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

Definition 2.1

[19] A mapping μ:ℳE → [0, ∞) is called a measure of noncompactness in E if it satisfies the following conditions:

1. μ(X) = 0 ⇔ X ∈ NE.

2. X ⊂Y ⇒ μ(X) ≤ μ(Y).

3. μ(X¯) = μ(convX) = μ(X).

4. μ(λX) = |λ| μ(X),forλ ∈ ℝ.

5. μ(X + Y) ≤ μ(X) + μ(Y).

6. μ(X∪ Y) = max{μ(X), μ(Y)}.

7. If Xn is a sequence of nonempty, bounded, closed subsets of E such that Xn+1⊂Xn,n=1,2,3,⋯, and limn→∞μ(Xn)=0, then the set X∞=∩n=1∞Xn is nonempty.

Lemma 2.1

[21, Theorem 1] Let X be a nonempty, bounded and compact in measure subset of an ideal regular space Y. Then,

Theorem 2.1

[19] Let Q be a nonempty, bounded, closed and convex subset of E and letV:Q → Qbe a continuous transformation which is a contraction with respect to the measure of noncompactnessμ, i.e., there existsk ∈ [0,1)such that

Definition 2.2

[22] 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

The operator Ff is said to be the superposition (Nemytskii) operator generated by the function f.

Lemma 2.2

[2, Theorem 17.5] Assume that a functionf: I×ℝ → ℝsatisfies Carathéodory conditions. Then,

Lemma 2.3

Assume that a functionf: I×ℝ → ℝsatisfies Carathéodory conditions. The superposition operatorFfmapsEϕ(I)→Eϕ(I)is continuous and bounded if and only if

Proof

Putting the N-functions M1=M2=ϕ and m(s)=ϕ−1(a(s)), where a∈L1(I) in [2, Theorem 17.6].□

Lemma 2.4

[6] Let X be a bounded subset ofLM(I). Assume that there is a family of measurable subsets(Ωl)0≤l≤b−aof the interval I such that measΩl=lfor everyl∈[0,b−a], and for everyx∈X, x(t1)≥x(t2),(t1∈Ωl,t2∉Ωl). Then, X is compact in measure inLM(I).

Proposition 3.1

1. Suppose the N-functionφsatisfies theΔ2-condition, thenFg, Ffi,i=1,2,3are bounded inEφ(I), and forx∈Eφwe have

   ∥(Ffi)(x)∥φ≤∥ai+bi|x| ∥φ≤∥ai∥φ+bi∥x∥φ,i=1,2,3

   and

   ∫ab|g(t,x(t))|pdt≤∫aba4(t)dt+b4∫abφ(|x(t)|)dt⇒∥(Fg)(x)∥Mp≤( ∥a4∥L1+b4∥x∥φ)1/p.

2. By assumption (iv), the operatorK0is continuous and the norm ofK0(x)is estimated by (cf. [10])

Theorem 3.1

Let assumptions (i)–(vi) be satisfied. If(b3 + b1b2∥K0∥L∞(∥a4∥L1+b4⋅r)1/p)<1, then equation (1) has at least one solutionx∈Eφ(I)which is a.e. nondecreasing on I.

Proof

1. Assumption (ii), assumption (iii) and Lemma 2.3 imply that each Ffi maps Eφ(I) into itself and is continuous and by Lemma 2.2 the operator Fg maps Eφ(I) into EMp(I). By assumption (iv), the operator KFg maps Eφ(I) into L∞(I) and is continuous. Thus, the operator A is a continuous mapping from Eφ(I) into itself. Finally, by assumption (i), we can deduce that the operator B =h+ Ff3 + A maps Eφ(I) into itself and is continuous.

2. Now, we will prove the operator B is bounded in Eφ(I). For x∈Eφ(I) with ∥x∥φ≤r≤1, and using Proposition 3.1, we have

   ∥B(x)∥φ≤∥h∥φ+∥Ff3(x)∥φ + ∥A(x)∥φ≤∥h∥φ+∥a3∥φ+b3∥x∥φ + ∥a2∥φ+b2∥Ff1(x)⋅KFg(x)∥φ≤∥h∥φ+∥a2∥φ+∥a3∥φ+b3∥x∥φ + b2∥Ff1(x)∥φ⋅∥KFg(x)∥L∞≤∥h∥φ+∥a2∥φ+∥a3∥φ+b3∥x∥φ + b2(∥a1∥φ+b1∥x∥φ)⋅∥K0∥L∞∥Fg(x)∥Mp≤∥h∥φ+∥a2∥φ+∥a3∥φ+b3∥x∥φ+b2(∥a1∥φ+b1∥x∥φ)⋅∥K0∥L∞( ∥a4∥L1+b4∥x∥φ)1/p.

