On nonlinear integral equations in the space of functions of bounded generalized φ-variation

Daria Bugajewska^{1}, Dariusz Bugajewski^{2} and Grzegorz Lewicki^{3}

October 11, 2006

IMUJ Preprint 2006/07

Abstract. The purpose of this paper is to deal with the superposition operator as well as with solutions to nonlinear integral equations in spaces of functions of bounded generalized φ-variation.

AMS Subject Classification: Primary 45G10, 26A45; Secondary 45D05.

Key words and phrases: Banach fixed point theorem; bounded generalized φ-variation, generalized φ-function, global and local solutions, Hammerstein integral equation, Leray-Schauder alternative for contractions, superposition operator, Volterra-Hammerstein integral equation.

1Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61 -614 Poznań, Poland

2Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61 -614 Poznań, Poland

3Department of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland

1. Introduction

The notion of φ-variation of a real function was introduced by L.C. Young [13] (see also [14]) in connection with investigation of the behaviour of Fourier series. This concept seems to be one of the most important generalizations of the classical variation in the sense of Jordan. It is worth to recall that the space of functions of bounded φ-variation from the point of view of functional analysis and some applications was studied by J. Musielak and W. Orlicz [11], and R. Leśniewicz and W. Orlicz [9]. Moreover, composing functions of bounded φ-variation was investigated by J. Ciemnoczołowski and W. Orlicz [6]; in particular they proved a generalization of the result by M. Josephy [8] concerning composing functions of bounded variation in the sense of Jordan.

Recall that basic results concerning the superposition operator in different spaces, in particular in the space of functions of bounded variation in the sense of Jordan as well as exhaustive references on this topic one can find in [2].

Recall also that the parameter t in φ(t, u) in connection with spaces of functions of bounded φ-variation was introduced and investigated in papers by S. Gniłka (see e.g. [7]). Such spaces are called spaces of functions of generalized bounded φ-variation.

In this paper we would like to pursue two purposes. First, we are interested in the superposition operator acting in the space of functions of generalized bounded φ-variation. In particular, for the large class of functions φ(t, u), we formulate the conditions which ensure the composition operator maps the space of functions of generalized bounded φ-variation into itself (see Corollary 1). Our results extend the results proved by Ciemnoczołowski and Orlicz [6].

Second, we are interested in solutions, in particular in continuous solutions, to nonlinear integral equations which are functions of generalized bounded φ-variation.

Our results generalize the previous ones from the papers [3], [4] and [5]. Let us draw a reader’s attention to Remark 1. In a sense, this remark explains the significance of our results. In particular, for some class of functions φ(t, u) we obtain solutions to equations under consideration, which are functions of bounded variation in the sense of Jordan, constant on each interval of continuity.

The paper is organized as follows; in Section 2 we collect a few definitions and facts which will be needed in the sequel. Section 3 contains results about acting of the autonomous superposition operator in the space of functions of generalized bounded φ-variation. Finally, in Sections 4-6 we deal with solutions to the nonlinear Hammerstein as well as the Volterra-Hammerstein integral equation which are functions of generalized bounded φ-variation. We prove a few existence results concerning local and global solutions to these equations.

The proofs of the theorems from Section 6 are based on the Leray-Schauder alternative for contractions from [12].

2. Preliminaries

In this section we collect some definitions and results which will be needed in the sequel. Throughout this paper we assume that φ : [0, a] × ℝ_{+} → ℝ_{+}, a < +∞ satisfies the following conditions:

(i) φ(t, u) is a continuous, nondecreasing function of u ≥ 0 for every t ∈ [0, a], φ(t, u) → +∞ as u → +∞;

(ii) φ(t, 0) = 0 for every t ∈ [0, a] and φ(0, u) = 0 implies u = 0.

Let X = {x : [0, a] → ℝ}. Recall that for a function x ∈ X, the number

V_{φ}(x) = sup_{π} ∑_{i=1}^{n} φ(s_{i}, |x(t_{i}) − x(t_{i−1})|),

where the supremum is taken over all partitions π : 0 = t_{0} < t_{1} < … < t_{n} = a with intermidiate points s_{i} ∈ [t_{i−1}, t_{i}], i = 1, …, n, is called the generalized φ-variation of the function x in [0, a]. Denote

BV_{φ} = BV_{φ}(I) = {x ∈ X : V_{φ}(λx) < +∞ for some λ > 0},

where I = [0, a]. It is well-known that, if φ satisfies the condition:

(iii) φ(t, u) is a convex function of u for all t ∈ [0, a],

then BV_{φ}(I) with the norm

∥x∥_{V_{φ}} = inf\{ε > 0 : V_{φ}(\frac{x}{ε}) ≤ 1\}

is a Banach space (see [10], Theorem 10.8, p.71 and Theorem 1.5, pp. 2-3). Elements of this space will be called generalized BV_{φ}-functions and solutions to integral equations belonging to this space will be called generalized BV_{φ}- solutions.

Let us denote ψ(u) = sup_{0≤s≤a} φ(s, u) and we will assume that the following

condition is satisfied

(iv) if ψ(u) = 0, then u = 0.

