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

Mohamed M. A. Metwali 1
  • 1 Department of Mathematics, Faculty of Sciences, Damanhour University, Damanhour, Egypt
Mohamed M. A. Metwali
  • Corresponding author
  • Department of Mathematics, Faculty of Sciences, Damanhour University, Damanhour, Egypt
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

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.

1 Introduction

This article is dedicated to examine the presence of solutions for the functional quadratic integral equation:

x(t)=h(t)+f3(t,x(t))+f2(t,f1(t,x(t))abK(t,s)g(s,x(s))ds),t[a,b].
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 fi 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
x(t)+Ik(t,s)expx(s)ds=0.

Additionally, the continuous solutions for the quadratic integral equation of Chandrasekhar type [4,5] seem to be inadequate for integral problems and lead to several restrictions on the considered functions, then discontinuous solutions are imperative (see also some comments in [6]). Let us also note that the solutions in Orlicz spaces are also studied in the case of partial differential equations (see [7,8] for instance).

Recall that, we started in [6] to solve quadratic integral equations in Banach-Orlicz algebras. It means that we have some additional properties of solutions, but with conditions stronger than that in this article. 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 Lφ for φ satisfying Δ2-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 Lp for p1 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)

dy(t)dt=h(t)+f3(t,dy(t)dt) + f2(t,f1(t, dy(t)dt)abK(t,s)g(s,dy(s)dt)ds)
with nonlocal condition
y(τ)=β y(ζ),τ[0,1),ζ(0,1],β1.
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.

2 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 Br stands for the closed ball centered at θ and with radius r and we will recall the space by the notation Br(E). If X is a subset of E, then X¯ and convX 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]). Let M and N be complementary N-functions, i.e., N(x)=supy0(x|y|M(x)), where M:[0,+)[0,+) is continuous, convex and even with limx0M(x)x=0, limxM(x)x= and M(x)>0 if x>0(M(u)=0u=0) ([2, p. 9]). The Orlicz class, denoted by OM, consists of measurable functions x:I for which ρ(x;M)=IM(x(t))dt<. We shall denote by LM(I) the Orlicz spaces of all measurable functions x:I for which

xM=infε>0{IM(x(s)ε)ds1}.
Let EM(I) be the closure in LM(I) of the set of all bounded functions. Note that EMLMOM.

If M satisfies the Δ2-condition, i.e., there exist ω, t00 such that for tt0, we have M(2t)ωM(t), then we have EM=LM=OM.

Now, we present the definition of a regular measure of noncompactness: we indicate by E the family of all nonempty and bounded subsets of E and NE its subfamily consisting of all relatively compact subsets.

Definition 2.1

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

  1. (i)μ(X) = 0  X  NE.
  2. (ii)X Y  μ(X)  μ(Y).
  3. (iii)μ(X¯) = μ(convX) = μ(X).
  4. (iv)μ(λX) = |λ| μ(X),forλ  .
  5. (v)μ(X + Y)  μ(X) + μ(Y).
  6. (vi)μ(X Y) = max{μ(X), μ(Y)}.
  7. (vii)If Xn is a sequence of nonempty, bounded, closed subsets of E such that Xn+1Xn,n=1,2,3,, and limnμ(Xn)=0, then the set X=n=1Xn is nonempty.

It is well-known that the Hausdorff measure of noncompactness [19] is defined by:

βH(X)=inf{r>0:thereexistsafinitesubsetYofEsuchthatXY+Br }
for nonempty and bounded subsets of XE.

For any ε > 0, let c be a measure of equiintegrability of the set X in LM(I) (cf. Definition 3.9 in [20] or [21]):

c(X) =limε0 {supmesDε[supxX{xχDLM(I)}]},
where χD denotes the characteristic function of a measurable subset DI.

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]).

Lemma 2.1

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

βH(X)=c(X).

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

μ(V(X))  kμ(X),
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

Ff(x)(t) = f(t,x(t)),t  I.
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×  satisfies Carathéodory conditions. Then,

M2(f(s,x))a(s)+bM1(x),
whereb0andaL1(I), if and only if the superposition operatorFfacts fromLM1(I)LM2(I).

Lemma 2.3

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

|f(s,x)|m(s)+b|x|,
whereb0andmLϕ(I)in which the N-functionϕ(x)satisfies theΔ2-condition.

Proof

Putting the N-functions M1=M2=ϕ and m(s)=ϕ1(a(s)), where aL1(I) in [2, Theorem 17.6].□

Let S=S(I) stand for the set of measurable (in Lebesgue sense) functions on I and let meas denote the Lebesgue measure in . Identifying the functions equal almost everywhere the set S furnished with the metric

d(x,y)=infa>0[a+meas{s:|x(s)y(s)|a}]
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”.

