Electronic Journal of Qualitative Theory of Differential Equations 2003, No. 20, 1-7; http://www.math.u-szeged.hu/ejqtde/

Perturbed integral equations in modular function spaces

By A.Hajji - E.Hanebaly

Abstract. We focus our attention on a class of perturbed integral equations in modular spaces, by using fixed point Theorem I.1 (see [1]).

Keywords. Modular space, Integral equation. A.M.S Subject Classifications: 46A80, 45G10

1 Introduction

In the present work, we focus our attention on a class of perturbed integral equation which can be written as

u(t) = exp(−tA)f_{0} + ∫_{0}^{t} exp((s − t)A)T u(s)ds (I)

in the modular space C^{φ} = C([0, b], L^{φ}) (see [1]), where L^{φ} is the Musielak-Orlicz space, f_{0} is a fixed element in L^{φ}, A : L^{φ} → L^{φ} is a linear operator and T : L^{φ} → L^{φ} is ρ − c-Lipschitz, i.e. there exists k > 0 such that ρ(c(T x − T y)) ≤ kρ(x − y) for any x, y in L^{φ} ( ρ being a modular ). Since ρ is not subadditive, then the sum of these operators is not necessarily ρ-Lipschitz and the convexity of the integral presents a more delicate problem. Therefore, it is natural in our study to introduce c_{0} constant c_{0} and assume some hypotheses on A, T, and b. For more details about the concepts of the above mentioned modular spaces, we refer the reader to the books by Musielak [4] and Kozlowski [3]. We begin by recalling the definition below.

Definition 1.1 Let X be an arbitrary vector space over K = (R or I C)

a) A functional ρ : X→[0, +∞] is called a pseudomodular if

i) ρ(0) = 0 . ii) ρ(αx) = ρ(x) for α ∈ K with |α| = 1, ∀x ∈ X.

iii) ρ(αx + βy) ≤ ρ(x) + ρ(y) for α, β ≥ 0 and α + β = 1. If in place of iii) there holds also: iii’) ρ(αx + βy) ≤ α^{s}ρ(x) + β^{s}ρ(y) for α, β ≥ 0 and α^{s} + β^{s} = 1 , with an s ∈ (0, 1[ , then the pseudomodular ρ is called s-convex. 1-convex pseudomodular are called convex. If besides i) there holds also. i’ )ρ(x) = 0 implies x = 0 , then ρ is called a modular.

b) If ρ is a pseudomodular in X, then . X_{ρ} = {x ∈ X/ρ(λx)→0 as λ→0} is called a modular space.

c) If ρ is a convex modular, then ∥x∥_{ρ} = inf\{u > 0, ρ(\frac{x}{u}) ≤ 1\} is called the Luxemburg norm.

Recall that ρ has the Fatou property if: ρ(x−y) ≤ lim inf ρ(x_{n}−y_{n}), whenever x_{n} →^{ρ} x

and y_{n} →^{ρ} y.

And we say that ρ satisfies the Δ_{2}-condition if: ρ(2x_{n})→0 as n→ + ∞ whenever ρ(x_{n})→0 as n→ + ∞, for any sequence (x_{n})_{n∈IN} in X_{ρ}.

2 Perturbed integral equation class

In this section, we will study the existence of solution of the following perturbed integral equation:

u(t) = exp (−tA)f_{0} + ∫_{0}^{t} exp ((s − t)A) T u(s)ds (I)

We present the general hypotheses of the equation (I). H_{1} ) Let ρ be a modular of the Musielak-Orlicz space L^{φ}, convex satisfying the Δ_{2}- condition and ρ_{a}(u) = sup_{t∈[0,b]} exp (−at)ρ(u(t)) is a modular of C([0, b], L^{φ}) with a > 0 ( see

[1]). H_{2} ) Let A : L^{φ} → L^{φ} be a linear application, assume that there exist α_{0} > max(e^{−1}, eb^{2}) and M > 0 such that ρ(α_{0}Ax) ≤ Mρ(x) for any x ∈ L^{φ}. H_{3} ) Let T : L^{φ} → L^{φ} be ρ − c-Lipschitz with c > 0, i.e there exists k > 0 such that ρ(c(T x − T y)) ≤ kρ(x − y) for any x, y ∈ L^{φ}. H_{4} ) Let f_{0} be fixed element in L^{φ}.