   Thus, B:Eφ(I)→Eφ(I).

   By our assumption (vi), it follows that there exists a positive solution r≤1 of the equation

   ∥h∥φ+∥a2∥φ+∥a3∥φ+b3⋅r+b2(∥a1∥φ+b1⋅r)⋅∥K0∥L∞( ∥a4∥L1+b4⋅r)1/p=r,

   which implies that B:Br(Eφ(I))→Br(Eφ(I)) is continuous.

3. Let Qr stand for the subset of Br(Eφ(I)) 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 Lφ(I). The set Qr is compact in measure in view of Lemma 2.4.

4. Now, we will show that B preserves the monotonicity of functions. Take x ∈ Qr, then x is a.e. nondecreasing on I and consequently Fg and Ffi,i=1,2,3, are also of the same type in virtue of assumption (ii). Furthermore, K0(x) 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 B(x)(t) =h + Ff3(x)(t) + A(x)(t) is also a.e. nondecreasing on I. This gives us that B:Qr→Qr and is continuous.

5. We will prove that B is a contraction with respect to a measure of strong noncompactness.

Assume that X⊂Qr is a nonempty and let the fixed constant ε>0 be arbitrary. Then, for an arbitrary x ∈ X and for an arbitrary measurable set D ⊂ I, meas D ≤ ε and t∈D we have

Hence, taking into account that h,a1,a2,a3∈Eφ(I), we have

Thus, by definition of c(x), we get

Since X⊂Qr is a nonempty, bounded and compact in measure subset of an ideal regular space Eφ(I), we can use Lemma 2.1 and get

Thus, we can use Darbo fixed point theorem 2.1, which completes the proof. Moreover, the set of solutions is compact in Eφ(I).□

Corollary 3.1

Assume thatp≥1and1p+1q=1.

1. h ∈ Lp(I)is nondecreasing a.e. on I.

2. g,fi: I×ℝ → ℝsatisfy Carathéodory conditions andg(t,x),fi(t,x)are assumed to be nondecreasing with respect to both variables t and x separately, fori=1,2,3.

3. There exist constantsbj≥0,j=1,⋯4and positive functionsa4∈Lq(I), ai∈Lp(I)such that

   |fi(t,x)|≤ai(t)+bi|x|,i=1,2,3

   and

   |g(t,x)|≤a4(t)+b4|x|p/q.

4. Assume that the function K is measurable in(t,s)and the linear integral operatorK0with kernelK(⋅,⋅)mapsLq(I)intoL∞(I)and is continuous, such that

   ∥K′0∥L∞=esssupt∈[a,b](∫ab|K(t,s)|q′ds)1/q′<∞.

5. ∫IK(t1,s)ds≥∫IK(t2,s)dsfort1,t2∈Iwitht1<t2.

6. Assume that there exists a positive numberr≤1such that

If(b3+b1b2∥K0∥L∞(∥a4∥q + b4⋅rp/q))<1,then equation (1) has at least one solutionx∈Lp(I)which is a.e. nondecreasing on I.

Definition 4.1

A function y=∫0tx(s)ds is called a solution of problem (2) with nonlocal condition (3), if x∈Eφ(I) is a solution of equation (1), i.e., belongs to an Orlicz-Sobolev space W1Lφ(I).

Theorem 4.1

Let assumptions of Theorem 3.1 be satisfied, then there exists a function

Proof

Let x be a solution of integral equation (1). Put

then by integrating both sides, we have

which yields

Using condition (3)

By omitting y(0) from the aforementioned equations and substitute in (4), we have

Since x∈Eφ (thanks to Theorem 3.1), then we deduce that y is a solution for IVP (2) with nonlocal condition (3), which completes the proof.□

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].

Remark 5.2

We obtain the same results, if we assume that Ff1:Lφ(I)→L∞(I) and the Hammerstein operator maps Lφ(I) into itself (see [10]), where f2(t,x)=x. 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 K0 is depending on the kernel K (cf. assumption (iv)). For example, the fractional integral operator

maps LMp→L∞(I) and is continuous for p<1α, but is not continuous at p=1α (cf. [28, Remark 4.1.2]).