumption

Measures of Noncompactness
Our operators need not be neither Lipschitz nor compact, in general. For quadratic integral equations the Darbo fixed-point theorem seems to an appropriate tool. It is based on a contraction property with respect to a measure of noncompactness. Although it seems to be known, we recall necessary notions for completeness.
By M E denote the family of all nonempty and bounded subsets of E and by N E its subfamily consisting of all relatively compact subsets. We denote the standard algebraic operations on sets by the symbols k · X and X + Y . If X is a subset of E, thenX and convX denote the closure and convex closure of X, respectively. We recall an axiomatic approach to the definition of measures of noncompactness.
An example of such a mapping is the following [23]: let X be a nonempty and bounded subset of E. The Hausdorff measure of noncompactness β H (X) is defined as We need one more interesting notion. For any ε > 0, let c be a measure of equiintegrability of the set X in L M (I) (cf. Definition 3.9 in [14] or [24,25]): where χ D denotes the characteristic function of D.
The following theorem clarifies the connections between different coefficients in Orlicz spaces. Since all Orlicz spaces L M (I) are regular, when M satisfies Δ 2 condition, then the Theorem 1 in [25] is read as follows: As a consequence, we obtain that bounded sets which are additionally compact in measure are compact in L M (I) iff they are equiintegrable in this space (i.e. have equiabsolutely continuous norms cf. [26], in particular X ⊂ E M (I)).
An importance of such a kind of functions can be clarified using the contraction property with respect to this measure instead of compactness in the Schauder fixed-point theorem. Namely, we have the Darbo theorem ( [23]):

Main Results
Denote by B the operator associated with the right-hand side of the Eq.
We will try to choose the domains of operators defined above in such a way to obtain the existence of solutions in a desired Orlicz space L ϕ (I). We stress on conditions allowing us to consider strongly nonlinear operators and simultaneously to cover both quadratic and classical integral equations.
Let us note that our assumptions on G are also referred to the case of standard quadratic integral equations (i.e. for G(x)(t) = q(t) · x(t)).
We need to distinguish two different cases. This allow us to obtain more general growth conditions on f (cf. [27][28][29][30] for non-quadratic equations). In every case we need to describe some assumptions on "intermediate" spaces being the images of L ϕ (I) for G and F (L ϕ1 (I) and L M (I), respectively) MJOM and the range for H (i.e. L ϕ2 (I)). This approach is based on a classical (nonquadratic) case as in [27][28][29]) and seems to be important in view of optimality of assumptions for every considered case. The main difference is in the part of the proof for continuity of integral operators.
6.1. The Case of ϕ Satisfying the Δ -condition Theorem 6.1. Assume, that ϕ, ϕ 1 , ϕ 2 are N -functions and that M and N are complementary N -functions. Moreover, put the following set of assumptions: (N1) there exists a constant k 1 > 0 such that for every u ∈ L ϕ1 (I) and w ∈ L ϕ2 (I) we have uw ϕ ≤ k 1 u ϕ1 w ϕ2 , (C1) g ∈ E ϕ (I) is nondecreasing a.e. on I, (C2) f : I ×R → R satisfies Carathéodory conditions and f (t, x) is assumed to be nondecreasing with respect to both variables t and x separately, nonnegative, nondecreasing, continuous function defined on R + , (C4) Let N satisfies the Δ -condition and suppose that there exist ω, γ, u 0 ≥ 0 for which Then there exists a number ρ > 0 such that for all λ ∈ R with |λ| < ρ and for all g with g ϕ < 1 there exists a solution x ∈ E ϕ (I) of (1.1) which is a.e. nondecreasing on I.
Proof. We need to divide the proof into a few steps.
I. The operator B is well defined from L ϕ (I) into itself and continuous on a domain depending on the considered case. II. We will construct an invariant ball B r for B in L ϕ (I). III. We construct a subset Q r of this ball which contains a.e. nondecreasing functions and investigate the properties Q r . IV. We check the continuity and monotonicity properties of B in Q r , so U : Q r → Q r . V. We prove that B is a contraction with respect to a measure of noncompactness. VI. We use the Darbo fixed-point theorem to find a solution in Q r .
