Abstract. We consider ordinary differential equations u

Wojciech. M. Kozlowski

On Nonlinear Differential Equations in Generalized Musielak-Orlicz Spaces

In tribute to Julian Musielak on his 85th birthday

Abstract. We consider ordinary differential equations u

(t) + (I − T )u(t) = 0, where an unknown function takes its values in a given modular function space being a generalization of Musielak-Orlicz spaces, and T is nonlinear mapping which is nonexpansive in the modular sense. We demonstrate that under certain natural assumptions the Cauchy problem related to this equation can be solved. We also show a process for the construction of such a solution. This result is then linked to the recent results of the fixed point theory in modular function spaces.

2010 Mathematics Subject Classification: Primary 34G20, Secondary 34K30, 65J15, 46E30, 47H09, 46B20, 47H10, 47H20, 47J25.

Key words and phrases: Ordinary differential equation, nonlinear equation, Cauchy problem, initial value problem, fixed point, nonexpansive mapping, modular function space, Orlicz space, Musielak-Orlicz space, convex modular.

1. Introduction. The purpose of this paper is to prove the existence of a

solution of a Cauchy problem given by a differential equation u ⁰ (t)+(I −T )u(t) = 0,

where an unknown function takes its values in a modular function space, and T is

nonlinear mapping which is nonexpansive in the modular sense, and not necessarily

in the norm sense. Theory of Musielak-Orlicz spaces grew out of the theory of Orlicz

and modular spaces in late 1950s, see [56, 57], and then achieved its established

status in a seminal book by Musielak [55] which laid solid foundations for the theory

and its applications. Modular function spaces are natural generalizations of both

function and sequence variants of many important, from applications perspective,

spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-

Lozanovskii spaces and many others, see the book by Kozlowski [42] for an extensive

list of examples and special cases. See also the paper by Cerda, Hudzik and Mastylo

[7] for the definition and geometrical properties of Calderon-Lozanovskii spaces.

The results and methods of fixed point theory, applied to spaces of measurable functions, have been used extensively in the field of differential and integral equ- ations. Since the 1930s many prominent mathematicians like Orlicz and Birnbaum recognized that using the methods of L ^p -spaces alone created many complications and in some cases did not allow to solve some non-power type integral equations, see [5, 59, 60]. They considered spaces of functions with some growth properties different from the power type growth control provided by the L ^p -norms. Orlicz and Birnbaum considered function spaces defined as follows:

L ^ϕ =



f : R → R; ∃λ > 0 : Z

R ϕ λ |f(x)|

dm(x) < ∞

 ,

where ϕ : [0, ∞) → [0, ∞) was assumed to be a convex function increasing to infinity, i.e. the function which to some extent behaves similarly to power functions ϕ(t) = t ^p . Let us mention two other typical examples of such functions: ϕ 1 (t) = e ^t − t − 1 or ϕ ² (t) = e ^t

− 1. The possibility of introducing the structure of a linear metric in L ^ϕ as well as the interesting properties of these spaces, later named Orlicz spaces, and many applications to differential and integral equations with kernels of nonpower types were among the reasons for the development of the theory of Orlicz spaces, their applications and generalizations. Consider for example the following Hammerstein nonlinear integral equation which plays an important role in the elasticity theory:

f (x) = Z 1

0

k(x, y)ϕ(f (y))dy,

where ϕ(u) is a function which increases more rapidly than an arbitrary power function. Krasnosel’skii and Rutickii, [52], showed that the Hammerstein operator defined by the right member of this integral equation does not operate in any of the L ^p spaces. And yet, they showed how to find an Orlicz space where the Hammerstein operator is well defined and posses properties allowing to use some fixed point theorems for solving the corresponding integral equation.

Many successful applications of Orlicz spaces led to several extensions and gene- ralizations. Using the apparatus of abstract modular spaces introduced by Nakano in [58] and then developed further by Musielak and Orlicz, see e.g. [56, 57], Mu- sielak developed a theory of generalized Orlicz spaces, known in the contemporary literature as Musielak-Orlicz spaces, see the book by Musielak [55].

Given a a nontrivial, σ-finite measure space (Ω, Σ, µ), and a real-valued function ϕ : Ω × [0, ∞) → [0, ∞) which satisfies the following conditions

(i) ϕ(ω, u) is a non-decreasing, continuous function of u such that ϕ(ω, 0) = 0, ϕ(ω, u) > 0 for u > 0, ϕ(ω, 0) → ∞ as u → ∞,

(ii) ϕ(ω, u) is a Σ-measurable function of ω for all u ≥ 0.

Denoting by M the space of all Σ-measurable, real-valued functions with equality µ-almost everywhere, we can define the Orlicz-Musielak modular on a function f ∈ M by

(1) ρ(f ) =

Z

Ω

ϕ(ω, |f(ω)|)dµ.

Keeping this in mind, the Musielak-Orlicz space is defined as a corresponding modular space, that is

(2) L ^ϕ = n

f ∈ M : Z

Ω

ϕ(ω, λ |f(ω)|)dµ → 0 as λ → 0 ⁺ o .