Theorem 2.1 Under these conditions H_{1} − H_{4} and for all b > 0, the perturbed integral equation (I) has a solution u ∈ C([0, b], L^{φ}).

Remark. If we restrict our attention to the Banach space (L^{φ}, ∥.∥_{ρ}). Then the equation (I) can be written as follows:

u′(t) + Au(t) = T u(t) (∗).

Thus, if A ≡ I then (∗) becomes

u′(t) + (I − T)u(t) = 0.

But the latter equation has been treated before in [1] and [4]. This let us to reduce the study to the case A ≢ I when (∗) can be written in the form below:

u′(t) + (I − [T + (I − A)])u(t) = 0.

Set B = I − A. It follows from the fact that ρ is not subadditive that T + B is not necessarily ρ-Lipschitz contrary to the situation in [1] and [2]. We cite first the theorem below which we shall use in the proof of Theorem 2.1.

Theorem 2.2 . (See [1]) Let X_{ρ} be a ρ-complete modular space. Assume that ρ is an s-convex, satisfying the Δ_{2}- condition and having the Fatou property. Let B be a ρ-closed subset of X_{ρ} and T : B → B a mapping such that (∗) ∃c, k ∈ R^{+} : c > max(1, k), ρ(c(T x − T y)) ≤ k^{s}ρ(x − y) for any x, y ∈ B. Then T has a fixed point.

Proof of Theorem 2.1. 1^{st}) step. We use the following property. Under the hypotheses of Theorem 2.1, the operator A is continuous from (L^{φ}, ∥.∥_{ρ}) to itself. Indeed, we have ρ(α_{0}Ax) ≤ Mρ(x) for any x ∈ L^{φ}. Let (x_{n})_{n∈IN} be a sequence in L^{φ} such that ∥x_{n}∥_{ρ} → 0 as n → +∞. So ρ(x_{n}) → 0 as n → +∞, which implies that ρ(α_{0}Ax_{n}) → 0 as n → +∞. By Δ_{2}-condition, ∥α_{0}Ax_{n}∥_{ρ} → 0 as n → +∞. Hence ∥Ax_{n}∥_{ρ} → 0 as n → +∞. Thus, there exists a constant c > 0 such that ∥Ax∥_{ρ} ≤ c∥x∥_{ρ}, for any x ∈ L^{φ}.

Therefore, exp (A)(x) = ∑_{m=0}^{+∞} \frac{A^{m}}{m!} (x) make a sense.

2^{end}) step. We claim that \frac{eb}{α_{0}} < \frac{1}{b}. Indeed, since α_{0} > max\{e^{−1}, eb^{2}\} we have:

a) If e^{−1} ≥ eb^{2} then e^{2}b^{2} ≤ 1 therefore \frac{eb}{α_{0}} < \frac{e^{2}b^{2}}{b} ≤ \frac{1}{b}.

b) If eb^{2} ≥ e^{−1} then e^{2}b^{2} ≥ 1 therefore \frac{eb}{α_{0}} < \frac{eb}{eb^{2}} = \frac{1}{b}. Hence in both cases we have \frac{eb}{α_{0}} < \frac{1}{b}, we choose c_{0} such that \frac{eb}{α_{0}} ≤ c_{0} < \frac{1}{b} and c = \frac{e}{c_{0}} . Then c_{0}b < 1. Let λ > 1 such that 1 < λ < \frac{1}{c_{0}b}. We consider S : C([0, b], L^{φ}) → C([0, b], L^{φ}) defined by. Su(t) = exp (−tA)f_{0} + ∫_{0}^{t} exp ((s − t)A) T u(s)ds for any u ∈ C([0, b], L^{φ}). It is clear that Su(t) ∈ L^{φ} for each t ∈ [0, b]. As Su is continuous from [0, b] into (L^{φ}, ∥.∥_{ρ}), then, Su is ρ-continuous from [0, b] into (L^{φ}, ρ). Let u, v ∈ C([0, b], L^{φ}), we have

λ(Su(t) − Sv(t)) = ∫_{0}^{t} λ exp ((s − t)A) (T u − T v)(s)ds . We put T u − T v = x.

