Integrodifferential Equations on Time Scales with Henstock-Kurzweil-Pettis Delta Integrals

Abstract

We prove existence theorems for integro-differential equations 𝑥^Δ(𝑡)=𝑓(𝑡,𝑥(𝑡),∫^𝑡₀𝑘(𝑡,𝑠,𝑥(𝑠))Δ𝑠), 𝑥(0)=𝑥₀, 𝑡∈𝐼_𝑎=[0,𝑎]∩𝑇, 𝑎∈𝑅₊, where 𝑇 denotes a time scale (nonempty closed subset of real numbers 𝑅), and 𝐼_𝑎 is a time scale interval. The functions 𝑓,𝑘 are weakly-weakly sequentially continuous with values in a Banach space 𝐸, and the integral is taken in the sense of Henstock-Kurzweil-Pettis delta integral. This integral generalizes the Henstock-Kurzweil delta integral and the Pettis integral. Additionally, the functions 𝑓 and 𝑘 satisfy some boundary conditions and conditions expressed in terms of measures of weak noncompactness. Moreover, we prove Ambrosetti's lemma.

1. Introduction

A time scale 𝑇 is a nonempty closed subset of real numbers 𝑅, with the subspace topology inherited from the standard topology of 𝑅.

The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see Kac and Cheung [1]), that is, when 𝑇=𝑅, 𝑇=𝑁, 𝑇=𝑞^𝑁0={𝑞^𝑡∶𝑡∈𝑁₀}, where 𝑞>1.

Time scale (or a measure chain) was introduced by Hilger in his Ph.D. thesis in 1988, [2]. Since the time Hilger formed the definitions of a derivative and integral on a time scale, several authors have extended on various aspects of the theory [3–11]. Time scales have been shown to be applicable to any field that can be described by means of discrete or continuous models.

In this paper we consider an integrodifferential equation. As is known, ordinary integrodifferential equations, an extreme case of integrodifferential equations on time scales [12–21], find many applications in various mathematical problems; see Corduneanu's book [22] and references therein for details.

In [23] the authors extend such results to the integrodifferential equations on time scales and therefore obtained corresponding criteria which can be employed to study the difference equation of Volterra type [21, 24], 𝑞-difference equations of Volterra type, and so forth.

In [25] the authors proved a new comparison result and develop the monotone iterative technique to show the existence of extremal solutions of the periodic boundary value problems of nonlinear integrodifferential equation on time scales.

We extend this result by proving the existence of a pseudosolution of the problem𝑥^Δ(𝑡)=𝑓(𝑡,𝑥(𝑡),^𝑡₀𝑘(𝑡,𝑠,𝑥(𝑠))Δ𝑠),𝑡∈𝐼_𝑎=[0,𝑎]∩𝑇,𝑎∈𝑅₊,𝑥(0)=𝑥₀, (1.1) where 𝑓∶𝐼_𝑎×𝐸×𝐸→𝐸,𝑘∶𝐼_𝑎×𝐼_𝑎×𝐸→𝐸,𝑥∶𝐼_𝑎→𝐸, 𝑇 denotes a time scale, 0∈𝑇,𝐼_𝑎 denotes a time scale interval, (𝐸,‖⋅‖) is a Banach space, and 𝑥^Δ denotes the pseudoΔ-derivative of 𝑥.

We use a new (Henstock-Kurzweil-Pettis-Δ-integral), more general concept of integral on time scales. This new integral generalizes the Henstock-Kurzweil delta integral, which has been studied by Peterson and Thompson in [26], Avsec et al. in [27] and the Pettis integral [28]. These integrals are important when we consider the weak topology on a Banach space.

The Henstock-Kurzweil delta integral contains the Riemann delta, Lebesgue delta and Bochner delta integrals as special cases. These integrals will enable time scale researchers to study more general dynamic equations. In [26] it is shown that there are highly oscillatory functions that are not delta integrable on a time scale, but are Henstock-Kurzweil delta integral.

Let us remark that the existence of the Henstock-Kurzweil integral over [𝑎,𝑏] implies the existence of such integrals over all subintervals of [𝑎,𝑏] but not for all measurable subsets of this interval, so the theory of such integrals on 𝑇 does not follows from general theory on 𝑅.

In [29] Cichoń introduce a definition of the Henstock-Kurzweil delta integral (HK-Δ-integral) and HL delta integral (HL-Δ-integral) on Banach spaces for checking the existence of solutions of differential (or dynamic) equations in Banach spaces. He presented also a new definition of the Henstock-Kurzweil-Pettis delta integral on time scales.

The study for weak solutions of Cauchy differential equations in Banach spaces was initiated by Szép [30] and theorems on the existence of weak solutions of this problem were proved by Cramer et al. [31], Kubiaczyk [32], Kubiaczyk and Szufla [33], Mitchell and Smith [34], Szufla [35], and Cichoń and Kubiaczyk [36]. There are also some existence theorems for the Volterra and Urysohn integral equations [37] on time scales.

Similar methods for solving existence problems for difference equations in Banach spaces equipped with its weak topology were studied, for instance, in [38–40]. In particular, the importance of conditions expressed in terms of the weak topology was considered in [38].

We will unify both cases and using the weak topology, we will obtain the first result for pseudosolutions of an integrodifferential dynamic problem. (This is new also for 𝑞-difference equations.)

Our result extends the existence of pseudosolutions not only to the discrete intervals with uniform step size (ℎZ) but also to the discrete intervals with nonuniform step size (𝐾_𝑞).

We note that (1.1) in its general form involves some different types of differential and difference equations depending on the choice of the time scale 𝑇. For example: (1)for 𝑇=𝑅, we have 𝜎(𝑡)=𝑡, 𝜇(𝑡)=0, and 𝑥^Δ(𝑡)=𝑥^′(𝑡), and (1.1) becomes the Cauchy integrodifferential equation: 𝑥^′(𝑡)=𝑓(𝑡,𝑥(𝑡),^𝑡₀𝑘(𝑡,𝑠,𝑥(𝑠))𝑑𝑠),𝑡∈𝑅,𝑥(0)=𝑥₀; (1.2)(2)for 𝑇=𝑁, we have 𝜎(𝑛)=𝑛+1, 𝜇(𝑛)=1, and 𝑥^Δ(𝑛)=Δ𝑥(𝑛)=𝑥(𝑛+1)−𝑥(𝑛), and (1.1) becomes the Cauchy problem: Δ𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),^𝑡₀𝑘(𝑡,𝑠,𝑥(𝑠))𝑑𝑠),𝑡∈𝑁,𝑥(0)=𝑥₀. (1.3)

We assume that the functions 𝑓,𝑘 are weakly-weakly sequentially continuous with values in a Banach space and satisfy some regularity conditions expressed in terms of the De Blasi measure of weak noncompactness. We introduce a weakly sequentially continuous operator associated to an integral equation which is equivalent to (1.1). There exist many important examples of mappings which are weakly sequentially continuous but not weakly continuous. The relations between weakly sequentially continuous and weakly continuous mappings are studied by Ball [41].