Musielak-Orlicz spaces have been proven to be very useful in application to integral and differential equations due to their flexibility and generality (e.g. they do not have to be symmetric spaces). They have been generalized to a more abstract and even more flexible setting of modular functions spaces introduced by Kozlowski in [40, 41, 42]. We refer the reader to the Section “Preliminaries” for the brief recollection of the fundamentals of the theory of modular function spaces.

There is a solid body of work on the subject of ordinary and partial differential equations in Orlicz and Musielak Orlicz spaces, especially in the context of Orlicz- Sobolev and Musielak-Orlicz-Sobolev spaces, see for instance [21, 64, 22, 26, 11], and more recent works often related to applications to modeling of “smart fluids”, see e.g. [63, 14, 25, 3, 23, 24, 4] and the papers referenced there. Frequently in these applications, a nonlinear extension of modular function spaces are used leading to the concepts of the modular metric spaces, see e.g. [9, 10, 1].

Typically these results use the classical techniques of differential equations with values in Banach spaces. Our approach is to use the modular notions, like the ρ- nonexpansiveness, whenever this is practical. Our results generalize the results of Khamsi from [32] obtained for norm-continuous, nonexpansive mappings acting in Musielak-Orlicz spaces L ^ϕ with ϕ satisfying the Musielak-Orlicz version of the ∆ 2

condition, in relation to the problem of existence of ρ-nonexpansive semigroups of mappings, which - on their own - extended classical Banach space results of [12, 62].

2. Preliminaries. Let us introduce basic notions related to modular function spaces and related notation which will be used in this paper. For further details we refer the reader to preliminary sections of the recent articles [35, 36, 13] or to the survey article [46]; see also [40, 41, 42] for the standard framework of modular function spaces.

Let Ω be a nonempty set and Σ be a nontrivial σ-algebra of subsets of Ω. Let P be a δ-ring of subsets of Ω, such that E ∩ A ∈ P for any E ∈ P and A ∈ Σ.

Let us assume that there exists an increasing sequence of sets K n ∈ P such that Ω = S

K n . By E we denote the linear space of all simple functions with supports from P. By M ∞ we will denote the space of all extended measurable functions, i.e. all functions f : Ω → [−∞, ∞] such that there exists a sequence {g ⁿ } ⊂ E,

|g ⁿ | ≤ |f| and g ⁿ (ω) → f(ω) for all ω ∈ Ω. By 1 ^A we denote the characteristic function of the set A.

Definition 2.1 Let ρ : M ∞ → [0, ∞] be a nontrivial, convex and even function.

We say that ρ is a regular convex function pseudomodular if:

(i) ρ(0) = 0;

(ii) ρ is monotone, i.e. |f(ω)| ≤ |g(ω)| for all ω ∈ Ω implies ρ(f) ≤ ρ(g), where

f, g ∈ M ∞ ;

(iii) ρ is orthogonally subadditive, i.e. ρ(f1 A∪B ) ≤ ρ(f1 ^A )+ρ(f 1 B ) for any A, B ∈ Σ such that A ∩ B 6= ∅, f ∈ M;

(iv) ρ has the Fatou property, i.e. |f n (ω) | ↑ |f(ω)| for all ω ∈ Ω implies ρ(f ⁿ ) ↑ ρ(f ) , where f ∈ M ∞ ;

(v) ρ is order continuous in E, i.e. g ⁿ ∈ E and |g ⁿ (ω) | ↓ 0 implies ρ(g ⁿ ) ↓ 0.

Similarly, as in the case of measure spaces, we say that a set A ∈ Σ is ρ-null if ρ(g1 A ) = 0 for every g ∈ E. We say that a property holds ρ-almost everywhere if the exceptional set is ρ-null. As usual we identify any pair of measurable sets whose symmetric difference is ρ-null as well as any pair of measurable functions differing only on a ρ-null set. With this in mind we define M = {f ∈ M ∞ : |f(ω)| <

∞ ρ − a.e}, where each element is actually an equivalence class of functions equal ρ -a.e. rather than an individual function.

Definition 2.2 We say that a regular function pseudomodular ρ is a regular co- nvex function modular if ρ(f) = 0 implies f = 0 ρ − a.e.. The class of all nonzero regular convex function modulars defined on Ω will be denoted by <.

Definition 2.3 ([40, 41, 42]) Let ρ be a convex function modular. A modular function space is the vector space L ρ = {f ∈ M : ρ(λf) → 0 as λ → 0}.

The following notions will be used throughout the paper.

Definition 2.4 Let ρ ∈ <.

(a) We say that {f n } is ρ-convergent to f and write f ⁿ → f (ρ) if and only if ρ(f n − f) → 0.

(b) A sequence {f ⁿ } where f ⁿ ∈ L ^ρ is called ρ-Cauchy if ρ(f n − f ^m ) → 0 as n, m → ∞.

(c) A set B ⊂ L ^ρ is called ρ-closed if for any sequence of f n ∈ B, the convergence f n → f (ρ) implies that f belongs to B.

(d) A set B ⊂ L ^ρ is called ρ-bounded if sup{ρ(f − g) : f ∈ B, g ∈ B} < ∞.

(e) A set B ⊂ L ^ρ is called strongly ρ-bounded if there exists β > 1 such that M β (B) = sup {ρ(β(f − g)) : f ∈ B, g ∈ B} < ∞.