For other basic concepts concerning modular spaces (as e.g. φ-function, s-convexity, the condition Δ_{2} for small u) a reader is refereed to [10].

3. Superposition operator in BV_{φ}(I)

We start with the following

Lemma 1 Let φ : [0, a] × ℝ_{+} → ℝ_{+} satisfy conditions (i) and (ii). Let F_{n} : ℝ → ℝ be a sequence of functions such that F_{n}(0) = 0. Assume that for any v > 0 there exists K_{v} > 0 such that for any u_{1}, u_{2} ∈ [−v, v] and n ∈ ℕ

|F_{n}(u_{1}) − F_{n}(u_{2})| ≤ K_{v}|u_{1} − u_{2}|.

Then for any x ∈ BV_{φ} there exists λ > 0 such that

sup_{n∈ℕ}V_{φ}(λ(F_{n} ∘ x)) < +∞;

P r o o f. Fix x ∈ BV_{φ} and λ > 0 with V_{φ}(λx) < +∞. By [10] Theorem 10.7, (a), p. 69, there exists v > 0 such that for any t ∈ [0, a], |x(t)| < v. By our assumptions, for any partition Π = {t_{o}, t_{1}, ..., t_{l}} of [0, a] and s_{i} ∈ [t_{i−1}, t_{i}], i = 1, ..., l, we have

∑_{i=1}^{l} φ(s_{i}, (λ/K_{v})|F_{n}(x(t_{i})) − F_{n}(x(t_{i−1}))|)

≤ ∑_{i=1}^{l} φ(s_{i}, λ|x(t_{i}) − x(t_{i−1})|) ≤ V_{φ}(λx) < +∞.

Hence sup_{n} V_{φ}((λ/K_{v})(F_{n} ∘ x)) < +∞, as required.

Before presenting next results, we introduce some notation. Let g : ℝ_{+} → ℝ_{+} be a continuous, nondecreasing function such that

(v) lim_{u→+∞} g(u) = +∞;

(vi) g(u) = 0 if and only if u = 0;

(vii) g satisfies Δ_{2} condition for small u.

Assume furthermore that there exist positive constants M, m and u_{o} > 0 such that for any u ∈ [0, u_{o}] and t ∈ [0, a]

mg(u) ≤ φ(t, u) ≤ Mg(u). (1)

Now we can state

Theorem 1 Let φ : [0, a]×ℝ_{+} → ℝ_{+} and g : ℝ_{+} → ℝ_{+} satisfy the conditions (i),(ii), (v)-(vii) and (1). Let F_{n} : ℝ → ℝ be a sequence of functions such that F_{n}(0) = 0. Then the following conditions are equivalent:

(a) For any x ∈ BV_{φ} there exists k > 0 such that

sup_{n∈ℕ}V_{φ}(k(F_{n} ∘ x)) < +∞;

(b) For any v > 0 there exists K_{v} > 0 such that for any u_{1}, u_{2} ∈ [−v, v]

and n ∈ ℕ

g(|F_{n}(u_{1}) − F_{n}(u_{2})|) ≤ K_{v}g(|u_{1} − u_{2}|).

P r o o f. Assume that condition (a) is satisfied and fix x ∈ BV_{φ}. By [10] Theorem 10.7, (b), p. 69, there exists M > 0 such that for any n ∈ ℕ and t ∈ [0, a], |F_{n}(x(t))| < M. (2) Hence there exists k_{1} > 0 such that k_{1}|F_{n}(x(t))| < u_{o}/2 for any t ∈ [0, a] and n ∈ ℕ. Without loss of generality, we can assume that k < 1 and k_{1} < 1. Note that for any partition P = {t_{o}, t_{1}, ..., t_{l}} of [0, a], by (1) and (a),

∑_{i=1}^{l} g(k_{1}k|F_{n}(x(t_{i})) − F_{n}(x(t_{i−1}))|)

≤ ∑_{i=1}^{l} φ(t_{i}, k_{1}k|F_{n}(x(t_{i})) − F_{n}(x(t_{i−1}))|)/m

≤ (sup_{n∈ℕ}V_{φ}(k(F_{n} ∘ x)))/m < +∞.

Consequently for any x ∈ BV_{φ}

sup_{n∈ℕ}V_{g}(k_{1}k(F_{n} ∘ x)) < +∞. (3)

Observe that by (1) and [10], Theorem 10.11, p. 74, BV_{φ} = BV_{g}. Hence for any x ∈ BV_{g} (3) holds true. Now we show that

sup_{n∈ℕ}V_{g}(F_{n} ∘ x) < +∞ (4)

for any x ∈ BV_{g}. Since g satisfies local Δ_{2} condition and g is nondecreasing, for any M > 0 and u ∈ [0, M], there exists L_{M} > 0

g(2u) ≤ L_{M}g(u).