Let K = {t_{0}, t_{1}, ......t_{n}} be any subdivision of [0, t]. ∑_{i=0}^{n−1} λ(t_{i+1} − t_{i}) exp((t_{i} − t)A)x(t_{i}) is

∥.∥_{ρ}-convergent, and consequently, ρ-convergent to ∫_{0}^{t} λ exp((s − t)A)x(s)ds in L^{φ} when, |K| = sup{|t_{i+1} − t_{i}|, i = 0, ...., n − 1} → 0 as n → +∞. By Fatou property we have

ρ(∫_{0}^{t} λ exp ((s − t)A)x(s)ds) ≤ lim inf ρ(∑_{i=0}^{n−1} λ(t_{i+1} − t_{i}) exp ((t_{i} − t)A)x(t_{i})).

And ∑_{i=0}^{n−1} λ(t_{i+1} − t_{i}) exp ((t_{i} − t)A)x(t_{i}) = ∑_{i=0}^{n−1} λ(t_{i+1} − t_{i})c_{0} \frac{1}{c_{0}} exp ((t_{i} − t)A)x(t_{i}).

Moreover ∑_{i=0}^{n−1} λ(t_{i+1} − t_{i})c_{0} ≤ λc_{0}b ≤ 1

Then ρ(∑_{i=0}^{n−1} λ(t_{i+1} − t_{i}) exp ((t_{i} − t)A)x(t_{i})) ≤ ∑_{i=0}^{n−1} λ(t_{i+1} − t_{i})c_{0}ρ(\frac{1}{c_{0}} exp ((t_{i} − t)A)x(t_{i})).

3^{rd} step. In this part, we show that

ρ(\frac{1}{c_{0}} exp ((t_{i} − t)A)x(t_{i})) ≤ exp (M − 1)ρ(\frac{e}{c_{0}} x(t_{i}))

We have \frac{1}{c_{0}} exp ((t_{i} − t)A)x(t_{i}) = ∑_{m=0}^{+∞} \frac{1}{c_{0}} \frac{(t − t_{i})^{m}}{m!} A^{m}((−1)^{m}x(t_{i})).

And since ∑_{m=0}^{+∞} \frac{exp(−1)}{m!} = 1, then ρ(\frac{1}{c_{0}} exp ((t − t_{i})A)x(t_{i})) ≤ ∑_{m=0}^{+∞} \frac{exp (−1)}{m!}ρ(\frac{e}{c_{0}} b^{m}A^{m}x(t_{i})).

We have α_{0} ≥ \frac{eb}{c_{0}} > 0, and since α_{0} > max(e^{−1}, eb^{2}), then α_{0} > b. Indeed,

i) if e^{−1} ≥ eb^{2}, then e^{2}b^{2} ≤ 1 which implies that eb ≤ 1. Therefore b ≤ e^{−1} < α_{0}. ii) if eb^{2} ≥ e^{−1}, then e^{2}b^{2} ≥ 1 which implies that eb ≥ 1. Therefore eb^{2} ≥ b and α_{0} > b. From the hypothesis ρ(α_{0}Ax(t_{i})) ≤ Mρ(x(t_{i})), we have

ρ(α_{0}bA^{2}x(t_{i})) ≤ Mρ(bAx(t_{i}))
≤ Mρ(α_{0}Ax(t_{i}))
≤ M^{2}ρ(x(t_{i}))

Which implies that ρ(\frac{e}{c_{0}} b^{m}A^{m}x(t_{i})) ≤ M^{m}ρ(x(t_{i})) ≤ M^{m}ρ(\frac{e}{c_{0}} x(t_{i})) for any m in IN^{∗}. Therefore,

ρ(\frac{1}{c_{0}} exp ((t_{i} − t)A)x(t_{i})) ≤ ∑_{m=0}^{+∞} \frac{exp(−1)M^{m}}{m!} ρ(\frac{e}{c_{0}} x(t_{i}))
= exp (M − 1)ρ(\frac{e}{c_{0}} x(t_{i})).