Since ρ fails in general the triangle identity, many of the known properties of limit

may not extend to ρ-convergence. For example, ρ-convergence does not necessarily

imply ρ-Cauchy condition. However, it is important to remember that the ρ-limit

is unique when it exists. The following proposition brings together few facts that

will be often used in the proofs of our results.

Proposition 2.5 Let ρ ∈ <.

(i) L ρ is ρ-complete.

(ii) ρ-balls B ρ (x, r) = {y ∈ L ^ρ : ρ(x − y) ≤ r} are ρ-closed and ρ-a.e. closed.

(iii) If ρ(αf n ) → 0 for an α > 0 then there exists a subsequence {g ⁿ } of {f ⁿ } such that g n → 0 ρ − a.e.

(iv) ρ(f) ≤ lim inf ρ(f ⁿ ) whenever f n → f ρ−a.e. (Note: this property is equivalent to the Fatou Property).

Definition 2.6 The following formula defines a norm in L ρ (frequently called the Luxemburg norm) :

kfk ^ρ = inf {α > 0 : ρ(f/α) ≤ 1}.

Remark 2.7 It is not difficult to prove that k·k ^ρ defines actually a norm such that kfk ^ρ ≤ kgk ^ρ whenever |f)| ≤ |g| ρ-a.e. It is also straightforward to demonstrate that kf ⁿ k ^ρ → 0 if and only if ρ(αf ⁿ ) → 0 for every α > 0. See Theorem 1.6 in [55].

The above remarks immediately implies the next proposition.

Proposition 2.8 Let ρ ∈ <. If C ⊂ L ^ρ is ρ-closed then C is closed with respect to the Luxemburg norm.

Since every ρ ∈ < is a left-continuous, convex modular we have the following result, see Theorems 1.5 and 1.8 in [55].

Proposition 2.9 Let ρ ∈ <. The following assertions are true:

(a) If kfk ^ρ < 1 then ρ(f) ≤ kfk ^ρ (b) kfk ρ ≤ 1 if and only if ρ(f) ≤ 1.

Using the definition of the Luxemburg norm, it is easy to prove the following proposition.

Proposition 2.10 Let ρ ∈ < and let f ∈ L ^ρ . Then kfk ^ρ > 1 implies ρ(f) ≥ kfk ^ρ . In the sequel, we will use the following important result being an immediate corollary to Proposition 2.10.

Proposition 2.11 Let ρ ∈ <. If C ⊂ L ^ρ is ρ-bounded then C is bounded with respect to the Luxemburg norm.

We will also need the definition of the ∆ 2 -property of a function modular, see

e.g. [42, 13].

Definition 2.12 Let ρ ∈ <. We say that ρ has the ∆ 2 -property if sup

n ρ(2f n , D k ) → 0 whenever D k ↓ ∅ and sup

n ρ(f n , D k ) → 0.

The next result is a straightforward consequence from Definitions 2.6 and 2.12, and from Remark 2.7.

Proposition 2.13 Let ρ ∈ <. The ρ-convergence is equivalent to the convergence with respect to the Luxemburg norm k · k ^ρ if and only if ρ has the ∆ 2 -property.

A specific subspace of L ρ will be of particular importance for our discussions in this paper.

Definition 2.14 Define L ⁰ ρ = {f ∈ L ^ρ ; ρ(f, ·) is order continuous} and E ^ρ = {f ∈ L ^ρ ; λf ∈ L ⁰ ρ f or every λ > 0 }.

The position of E ρ with respect to L ρ is characterized in the following theorem.

Theorem 2.15 ([40, 41, 42]) Let ρ ∈ <.

(a) L ρ ⊃ L ⁰ ρ ⊃ E ^ρ ,

(b) E ρ has the Lebesgue property, i.e. ρ(αf, D k ) → 0 for α > 0, f ∈ E ^ρ and D k ↓ ∅.

(c) E ρ is the closure of E (in the sense of k · k ρ ).

(d) E ρ = L ρ if and only if ρ has the ∆ 2 -property.

An extremal flexibility gained by using the apparatus of the modular function spaces can be illustrated as follows: the operator itself is used for the construction of a modular and hence a space in which this operator has required properties. Let us consider for instance the following Uryshon integral operator, being a generalization of the Hammerstein operator:

T (f )(x) = Z 1

0

k(x, y, |f(y)|)dy + f ⁰ (x),

where f 0 is a fixed function and f : [0, 1] → R is Lebesgue measurable. For the kernel k we assume that

(a) k : [0, 1] × [0, 1] × R + → R ⁺ is Lebesgue measurable, (b) k(x, y, 0) = 0,

(c) k(x, y, .) is continuous, convex and increasing to +∞, (d) Z 1

0

k(x, y, t)dx > 0 for t > 0 and y ∈ (0, 1),

Assume in addition that for almost all t ∈ [0, 1] and measurable functions f, g there holds

Z 1 0

n Z ¹

0

k(t, u, |k(u, v, |f(v)|) − k(u, v, |g(v)|)|)dv o du ≤

Z 1 0

k(t, u, |f(u) − g(u)|)du.