I. First of all observe that under the assumptions (C2) and (C3) by Lemma 3.3 the superposition operator F acts from L ϕ (I) to L N (I).
In this case we will prove that U is a continuous mapping from the unit ball in E ϕ (I) into the space E ϕ (I).
Let us recall that x ∈ E ϕ (I) iff for arbitrary ε > 0 there exists δ > 0 such that xχ T ϕ < ε for every measurable subset T of I with the Lebesgue Vol. 12 (2015) On Solutions of Quadratic Integral Equations 911 measure smaller that δ (i.e. x has absolutely continuous norm). First, let us observe that in view of Lemma 3.6, it is sufficient to check this property for the operator A = H • F . Since N is an N -function satisfying Δ -condition and by (C3), we are able to use [11,Lemma 19.1]. From this there exists a constant C (not depending on the kernel) such that for any measurable subset T of I and x ∈ L ϕ (I), x ϕ ≤ 1 we have Now, by the Hölder inequality and the assumption (C2) we get for t, s ∈ I. Put k(t) = 2 K(t, ·) M for t ∈ I. As K ∈ E M (I 2 ) this function is integrable on I. By the assumptions (K1) and (K2) about the kernel K of the operator H (cf. [29]) we obtain that Whence for arbitrary measurable subset T of I and x ∈ E ϕ (I) Finally if t is such that K(t, ·) ∈ E M (I) and x ∈ E ϕ (I) we have for a.e. t ∈ I. From this it follows that A maps B 1 (E ϕ (I)) into E ϕ2 (I).
We are in a position to prove the continuity of A as a mapping from the unit ball B 1 (E ϕ (I)) into the space E ϕ2 (I). Let x n , x 0 ∈ B 1 (E ϕ (I)) be such that x n − x 0 ϕ → 0 as n tends to ∞. Suppose, contrary to our claim, that A is not continuous and the A(x n ) − A(x 0 ) ϕ2 does not converge to zero. Then, there exists ε > 0 and a subsequence (x n k ) such that A(x n k ) − A(x 0 ) ϕ2 > ε for k = 1, 2, . . . (6.2) and the subsequence is a.e. convergent to x 0 . Since (x n ) is a subset of the ball the sequence ( b a ϕ(|x n (t)|)dt) is bounded. As the space E ϕ2 (I) is regular the balls are norm-closed in L 1 (I) so the sequence ( b a |x n (t)|dt) is also bounded. Moreover, by (C3) and (C4) there exist ω, γ, u 0 > 0, s.t. (cf. [11, p. 196 and then the sequence ( K(t, s)f (s, x n (s)) ) is equiintegrable on I for a.e. t ∈ I. By the continuity of f (t, ·) we get lim k→∞ K(t, s)f (s, x n k (s)) = K(t, s)f (s, x 0 (s)) for a.e. s ∈ I. Now, applying the Vitali convergence theorem we obtain that But the Eq. (6.1) implies that A(x n k ) is a subset of E ϕ2 (I) and then lim k→∞ A(x n k )(t) = A(x 0 )(t) which contradicts the inequality (6.2). Since A is continuous between indicated spaces, By our assumption (G1) the operator G is continuous from B 1 (E ϕ (I)) into E ϕ1 (I) and then by (N1) the operator U has the same property and then U is a continuous mapping from B 1 (E ϕ (I)) into the space E ϕ (I). Finally, by the assumption (C1) B maps B 1 (E ϕ (I)) into E ϕ (I) continuously. II. We will prove the boundedness of the operator U , namely we will construct the invariant ball for this operator. By B we will denote the righthand side of our integral equation, i.e. B = g + U .
Set r ≤ 1 and let Let x be an arbitrary element from B 1 (E ϕ (I)). Then using the above consideration, the assumption (C3), the formula (6.1) and Proposition 3.6 for sufficiently small λ (i.e. |λ| < ρ) we obtain Then we have B : B r (E ϕ (I)) → B r (E ϕ (I)). Moreover, B is continuous on B r (E ϕ (I)) (see the part I of the proof).