4^{th} Step. We have

ρ(λ(Su(t) − Sv(t))) ≤ lim inf(∑_{i=0}^{n−1} λ(t_{i+1} − t_{i})c_{0} exp (M − 1)kρ(u − v)(t_{i}))
≤ kλ exp (M − 1) lim inf(∑_{i=0}^{n−1}(t_{i+1} − t_{i})c_{0} exp (at_{i}))ρ_{a}(u − v)
= λk exp (M − 1) ∫_{0}^{t} c_{0} exp (as)ds ρ_{a}(u − v)

therefore

exp (−at)ρ(λ(Su(t) − Sv(t))) ≤ kλ exp (M − 1) ∫_{0}^{t} c_{0} exp (a(s − t))ds ρ_{a}(u − v)

Hence,

ρ_{a}(λ(su − sv)) ≤ kλ exp (M − 1)\frac{c_{0}}{a} (1 − e^{−ab})ρ_{a}(u − v).

It suffices to take a > ke^{M−1}c_{0}, then we have λk exp (M − 1)\frac{c_{0}}{a} (1 − e^{−ab}) < λ . By Theorem 2.2, S has a fixed point which is a solution of the equation (I). Remark In third step, instead of the combination convex ∑_{m=0}^{∞} \frac{e^{−1}}{m!} = 1 , we may choose the combi-

nation convex ∑_{m=0}^{∞} \frac{e^{−1}b^{m}}{m!} = 1, which gives the conclusion of theorem under the following hypotheses: H′_{2} A : L^{φ} → L^{φ} is a linear application , and there exists M > 0 such that : ρ(Ax) ≤ Mρ(x) for any x ∈ L^{φ}. H′_{3} T : L^{φ} → L^{φ} is an application and for α_{0} = \frac{exp (b)}{c_{0}} with c_{0}b < 1 there exists k > 0 such that: ρ(α_{0}(T x − T y)) ≤ kρ(x − y). Consider now the following perturbed integral equation.

u(t) = exp (−t) exp (−tA)f_{0} + ∫_{0}^{t} exp (s − t) exp ((s − t)A) T u(s)ds (II).

The same techniques than in the proof of Theorem 2.1 are used to establish Theorem 2.3 below by taking care of the choose of λ in (1, \frac{1}{1−e^{−b}}] , which gives

ρ(∫_{0}^{t} λe^{s−t}e^{(s−t)A}x(s)ds) ≤ lim inf(∑_{i=0}^{n−1} λ(t_{i+1} − t_{i})e^{t_{i}−t}ρ(e^{(t_{i}−t)A}x(t_{i})) and

∑_{i=0}^{n−1} λ(t_{i+1} − t_{i})e^{t_{i}−t} ≤ λ ∫_{0}^{t} e^{s−t}ds ≤ 1.

Theorem 2.3 Assume that for α_{1} ≥ eb, there exists M > 0 such that ρ(α_{1}Ax) ≤ Mρ(x) for any x ∈ L^{φ} and there exists k > 0 such that ρ(e(T x − T y)) ≤ kρ(x − y) for any x, y in L^{φ}. Then, the perturbed integral equation (II) has a solution u ∈ C([0, b], L^{φ}).

Remark. By using the same technics as in the proof of Theorem 2.3, we can prove the existence of a solution of the equation below:

u(t) = e^{−t}f_{0} + ∫_{0}^{t} φ(s − t)e^{(s−t)}T u(s)ds,

where φ : R → R_{∗}^{+} is a continuous function satisfying ∫_{0}^{b} φ(−s)ds < 1.

Conclusion Concerning the equations (I) and (II), Theorem 2.1 and Theorem 2.3 give local solutions because of the constraint on b. In this frame, we notice that if A is ρ-Lipschitz i.e. if there exists M > 0 such that ρ(Ax) ≤ Mρ(x) for any x ∈ L^{φ} , then the equation (I) and the equation (II) have a solution in [0, \frac{1}{e}].