Lemma 2.4

[6] Let X be a bounded subset ofLM(I). Assume that there is a family of measurable subsets(Ωl)0lbaof the interval I such that measΩl=lfor everyl[0,ba], and for everyxX, x(t1)x(t2),(t1Ωl,t2Ωl). Then, X is compact in measure inLM(I).

3 Main results

Rewrite equation (1) as

x(t)=B(x)(t),
where
B(x)=h+Ff3(x)+A(x),A(x)(t)=Ff1Ff2KFg(x)(t)
such that
Fg(x)(t)=g(t,x(t)),Ffi=fi(s,x(s)),i=1,2,3andK0x(t)=abK(t,s)x(s)ds.
Note that Lp-spaces can be treated as Orlicz spaces Lp=EMp, where the N-function Mp=|x|pp satisfies the Δ2-condition, which will be useful in the next theorem.

First, equation (1) shall be examined under the following assumptions.

Let 1p+1q=1 and assume that φ is an N-function which satisfies Δ2-condition and that:

  1. (i)h  Eφ(I) is nondecreasing a.e. on I;
  2. (ii)g,fi: I×   satisfy Carathéodory conditions and g(t,x),fi(t,x) are assumed to be nondecreasing with respect to both variables t and x separately, for i=1,2,3;
  3. (iii)There exist constants bj0,j=1,4 and positive functions a4L1(I),aiLφ(I) such that
    |fi(t,x)|ai(t)+bi|x|,i=1,2,3
    and
    |g(t,x)|pa4(t)+b4φ(|x|);
  4. (iv)Assume that function K is measurable in (t,s) and assume that the linear integral operator K0 with kernel K(t,s) maps LMp(I) into L(I), s|K(,s)|LMq(I), k(t)=|K(t,)|L(I) and K0 is continuous with a norm
    K0LesssuptI(I|K(t,s)|qds)1/q;
  5. (v)IK(t1,s)dsIK(t2,s)ds for t1,t2I with t1<t2;
  6. (vi)Let r1 be a positive solution of the equation
hφ+a2φ+a3φ+b3r+b2(a1φ+b1r)K0L(a4L1+b4r)1/p=r.

Proposition 3.1

  1. (a)Suppose the N-functionφsatisfies theΔ2-condition, thenFg, Ffi,i=1,2,3are bounded inEφ(I), and forxEφwe have
    (Ffi)(x)φai+bi|x| φaiφ+bixφ,i=1,2,3
    and
    ab|g(t,x(t))|pdtaba4(t)dt+b4abφ(|x(t)|)dt(Fg)(x)Mp(a4L1+b4xφ)1/p.
  2. (b)By assumption (iv), the operatorK0is continuous and the norm ofK0(x)is estimated by (cf. [10])
K(x)K0LxMp.

Now, we are ready to proclaim our main results.

Theorem 3.1

Let assumptions (i)–(vi) be satisfied. If(b3 + b1b2K0L(a4L1+b4r)1/p)<1, then equation (1) has at least one solutionxEφ(I)which is a.e. nondecreasing on I.

Proof

  1. I.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. II.Now, we will prove the operator B is bounded in Eφ(I). For xEφ(I) with xφr1, and using Proposition 3.1, we have
    B(x)φhφ+Ff3(x)φ + A(x)φhφ+a3φ+b3xφ + a2φ+b2Ff1(x)KFg(x)φhφ+a2φ+a3φ+b3xφ + b2Ff1(x)φKFg(x)Lhφ+a2φ+a3φ+b3xφ + b2(a1φ+b1xφ)K0LFg(x)Mphφ+a2φ+a3φ+b3xφ+b2(a1φ+b1xφ)K0L(a4L1+b4xφ)1/p.
    Thus, B:Eφ(I)Eφ(I).
    By our assumption (vi), it follows that there exists a positive solution r1 of the equation
    hφ+a2φ+a3φ+b3r+b2(a1φ+b1r)K0L(a4L1+b4r)1/p=r,
    which implies that B:Br(Eφ(I))Br(Eφ(I)) is continuous.
  3. III.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. IV.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:QrQr and is continuous.
  5. V.We will prove that B is a contraction with respect to a measure of strong noncompactness.