Setting ρ(f) = Z 1 0

n Z ¹

0

k(x, y, |f(y)|)dy o

dx and using Jensen’s inequality it is easy to show that ρ is a nonnegative, even, convex nonlinear functional on the space of measurable functions L ρ = {f : [0, 1] → R : ∃λ > 0, ρ(λf) < ∞}, and that ρ(T (f ) − T (g)) ≤ ρ(f − g), that is, T is nonexpansive with respect to ρ. We will come back to this example towards the end of this paper, see Example 4.8.

An additional importance for applications of modular function spaces consists in the richness of structure of modular function spaces, that - besides being Banach spaces (or F-spaces in a more general settings) - are equipped with modular equ- ivalents of norm or metric notions, and also are equipped with almost everywhere convergence and convergence in submeasure. As the above example of the Urysohn operator vividly demonstrated, in many situations in differential and integral equ- ations, approximation and fixed point theory, modular type conditions are much more natural and modular type assumptions can be more easily verified than their metric or norm counterparts. There are also important results that can be proved only using the apparatus of modular function spaces.

3. Vitali Property. According to Proposition 2.13, the ρ-convergence and the k · k ^ρ -convergence are in general equivalent if only if ρ satisfies ∆ 2 . However, it is legitimate to ask on what subsets of L ρ such equivalence may hold even ρ does not have ∆ 2 . In this paper we introduce a new concept of sets with the Vitali property that will play this role. The reference to Giuseppe Vitali is justified by the following version of the Vitali Convergence Theorem which was proved in the context of modular function spaces in [42], Theorem 2.4.3.

Theorem 3.1 ([42]) Let ρ ∈ <. Let f ⁿ ∈ E ^ρ , f ∈ L ^ρ and f n → f ρ-a.e. Then the following conditions are equivalent:

(i) f ∈ E ^ρ and kf ⁿ − fk ^ρ → 0.

(ii) for every α > 0 the subadditive measures ρ(αf n , ·) are order equicontinuous, that is, if E k ∈ Σ are such that E ^k ↓ ∅ then lim _k

→∞ sup

n ∈N

ρ(αf n , E k ) = 0.

Definition 3.2 A set C ⊂ L ρ is said to posses the Vitali property if C ⊂ E ρ , and for any g ∈ L ρ and g n ∈ C with ρ(g ⁿ − g) → 0 there exists a subsequence {g ⁿ

} of {g n } such that for every α > 0 the subadditive measures ρ(αg ⁿ

, ·) are order equicontinuous.

Our next result characterizes sets with the Vitali property as those subsets of

E ρ on which the ρ-convergence and the k · k ^ρ -convergence are indeed equivalent.

Theorem 3.3 Let ρ ∈ <. A set C ⊂ L ρ has the Vitali property if and only the following two conditions are satisfied:

(i) C ⊂ E ρ .

(ii) If g ∈ L ^ρ and g n ∈ C with ρ(g ⁿ − g) → 0 then kg ⁿ − gk ^ρ → 0.

Proof Let us assume first that C ⊂ L ^ρ has the Vitali property. Hence, by de- finition, the condition (i) is satisfied. To prove (ii), take a sequence {g ⁿ }, and a function g ∈ L ^ρ such that ρ(g n − g) → 0. By Proposition 2.5 there exists a subsequence {g ⁿ

} such that

(3) g n

→ g ρ − a.e.

Because C has the Vitali property, there exists a subsequence {g ⁿ

} such that for every α > 0 the submeasures ρ(αg n

, ·) are order equicontinuous. Hence, using (3) and the Vitali Theorem (Theorem 3.1), we conclude that kg ⁿ

− gk ^ρ → 0. Since this reasoning can be repeated for any subsequence of {g ⁿ }, every subsequence of {g ⁿ } contains a subsequence converging in k · k ^ρ to g, proving (ii).

Conversely, let us assume that (i) and (ii) are satisfied. Assume to the contrary that C does not have the Vitali property. Then there exists a sequence {g k } of elements of C, g ∈ L ρ and α > 0 such that the submeasures ρ(αg n , ·) are not order equicontinuous while ρ(g k − g) → 0. The latter fact implies by (ii) that kg ⁿ −gk ^ρ → 0, which also implies, via closedness of E ^ρ that g ∈ E ρ . By Proposition 2.5 we can assume, without loosing generality, that g n

→ g ρ-a.e. From Theorem 3.1 we conclude that kg n − gk ^ρ cannot tend to zero. Contradiction completes the

proof. _

Remark 3.4 Combining Proposition 2.8 with Theorem 3.3, we can easily see that a set with the Vitali property is ρ-closed if and only if it is k · k ^ρ -closed.

Remark 3.5 Let C ⊂ L ^ρ be a set with the Vitali property and let a, b ∈ R. Let u : [a, b] → C be a ρ-continuous function, that is, ρ(u(t ⁿ ) − u(t)) → 0 provided t n → t. It follows immediately from Theorem 3.3 that u is k · k ^ρ -continuous.

As an immediate corollary to Remark 3.5 we obtain the following important result.

Remark 3.6 Let Z be a separable linear subspace of (E ρ , k · k ^ρ ) and let C ⊂ Z have the Vitali property. Assume that the function u : [a, b] → C is ρ-continuous.