Fix M > 0 satisfying (2) and w ∈ ℕ such that 2^{−w} < kk_{1}. Note that for any partition P = {t_{o}, t_{1}, ..., t_{l}} of [0, a] and n ∈ ℕ,

∑_{i=1}^{l} g(|F_{n}(x(t_{i})) − F_{n}(x(t_{i−1}))|)

≤ (L_{M})^{w} ∑_{i=1}^{l} g(2^{−w}|F_{n}(x(t_{i})) − F_{n}(x(t_{i−1}))|)

≤ (L_{M})^{w}(∑_{i=1}^{l} g(kk_{1}|F_{n}(x(t_{i})) − F_{n}(x(t_{i−1}))|))

< (L_{M})^{w} sup_{n∈ℕ}V_{g}(kk_{1}(F_{n} ∘ x)) < +∞,

which shows our claim. By (4) and [12], Theorem 1 applied to g, for any v > 0 there exists K_{v} > 0 such that for any u_{1}, u_{2} ∈ [−v, v] and n ∈ ℕ

g(|F_{n}(u_{1}) − F_{n}(u_{2})|) ≤ K_{v}g(|u_{1} − u_{2}|), (5)

which shows (b). Now assume that (b) is satisfied and fix x ∈ V_{φ}. Let k > 0 be so chosen that V_{φ}(kx) < +∞. By [10], Theorem 10.7 a), p. 69 x is a bounded function. Choose v > 0 such that |x(t)| < v for any t ∈ [0, a]. Since g is nondecreasing, lim_{u→+∞} g(u) = +∞, and F_{n}(0) = 0, by (5),

sup{|F_{n}(u)| : n ∈ ℕ, u ∈ [−v, v]} < +∞.

Hence making k smaller, if necessary, we can assume that k|x(t)| < u_{o} and k|F_{n}(x(t))| ≤ u_{o}/2 for any t ∈ [0, a] and n ∈ ℕ. Since g satisfies local Δ_{2} condition, by (b) there exists L_{v} > 0 such that for any u_{1}, u_{2} ∈ [−v, v] and n ∈ ℕ, g(k|F_{n}(u_{1}) − F_{n}(u_{2})|) ≤ L_{v}g(k|u_{1} − u_{2}|). (6)

Note that for any partition P = {t_{o}, t_{1}, ..., t_{l}} of [0, a] and s_{i} ∈ [t_{i−1}, t_{i}], i = 1, ..., l, by (1) and (6),

∑_{i=1}^{l} φ(s_{i}, k|F_{n}(x(t_{i})) − F_{n}(x(t_{i−1}))|) ≤ M(∑_{i=1}^{l} g(k|F_{n}(x(t_{i})) − F_{n}(x(t_{i−1}))|))

≤ (ML_{v})(∑_{i=1}^{l} g(k|x(t_{i})) − x(t_{i−1}))|))

≤ (ML_{v}/m)(∑_{i=1}^{l} φ(s_{i}, k|x(t_{i}) − x(t_{i−1})|)) ≤ (ML_{v}/m)V_{φ}(kx) < +∞.

Hence

sup_{n} V_{φ}(kF_{n} ∘ x) < +∞,

which completes the proof.

Theorem 2 Let φ and g be as in Theorem 1. Assume furthermore that g is s-convex for some s ∈ (0, 1] or there exists t ∈ [0, a] such that φ(t, .) is s-convex for some s ∈ (0, 1]. Then (a) is equivalent to

(c) For any v > 0 there exists K_{v} > 0 such that for any u_{1}, u_{2} ∈ [−v, v]

and n ∈ ℕ

|F_{n}(u_{1}) − F_{n}(u_{2})| ≤ K_{v}|u_{1} − u_{2}|.

P r o o f. First assume that g is s-convex. By Theorem 1, (a) implies (b). Fix v > 0 and positive constant L_{v} corresponding to v by (b). Without loss of generality we can assume that L_{v} > 1. By s-convexity,

g(|F_{n}(u_{1}) − F_{n}(u_{2})|/(L_{v})^{1/s}) ≤ g(|F_{n}(u_{1}) − F_{n}(u_{2})|)/L_{v} ≤ g(|u_{1} − u_{2}|).

Since g is nondecreasing, this implies that

|F_{n}(u_{1}) − F_{n}(u_{2})| ≤ (L_{v})^{1/s}|u_{1} − u_{2}|

as required. Now assume that φ(t, .) is an s-convex function for some t ∈ [0, a] and s ∈ (0, 1]. We show that (b) implies (c). Fix v > 0. Reasoning as in the proof of Theorem 1 we can show that

sup{|F_{n}(u)| : u ∈ [−v, v], n ∈ ℕ} < +∞.

Hence we can find k ∈ (0, 1) such that k|F_{n}(u)| < u_{o}/2 and k|u| < u_{o}/2 for any n ∈ ℕ and u ∈ [−v, v]. Since g satisfies local Δ_{2} condition, by (b) and (1)

φ(t, k|F_{n}(u_{1}) − F_{n}(u_{2})|) ≤ Mg(k|F_{n}(u_{1}) − F_{n}(u_{2})|) ≤ (L_{v}M)g(|u_{1} − u_{2}|)

≤ M_{v}g(k|u_{1} − u_{2}|) ≤ (M_{v}/m)φ(t, k|u_{1} − u_{2}|)

with some constant M_{v} > 0. Since φ(t, .) is s-convex, reasoning as in the previous case, we get

k|F_{n}(u_{1}) − F_{n}(u_{2})|/(M_{v}M/m)^{1/s} ≤ k|u_{1} − u_{2}|,

which immediately give us (c) with the constant (M_{v}M/m)^{1/s}. By Lemma 1,

(c) implies (a). The proof is complete.