Assume that XQr 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 tD we have

B(x)χDφhχDφ+Ff3(x)χDφ + A(x)χDφhχDφ+(a3+b3|x|)χDφ + (a2+b2 Ff1(x)KFg(x))χD φhχDφ+a3χDφ+b3xχDφ + a2χDφ+b2 Ff1(x)χDφKFg(x))hχDφ+a3χDφ+b3xχDφ + a2χDφ+b2 (a1χDφ+b1xχDφ )K0L(a4L1+b4xφ)1/phχDφ+a3χDφ+b3xχDφ + a2χDφ+b2 (a1χDφ+b1xχDφ )K0L(a4L1+b4r)1/p.
Hence, taking into account that h,a1,a2,a3Eφ(I), we have
limε0{supmesDε[supxX{hχDφ}]}=0,limε0{supmesDε[supxX{aiχDφ}]}=0,i=1,2,3.
Thus, by definition of c(x), we get
c(B(X))(b3 + b1b2K0L(a4L1+b4r)1/p)c(X).
Since XQr is a nonempty, bounded and compact in measure subset of an ideal regular space Eφ(I), we can use Lemma 2.1 and get
βH(B(X))(b3 + b1b2K0L(a4L1+b4r)1/p)βH(X).
Thus, we can use Darbo fixed point theorem 2.1, which completes the proof. Moreover, the set of solutions is compact in Eφ(I).□

Next, we present Lp-solutions for equation (1), which is still a more general result than the earlier ones.

Corollary 3.1

Assume thatp1and1p+1q=1.

  1. (i)h  Lp(I)is nondecreasing a.e. on I.
  2. (ii)g,fi: I×  satisfy 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. (iii)There exist constantsbj0,j=1,4and positive functionsa4Lq(I), aiLp(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. (iv)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
    K0L=esssupt[a,b](ab|K(t,s)|qds)1/q<.
  5. (v)IK(t1,s)dsIK(t2,s)dsfort1,t2Iwitht1<t2.
  6. (vi)Assume that there exists a positive numberr1such that
hp+a2p+a3p +b3r+ b2K0L(a1p + b1r )( a4q + b4rp/q) = r.

If(b3+b1b2K0L(a4q + b4rp/q))<1,then equation (1) has at least one solutionxLp(I)which is a.e. nondecreasing on I.

4 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.

Definition 4.1

A function y=0tx(s)ds is called a solution of problem (2) with nonlocal condition (3), if xEφ(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

y(t)=β1β0ζx(s)ds11β0τx(s)ds+0tx(s)ds,forxEφ(I),
which satisfies IVP (2) with nonlocal condition (3).

Proof

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

Dy(t)=x(t),D=ddt,
then by integrating both sides, we have
0tDy(s)ds=0tx(s)ds,
which yields
y(t)=y(0)+0tx(s)ds.
Using condition (3)
y(τ)=y(0)+0τx(s)dsandy(ζ)=y(0)+0ζx(s)ds.
By omitting y(0) from the aforementioned equations and substitute in (4), we have
y(t)=β1β0ζx(s)ds11β0τx(s)ds+0tx(s)ds.
Since xEφ (thanks to Theorem 3.1), then we deduce that y is a solution for IVP (2) with nonlocal condition (3), which completes the proof.□

5 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].

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

Jαx(s)=1Γ(α)as(st)α1x(t)dt,s[a,b]
maps LMpL(I) and is continuous for p<1α, but is not continuous at p=1α (cf. [28, Remark 4.1.2]).

Acknowledgments

The authors are grateful to the referee for his/her helpful suggestions and comments on this article which greatly improved this article.

References

  • [1]

    I.-Y. S. Cheng and J. J. Kozak, Application of the theory of Orlicz spaces to statistical mechanics, I. Integral equations, J. Math. Phys. 13 (1972), no. 51, 51–58, .

    • Crossref
    • Export Citation
  • [2]

    M. A. Krasnoseliskii and Yu. Rutitskii, Convex Functions and Orlicz Spaces, Noordhoff, Gröningen, 1961.

  • [3]

    W. A. Majewski and L. E. Labuschagne, On applications of Orlicz spaces to statistical physics, Ann. Henri Poincaré 15 (2014), 1197–1221, .

    • Crossref
    • Export Citation
  • [4]

    J. Caballero, A. B. Mingarelli, and K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electron. J. Differ. Equ. 57 (2006), 1–11.

  • [5]

    S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960.

  • [6]

    M. Cichoń and M. Metwali, On quadratic integral equations in Orlicz spaces, J. Math. Anal. Appl. 387 (2012), no. 1, 419–432, .