Then u is the Bochner integrable function with respect the the Lebesgue measure m on [a, b], i.e. u ∈ L ¹ (Ω, Z, m).

In this context, let us discuss the separability of (E ρ , k · k ^ρ ). First, we need the

following definition.

Definition 3.7 The function modular ρ ∈ < is called separable if kf 1 ( ·) k ^ρ ) is a separable set function for each f ∈ E, which means that there exists a countable A ⊂ P such that to every A ∈ P there corresponds a sequence {A ^k } of elements of A with

(4) ρ(αf, A∆A k ) → 0

for every α > 0, where ∆ denotes the symmetric set difference.

We are now ready to formulate the following characterization of separable (E ρ , k · k ^ρ ) spaces, see Theorem 2.5.4 in [42].

Theorem 3.8 ([42]) Let ρ ∈ <. The space (E ρ , k · k ^ρ ) is a separable Banach space if and only ρ is separable.

Finally, we can combine the last two results into the following very useful state- ment.

Proposition 3.9 Let ρ ∈ < be a separable function modular. If u : [a, b] → C is ρ -continuous, where C ⊂ E ^ρ has the Vitali property, then u ∈ L ¹ (Ω, Z, m).

Let us finish this section with few examples of sets with the Vitali property.

Example 3.10 If ρ has ∆ 2 property then every set C ⊂ L ^ρ has the Vitali property.

Example 3.11 Let C ⊂ E ^ρ . If there exists g ∈ E ^ρ such that |f(ω) ≤ |g(ω)| ρ-a.e for every f ∈ C then C has the Vitali property.

Example 3.12 Let C ⊂ E ^ρ be k · k ^ρ -conditionally compact. Then C has the Vitali property (see Theorem 2.5.1 in [42]) .

4. Solution of the Initial Value Problems in Modular Function Spaces.

To the end of this paper we will be considering the following initial value problem for an unknown function u : [0, A] → C, where C ⊂ E ^ρ :

(5)

( u(0) = f

u ⁰ (t) + (I − T )u(t) = 0,

where f ∈ C and A > 0 are fixed and T : C → C is ρ-nonexpansive.

To start with, let us recall the definition of ρ-nonexpansive mappings.

Definition 4.1 Let ρ ∈ < and let C ⊂ L ^ρ be nonempty. A mapping T : C → C

is called a ρ-nonexpansive mapping if ρ(T (f) − T (g)) ≤ ρ(f − g), for all f, g ∈ C.

Let us first consider the following question: Are the ρ-nonexpansive mappings really different from the mappings nonexpansive with respect to the Luxemburg norm associated with the modular ρ? First we will show the following simple result.

Proposition 4.2 Let ρ ∈ <. If for every λ > 0

(6) ρ (λ (T (f ) − T (g))) ≤ ρ (λ(f − g)) then, kT (f) − T (g)k ^ρ ≤ kf − gk ^ρ .

Proof Assume to the contrary that there exist f, g ∈ L ^ρ and α > 0 such that kf − gk ^ρ < α < kT (f) − T (g)k ^ρ .

Then, f − g

α

ρ

< 1 , which by Proposition 2.9 part (a) implies that ρ

 f − g α



< 1 . It also implies that

1 <

T (f ) − T (g) α

ρ

,

which, by Proposition 2.9 part (b), yields 1 < ρ

 T (f ) − T (g) α



. Finally, setting λ = α ⁻¹ , we obtain

ρ (λ(f − g)) < 1 < ρ (λ (T (f) − T (g))) .

Contradiction completes the proof. 

In view of Proposition 4.2, we need to ask whether the inequality (6) needs to hold for every λ > 0 in order to ensure the norm nonexpansiveness? If we knew that it sufficed to assume it merely for λ = 1, then there would be no real reason to consider ρ-nonexpansiveness. The answer to this question can be found in the following simple example of a mapping which is ρ-nonexpansive but it is not k.k ^ρ - nonexpansive.

Example 4.3 [37] Let X = (0, ∞) and Σ be the σ-algebra of all Lebesgue measu- rable subsets of X. Let P denote the δ-ring of subsets of finite measure. Define a function modular by

ρ(f ) = 1 e ²

Z ∞

0 |f(x)| ^x+1 dm(x).

Let B be the set of all measurable functions f : (0, ∞) → R such that 0 ≤ f(x) ≤ 1/2. Consider the map

T (f )(x) =

 f (x − 1), for x ≥ 1, 0, for x ∈ [0, 1].

Clearly, we have T (B) ⊂ B. For every f, g ∈ B and λ ≤ 1, we have

ρ (λ (T (f ) − T (g))) ≤ λρ (λ(f − g)) ,

which implies that T is ρ-nonexpansive. On the other hand, if we take f = 1 [0,1] , then

kT (f)k ^ρ > e ≥ kfk ^ρ ,

which clearly implies that T is not k.k ρ -nonexpansive. Note that T is linear.

Returning to our Cauchy problem (5), the meaning of the above considerations is that for such mappings T the classical methods of differential equations for functions with values in Banach spaces would not work. It is also worthwhile mentioning that, due to the indirect definition of the Luxemburg norm, quite often it is much more convenient to evaluate formulas expressed only in terms of a modular which is, in applications, typically given by a direct formula allowing numerical computations.