Corollary 1 Let φ, g be as in Theorem 2. Assume that F : ℝ → ℝ satisfies F (0) = 0. Then the composition operator x → F ∘ x maps BV_{φ} into BV_{φ} if and only if F satisfies local Lipschitz condition.

P r o o f. Follows immediately from Lemma 1 and Theorem 2, taking F_{n} = F for any n ∈ ℕ.

Corollary 2 Let φ, g and F_{n} be as in Theorem 2. Assume that there exists s ∈ (0, 1] such that φ(t, ·) is an s-convex function for any t ∈ [0, a]. Then (a) is equivalent to

(d) There exists C > 0 such that

sup{∥F_{n} ∘ x∥_{φ,s} : x ∈ V_{φ}, ∥x∥_{s,φ} = 1, n ∈ ℕ} ≤ C.

P r o o f. By Theorem 2, (a) implies (c). By s-convexity of, φ(t, ·) (c) implies (d). Conversely, (d) implies

(e) For any x ∈ V_{φ}, there exists C_{x} > 0 with sup_{n} ∥F_{n} ∘ x∥_{φ,s} < C_{x}.

It is clear that (e) implies (a), which completes the proof.

4. Hammerstein integral equation

For simplicity assume that a = 1. Assume also that φ satisfies (i)-(iii) from Section 2. Consider the Hammerstein integral equation

x(t) = g(t) + ν ∫_{I} K(t, s)f(x(s))ds for t ∈ I, ν ∈ ℝ. (7)

Assume that

1^{0} g : I → ℝ is a generalized BV_{φ}-function (g(0) = 0);

2^{0} f : ℝ → ℝ is a locally Lipschitz function;

3^{0} K : I × I → ℝ is a function such that K(t, ·) is integrable in the Lebesgue sense (briefly: L-integrable) for every t ∈ I, K(0, s) = 0 and there exists a number α > 0 such that

⋁_{0}^{1} _{φ} (\frac{K(·,s)}{α}) ≤ M(s) for a.e. s ∈ I, where M : I → ℝ_{+} is an L-

integrable function.

Theorem 3 Under the above assumptions there exists a number ρ > 0 such that for every ν with |ν| < ρ, equation (7) has a unique generalized BV_{φ}- solution, defined on I.

P r o o f. First, let us observe that from 3^{0} it follows that

inf\{ε > 0 : ∫_{I} V_{φ} (\frac{K(·, s)}{ε}) ds ≤ 1\} =: c < +∞.

Indeed, by 3^{0} we have ∫_{I} V_{φ} (\frac{K(·,s)}{α}) ds < +∞. Let β = ∫_{I} V_{φ} (\frac{K(·,s)}{α}) ds and

γ = max(1, β)α. Now, we have

∫_{I} V_{φ} (\frac{K(·, s)}{γ}) ds ≤ \frac{1}{max(1, β)} ∫_{I} V_{φ} (\frac{K(·, s)}{α}) ds = \frac{β}{max(1, β)} ≤ 1,

so ∫_{I} V_{φ} (\frac{K(·,s)}{γ}) ds ≤ 1. Thus inf\{ε > 0 : ∫_{I} V_{φ} (\frac{K(·,s)}{ε}) ds ≤ 1\} =: c ≤ γ.

Choose a positive number r > 0 such that ∥g∥_{V_{φ}} < r. Denote by L_{r} the Lipschitz constant which corresponds to the function f and the interval [−r, r]. Choose a number ρ > 0 such that ∥g∥_{V_{φ}} + ρ < sup_{[−r,r]} |f(t)| < r and

ρL_{r}cc̃ < 1, where c̃ > 0 is the infimum of all positive numbers ˜ c̃ such that ∥x∥_{sup} ≤ ˜ c̃∥x∥_{V_{φ}}. The existence such a number c̃ follows from [10], 10.7c, p. 69 and [1], Theorem 4.1, p. 119. Let B̄_{r} denote the closed ball of center zero and radius r in the space BV_{φ}(I). Fix ν such that |ν| < ρ. Define the operators

F (x)(t) = ∫_{I} K(t, s)f(x(s))ds,

G(x)(t) = g(t) + νF (x)(t),

where x ∈ B̄_{r} and t ∈ I. By Lemma 1, f(x) ∈ BV_{φ}(I), so in view of [10], 10.7a, p.69 and Theorem 10.9, p.71 it is Lebesgue measurable and bounded. Thus the mappings F and G are well defined. Now, we verify that G maps B̄_{r} into itself. Indeed, for any x ∈ B̄_{r}, we have

∥G(x)∥_{V_{φ}} ≤ ∥g∥_{V_{φ}} + ∥νF (x)∥_{V_{φ}} = ∥g∥_{V_{φ}} + inf\{ε > 0 : V_{φ} (\frac{νF (x)}{ε}) ≤ 1\}.