Adopting the fixed point theorem for weakly sequentially continuous mappings given by Kubiaczyk [42] and the properties of measures of weak noncompactness, we are able to study the existence results for problem (1.1)

2. Preliminaries

Let (𝐸,‖⋅‖) be a Banach space and let 𝐸^∗ be the dual space. Moreover, let (𝐶(𝐼_𝑎,𝐸),𝜔) denote the space of all continuous functions from 𝐼_𝑎 to 𝐸 endowed with the topology 𝜎(𝐶(𝐼_𝑎,𝐸),𝐶(𝐼_𝑎,𝐸)^∗) and 𝐶_rd(𝐼_𝑎,𝐸) denotes the space of all rd-continuous functions from the time scale interval 𝐼_𝑎 to 𝐸.

By 𝜇_Δ we denote the Lebesgue measure on 𝑇. For a precise definition and basic properties of this measure we refer the reader to [43].

We now gather some well-known definitions and results from the literature, which we will use throughtout this paper.

Definition 2.1 (see [44]). A family 𝔽 of functions 𝐹 is said to be uniformly absolutely continuous in the restricted sense on 𝐴⊆[𝑎,𝑏] or in short uniformly 𝐴𝐶_∗(𝐴) if for every 𝜀>0 there exists 𝜂>0 such that for every 𝐹 in 𝔽 and for every finite or infinite sequence of nonoverlapping intervals {[𝑎_𝑖,𝑏_𝑖]} with 𝑎_𝑖,𝑏_𝑖∈𝐴 and satisfying ∑_𝑖𝜇_Δ([𝑎_𝑖,𝑏_𝑖])<𝜂, one has ∑_𝑖𝜔(𝐹,[𝑎_𝑖,𝑏_𝑖])<𝜀 where 𝜔 denotes the oscillation of 𝐹 over [𝑎_𝑖,𝑏_𝑖] (i.e., 𝜔(𝐹,[𝑎_𝑖,𝑏_𝑖])=sup{|𝐹(𝑟)−𝐹(𝑠)|∶𝑟,𝑠∈[𝑎_𝑖,𝑏_𝑖]}).

A family 𝔽 of functions 𝐹 is said to be uniformly generalized absolutely continuous in the restricted sense on [𝑎,𝑏] or uniformly ACG_∗ if [𝑎,𝑏] is the union of a sequence of closed sets 𝐴_𝑖 such that on each 𝐴_𝑖 the function 𝐹 is uniformly 𝐴𝐶_∗(𝐴_𝑖).

Definition 2.2. A function 𝑓∶𝐼_𝑎→𝐸 is said to be weakly continuous if it is continuous from 𝐼_𝑎 to 𝐸 endowed with its weak topology. A function 𝑔∶𝐸→𝐸₁, where 𝐸 and 𝐸₁ are Banach spaces, is said to be weakly-weakly sequentially continuous if for each weakly convergent sequence (𝑥_𝑛)⊂𝐸, the sequence (𝑔(𝑥_𝑛)) is weakly convergent in 𝐸₁. If a sequence 𝑥_𝑛 tends weakly to 𝑥₀ in 𝐸 we will denote it by 𝑥−→_𝑛^𝜔𝑥₀.

Theorem 2.3 (see [42]). Let 𝐸 be a metrizable locally convex topological vector space. Let 𝐷 be a closed convex subset of 𝐸, and let 𝐹 be a weakly-weakly sequentially continuous map of 𝐷 into itself. If for some 𝑥∈𝐷 the implication 𝑉=𝑐𝑜𝑛𝑣({𝑥}∪𝐹(𝑉))⇒𝑉isrelativelyweaklycompact, (2.1) holds for every subset 𝑉 of 𝐷, then 𝐹 has a fixed point.

(I)To understand the so-called dynamic equations and follow this paper easily, we present some preliminary definitions and notations of time scales which are very common in the literature (see [3–5, 7–11] and references therein).

A time scale 𝑇 is a nonempty closed subset of real numbers 𝑅, with the subspace topology inherited from the standard topology of 𝑅. If 𝑎,𝑏 are points in 𝑇, then a time scale interval 𝐼 we denote by 𝐼=[𝑎,𝑏]={𝑡∈𝑇∶𝑎≤𝑡≤𝑏} and 𝐼_𝑎={𝑡∈𝑇∶0≤𝑡≤𝑎}. Other types of intervals are approached similarly. By a subinterval 𝐼_𝑏 of 𝐼_𝑎 we mean the time scale subinterval.

Definition 2.4. The forward jump operator 𝜎∶𝑇→𝑇 and the backward jump operator 𝜌∶𝑇→𝑇 are defined by 𝜎(𝑡)=inf{𝑠∈𝑇∶𝑠>𝑡} and 𝜌(𝑡)=sup{𝑠∈𝑇∶𝑠<𝑡}, respectively. We put inf∅=sup𝑇 (i.e., 𝜎(𝑀)=𝑀 if 𝑇 has a maximum 𝑀) and sup∅=inf𝑇 (i.e., 𝜌(𝑚)=𝑚 if 𝑇 has a minimum 𝑚). The jump operators 𝜎 and 𝜌 allow the classification of points in time scale in the following way: 𝑡 is called right dense, right scattered, left dense, left scattered, dense and isolated if 𝜎(𝑡)=𝑡,𝜎(𝑡)>𝑡,𝜌(𝑡)=𝑡,𝜌(𝑡)<𝑡,𝜌(𝑡)=𝑡=𝜎(𝑡) and 𝜌(𝑡)<𝑡<𝜎(𝑡), respectively.

Definition 2.5. One says that 𝑘∶𝑇→𝐸 is right-dense continuous (rd-continuous) if 𝑘 is continuous at every right-dense point 𝑡∈𝑇 and lim_{𝑠→𝑡−}𝑘(𝑠) exists and is finite at every left-dense point 𝑡∈𝑇.

Definition 2.6. Fix 𝑡∈𝑇. Let 𝑓∶𝐼→𝐸. Then one defines the Δ-derivative 𝑓^Δ(𝑡) by 𝑓^Δ(𝑡)=lim_𝑠→𝑡𝑓(𝜎(𝑡))−𝑓(𝑠).𝜎(𝑡)−𝑠 (2.2)

Remark 2.7. The Δ-derivative satisfies (1)𝑓^Δ=𝑓^′ is the usual derivative if 𝑇=𝑅,(2)𝑓^Δ=Δ𝑓 is the usual forward difference operator if 𝑇=𝑍,(3)𝑓^Δ=𝐷_𝑞𝑓 is the 𝑞-derivative if 𝑇=𝑞^𝑁0={𝑞^𝑡∶𝑡∈𝑁₀},𝑞>1. Hence, time scales allows us unify the treatment of differential and difference equations (and many more time scales).

(II) We need to define some integrals which are important, when we consider the weak topology on a Banach space 𝐸.

We will use the notation 𝜂(𝑡)∶=𝜎(𝑡)−𝑡, where 𝜂 is called the graininess function and 𝜗(𝑡)∶=𝑡−𝜌(𝑡), where 𝜗 is called the left-graininess function.