Let us introduce the following convenient notations which will be used thro- ughout this paper. For any t > 0 we define

(7) K(t) = 1 − e ^−t =

Z t 0

e ^{s −t} ds.

We define the ρ-diameter of a set C ⊂ L ρ as

(8) δ ρ (C) = sup

f,g∈C ρ(f − g).

Observe that δ ρ (C) < ∞ whenever the set C is ρ-bounded.

For any Bochner measurable function v : [0, A] → L ρ , where A > 0, and any t ∈ [0, A], we define

(9) S(v)(t) =

Z t 0

e ^{s −t} v(s)ds.

Similarly, we will denote

(10) S τ (v)(t) =

n X −1 i=0

(t i+1 − t ⁱ )e ^t

^−t v(t i ), for any τ = {t 0 , . . . , t n }, a subdivision of the interval [0, A].

Let us start with the following technical lemma.

Lemma 4.4 Let ρ ∈ < be separable. Let x, y : [0, A] → L ^ρ be two Bochner-integrable k · k ^ρ -bounded functions, where A > 0. Then for every t ∈ [0, A] we have

(11) ρ 

e ^−t y(t) + Z t

0

e ^{s −t} x(s)ds 

≤ e ^−t ρ(y(t)) + K(t) sup

s∈[0,t]

ρ(x(s)).

Proof Without any loss of generality we can assume that sup

s ∈[0,t]

ρ(x(s)) < ∞. Let τ = {t ⁰ , . . . , t n } be a subdivision of the interval [0, A]. Define

(12) Z τ (t) = e ^−t y(t) + S τ (x)(t) = e ^−t y(t) +

n X −1 i=0

(t i+1 − t ⁱ )e ^t

^−t x(t i ).

It follows from the Lebesgue Dominated Convergence Theorem for Bochner in- tegrals, that for every t ∈ [0, A], Z τ (t) converges in k.k ρ to Z(t) defined by

(13) Z(t) = e ^−t y(t) + S(x)(t) = e ^−t y(t) + Z t

0

e ^{s −t} x(s)ds, whenever |τ| = sup

0 ≤i≤n |t ⁱ⁺¹ −t ⁱ | → 0. Using the Fatou Property of ρ and Proposition 2.5, we can easily see that

(14) ρ(Z(t)) ≤ lim inf

|τ|→0 ρ(Z τ (t)).

Since

(15) e ^−t +

n X −1 i=0

(t i+1 − t ⁱ )e ^t

^−t ≤ 1,

and ρ is convex, it follows that

(16) ρ(Z τ (t)) ≤ e ^−t ρ(y(t)) +  ⁿ X ⁻¹

i=0

(t i+1 − t ⁱ )e ^t

^−t  sup

s∈[0,t]

ρ(x(s)).

By combining (16) with (14) and letting |τ| → 0 we get the desired inequality

(11). _

We are now ready to prove our main result.

Theorem 4.5 Let ρ ∈ < be separable. Let C ⊂ E ρ be a nonempty, convex, ρ- bounded, ρ-closed set with the Vitali property. Let T : C → C be a ρ-nonexpansive mapping. Let us fix f ∈ C and A > 0 and define the sequence of functions u n : [0, A] → C by the following inductive formula:

(17)

( u 0 (t) = f

u n+1 (t) = e ^−t f + R t

0 e ^s−t T (u n (s))ds.

Then for every t ∈ [0, A] there exists u(t) ∈ C such that

(18) ρ(u n (t) − u(t)) → 0

and the function u : [0, A] → C defined by (18) is a solution of the Initial Value Problem (5).

Proof We will show that u n (t) ∈ C for every n ∈ N and every t ∈ [0, A]. Indeed, for a given subdivision τ of [0, A], define

u ^τ _n+1 (t) = e ^−t f + S τ

 T (u n (t i )) 

(t).

Since C is a a convex set, it is easy to prove by induction that u ^τ n (t) ∈ C. Note that, by the properties of the Bochner integral, ku ^τ n (t) − u ⁿ (t) k ^ρ → 0. Since C is k · k ^ρ -closed it is also ρ-closed. Hence u n (t) ∈ C as claimed.

Now we are going to prove that

(19) ρ(u n+p (t) − u ⁿ (t)) ≤ K ⁿ⁺¹ (A)δ ρ (C)

for all t ∈ [0, A] and every n, p ∈ N. We will prove (19) by induction with respect to n. For n = 0 we have

(20) u p (t) − u ⁰ (t) =

Z t 0

e ^{s −t} T (u p −1 (s))ds.

Applying Lemma 4.4 with y(t) = 0 and x(t) = T (u p −1 (s)) − f we have (21) ρ(u p (t) − u ⁰ (t)) ≤ K(t) sup

s ∈[0,t]

ρ(T (u p −1 (s)) − f) ≤ K(A)δ ^ρ (C),

because both f, and T (u p−1 (s)) belong to C. Assume now that (19) holds for an n ∈ N. Then

(22) u n+1+p (t) − u ⁿ⁺¹ (t) = Z t

0

e ^s−t 

T (u n+p (s)) − T (u ⁿ (s))  ds.