The sign ” sup_{π,{s_{i}}}” below denotes that the supremum is taken over all partitions

π with all possible intermidiate points s_{i} ∈ [t_{i−1}, t_{i}], i = 1, …, n. By the Jensen inequality, we have

V_{φ} (\frac{νF (x)}{ε}) = sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, \frac{|ν|}{ε}|F (x)(t_{i}) − F (x)(t_{i−1})|)

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, ∫_{0}^{1} \frac{|ν|}{ε}|K(t_{i}, s) − K(t_{i−1}, s)||f(x)(s)|ds)

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, ∫_{0}^{1} \frac{|ν|}{ε} sup_{s∈I} |f(x)(s)||K(t_{i}, s) − K(t_{i−1}, s)|ds)

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, ∫_{0}^{1} \frac{|ν|}{ε} sup_{t∈[−r,r]} |f(t)||K(t_{i}, s) − K(t_{i−1}, s)|ds)

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} ∫_{0}^{1} φ(s_{i}, \frac{|ν|}{ε} sup_{t∈[−r,r]} |f(t)||K(t_{i}, s) − K(t_{i−1}, s)|ds)

≤ ∫_{0}^{1} φ(|ν| sup_{t∈[−r,r]} |f(t)|\frac{K(·, s)}{ε})ds

and

inf\{ε > 0 : V_{φ} (\frac{νF (x)}{ε}) ≤ 1\}

≤ inf\{ε > 0 : ∫_{0}^{1} V_{φ} (|ν| sup_{t∈[−r,r]} |f(t)|\frac{K(·, s)}{ε}) ds ≤ 1\}

= |ν| sup_{t∈[−r,r]} |f(t)| inf\{ε > 0 : ∫_{0}^{1} V_{φ} (\frac{K(·, s)}{ε}) ds ≤ 1\}

= |ν| sup_{t∈[−r,r]} |f(t)|c.

Therefore, we conclude that

∥G(x)∥_{V_{φ}} ≤ ∥g∥_{V_{φ}} + |ν| sup_{[−r,r]} |f(t)|c < r,

which means that G maps B̄_{r} into itself. Similarly, for any x, y ∈ B̄_{r} we have

V_{φ} (\frac{ν(F (x) − F (y))}{ε})

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, ∫_{0}^{1} \frac{|ν|}{ε}|K(t_{i}, s) − K(t_{i−1}, s)||f(x(s)) − f(y(s))|ds)

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} ∫_{0}^{1} φ(s_{i}, \frac{|ν|}{ε} sup_{s∈I} |f(x(s)) − f(y(s))||K(t_{i}, s) − K(t_{i−1}, s)|)ds

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} ∫_{0}^{1} φ(s_{i}, \frac{|ν|}{ε}L_{r} sup_{s∈I} |x(s) − y(s)||K(t_{i}, s) − K(t_{i−1}, s)|)ds

≤ ∫_{0}^{1} V_{φ}(|ν|L_{r} sup_{s∈I} |x(s) − y(s)|\frac{K(·, s)}{ε})ds,

and

∥G(x) − G(y)∥_{V_{φ}} = inf\{ε > 0 : V_{φ} (\frac{ν(F (x) − F (y))}{ε}) ≤ 1\}

≤ inf\{ε > 0 : ∫_{0}^{1} V_{φ}(|ν|L_{r} sup_{s∈I} |x(s) − y(s)|\frac{K(·, s)}{ε})ds ≤ 1\}

= |ν|L_{r}cc̃∥x − y∥_{V_{φ}},

so G is a contraction. Now, applying the Banach contraction principle we infer that G has a unique fixed point in B̄_{r}, which is a generalized BV_{φ}- solution to equation (7).

Remark 1 1) Let φ be a φ-function without parameter. Then Theorem 3 covers with Theorem 1 from [3].

2) Again, let φ be a φ-function without parameter satisfying the conditions u^{−1}φ(u) → +∞ as u → 0+ and φ(u_{1} + … + u_{n}) ≤ k(φ(λu_{1}) + … + φ(λu_{n})) for n ∈ ℕ with some constants k, λ > 0. Let us denote

s_{x}(t) = x(0 + 0) − x(0) + ∑_{t_{i}<t}(x(t_{i} + 0) − x(t_{i} − 0)) + x(t) − x(t − 0)

for 0 < x ≤ a for every x ∈ V_{φ}, where t_{1}, t_{2}, … are all points of discontinuity of x. It can be shown that in this case x(t) = s_{x}(t) for every t ∈ [0, a] (see [10], pp. 73-74 for details).

3) Now, assume that φ(t, u) satisfies the condition uψ^{−1}(u) → +∞, as u → 0+, where ψ is the function defined in (iv) and let x ∈ V_{φ}. One can easily verify then, that the function x is of bounded variation in [0, a], in the usual sense. Moreover, it can be shown that x is constant in each interval of continuity (see again [10], p. 73 for details).