    • Crossref
    • Export Citation
  • [7]

    A. Benkirane and A. Elmahi, An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces, Nonlinear Anal. 36 (1999), no. 1, 11–24.

    • Crossref
    • Export Citation
  • [8]

    J. Berger and J. Robert, Strongly nonlinear equations of Hammerstein type, J. Lond. Math. Soc. 15 (1977), no. 2, 277–287.

  • [9]

    M. Cichoń and M. Metwali, On solutions of quadratic integral equations in Orlicz spaces, Mediterr. J. Math. 12 (2015), no. 3, 901–920, .

    • Crossref
    • Export Citation
  • [10]

    M. Cichoń and M. Metwali, On the existence of solutions for quadratic integral equations in Orlicz space, Math. Slovaca 66 (2016), no. 6, 1413–1426, .

    • Crossref
    • Export Citation
  • [11]

    M. Cichoń and M. Metwali, Existence of monotonic Lϕ-solutions for quadratic Volterra functional-integral equations, Electron. J. Qual. Theory Differ. Equ. 13 (2015), 1–16.

  • [12]

    R. Płuciennik and S. Szufla, Nonlinear Volterra integral equations in Orlicz spaces, Demonstr. Math. 17 (1984), no. 2, 515–532.

  • [13]

    C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, Walter de Gruyter, Berlin, New York, 2003.

  • [14]

    D. O’Regan, Solutions in Orlicz spaces to Urysohn integral equations, Proc. R. Irish Acad., Sect. A 96 (1996), 67–78.

  • [15]

    S. K. Ntouyas, Nonlocal initial and boundary value problems: a survey, in: A. Cañada, P. Drábek, and A. Fonda (Eds.), Handbook of Differential Equations: Ordinary Differential Equations, Elsevier, Amsterdam, Vol. II, 2005, pp. 461–557.

  • [16]

    Y. Raffoul, Positive solutions of three-point nonlinear second order boundary value problem, Electron. J. Qual. Theory Differ. Equ. 15 (2002), 1–11, .

    • Crossref
    • Export Citation
  • [17]

    S. Timoshenko, Theory of Elastic Stability, McGraw-Hill, New York, 1961.

  • [18]

    M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Marcel Dekker, New York, 2002.

  • [19]

    J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes Math. 60, M. Dekker, New York-Basel, 1980.

  • [20]

    M. Väth, Volterra and Integral Equations of Vector Functions, Marcel Dekker, New York-Basel, 2000.

  • [21]

    N. Erzakova, Compactness in measure and measure of noncompactness, Sib. Math. J. 38 (1997), no. 5, 926–928, .

    • Crossref
    • Export Citation
  • [22]

    J. Appell and P. P. Zabreiko, Nonlinear Superposition Operators, Cambridge University Press, Cambridge, 1990.

  • [23]

    I. K. Argyros, On a class of quadratic integral equations with perturbations, Funct. Approx. Comment. Math. 20 (1992), 51–63.

  • [24]

    W. Orlicz and S. Szufla, On some classes of nonlinear Volterra integral equations in Banach spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), 239–250.

  • [25]

    A. Sołtysiak and S. Szufla, Existence theorems for Lϕ-solutions of the Hammerstein integral equation in Banach spaces, Comment. Math. Prace Mat. 30 (1990), 177–190.

  • [26]

    I. V. Shragin, On the boundedness of the Nemytskii operator in Orlicz spaces, Kišinev. Gos. Univ. Učen. Zap. 50 (1962), 119–122.

  • [27]

    M. M. Vainberg and I. V. Shragin, Nonlinear operators and the Hammerstein equation in Orliez spaces, Dokl. Akad. Nauk SSSR, 128 (1959), no. 1, 9–12.

  • [28]

    R. Gorenflo and S. Vessela, Abel Integral Equations, Lect. Notes Math. 1461, Springer, Berlin-Heidelberg, 1991.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    I.-Y. S. Cheng and J. J. Kozak, Application of the theory of Orlicz spaces to statistical mechanics, I. Integral equations, J. Math. Phys. 13 (1972), no. 51, 51–58, .

    • Crossref
    • Export Citation
  • [2]

    M. A. Krasnoseliskii and Yu. Rutitskii, Convex Functions and Orlicz Spaces, Noordhoff, Gröningen, 1961.

  • [3]

    W. A. Majewski and L. E. Labuschagne, On applications of Orlicz spaces to statistical physics, Ann. Henri Poincaré 15 (2014), 1197–1221, .