We say that 𝛿=(𝛿_𝐿,𝛿_𝑅) is a Δ-gauge for time scale interval [𝑎,𝑏] provided 𝛿_𝐿(𝑡)>0 on (𝑎,𝑏], 𝛿_𝑅(𝑡)>0 on [𝑎,𝑏), 𝛿_𝐿≥0, 𝛿_𝑅≥0 and 𝛿_𝑅≥𝜂(𝑡) for all 𝑡∈[𝑎,𝑏).

We say that a partition 𝐷 for a time scale interval [𝑎,𝑏] given by𝐷={𝑎=𝑡₀≤𝜉₁≤𝑡₁≤⋯≤𝑡_𝑛−1≤𝜉_𝑛≤𝑡_𝑛=𝑏} (2.3) with 𝑡_𝑖>𝑡_𝑖−1 for 1≤𝑖≤𝑛 and 𝑡_𝑖,𝜉_𝑖∈𝑇 is 𝛿-fine if𝜉_𝑖−𝛿_𝐿(𝜉_𝑖)≤𝑡_𝑖−1<𝑡_𝑖≤𝜉_𝑖+𝛿_𝑅(𝜉_𝑖)for1≤𝑖≤𝑛. (2.4)

Definition 2.8 (see [29]). A function 𝑓∶[𝑎,𝑏]→𝐸 is Henstock-Kurzweil-Δ-integrable on [𝑎,𝑏](Δ-HK integrable in short) if there exists a function 𝐹∶[𝑎,𝑏]→𝐸, defined on the subintervals of [𝑎,𝑏], satisfying the following property: given 𝜀>0 there exists a positive function 𝛿 on [𝑎,𝑏] such that 𝐷={[𝑢,𝑣],𝜉} is 𝛿-fine division of a [𝑎,𝑏], one has ‖_𝐷𝑓(𝜉)(𝑣−𝑢)−(𝐹(𝑣)−𝐹(𝑢))‖<𝜀. (2.5)

Definition 2.9 (see [29]). A function 𝑓∶[𝑎,𝑏]→𝐸 is Henstock-Lebesgue-Δ-integrable on [𝑎,𝑏](Δ-HL integrable in short) if there exists a function 𝐹∶[𝑎,𝑏]→𝐸, defined on the subintervals of [𝑎,𝑏], satisfying the following property: given 𝜀>0 there exists a positive function 𝛿 on [𝑎,𝑏] such that 𝐷={[𝑢,𝑣],𝜉} is 𝛿-fine division of a [𝑎,𝑏], one has _𝐷‖𝑓(𝜉)(𝑣−𝑢)−(𝐹(𝑣)−𝐹(𝑢))‖<𝜀. (2.6)

Remark 2.10. We note that, by the triangle inequality, if 𝑓 is Δ-HL integrable it is also Δ-HK integrable. In general, the converse is not true. For real-valued functions, the two integrals are equivalent.

Definition 2.11 (see [28, 29]). The function 𝑓∶𝐼_𝑎→𝐸 is Pettis-Δ-integrable ((𝑃) integrable for short) if (1)∀𝑥^∗∈𝐸^∗,𝑥^∗𝑓 is Lebesgue-Δ-integrable on 𝐼_𝑎,(2)∀𝐴⊂𝐼,∃𝑔∈𝐸,∀𝑥^∗∈𝐸^∗,𝑥^∗𝑔=(Δ−𝐿)∫_𝐴𝑥^∗𝑓(𝑠)Δ𝑠.

Now, we present a new definition of the integral on time scales which is a generalization for both the Pettis-Δ-integral and the Henstock-Kurzweil-Δ-integral.

Definition 2.12 (see [29]). The function 𝑓∶𝐼_𝑎→𝐸 is Henstock-Kurzweil-Pettis-Δ-integrable (HKP-Δ-integrable for short) if (1)∀𝑥^∗∈𝐸^∗,𝑥^∗𝑓 is Henstock-Kurzweil-Δ-integrable on 𝐼_𝑎,(2)∀𝑡∈𝐼_𝑎∀𝑥^∗∈𝐸^∗,𝑥^∗𝑔(𝑡)=(Δ-HK)∫^𝑡₀𝑥^∗𝑓(𝑠)Δ𝑠.The function 𝑔 will be called a primitive of 𝑓 and by 𝑔(𝑡)=(Δ-HKP)∫^𝑡₀𝑓(𝑠)Δ𝑠 we will denote the Henstock-Kurzweil-Pettis-Δ-integral of 𝑓 on the interval 𝐼_𝑎.

In [29] the authors give examples of Henstock-Kurzweil-Pettis-Δ-integrable functions which are not integrable in the sense of Pettis and Henstock-Kurzweil on time scales.

Theorem 2.13 (see [29] mean value theorem). For each Δ-subinterval [𝑐,𝑑]⊂[𝑎,𝑏], if the integral (Δ-HKP)∫^𝑑_𝑐𝑦(𝑠)Δ𝑠 exists, then one has (Δ-HKP)^𝑑_𝑐𝑦(𝑠)Δ𝑠∈𝜇_Δ([𝑐,𝑑])⋅𝑐𝑜𝑛𝑣𝑦([𝑐,𝑑]), (2.7) where 𝑐𝑜𝑛𝑣𝑦([𝑐,𝑑]) denotes the close convex hull of the set 𝑦([𝑐,𝑑]).

Theorem 2.14 (see [45]). Let 𝑓_𝑛,𝑓∶𝐼_𝑎→𝐸 and assume that 𝑓_𝑛∶𝐼_𝑎→𝐸 are Δ-HKP integrable on 𝐼_𝑎. Let 𝐹_𝑛 be a primitive of 𝑓_𝑛. If one assumes that: (1)∀𝑥^∗∈𝐸^∗,𝑥^∗𝑓_𝑛(𝑡)→𝑥^∗𝑓(𝑡)𝜇_Δ a.e. on 𝐼_𝑎,(2)for each 𝑥^∗∈𝐸^∗ the family 𝐺={𝑥^∗𝐹_𝑛∶𝑛=1,2,…} is uniformly ACG_∗ on 𝐼_𝑎 (i.e., weakly uniformly ACG_∗ on 𝐼_𝑎), (3)for each 𝑥^∗∈𝐸^∗ the set 𝐺 is equicontinuous on 𝐼_𝑎,then 𝑓 is Δ-HKP integrable on 𝐼_𝑎 and ∫^𝑡₀𝑓_𝑛(𝑠)Δ𝑠 tends weakly in 𝐸 to ∫^𝑡₀𝑓(𝑠)Δ𝑠 for each 𝑡∈𝐼_𝑎.

Theorem 2.15 (see [5] Gronwall's inequality). Suppose that 𝑢,𝑔,ℎ∈𝐶_rd(𝐼_𝑎,𝐸) and ℎ≥0. Then 𝑢(𝑡)≤𝑔(𝑡)+^𝑡₀ℎ(𝜏)𝑢(𝜏)Δ𝜏,foreach𝑡∈𝐼_𝑎 (2.8) implies that 𝑢(𝑡)≤(𝑔(𝑡)+^𝑡₀𝑔(𝜏)ℎ(𝜏)Δ𝜏)exp(^𝑡₀ℎ(𝜏)Δ𝜏),foreach𝑡∈𝐼_𝑎. (2.9)

(III)The De Blasi measure of weak noncompactness is one of our fundamental tool in this paper (see [46]).