5. Volterra-Hammerstein integral equation

Throughout this section we assume that φ-function φ is convex and satisfies following Δ_{2}-condition:

φ(t, 2u) ≤ kφ(t, u) for 0 ≤ u ≤ u_{0}, t ∈ [0, a],

where u_{0} > 0 is fixed and k is a positive constant.

For x ∈ X, we shall denote by ⋁_{s}^{1}_{φ}(x) the φ-variation of x on the interval

[s, 1], where 0 ≤ s < 1.

Consider the following Volterra-Hammerstein integral equation

x(t) = g(t) + ∫_{0}^{t} K(t, s)f(x(s))ds for t ∈ I. (8)

In what follows we shall need the following assumption

4^{0} Let T = {(t, s) : 0 ≤ t ≤ 1, 0 ≤ s ≤ t} and K : T → ℝ be a functin such that K(t, ·) is an L-integrable on [0, t] for every t ∈ I, and there

exists a number α > 0 such that sup_{{0≤w≤1 \atop s≤t≤1}} φ (w, \frac{|K(t,s)|}{α}) + ⋁_{s}^{1}_{φ} (\frac{K(·,s)}{α}) ≤

m(s) for a.e. s ∈ I, where m : I → ℝ_{+} is an L-integrable function.

Now, we prove the following

Theorem 4 Suppose conditions 1^{0}, 2^{0} and 4^{0} are satisfied. Then there exists an interval J ⊂ I such that the equation (8) has a unique generalized BV_{φ}- solution, defined on J.

P r o o f. Let r, L_{r} and c̃ be as in the proof of Theorem 3. Choose a positive integer N such that sup_{t∈[−r,r]} |f(t)|\frac{α}{2^{N}} + ∥g∥_{V_{φ}} < r and L_{r}c̃\frac{α}{2^{N}} < 1.

Further, let 0 < d ≤ min{u_{0}, 1} be such that

∫_{0}^{d}[ sup_{{0≤w≤d \atop s≤t≤d}} φ (w, \frac{2^{N}|K(t, s)|}{α}) + ⋁_{s}^{d}_{φ} (\frac{2^{N}K(·, s)}{α})]ds (9)

≤ K^{N} ∫_{0}^{d} m(s)ds ≤ 1.

Indeed, by 4^{0} and the absolute continuity of the Lebesgue integral, there exists 0 < d ≤ min{u_{0}, 1} such that

∫_{0}^{d}[ sup_{{0≤w≤d \atop s≤t≤d}} φ (w, \frac{|K(t, s)|}{α}) + ⋁_{s}^{d}_{φ} (\frac{K(·, s)}{α})]ds ≤ ∫_{0}^{d} m(s)ds ≤ 1.

In view of the Δ_{2}-condition, we obtain

∫_{0}^{d}[ sup_{{0≤w≤d \atop s≤t≤d}} φ (w, \frac{2|K(t, s)|}{α}) + ⋁_{s}^{d}_{φ} (\frac{2K(·, s)}{α})]ds ≤ K ∫_{0}^{d} m(s)ds,

so one can choose d such that K ∫_{0}^{d} m(s)ds ≤ 1. Arguing in such a way as above we deduce that for every N ∈ ℕ there exists a number 0 < d ≤ min{u_{0}, 1} which satisfies (9). From (9) we get the inequality

(10)

inf\{ε > 0 : ∫_{0}^{d}[ sup_{{0≤w≤d \atop s≤t≤d}} φ (w, \frac{|K(t, s)|}{ε}) + ⋁_{s}^{d}_{φ} (\frac{K(·, s)}{ε})]ds ≤ 1\} ≤ \frac{α}{2^{N}} .

Define K̃(t, s) = \{ \begin{matrix}K(t, s) , & 0 ≤ s ≤ t, \\ 0 , & s < t ≤ d, \end{matrix} J = [0, d] and G(x)(t) = g(t) +

F (x)(t), where

F (x)(t) = ∫_{0}^{t} K(t, s)f(x(s))ds, for x ∈ B̄_{r}, t ∈ J,

(B̄_{r} denotes here the same ball as in the proof of Theorem 3). Now, we verify that G maps B̄_{r} into itself. We have obviously

∥G(x)∥_{V_{φ}} ≤ ∥g∥_{V_{φ}} + ∥F (x)∥_{V_{φ}} = ∥g∥_{V_{φ}} + inf\{ε > 0 : ⋁_{0}^{d}_{φ}(\frac{F (x)}{ε}) ≤ 1\}.