    • Crossref
    • Export Citation
  • [4]

    J. Caballero, A. B. Mingarelli, and K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electron. J. Differ. Equ. 57 (2006), 1–11.

  • [5]

    S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960.

  • [6]

    M. Cichoń and M. Metwali, On quadratic integral equations in Orlicz spaces, J. Math. Anal. Appl. 387 (2012), no. 1, 419–432, .

    • Crossref
    • Export Citation
  • [7]

    A. Benkirane and A. Elmahi, An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces, Nonlinear Anal. 36 (1999), no. 1, 11–24.

    • Crossref
    • Export Citation
  • [8]

    J. Berger and J. Robert, Strongly nonlinear equations of Hammerstein type, J. Lond. Math. Soc. 15 (1977), no. 2, 277–287.

  • [9]

    M. Cichoń and M. Metwali, On solutions of quadratic integral equations in Orlicz spaces, Mediterr. J. Math. 12 (2015), no. 3, 901–920, .

    • Crossref
    • Export Citation
  • [10]

    M. Cichoń and M. Metwali, On the existence of solutions for quadratic integral equations in Orlicz space, Math. Slovaca 66 (2016), no. 6, 1413–1426, .

    • Crossref
    • Export Citation
  • [11]

    M. Cichoń and M. Metwali, Existence of monotonic Lϕ-solutions for quadratic Volterra functional-integral equations, Electron. J. Qual. Theory Differ. Equ. 13 (2015), 1–16.

  • [12]

    R. Płuciennik and S. Szufla, Nonlinear Volterra integral equations in Orlicz spaces, Demonstr. Math. 17 (1984), no. 2, 515–532.

  • [13]

    C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, Walter de Gruyter, Berlin, New York, 2003.

  • [14]

    D. O’Regan, Solutions in Orlicz spaces to Urysohn integral equations, Proc. R. Irish Acad., Sect. A 96 (1996), 67–78.

  • [15]

    S. K. Ntouyas, Nonlocal initial and boundary value problems: a survey, in: A. Cañada, P. Drábek, and A. Fonda (Eds.), Handbook of Differential Equations: Ordinary Differential Equations, Elsevier, Amsterdam, Vol. II, 2005, pp. 461–557.

  • [16]

    Y. Raffoul, Positive solutions of three-point nonlinear second order boundary value problem, Electron. J. Qual. Theory Differ. Equ. 15 (2002), 1–11, .

    • Crossref
    • Export Citation
  • [17]

    S. Timoshenko, Theory of Elastic Stability, McGraw-Hill, New York, 1961.

  • [18]

    M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Marcel Dekker, New York, 2002.

  • [19]

    J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes Math. 60, M. Dekker, New York-Basel, 1980.

  • [20]

    M. Väth, Volterra and Integral Equations of Vector Functions, Marcel Dekker, New York-Basel, 2000.

  • [21]

    N. Erzakova, Compactness in measure and measure of noncompactness, Sib. Math. J. 38 (1997), no. 5, 926–928, .

    • Crossref
    • Export Citation
  • [22]

    J. Appell and P. P. Zabreiko, Nonlinear Superposition Operators, Cambridge University Press, Cambridge, 1990.

  • [23]

    I. K. Argyros, On a class of quadratic integral equations with perturbations, Funct. Approx. Comment. Math. 20 (1992), 51–63.

  • [24]

    W. Orlicz and S. Szufla, On some classes of nonlinear Volterra integral equations in Banach spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), 239–250.

  • [25]

    A. Sołtysiak and S. Szufla, Existence theorems for Lϕ-solutions of the Hammerstein integral equation in Banach spaces, Comment. Math. Prace Mat. 30 (1990), 177–190.

  • [26]

    I. V. Shragin, On the boundedness of the Nemytskii operator in Orlicz spaces, Kišinev. Gos. Univ. Učen. Zap. 50 (1962), 119–122.

  • [27]

    M. M. Vainberg and I. V. Shragin, Nonlinear operators and the Hammerstein equation in Orliez spaces, Dokl. Akad. Nauk SSSR, 128 (1959), no. 1, 9–12.

  • [28]

    R. Gorenflo and S. Vessela, Abel Integral Equations, Lect. Notes Math. 1461, Springer, Berlin-Heidelberg, 1991.

OPEN ACCESS

Journal + Issues

Demonstratio Mathematica, founded in 1969, is a fully peer-reviewed, open access journal that publishes original and significant research works and review articles devoted to functional analysis, approximation theory, and related topics. The journal provides the readers with free, instant, and permanent access to all content worldwide (all 53 volumes are available online!)

Search