Applying Lemma 4.4 again, with y(t) = 0 and x(t) = T (u n+p (s)) − T (u ⁿ (s)) we have

ρ(u n+1+p (t) − u ⁿ⁺¹ (t)) ≤ K(A) sup

s∈[0,t]

ρ 

T (u n+p (s)) − T (u ⁿ (s))  (23)

≤ K(A) sup

s ∈[0,t]

ρ(u n+p (s) − u ⁿ (s))

≤ K ⁿ⁺¹ (A)δ ρ (C), where we used the ρ-nonexpansiveness of T and the inductive assumption.

It follows directly from (19) that for every t ∈ [0, A], the sequence {u n (t) } is a ρ-Cauchy sequence the latter implies that

(24) ku ⁿ (t) − u(t)k ^ρ → 0.

Rememeber that T is ρ-nonexpansive and hence it is ρ-continuous in C, which - via the Vitali property - implies that T is k · k ^ρ -continuous in C. Using this fact, it is easy to prove by induction that u n is a k · k ^ρ -continuous mapping from [0, A]

to C. Consequently, functions λ n : [0, A] → C defined by λ ⁿ (s) = T (u n (s)) are k · k ^ρ -continuous and bounded. Hence, it follows from the Lebesgue Dominated Convergence Theorem for Bochner integrals that

(25) lim

|τ|→0 kS ^τ (T (u k (t)) − Z t

0

e ^s−t T (u k (s))ds k ^ρ → 0,

for every k ∈ N. Using the ρ-nonexpansiveness of T and the fact that ρ(u n (t) − u(t)) → 0, we have the following

ρ 

S τ (T (u(t)) − S ^τ (T (u n (t))  (26)

≤

n−1 X

i=0

(t i+1 − t ⁱ )e ^t

^−t ρ 

T (u(t)) − T (u ⁿ (t)) 

≤ Ae ^{A −t} ρ(u(t) − u ⁿ (t)) → 0, which, via the Vitali property, implies that

(27) lim

n →∞ kS ^τ (T (u(t)) − S ^τ (T (u n (t)) k ^ρ = 0.

For every t ∈ [0, A] we have now

kS ^τ (T (u(t)) − u(t) + e ^−t f k ^ρ ≤ kS ^τ (T (u(t)) − S ^τ (T (u n (t)) k ^ρ (28)

+ kS ^τ (T (u n (t)) − Z t

0

e ^s−t T (u n (s))ds k ^ρ

+ k Z t

0

e ^{s −t} T (u n (s))ds − u(t) + e ^−t f k ^ρ . Observing that

(29) Z t

0

e ^s−t T (u n (s))ds = u n+1 − e ^−t f and using (27), (25) and (24) we obtain that

(30) lim

|τ|→0 kS ^τ (T (u(t)) − u(t) + e ^−t f k ^ρ = 0.

Hence the function s 7→ e ^s−t T (u(s) is Bochner-integrable on [0, A] and

(31) Z t

0

e ^s−t T (u(s))ds = u(t) − e ^−t f.

Using (31) it can be easily proved, via the standard methods of Bochner inte- gration, that u is differentiable and is a solution of the Initial Value Problem (5).

The proof of Theorem 4.5 is now complete. 

Remark 4.6 It is clear that the solution of the the Initial Value Problem (5) can

be extended to a solution u(t) defined on [0, +∞) such that its restriction to the

interval [0, A] is the ρ-limit of the sequence {u n } defined as in the proof of Theorem

4.5.

Remark 4.7 Theorem 4.5 extends results of [32] proven for norm continuous, no- nexpansive mappings acting in Musielak-Orlicz spaces with the ∆ 2 property.

Example 4.8 Let us go back to the example of the Urysohn operator T from the Preliminary section of this paper:

T (f )(x) = Z 1

0

k(x, y, |f(y)|)dy + f ⁰ (x),

As we saw, T i a ρ-nonexpansive mapping where the convex function modular ρ is defined as

(32) ρ(f ) =

Z 1 0

n Z ¹

0

k(x, y, |f(y)|)dy o dx.

Let us fix an r > 0 and set C = {f ∈ E ρ : ρ(f − f ⁰ ) ≤ r}. It is easy to see that T : C → C. If we assume additionally that there exists a constant M > 0 and a Bochner-integrable function h : [0, 1] × [0, 1] → [0, ∞) such that for every u ≥ 0 and x, y ∈ [0, 1]

(33) k(x, y, 2u) ≤ Mk(x, y, u) + h(x, y),

then the modular ρ has the property ∆ 2 in the sense of Definition 2.12. Using the Vitali property of C and Theorem 4.5 it is then easy to see that the corresponding Initial Value Problem

(34)

( u(0) = f 0

u ⁰ (t) + (I − T )u(t) = 0,

has a solution in C that can be calculated as the ρ-limit of the sequence {u ⁿ } as defined in Theorem 4.5.

Still in the context of Theorem 4.5, let us introduce the following notation: for f ∈ C we write u ^f (t) ≡ f for all t ≥ 0. Let ask a question for which f ∈ C, u ^f are the solutions of (5). The answer will be provided in our next result. First, let us denote by F (T ) the set of all fixed points of T , that is, F (t) = {f ∈ C : T (f) = f}.