III. Let Q r stand for the subset of B r (E ϕ (I)) consisting of all functions which are a.e. nondecreasing on I. Similarly as claimed in [31] this set is Vol. 12 (2015) On Solutions of Quadratic Integral Equations 913 nonempty, bounded (by r) and convex (direct calculation from the definition). It is also a closed set in L ϕ (I). Indeed, let (y n ) be a sequence of elements in Q r convergent in L ϕ (I) to y. Then, the sequence is convergent in measure and as a consequence of the Vitali convergence theorem for Orlicz spaces and of the characterization of convergence in measure (the Riesz theorem) we obtain the existence of a subsequence (y n k ) of (y n ) which converges to y almost uniformly on I (cf. [27]). Moreover, y is still nondecreasing a.e. on I which means that y ∈ Q r and so the set Q r is closed. Now, in view of Lemma 3.2 the set Q r is compact in measure.
IV. Now, we will show that B preserve the monotonicity of functions. Take x ∈ Q r , then x is a.e. nondecreasing on I and consequently F (x) is also of the same type in virtue of the assumption (C2) and Lemma 4.2. Further, is a.e. nondecreasing on I thanks for the assumption (K3). Since the pointwise product of a.e. monotone functions is still of the same type and by (G1), the operator U is a.e. nondecreasing on I.
Moreover, the assumption (C1) permits us to deduce that Bx(t) = g(t) + U (x)(t) is also a.e. nondecreasing on I. This fact together with the assertion that B : B r (E ϕ (I)) → B r (E ϕ (I)) gives us that B is also a selfmapping of the set Q r . From the above considerations it follows that B maps continuously Q r into Q r .
V. We will prove that B is a contraction with respect to the measure of noncompactness μ. Assume that X is a nonempty subset of Q r and let the fixed constant > 0 be arbitrary. Then for an arbitrary x ∈ X and for a set D ⊂ I, measD ≤ ε we obtain Hence, taking into account that Thus, by definition of c(x) and by taking the supremum over all x ∈ X and all measurable subsets D with measD ≤ ε we get c(B(X)) ≤ 2k 1 · |λ| · G 0 · K M · R(1) · c(X).
Since X ⊂ Q r is a nonempty, bounded and compact in measure subset of an ideal regular space E ϕ , we can use Proposition 5.2 and get The inequality obtained above together with the properties of the operator B and the set Q r established before and the inequality from the Assumption (K2) allows us to apply the Darbo Fixed-Point Theorem 5.3, which completes the proof.

The Case of ϕ Satisfying the Δ 3 -Condition
Let us consider the case of N -functions with the growth essentially more rapid than a polynomial. In fact, we will consider N -functions satisfying Δ 3condition. This is very large and important class, especially from an application point of view (cf. [32][33][34][35]). An extensive description of this class can be found in [34,Section 2.5]. Recall, that an N -function M determines the properties of the Orlicz space L M (I) and then the less restrictive rate of the growth of this function implies the "worser" properties of the space. By ϑ we will denote the norm of the identity operator from L ϕ (I) into L 1 (I), i.e. sup{ x 1 : x ∈ B 1 (L ϕ (I))}. For the discussion about the existence of ϕ which satisfies our conditions see [17, p. 61].
Then, there exist a number ρ > 0 and a number > 0 such that for all λ ∈ R with |λ| < ρ and for all g ∈ E ϕ (I) with g ϕ < there exists a solution x ∈ E ϕ (I) of (1.1) which is a.e. nondecreasing on I.
Proof. We will indicate only the points of the proof if they differ from the previous case.
I. In this case the operator B can be considered as continuous when acting on the whole E ϕ (I).
By [17,Lemma 15.1 and Theorem 19.2] and the assumption (K4): 3) for arbitrary x ∈ L ϕ (I) and arbitrary measurable subset T of I.
Thus for x ∈ L ϕ (I) The remaining estimations can be derived as in the first main theorem and then we obtain that A : E ϕ (I) → E ϕ2 (I), so by the properties of G we get B : E ϕ (I) → E ϕ (I). II. Put .