Example of the equation (I). Let φ be a Musielak-Orlicz function on a measurable space ([0, 1], A, μ), ρ_{φ} be a modular defined by

ρ_{φ}(u) = ∫_{0}^{1} φ(s, |u(s)|)ds,

for any u ∈ L^{φ} and α_{0} > max(e^{−1}, eb^{2}), c_{0} ∈ [\frac{eb}{α_{0}} , \frac{1}{b}[. Assume that ρ_{φ} is convex satisfying the Δ_{2}-condition. In this example, we study the existence of a solution of the following integral equation

u(t) = exp(−tA)f_{0} + ∫_{0}^{t} exp[(s − t)A](∫_{0}^{1} K_{1}(ξ, u(s))dξ)ds (I′),

where K_{1} : [0, 1] × L^{φ} → L^{φ} is a measurable function satisfying

1) lim_{λ→0^{+}} ∫_{0}^{1} φ(ξ, λ|(∫_{0}^{1} K_{1}(s, u)ds).ξ|)dξ = 0 for any u ∈ L^{φ}.

2) |(∫_{0}^{1}(K_{1}(ξ, u(s)) − K_{1}(ξ, v(s)))dξ)|) ≤ k|(u − v)(s)|, for any u, v in L^{φ}, with k ∈]0, 1[. f_{0} is a fixed element in L^{φ} and the operator A is equal to k_{0}I, where I is the identity function of L^{φ} with k_{0} ≤ \frac{1}{α_{0}} . Let T be a mapping from L^{φ} into L^{φ} defined by

T u = ∫_{0}^{1} \frac{c_{0}}{e} K_{1}(s, u)ds.

Hence, we have ρ_{φ}(α_{0}k_{0}x) ≤ α_{0}k_{0}ρ_{φ}(x) for any x ∈ L^{φ}, .i.e. ρ(α_{0}Ax) ≤ α_{0}k_{0}ρ(x) for any x ∈ L^{φ}. Now, we show that T is ρ − \frac{e}{c_{0}} -Lipschitz. At first, by 1), we have ∫_{0}^{1} φ(ξ, λ|T u(ξ)|)dξ → 0 as λ → 0^{+}. Hence, by the definition of L^{φ}, T u ∈ L^{φ} for any u ∈ L^{φ}. On the other hand, let x, y ∈ L^{φ}

ρ_{φ}(\frac{e}{c_{0}} (T x − T y)) = ∫_{0}^{1} φ(s, \frac{e}{c_{0}} |(T x − T y).(s)|)ds
= ∫_{0}^{1} φ(s, | ∫_{0}^{1}(K_{1}(ξ, x(s)) − K_{1}(ξ, y(s)))dξ)|)ds

Therefore, by 2)

ρ_{φ}(\frac{e}{c_{0}} (T x − T y)) ≤ ∫_{0}^{1} φ(s, k|(x − y)(s)|)ds
= ρ_{φ}(k(u − v))
= kρ_{φ}(u − v).

Hence T is ρ − \frac{e}{c_{0}} -Lipschitz. So by Theorem 2.1 the equation (I′) has a solution in C([0, b], L^{φ}).

References

[1] Ait Taleb.A, Hanebaly.E. A fixed point theorem and its application to integral equations in modular function spaces.Proc. Amer. Math. Soc.127, no 8, 2335-2342 (1999) 128 , no. 2, 419-426 (2000).

[2] Khamsi, M.A. Nonlinear Semigroups in Modular Function Spaces. Thèse d’état. Département de Mathématique, Rabat (1994).

[3] Kozlowski.W.M. Modular Function Spaces. Dekker New-York (1988).

[4] Musielak.J. Orlicz Spaces and Modular Spaces. L.N vol. 1034, S.P. (1983).

Department of Mathematics And Informatic BP. 1014, Rabat Morocco E-mail: hajid2@yahoo.fr E-mail: hanebaly@fsr.ac.ma