Let 𝐴 be a bounded nonempty subset of 𝐸.

The measure of weak noncompactness 𝛽(𝐴) is defined by𝛽(𝐴)=inf{𝑡>0∶thereexists𝐶∈𝐾^𝜔suchthat𝐴⊂𝐶+𝑡𝐵₀}, (2.10) where 𝐾^𝜔 is a set of weakly compact subsets of 𝐸 and 𝐵₀ is a norm unit ball in 𝐸.

Some properties of the measure of weak noncompactness 𝛽(𝐴) are known [46]: (1)if 𝐴⊂𝐵, then 𝛽(𝐴)≤𝛽(𝐵); (2)𝛽(𝐴)=𝛽(𝐴^𝑤), where 𝐴^𝑤 denotes the weak closure of 𝐴; (3)𝛽(𝐴)=0 if and only if 𝐴 is relatively weakly compact; (4)𝛽(𝐴∪𝐵)=max{𝛽(𝐴),𝛽(𝐵)}; (5)𝛽(𝜆𝐴)=|𝜆|𝛽(𝐴), (𝜆∈𝑅); (6)𝛽(𝐴+𝐵)≤𝛽(𝐴)+𝛽(𝐵); (7)𝛽(𝑐𝑜𝑛𝑣(𝐴))=𝛽(𝑐𝑜𝑛𝑣(𝐴))=𝛽(𝐴), where 𝑐𝑜𝑛𝑣(𝐴) denotes the convex hull of 𝐴.

We will need the following lemmas.

Lemma 2.16 (see [47]). Let 𝐸₁,𝐸₂ be bounded subsets of the Banach space 𝐸. If ‖𝐸₁‖=sup{‖𝑥‖∶𝑥∈𝐸₁}<1, then 𝛽(𝐸₁+𝐸₂)≤𝛽(𝐸₂)+‖𝐸₁‖𝛽(𝐾(𝐸₂,1)), (2.11) where 𝐾(𝐸₂,1)={𝑥∶𝐷(𝐸₂,𝑥)<1} and 𝐷(𝐸₂,𝑥)=inf{‖𝑥−𝑦‖∶𝑦∈𝐸₂}.

The lemma below is an adaptation of the corresponding result of Ambrosetti (see [48]).

Lemma 2.17. Let 𝐻⊂𝐶(𝐼_𝑎,𝐸) be a family of strongly equicontinuous functions. Let 𝐻(𝑡)={ℎ(𝑡)∈𝐸,ℎ∈𝐻}, for 𝑡∈𝐼_𝑎 and 𝐻(𝐼_𝑎)=⋃_{𝑡∈𝐼𝑎}𝐻(𝑡). Then 𝛽_𝐶(𝐻)=sup_{𝑡∈𝐼𝑎}𝛽(𝐻(𝑡))=𝛽(𝐻(𝐼_𝑎)), (2.12) where 𝛽_𝐶(𝐻) denotes the measure of weak noncompactness in 𝐶(𝐼_𝑎,𝐸) and the function 𝑡↦𝛽(𝐻(𝑡)) is continuous.

Proof. (I) First we prove the equality: sup_{𝑡∈𝐼𝑎}𝛽(𝐻(𝑡))=𝛽(𝐻(𝐼_𝑎)).
Since 𝐻(𝑡)⊂𝐻(𝐼_𝑎) by the first property of the measure of noncompactness 𝛽(𝐻(𝑡))≤𝛽(𝐻(𝐼_𝑎)) and consequentlysup_{𝑡∈𝐼𝑎}𝛽(𝐻(𝑡))≤𝛽(𝐻(𝐼_𝑎)). (2.13) By strong equicontinuity of 𝐻 we deduce that for 𝜀>0, there exists 𝛿>0 such that |𝑡−𝑠|<𝛿⇒‖𝑢(𝑡)−𝑢(𝑠)‖<𝜀 for 𝑡,𝑠∈𝐼_𝑎 and for all 𝑢∈𝐻.
We divide the interval 𝐼_𝑎={𝑡∈𝑇∶0≤𝑡≤𝑎} in the following way: 𝑡₀=0,𝑡₁=sup_{𝑠∈𝐼𝑎}{𝑠∶𝑠≥𝑡₀,𝑠−𝑡₀𝑡<𝛿},₂=sup_{𝑠∈𝐼𝑎}{𝑠∶𝑠≥𝑡₁,𝑠−𝑡₁⋮𝑡<𝛿},_𝑛=sup_{𝑠∈𝐼𝑎}{𝑠∶𝑠≥𝑡_𝑛−1,𝑠−𝑡_𝑛−1<𝛿}. (2.14) Since 𝑇 is closed, we have 𝑡_𝑖∈𝐼_𝑎. If some 𝑡_𝑖+1=𝑡_𝑖 then 𝑡_𝑖+2=inf{𝑡∈𝑇∶𝑡≥𝑡_𝑖+1}.
As𝑢(𝑡)=𝑢(𝑡_𝑖)+𝑢(𝑡)−𝑢(𝑡_𝑖)∈𝑢(𝑡_𝑖)+𝜀𝐾(0,1), (2.15) where 𝐾(0,1)={𝑥∶‖𝑥‖<1}, we have 𝑢(𝑡)∈^𝑛_𝑖=1𝐻(𝑡_𝑖)+𝜀𝐾(0,1),𝐻(𝐼_𝑎)⊂^𝑛_𝑖=1𝐻(𝑡_𝑖)+𝜀𝐾(0,1). (2.16)
By the properties of the measure of weak noncompactness and Lemma 2.16, we obtain𝛽(𝐻(𝐼_𝑎))≤𝛽(^𝑛_𝑖=1𝐻(𝑡_𝑖))+‖𝜀𝐾(0,1)‖⋅𝛽(𝐾(^𝑛_𝑖=1𝐻(𝑡_𝑖),1))<sup_{𝑡𝑖∈𝐼𝑎}𝛽(𝐻(𝑡_𝑖))+𝜀𝛽(𝐾(𝐻(𝐼_𝑎),1))≤sup_{𝑡∈𝐼𝑎}𝛽(𝐻(𝑡))+𝜀𝛽(𝐾(𝐻(𝐼_𝑎),1)). (2.17) Since the above inequality holds, for any 𝜀>0, we have 𝛽(𝐻(𝐼_𝑎))≤sup_{𝑡∈𝐼𝑎}𝛽(𝐻(𝑡)). (2.18) Hence, from (2.13) and (2.18), we conclude that 𝛽(𝐻(𝐼_𝑎))=sup_{𝑡∈𝐼𝑎}𝛽(𝐻(𝑡)).
(II) The proof of the equality 𝛽_𝐶(𝐻)=sup_{𝑡∈𝐼𝑎}𝛽(𝐻(𝑡)) is similar to the proof of Lemma 2.1 of Ambrosetti (see [48]), where we choose points 𝑡_𝑖 as in part (I) of our proof.
(III) Now we prove that the function 𝑡↦𝛽(𝐻(𝑡)) is continuous.
Let 𝑣(𝑡)=𝛽(𝐻(𝑡)) because 𝐻(𝑡)⊂𝐻(𝑡)−̇𝐻(𝑠)+̇𝐻(𝑠)⊂𝐻(𝑡)−̇𝐻(𝑠)+𝐻(𝑠), where 𝐻(𝑡)−̇𝐻(𝑠)+̇𝐻(𝑠)={𝑦(𝑡)∶𝑦(𝑡)=𝑦(𝑡)−𝑦(𝑠)+𝑦(𝑠)∶𝑦∈𝐻} and the sum is taken in the Minkowski sense.
By the property (vi) of the measure of weak noncompactness, we have𝛽(𝐻(𝑡))≤𝛽(𝐻(𝑡)−̇𝐻(𝑠))+𝛽(𝐻(𝑠)). (2.19) This implies that |𝛽(𝐻(𝑡))−𝛽(𝐻(𝑠))|≤𝛽(𝐻(𝑡)−̇𝐻(𝑠))≤𝛿(𝐻(𝑡)−̇𝐻(𝑠))=sup_{𝑥,𝑦∈𝐻}{‖(𝑥(𝑡)−𝑥(𝑠))−(𝑦(𝑡)−𝑦(𝑠))‖}≤sup_{𝑥,𝑦∈𝐻}{‖𝑥(𝑡)−𝑥(𝑠)‖+‖𝑦(𝑡)−𝑦(𝑠)‖}. (2.20) By equicontinuity of 𝐻 we obtain the continuity of 𝑣(𝑡).