Theorem 4.9 Under the assumptions of Theorem 4.5, u f is a solution of (5) if and only if f ∈ F (T ).

Proof It is obvious that for every fixed point f ∈ F (T ), u f is a solution of (5).

Conversely, let us assume that u f is a solution of (5). Arguing like in the proof of Theorem 4.5, we obtain that

(35) ρ(f − u ⁿ (t)) ≤ K ⁿ⁺¹ (A)δ ρ (C) for any n ∈ N, any A > 0 and any t ∈ [0, A]. Noting that (36) e ^−t f + K(t)T (f ) − u ⁿ⁺¹ (t) =

Z t 0

e ^{s −t} 

T (f ) − T (u ⁿ (s)) 

ds

and applying Lemma 4.4 we obtain (37) ρ 

e ^−t f + K(t)T (f ) − u ⁿ⁺¹ (t) 

≤ K(t) sup

s ∈[0,t]

ρ 

T (f ) − T (u ⁿ (s)  .

Using the fact that T is ρ-nonexpansive and inequality (35) we get

(38) ρ 

e ^−t f + K(t)T (f ) − u ⁿ⁺¹ (t) 

≤ K ⁿ⁺² (A)δ ρ (C),

hence {u n (t) } ρ-converges to e ^−t f + K(t)T (f ) for any t ≥ 0. The uniqueness of the ρ-limit implies that

(39) f = u f (t) = e ^−t f + K(t)T (f ),

which yields T (f) = f, as claimed. _

The above-mentioned result establishes a connection between the theory of diffe- rential equations in modular function spaces and the fixed point theory for nonlinear mappings acting in modular function spaces. The latter theory has been a subject to intensive study since 1990s, see e.g. [37, 38, 33, 34, 15, 16, 17, 35, 36, 45, 46, 13, 49, 51]. The fixed point theorem which is most relevant for the considerations in this paper is a result by Khamsi and Kozlowski, [36]. First we need to recall a notion of uniformly convex function modulars.

Definition 4.10 Let ρ ∈ <. We define the following uniform convexity type properties of the function modular ρ:

(i) Let r > 0, ε > 0. Define

D 1 (r, ε) = {(f, g); f, g ∈ L ^ρ , ρ(f ) ≤ r, ρ(g) ≤ r, ρ(f − g) ≥ εr}.

Let

δ 1 (r, ε) = inf n 1 − 1

r ρ  f + g 2

 ; (f, g) ∈ D ¹ (r, ε) o

, if D 1 (r, ε) 6= ∅, and δ 1 (r, ε) = 1 if D 1 (r, ε) = ∅.

(ii) We say that ρ is uniformly convex (UUC1) if for every s ≥ 0, ε > 0 there exists

η 1 (s, ε) > 0 depending on s and ε such that

δ 1 (r, ε) > η 1 (s, ε) > 0 f or r > s.

Example 4.11 It is known that in Orlicz spaces, the Luxemburg norm is uniformly convex if and only ϕ is uniformly convex and ∆ 2 property holds; this result can be traced to early papers by Luxemburg [53], Milnes [54], Akimovic [2], and Kamin- ska [31]. It is also known that, under suitable assumptions, the modular uniform convexity in Orlicz spaces is equivalent to the very convexity of the Orlicz function [38, 8]. Remember that the function ϕ is called very convex if or every ε > 0 and any x 0 > 0, there exists δ > 0 such that

ϕ

 1 2 (x − y)



≥ ε

2 (ϕ(x) + ϕ(y)) ≥ εϕ(x ⁰ ), implies

ϕ

 1 2 (x + y)



≤ 1

2 (1 − δ) (ϕ(x) + ϕ(y)) .

Typical examples of Orlicz functions that do not satisfy the ∆ 2 condition but are very convex are: ϕ 1 (t) = e ^|t| − |t| − 1 and ϕ ² (t) = e ^t

− 1, [54, 52]. Therefore, these are the examples of Orlicz spaces that are not uniformly convex in the norm sense and hence the classical Kirk theorem cannot be applied. However, these spaces are uniformly convex in the modular sense, and respective modular fixed point results can be applied.

Theorem 4.12 ([36] ) Assume ρ ∈ < is uniformly convex (UUC1). Let C be a ρ-closed ρ-bounded convex nonempty subset of L ρ . Then any T : C → C ρ- nonexpansive mapping has a fixed point. Moreover, the set of all fixed points F (T ) is convex and ρ-closed.

Note that the statement of Theorem 4.12 is completely parallel to that of the Browder/Gohde/Kirk classic fixed point theorem but formulated purely in terms of function modulars without any reference to norms. Also, note that the results in [36] actually extend outside nonexpansiveness and assumes merely asymptotic pointwise ρ-nonexpansiveness of the mapping T . Therefore, Theorem 4.12 can be actually understood as the modular equivalent of the theorem by Kirk and Xu [28], see also [20, 61, 19, 68, 65, 6, 66, 67, 29, 30, 18, 39, 27, 43, 44, 47, 48, 50] and the literature referenced there.

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Wojciech. M. Kozlowski

School of Mathematics and Statistics, University of New South Wales Sydney, NSW 2052, Australia

E-mail: w.m.kozlowski@unsw.edu.au

(Received: 19.08.2013)