Fix λ with |λ| < ρ. Choose a positive number r in such a way that As a domain for the operator B we will consider the ball B r (E ϕ (I)). Let us remark that the above inequality is of the form a + (b + vr)cr ≤ r with a, b, c, v > 0. Then, vc > 0 and if we assume that bc − 1 < 0 and that the discriminant is positive, then Viète's formulas imply that the quadratic equation has two positive solutions r 1 < r 2 for sufficiently small λ. By the definition of ρ it is clear that our assumptions guarantee the above requirements, so there exists a positive number r satisfying this inequality.
Put C = (2 + (b − a)(1 + ϕ(1))). Let us note, in view of the above considerations, that the assumption about the discriminant which implies that the existence of solutions for the above problem is of the form: i.e. For x ∈ B r (E ϕ (I)) we have the following estimation: Then B : B r (E ϕ (I)) → B r (E ϕ (I)). Note, that the parts III. and IV. of the previous proof are similar to those from the first theorem, so we omit the details.
V. We will prove that B is a contraction with respect to a measure of noncompactness. Assume that X is a nonempty subset of Q r and let the fixed constant ε > 0 be arbitrary. Then, for an arbitrary x ∈ X and for a set D ⊂ I, measD ≤ ε we obtain Vol. 12 (2015) On Solutions of Quadratic Integral Equations 917 Let us note that r 0 is an upper bound for solutions of (6.4).
Similarly as in the previous theorem we get The inequality obtained above together with the properties of the operator B and the set Q r established before and then the assumption (K4) allow us to apply the Theorem 5.3, which completes the proof.
We need to stress on some aspects of our results. First of all we can observe that our solutions are not necessarily continuous as in previously investigated cases. In particular, we need not to assume that the Hammerstein operator transforms the space C(I) into itself. For the examples and conditions related to Hammerstein operators in Orlicz spaces we refer the readers to [34, Chapter VI.6.1., Corollary 6 and Example 7].
We have two more information about the set of solutions: it is included in E ϕ (I) and in view of Theorem 5.3 it can be proved that this set is compact as a subset of L ϕ (I).
Let X, Y be ideal spaces. A superposition operator F : X → Y is called improving if it takes bounded subsets of X into the subsets of Y with equiabsolutely continuous norms. The applications of such operators are based on the observation that large classes of linear integral operators Hy(t) = λ D k(t, s)y(s)ds, although not being compact, map sets with equiabsolutely continuous norms into precompact sets. In contrast to the classical (non-quadratic) case, for quadratic integral equations even such a nice assumption is not sufficient for using the Schauder fixed-point theorem.
Moreover, we assume in our main theorem that G maps sets with equiabsolutely continuous norms into the same family, but we do not need to assume, that it is an improving operator. Conversely, any improving operator G can be considered in our results. Let us note that for operators from Lebesgue spaces L p (I) into L r (I) (i.e. Orlicz spaces with p(x) = x p and r(x) = x r , respectively), the characterization of improving operators is known [36] = ∞ and such that G(x)(t) = M (f (t, x(t))) is also an operator from L p (I) into L r (I) (for an appropriate growth condition of f see [36]).
The aspect of applicability of our results deals also with the technique of Orlicz spaces for partial differential equations, so for an appropriate class of MJOM integral equations. In this context one can consider more singular equations than in a classical case. Motivated by previously considered equations (see [10,32,33,35] or [27,29]) we extend this method to the case of quadratic integral equations.
It should be recalled that our method of the proof can be also adapted to classical equations considered in [10,11,32,35]. For more information we refer the readers to the Chapter IX "Nonlinear PDEs and Orlicz spaces" in [37].
Finally, let us remark that our results can be applied also for Lebesgue spaces L p (I) (p ≥ 1) (cf. [38,39]). As mentioned above this class of spaces is also included into the class of Orlicz spaces. But even in this case we allow for f or K to be strongly nonlinear. The simplest case is that when Let us also pay attention to the particular case of our problem with G(x) = a(t)x(t): x(t) = g(t) + λa(t) · x(t) 1 0 K(t, s)f (s, x(s))ds.
Since we are motivated by some study on quadratic integral equations, this is of our particular interest. Note, that a full description for acting and continuity conditions for G(x) = a(t)x(t) can be found in [11,Theorem 18.2].
Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.