3. Main Results

Now, we will consider equivalently integral problem𝑥(𝑡)=𝑥₀+^𝑡₀𝑓(𝑧,𝑥(𝑧),^𝑧₀𝑘(𝑧,𝑠,𝑥(𝑠))Δ𝑠)Δ𝑧,𝑡∈𝐼_𝑎=[0,𝑎]∩𝑇 (3.1) where 𝑓∶𝐼_𝑎×𝐸×𝐸→𝐸,𝑘∶𝐼_𝑎×𝐼_𝑎×𝐸→𝐸,𝑥∶𝐼_𝑎→𝐸, 𝑇 denotes a time scale, 0∈𝑇, 𝐼_𝑎 denotes a time scale interval, (𝐸,‖⋅‖) is a Banach space, and integrals are taken in the sense of Δ-HKP integrals.

Fix 𝑥^∗∈𝐸^∗and consider the problem(𝑥^∗𝑥)^Δ=𝑥^∗(𝑓(𝑡,𝑥(𝑡),^𝑡₀𝑘(𝑡,𝑠,𝑥(𝑠)))Δ𝑠). (3.2) Let us introduce a definition.

Definition 3.1. Let 𝐹∶𝐼→𝐸 and let 𝐴⊂𝐼. The function 𝑓∶𝐴→𝐸 is a pseudoΔ-derivative of 𝐹 on 𝐴 if for each 𝑥^∗∈𝐸^∗ the real-valued function 𝑥^∗𝐹 is Δ-differentiable 𝜇_Δ almost everywhere on 𝐴 and (𝑥^∗𝐹)^Δ=𝑥^∗𝑓𝜇_Δ almost everywhere on 𝐴.

Regarding the above definition it is clear that the left-hand side of (3.2) can be rewritten to the form 𝑥^∗(𝑥^Δ), where 𝑥^Δ denotes the pseudoΔ-derivative.

To obtain the existence result for our problem it is necessary to define a notion of a solution.

Definition 3.2. A function 𝑥∶𝐼_𝑎→𝐸 is said to be a pseudosolution of the problem (1.1) if it satisfies the following conditions: (1)𝑥(⋅) is ACG_∗ function, (2)𝑥(0)=𝑥₀, (3)for each 𝑥^∗∈𝐸^∗ there exists a set 𝐴(𝑥^∗) with 𝜇_Δ measure zero, such that for each 𝑡∉𝐴(𝑥^∗), (𝑥^∗𝑥)^Δ(𝑡)=𝑥^∗(𝑓(𝑡,𝑥(𝑡),^𝑡₀𝑘(𝑡,𝑠,𝑥(𝑠))Δ𝑠)). (3.3) A continuous function 𝑥∶𝐼_𝑎→𝐸 is said to be a solution of the problem (3.1) if it satisfies 𝑥(𝑡)=𝑥₀+^𝑡₀𝑓(𝑧,𝑥(𝑧),^𝑧₀𝑘(𝑧,𝑠,𝑥(𝑠))Δ𝑠)Δ𝑧,forevery𝑡∈𝐼_𝑎. (3.4)

Because we consider a new type of integral and a new type of solutions is necessary to prove that each solution 𝑥 of problem (1.1) is the solution of problem (3.1).

Let 𝑥 is a continuous solution of (3.1). Fix 𝑥^∗∈𝐸^∗. By definition, 𝑥 is ACG_∗ function and 𝑥(0)=𝑥₀. Since, for each 𝑥^∗∈𝐸^∗ and 𝜇_Δ a.e 𝑡∈𝐼_𝑎, (𝑥^∗𝑥(𝑡))^Δ=𝑥^∗(𝑥^Δ(𝑡))=𝑥^∗(𝑓(𝑡,𝑥(𝑡),∫^𝑡₀𝑘(𝑡,𝑠,𝑥(𝑠))Δ𝑠)) and the last is Δ-HK integrable, so is differentiable 𝜇_Δ a.e.

Moreover, ∫^𝑡₀𝑥^∗(𝑓(𝑧,𝑥(𝑧),∫^𝑧₀𝑘(𝑧,𝑠,𝑥(𝑠)Δ𝑠))Δ𝑧=∫^𝑡₀(𝑥^∗𝑥(𝑠))^ΔΔ𝑠=𝑥^∗(𝑥(𝑡)−𝑥₀). Thus 𝑥(𝑡) satisfies (3.1).

Now assume that 𝑦 is ACG_∗ function and 𝑦(0)=𝑥₀. By the definition of Δ-HKP integrals there exists an ACG_∗ function 𝐺 such that 𝐺(0)=𝑥₀ and𝑥^∗(𝐺^Δ(𝑡))=𝑥^∗(𝑓(𝑡,𝑦(𝑡),^𝑡₀𝑘(𝑡,𝑠,𝑦(𝑠))Δ𝑠))𝜇_Δa.e. (3.5) Hence𝑥^∗(𝑦(𝑡))=𝑥₀+^𝑡₀𝑥^∗(𝑓(𝑧,𝑦(𝑧),^𝑧₀𝑘(𝑧,𝑠,𝑦(𝑠))Δ𝑠)Δ𝑧)=𝑥₀+^𝑡₀𝑥^∗(𝐺^Δ)(𝑠)Δ𝑠=𝑥₀+𝑥^∗(𝐺(𝑡))−𝑥^∗(𝐺(0))=𝑥^∗(𝐺(𝑡)). (3.6) We obtain 𝑦=𝐺,𝜇_Δ-a.e. and then 𝑦^Δ(𝑡)=𝑓(𝑡,𝑦(𝑡),∫^𝑡₀𝑘(𝑡,𝑠,𝑦(𝑠))Δ𝑠).