Since

⋁_{0}^{d}_{φ} (\frac{F (x)}{ε}) = sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, \frac{1}{ε}|F (x)(t_{i}) − F (x)(t_{i−1})|)

= sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, | ∫_{0}^{t_{i}} \frac{1}{ε}K(t_{i}, s)f(x(s))ds − ∫_{0}^{t_{i−1}} \frac{1}{ε}K(t_{i−1}, s)f(x(s))ds|)

= sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, | ∫_{0}^{d} \frac{1}{ε}(K̃(t_{i}, s) − K̃(t_{i−1}, s))f(x(s))ds|)

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, \frac{1}{d} ∫_{0}^{d} d\frac{1}{ε} sup_{s∈J} |f(x(s))||K̃(t_{i}, s) − K̃(t_{i−1}, s)|ds)

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} \frac{1}{d} ∫_{0}^{d} φ(s_{i}, \frac{d}{ε} sup_{t∈[−r,r]} |f(t)||K̃(t_{i}, s) − K̃(t_{i−1}, s)|ds)

≤ ∫_{0}^{d} ⋁_{0}^{d}_{φ}( sup_{t∈[−r,r]} |f(t)|\frac{K̃(·, s)}{ε})ds,

so

inf\{ε > 0 : ⋁_{0}^{d}_{φ} (\frac{F (x)}{ε}) ≤ 1\}

≤ inf\{ε > 0 : ∫_{0}^{d} ⋁_{0}^{d}_{φ} ( sup_{t∈[−r,r]} |f(t)|\frac{K̃(·, s)}{ε}) ds ≤ 1\}

= sup_{t∈[−r,r]} |f(t)| inf\{ε > 0 : ∫_{0}^{d} ⋁_{0}^{d}_{φ} (\frac{K̃(·, s)}{ε}) ds ≤ 1\}.

Further, since

∫_{0}^{d} ⋁_{0}^{d}_{φ} (\frac{K̃(·, s)}{ε}) ds ≤ ∫_{0}^{d}[ sup_{{0≤w≤d \atop s≤t≤d}} φ(w, \frac{|K(t, s)|}{ε}) + ⋁_{s}^{d}_{φ}(\frac{K(·, s)}{ε})]ds,

by (10), we get

inf\{ε > 0 : ⋁_{0}^{d}_{φ}(\frac{F (x)}{ε}) ≤ 1\} ≤ sup_{t∈[−r,r]} |f(t)|\frac{α}{2^{N}} .

Thus ∥G(x)∥_{V_{φ}} < r, which means that G(B̄_{r}) ⊂ B̄_{r}. Now, for any x, y ∈ B̄_{r} we have

∥G(x) − G(y)∥_{V_{φ}} = inf\{ε > 0 : ⋁_{0}^{d}_{φ}(\frac{F (x) − F (y)}{ε}) ≤ 1\}

and

⋁_{0}^{d}_{φ}(\frac{F (x) − F (y)}{ε})

= sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, \frac{1}{ε}|F (x)(t_{i}) − F (x)(t_{i−1}) − F (y)(t_{i}) + F (y)(t_{i−1})|)

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} φ(s_{i}, ∫_{0}^{d} \frac{1}{ε}|K̃(t_{i}, s) − K̃(t_{i−1}, s)||f(x(s)) − f(y(s))|ds)

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} ∫_{0}^{d} φ(s_{i}, \frac{1}{ε} sup_{s∈J} |f(x(s)) − f(y(s))||K̃(t_{i}, s) − K̃(t_{i−1}, s)|)ds

≤ sup_{π,\{s_{i}\}} ∑_{i=1}^{n} ∫_{0}^{d} φ(s_{i}, \frac{1}{ε}L_{r} sup_{s∈J} |x(s) − y(s)||K̃(t_{i}, s) − K̃(t_{i−1}, s)|)ds

≤ ∫_{0}^{d} ⋁_{0}^{d}_{φ}(L_{r} sup_{s∈J} |x(s) − y(s)|\frac{K̃(·, s)}{ε})ds,

so, by (10), we get

∥G(x) − G(y)∥_{V_{φ}}

≤ inf\{ε > 0 : ∫_{0}^{d} ⋁_{0}^{d}_{φ}(L_{r} sup_{s∈J} |x(s) − y(s)|\frac{K̃(·, s)}{ε})ds ≤ 1\}

≤ L_{r} sup_{s∈J} |x(s) − y(s)|\frac{α}{2^{N}} ≤ L_{r}c̃\frac{α}{2^{N}} ∥x − y∥_{V_{φ}}.

By the Banach contraction principle, we infer that G has a unique fixed point in B̄_{r}, which is a generalized BV_{φ}-solution of the equation (8).

6. Global solutions of equations (7) and (8)

Let us begin with the Hammerstein integral equation of the form

x(t) = g(t) + ∫_{I} K(t, s)f(x(s))ds, for t ∈ I, (11)

where I = [0, 1] for simplicity. Assume that

5^{0} f : ℝ → ℝ;

6^{0} there exists Ψ : [0, +∞) → [0, +∞) with Ψ(u) > 0 for u > 0 and

sup_{s∈[0,1]} |f(x(s))| ≤ Ψ(∥x∥_{V_{φ}}) for any x ∈ BV_{φ}(I);

7^{0} there exists M_{0} > 0 with \frac{M_{0}}{∥g∥_{Vφ}+Ψ(M_{0})c} > 1, where c is the constant defined in the proof of Theorem 3;

8^{0} there exists a continuous and nondecreasing function φ_{M_{0}} : [0, +∞) → [0, +∞) such that cφ_{M_{0}}(c̃z) < z for z > 0 and |f(x)−f(y)| < φ_{M_{0}}(|x− y|), for |x|, |y| ≤ M_{0}, where c̃ is the constant defined in the proof of Theorem 3.