Let𝐵={𝑥∈𝐸∶‖𝑥‖≤‖𝑥₀‖+𝑝,𝑝>0},𝐵={𝑥∈(𝐶(𝐼_𝑎,𝐸),𝜔)∶𝑥(0)=𝑥₀,‖𝑥‖≤‖𝑥₀‖+𝑝,𝑝>0}. (3.7) Moreover, let 𝐹(𝑥)(𝑡)=𝑥₀+∫^𝑡₀𝑓(𝑧,𝑥(𝑧),∫^𝑧₀𝑘(𝑧,𝑠,𝑥(𝑠))Δ𝑠)Δ𝑧, 𝑡∈𝐼_𝑎, 𝐾={𝐹(𝑥)∶𝑥∈𝐵}, 𝐾₁={∫^𝑧₀𝑘(𝑧,𝑠,𝑥(𝑠))Δ𝑠∶𝑧∈𝐼_𝑡=[0,𝑡]∩𝑇,𝑡∈𝐼_𝑎,𝑥∈𝐵}.

Theorem 3.3. Assume, that for each uniformly ACG_∗ function 𝑥∶𝐼_𝑎→𝐸 the functions: 𝑘(⋅,𝑠,𝑥(𝑠)),𝑓(⋅,𝑥(⋅),∫₀^(⋅)𝑘(⋅,𝑠,𝑥(𝑠))Δ𝑠) are Δ-HKP integrable, 𝑓 and 𝑘 are weakly-weakly sequentially continuous functions. Suppose that there exist constants 𝑑₁,𝑑₂,𝑑₃>0 such that 𝛽(𝑓(𝐼×𝐴×𝐶))≤𝑑₁⋅𝛽(𝐴)+𝑑₂⋅𝛽(𝐶), (3.8) for each subset 𝐴,𝐶 of 𝐵,𝐼⊂𝐼_𝑎, 𝛽(𝑘(𝐼×𝐼×𝑋))≤𝑑₃⋅𝛽(𝑋) (3.9) for each subset 𝑋 of 𝐵,𝐼⊂𝐼_𝑎, where 𝑓(𝐼×𝐴×𝐶)={𝑓(𝑡,𝑥₁,𝑥₂)∶(𝑡,𝑥₁,𝑥₂)∈𝐼×𝐴×𝐶},𝑘(𝐼×𝐼×𝑋)={𝑘(𝑡,𝑠,𝑥)∶(𝑡,𝑠,𝑥)∈𝐼×𝐼×𝑋}.
Moreover, let 𝐾 and 𝐾₁ be equicontinuous, equibounded and uniformly ACG_∗ on 𝐼_𝑎. Then there exists a pseudosolution of the problem (1.1) on 𝐼_𝑐, for some 𝑐∈𝑇,0<𝑐≤𝑎 and 0<𝑐⋅𝑑₁+𝑐²⋅𝑑₂⋅𝑑₃<1.

Proof. Fix an arbitrary 𝑝≥0. Recall, that a set 𝐾 of continuous functions 𝐹(𝑥)∈𝐾 defined on a time scale interval 𝐼_𝑎 is equicontinuous on 𝐼_𝑎 if for each 𝜀>0 there exists 𝛿>0 such that ‖𝐹(𝑥)(𝑡)−𝐹(𝑥)(𝜏)‖<𝜀 for all 𝑥∈𝐵 whenever |𝑡−𝜏|<𝛿,𝑡,𝜏∈𝐼_𝑎, for each 𝐹(𝑥)∈𝐾. Thus, for each 𝜀>0 there exists 𝛿>0 such that ‖∫^𝑡_𝜏𝑓(𝑧,𝑥(𝑧),∫^𝑧₀𝑘(𝑧,𝑠,𝑥(𝑠))Δ𝑠)Δ𝑧‖<𝜀 for all 𝑥∈𝐵 whenever |𝑡−𝜏|<𝛿 and 𝑡,𝜏∈𝐼_𝑎. As a result, there exists a number 𝑐, 0<𝑐≤𝑎 such that ‖^𝑡₀𝑓(𝑧,𝑥(𝑧),^𝑧₀𝑘(𝑧,𝑠,𝑥(𝑠))Δ𝑠)Δ𝑧‖≤𝑝,𝑡∈𝐼_𝑐,𝑥∈𝐵. (3.10) We will show that the operator 𝐹 is well defined and maps 𝐵 into 𝐵.
To see this note for any 𝑥^∗∈𝐸^∗, such that ‖𝑥^∗‖≤1, for any 𝑥∈𝐵 and 𝑡∈𝐼_𝑐 we have|𝑥^∗𝐹(𝑥)(𝑡)|≤|𝑥^∗𝑥₀|+|𝑥^∗𝑡₀𝑓(𝑧,𝑥(𝑧),^𝑧₀𝑘(𝑧,𝑠,𝑥(𝑠))𝑑𝑠)Δ𝑧|≤‖𝑥^∗‖‖𝑥₀‖+‖𝑥^∗‖‖^𝑡₀𝑓(𝑧,𝑥(𝑧),^𝑧₀𝑘(𝑧,𝑠,𝑥(𝑠))𝑑𝑠)Δ𝑧‖≤‖𝑥₀‖+𝑝, (3.11) so sup{|𝑥^∗𝐹(𝑥)(𝑡)|∶𝑥^∗∈𝐸^∗,‖𝑥^∗‖≤1}≤‖𝑥₀‖+𝑝 (3.12) and as a result ‖𝐹(𝑥)(𝑡)‖≤‖𝑥₀‖+𝑝. (3.13) Thus 𝐹(𝑥)(𝑡)∈𝐵.
We will show, that the operator 𝐹 is weakly-weakly sequentially continuous. By Lemma 9 of [34] a sequence 𝑥_𝑛(⋅) is weakly convergent in 𝐶(𝐼_𝑐,𝐸) to 𝑥(⋅) if and only if 𝑥_𝑛(𝑡) tends weakly to 𝑥(𝑡) for each 𝑡∈𝐼_𝑐, so if 𝑥−→_𝑛^𝜔𝑥 in 𝐶(𝐼_𝑐,𝐸) then 𝑘(𝑡,𝑠,𝑥_𝑛(𝑠))−→^𝜔𝑘(𝑡,𝑠,𝑥(𝑠)) in 𝐸 for 𝑡∈𝐼_𝑐 and by Theorem 2.14 (see our assumptions on 𝐾₁) we have lim_𝑛→∞∫^𝑡₀𝑘(𝑧,𝑠,𝑥_𝑛(𝑠))𝑑𝑠=∫^𝑡₀𝑘(𝑧,𝑠,𝑥(𝑠))𝑑𝑠 weakly in 𝐸, for each 𝑡∈𝐼_𝑐. Moreover, because 𝑓 is weakly-weakly sequentially continuous, we have𝑓(𝑡,𝑥_𝑛(𝑡),^𝑡₀𝑘(𝑡,𝑠,𝑥_𝑛(𝑠))Δ𝑠)−→^𝜔𝑓(𝑡,𝑥(𝑡),^𝑡₀𝑘(𝑡,𝑠,𝑥(𝑠))Δ𝑠) (3.14) in 𝐸, for each 𝑡∈𝐼_𝑐. Thus Theorem 2.14 (see our assumptions on 𝐾) implies 𝐹(𝑥_𝑛)(𝑡)→𝐹(𝑥)(𝑡) weakly in 𝐸 for each 𝑡∈𝐼_𝑐 so Lemma 9 of [34] guarantees that 𝐹(𝑥_𝑛)→𝐹(𝑥) in 𝐶(𝐼_𝑐,𝐸) with its weak topology.
Suppose that 𝑉⊂𝐵 satisfies the condition 𝑉=𝑐𝑜𝑛𝑣({𝑥}∪𝐹(𝑉)). We will prove that 𝑉 is relatively weakly compact and so (2.1) is satisfied. Since 𝑉⊂𝐵,𝐹(𝑉)⊂𝐾. Then 𝑉⊂𝑉=𝑐𝑜𝑛𝑣({𝑥}∪𝐹(𝑉)) is equicontinuous. By Lemma 2.17, 𝑡↦𝑣(𝑡)=𝛽(𝑉(𝑡)) is continuous on 𝐼_𝑐.
For fixed 𝑡∈𝐼_𝑐 we divide the interval [0,𝑡] into 𝑚 parts in the following way 𝑡₀=0,𝑡₁=sup_{𝑠∈𝐼𝑎}{𝑠∶𝑠≥𝑡₀,𝑠−𝑡₀𝑡<𝛿},₂=sup_{𝑠∈𝐼𝑎}{𝑠∶𝑠≥𝑡₁,𝑠−𝑡₁⋮𝑡<𝛿},_𝑛=sup_{𝑠∈𝐼𝑎}{𝑠∶𝑠≥𝑡_𝑛−1,𝑠−𝑡_𝑛−1<𝛿}. (3.15) Since 𝑇 is closed, we have 𝑡_𝑖∈𝐼_𝑎. If some 𝑡_𝑖+1=𝑡_𝑖 then 𝑡_𝑖+2=inf{𝑡∈𝑇∶𝑡≥𝑡_𝑖+1}.
For fixed 𝑧∈[0,𝑡] we divide the interval [0,𝑧] into 𝑚 parts: 𝑧₀=0,𝑧₁=sup_{𝑠∈[0,𝑡]}{𝑠∶𝑠≥𝑧₀,𝑠−𝑧₀𝑧<𝛿},₂=sup_{𝑠∈[0,𝑡]}{𝑠∶𝑠≥𝑧₁,𝑠−𝑧₁⋮𝑧<𝛿},_𝑛=sup_{𝑠∈[0,𝑡]}{𝑠∶𝑠≥𝑧_𝑛−1,𝑠−𝑧_𝑛−1<𝛿}suchthat𝜇_Δ(𝐼_𝑗)=𝑗𝑧𝑚, (3.16)𝑗=0,1,…,𝑚, 𝐼_𝑗=[𝑧_𝑗,𝑧_𝑗+1].
Let 𝑉([𝑧_𝑗,𝑧_𝑗+1])={𝑢(𝑠)∶𝑢∈𝑉,𝑧_𝑗≤𝑠≤𝑧_𝑗+1,𝑗=0,1,…,𝑚−1}. By Lemma 2.17 and the continuity of 𝑣 there exists 𝑠_𝑗∈𝐼_𝑗=[𝑧_𝑗,𝑧_𝑗+1], such that𝛽(𝑉([𝑧_𝑗,𝑧_𝑗+1]))=sup{𝛽(𝑉(𝑠))∶𝑧_𝑗≤𝑠≤𝑧_𝑗+1}∶=𝑣(𝑠_𝑗). (3.17) By Theorem 2.13 and the properties of the Δ-HKP integral we have for 𝑥∈𝑉 that 𝐹(𝑥)(𝑡)=𝑥₀+^𝑚−1_{𝑖=0𝐽𝑖}𝑓(𝑧,𝑥(𝑧),^𝑚−1_{𝑗=0𝐼𝑗}𝑘(𝑧,𝑠,𝑥(𝑠))Δ𝑠)Δ𝑧∈𝑥₀+^𝑚−1_𝑖=0𝜇_Δ(𝐽_𝑖)𝑐𝑜𝑛𝑣𝑓(𝐽_𝑖,𝑉(𝐽_𝑖),^𝑚−1_𝑗=0(𝑧_𝑗+1−𝑧_𝑗)𝑐𝑜𝑛𝑣𝑘(𝐼_𝑗,𝐼_𝑗,𝑉(𝐼_𝑗))), (3.18) where 𝐽_𝑖=[𝑡_𝑖,𝑡_𝑖+1], 𝑖=0,1,…,𝑚−1.
Using (3.8), (3.9) and properties of the measure of noncompactness we obtain𝛽(𝐹(𝑉)(𝑡))≤^𝑚−1_𝑖=0𝜇_Δ(𝐽_𝑖)𝛽(𝑓(𝐽_𝑖,𝑉(𝐽_𝑖),^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)𝑐𝑜𝑛𝑣𝑘(𝐼_𝑗,𝐼_𝑗,𝑉(𝐼_𝑗≤))))_{𝑚−1𝑖=0}𝜇_Δ(𝐽_𝑖)𝛽(𝑓(𝐽_𝑖,𝑉(𝐽_𝑖),^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)𝑐𝑜𝑛𝑣𝑘(𝐼_𝑗,𝐼_𝑗,𝑉(𝐼_𝑗≤))))_{𝑚−1𝑖=1}𝜇_Δ(𝐽_𝑖)⋅𝑑₁⋅𝛽(𝑉(𝐽_𝑖))+^𝑚−1_𝑖=1𝜇_Δ(𝐽_𝑖)⋅𝑑₂⋅𝛽(^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)⋅𝑐𝑜𝑛𝑣𝑘(𝐼_𝑗,𝐼_𝑗,𝑉(𝐼_𝑗≤)))_{𝑚−1𝑖=1}𝜇_Δ(𝐽_𝑖)⋅𝑑₁⋅𝛽(𝑉(𝐼_𝑐))+^𝑚−1_𝑖=1𝜇_Δ(𝐽_𝑖)⋅𝑑₂⋅^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)⋅𝛽(𝑘(𝐼_𝑗,𝐼_𝑗,𝑉(𝐼_𝑗)))≤𝛽(𝑉(𝐼_𝑐))⋅𝑑₁⋅𝑐+^𝑚−1_𝑖=1𝜇_Δ(𝐽_𝑖)⋅𝑑₂⋅^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)⋅𝑑₃⋅𝛽(𝑉(𝐼_𝑗))≤𝛽(𝑉(𝐼_𝑐))⋅𝑑₁⋅𝑐+𝛽(𝑉(𝐼_𝑐))⋅𝑑₂⋅𝑑₃⋅𝑐²=𝛽(𝑉(𝐼_𝑐))(𝑑₁⋅𝑐+𝑑₂⋅𝑑₃⋅𝑐²). (3.19) Since 𝑉=𝑐𝑜𝑛𝑣({𝑥}∪𝐹(𝑉)), 𝛽(𝑉(𝑡))=𝛽(𝑐𝑜𝑛𝑣({𝑥}∪𝐹(𝑉(𝑡)))) so 𝛽(𝑉(𝑡))≤𝛽(𝑉(𝐼_𝑐))(𝑑₁⋅𝑐+𝑑₂⋅𝑑₃⋅𝑐²), for 𝑡∈𝐼_𝑐.
Using Lemma 2.17 we obtain𝛽(𝑉(𝐼_𝑐))≤𝛽(𝑉(𝐼_𝑐))(𝑑₁⋅𝑐+𝑑₂⋅𝑑₃⋅𝑐²). (3.20) Since 0<𝑐⋅𝑑₁+𝑐²⋅𝑑₂⋅𝑑₃<1 we obtain 𝑣(𝑡)=𝛽(𝑉(𝑡))=0 for 𝑡∈𝐼_𝑐.