Now we prove the following existence result for equation (11).

Theorem 5 Under the assumptions 1^{0}, 3^{0}, 5^{0}-8^{0}, equation (11) has a generalized BV_{φ}-solution, defined on I.

P r o o f. Let B̄_{M_{0}} denote the closed ball of center zero and radius M_{0} in the space BV_{φ}(I). Define

G(x)(t) = g(t) + ∫_{I} K(t, s)f(x(s))ds, for x ∈ B̄_{M_{0}} and t ∈ I.

For any x, y ∈ B̄_{M_{0}} we have

∥G(x) − G(y)∥_{V_{φ}}

= inf\{ε > 0 : ∫_{I} V_{φ} (\frac{K(·, s)}{ε} sup_{s∈I} φ_{M_{0}}(|x(s) − y(s)|)) ds ≤ 1\}

≤ cφ_{M_{0}}(c̃∥x − y∥_{V_{φ}}).

From the above inequality it follows, in particular, that G(B̄_{M_{0}}) is a bounded set. Now suppose that x ∈ BV_{φ}(I) with ∥x∥_{V_{φ}} = M_{0} is a solution of

x(t) = λ(g(t) + ∫_{I} K(t, s)f(x(s))ds) for t ∈ I,

where λ ∈ (0, 1]. By 6^{0} and 7^{0}, we have

∥x∥_{V_{φ}} ≤ ∥g∥_{V_{φ}} + sup_{s∈I} |f(x(s))| · c ≤ ∥g∥_{V_{φ}} + cΨ(∥x∥_{V_{φ}}),

so

\frac{∥x∥_{V_{φ}}}{∥g∥_{V_{φ}} + cΨ(∥x∥_{V_{φ}})} ≤ 1. (12)

Since ∥x∥_{V_{φ}} = M_{0}, (12) implies that

\frac{M_{0}}{∥g∥_{V_{φ}} + cΨ(M_{0})} ≤ 1

which contradicts 7^{0}. Applying the nonlinear alternative of Leray-Schauder type (see [12]) we infer that G has a fixed point in the open ball B_{M_{0}}, which obviously is a global generalized BV_{φ} solution of (11).

Now, consider again equation (8). Define

K̃(t, s) = \{ \begin{matrix}K(t, s) , & 0 ≤ s ≤ t, \\ 0 , & t < s ≤ 1, \end{matrix}

and write (8) in the following form

x(t) = g(t) + ∫_{I} K̃(t, s)f(x(s))ds, for t ∈ I. (13)

Hence, as a corollary from Theorem 5 we obtain the following result for equation (8).

Theorem 6 Suppose 1^{0}, 4^{0}, 5^{0} and 6^{0} are satisfied. Moreover, assume that

9^{0} there exists M_{0} > 0 with \frac{M_{0}}{∥g∥_{Vφ}+Ψ(M_{0})c̄} > 1, where c̄ = inf\{ε > 0 :

∫_{0}^{1}[ sup_{{0≤w≤1 \atop s≤t≤1}} φ(w, \frac{|K(t,s)|}{ε}) + ⋁_{s}^{1}_{φ}(\frac{K(·,s)}{ε})]ds ≤ 1\}

and condition 8^{0} with c̄ instead of c holds. Then equation (13) has a generalized BV_{φ}-solution, defined on I.

P r o o f. Indeed, the results follows from Theorem 5. We have

⋁_{0}^{1}_{φ}(\frac{K̃(·, s)}{α}) ≤ sup_{{0≤w≤1 \atop s≤t≤1}} φ(w, \frac{|K(t, s)|}{α}) + ⋁_{s}^{1}_{φ}(\frac{K(·, s)}{α}) ≤ m(s)

for a.e. s ∈ I, so K̃ satisfies 3^{0}. Moreover,

inf\{ε > 0 : ∫_{I} ⋁_{0}^{1}_{φ}(\frac{K̃(·, s)}{ε})ds ≤ 1\}

≤ inf\{ε > 0 : ∫_{I} [ sup_{{0≤w≤1 \atop s≤t≤1}} φ(w, \frac{|K(t, s)|}{ε}) + ⋁_{s}^{1}_{φ}(\frac{K(·, s)}{ε})]ds ≤ 1\} = c̄

and thus we can take here c = c̄.

Remark 2 Note that the inequality ∥x∥_{sup} ≤ ˜ c̃∥x∥_{V_{φ}}, x ∈ BV_{φ}(I), mentioned in the proof of Theorem 3, in particular implies that continuous functions of bounded generalized φ-variation form a closed subspace of the space BV_{φ}(I). Therefore, it is clear that assuming additionally that g is continuous and imposing a suitable continuity assumption on the kernel K, one can obtain the existence and uniqueness results concerning continuous generalized BV_{φ} solutions to equations (7) and (8).

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