Using Ascoli`s theorem we have that 𝑉 is relatively weakly compact.

By Theorem 2.3 the operator 𝐹 has a fixed point. This means that there exists a pseudosolution of the problem (1.1).

Theorem 3.4. Assume, that for each uniformly ACG^∗ function 𝑥∶𝐼_𝑎→𝐸, the functions 𝑘(⋅,𝑠,𝑥(𝑠)),𝑓(⋅,𝑥(⋅),∫₀^(⋅)𝑘(⋅,𝑠,𝑥(𝑠))𝑑𝑠) are Δ-HKP integrable and 𝑘,𝑓 are weakly-weakly sequentially continuous. Suppose that there exists a constant 𝑑>0 and a continuous function 𝑐₁∶𝐼_𝑎→𝑅₊ such that 𝛽(𝑓(𝐼,𝐴,𝐶))≤𝑑⋅𝛽(𝐶),foreach𝐴,𝐶⊂𝐵,𝐼⊂𝐼_𝑎,𝛽(𝑘(𝐼,𝐼,𝑋))≤sup_𝑠∈𝐼𝑐₁(𝑠)𝛽(𝑋),foreach𝑋⊂𝐵,𝐼⊂𝐼_𝑎, (3.21) where 𝑓(𝐼,𝐴,𝐶)={𝑓(𝑡,𝑥₁,𝑥₂)∶(𝑡,𝑥₁,𝑥₂)∈𝐼×𝐴×𝐶},𝑘(𝐼,𝐼,𝑋)={𝑘(𝑡,𝑠,𝑥)∶(𝑡,𝑠,𝑥)∈𝐼×𝐼×𝑋}. (3.22) Moreover, let 𝐾 and 𝐾₁ be equicontinuous and uniformly ACG^∗ on 𝐼_𝑎. Then there exists a pseudo solution of the problem (1.1) on 𝐼_𝑐, for some 𝑐∈𝑇,0<𝑐≤𝑎.

Proof. The first part of the proof is the same as that of the proof of the previous theorem. It remains to show the relative compactness of 𝑉, where 𝑉 is defined in Theorem 3.3. In this case notice for 𝑡∈𝐼_𝑐 and 𝑧_𝑗 as in Theorem 3.3 we have 𝛽(𝑉(𝑡))≤^𝑚−1_𝑖=0𝜇_Δ(𝐽_𝑖)⋅𝑑⋅𝛽(^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)⋅𝑐𝑜𝑛𝑣𝑘(𝐼_𝑗,𝐼_𝑗,𝑉(𝐼_𝑗≤)))_{𝑚−1𝑖=0}𝜇_Δ(𝐽_𝑖)⋅𝑑⋅^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)⋅𝛽(𝑘(𝐼_𝑗,𝐼_𝑗,𝑉(𝐼_𝑗≤)))_{𝑚−1𝑖=0}𝜇_Δ(𝐽_𝑖)⋅𝑑⋅^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)⋅sup_{𝑠∈𝐼𝑗}𝑐₁(𝑠)𝛽(𝑉(𝐼_𝑗)). (3.23) Let sup_{𝑠∈𝐼𝑗}𝑐₁(𝑠)=𝑐₁(𝑝_𝑗), 𝑝_𝑗∈𝐼_𝑗. Then 𝛽(𝑉(𝑡))≤𝑐⋅𝑑⋅^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)⋅𝑐₁(𝑝_𝑗)𝑣(𝑠_𝑗)=𝑐⋅𝑑(^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)⋅𝑐₁(𝑝_𝑗)𝑣(𝑝_𝑗)+^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)(𝑐₁(𝑝_𝑗)(𝑣(𝑠_𝑗)−𝑣(𝑝_𝑗)))). (3.24) Fix 𝜀>0. From the equicontinuity of 𝑉 we may choose 𝑚 large enough so that 𝑣(𝑠_𝑗)−𝑣(𝑝_𝑗)<𝜀 and so 𝛽(𝑉(𝑡))≤𝑐⋅𝑑(^𝑚−1_𝑗=0𝜇_Δ(𝐼_𝑗)⋅𝑐₁(𝑝_𝑗)𝑣(𝑝_𝑗)+^𝑚−1_𝑗=0𝑧𝑚𝑐₁(𝑝_𝑗)⋅𝜀)≤𝑐⋅𝑑(^𝑚−1_𝑗−0𝜇_Δ(𝐼_𝑗)𝑐₁(𝑝_𝑗)𝑣(𝑝_𝑗)+𝑧⋅𝜀⋅max_{0≤𝑘≤𝑚−1}𝑐₁(𝑝_𝑘)). (3.25) Since 𝜀>0 is arbitrary small and 𝑧⋅max_{0≤𝑘≤𝑚−1}𝑐₁(𝑝_𝑘) is bounded, we have that 𝑧⋅𝜀⋅max_{0≤𝑘≤𝑚−1}𝑐₁(𝑝_𝑘) is arbitrary small. Therefore 𝑣(𝑡)=𝛽(𝑉(𝑡))≤𝑐⋅𝑑⋅^𝑡₀𝑐₁(𝑠)𝑣(𝑠)Δ𝑠,𝑡∈𝐼_𝑐. (3.26) Thus, by Gronwall's inequality for Δ-integrals [5] we have that 𝑣(𝑡)=𝛽(𝑉(𝑡))=0,for𝑡∈𝐼_𝑐. (3.27) Using Ascoli`s theorem we deduce that 𝑉 is relatively weakly compact.

By Theorem 2.14 the operator 𝐹 has a fixed point. This means that there exists a pseudo solution of the problem (1.1).

Remark 3.5. The conditions in Theorems 3.3 and 3.4 can be also generalized to the Sadovskii condition [28], the Szufla condition [49], and the others and 𝛽 can be replaced by some axiomatic measure of weak noncompactness.