Noncompact Perturbation of Sweeping Process with Delay in Banach Spaces

Abstract

We have proven an existence theorem concerning the existence of solutions for a functional evolution inclusion governed by sweeping process with closed convex sets depending on time and state and with a noncompact nonconvex perturbation in Banach spaces. This work extends some recent existence theorems concerning sweeping processes from Hilbert spaces setting to Banach spaces setting. Moreover, it improves some recent existence results for sweeping processes in Banach spaces.

1. Introduction

Differential inclusions represent an important generalization of differential equations. Moreover, they have several applications in different branches of mathematical sciences such as Control Theory, Viscosity, Optimization, and Mechanical problems (see [1–4]). In his leading paper, Moreau [5] proposed and studied the following differential inclusion governed by sweeping process of first order: −𝑢^′(𝑡)∈𝑁_𝐺(𝑡)(𝑢(𝑡[],𝑢(0)))a.e.on𝐼=0,𝑇=𝑢₀, (1) where 𝐺  is a set-valued function from the interval 𝐼=[0,𝑇] to the family of nonempty closed convex subsets of a Hilbert space 𝐻 and 𝑁_𝐺(𝑡)(𝑢(𝑡)) is the normal cone of the subset 𝐺(𝑡) at the point 𝑢(𝑡). Problem (1) corresponds to several important mechanical problems. For more details concerning the applications of (1), we refer to [5].

Since then, important improvements have been developed by weakening assumptions in order to obtain the most general result of existence for sweeping processes.

Let 𝑋 be a Banach space, 𝑟 a positive real number, 𝐶_𝑟=𝐶_𝑋([−𝑟,0])the Banach space of continuous functions from [−𝑟,0] to 𝑋 endowed with the uniform norm ‖𝑢‖_∞=Sup{‖𝑢(𝑡)‖∶𝑡∈[−𝑟,0]}, 𝐶 a multifunction from 𝐼×𝑋 to the family of nonempty closed convex subsets of 𝑋, and 𝐹 a multifunction defined on [0,𝑇]×𝐶_𝑋([−𝑟,0])×𝐶_𝑋([−𝑟,0]) and with nonconvex noncompact values in the dual space, 𝑋^∗, of 𝑋. The aim of this paper is to find the sufficient conditions that guarantee the existence of two continuous functions 𝑢∶[−𝑟,𝑇]→𝑋^∗and 𝑤∶[−𝑟,𝑇]→𝑋 such that the following perturbed sweeping process is satisfied: 𝑢(𝑡)𝜑(𝑡)(𝑡)(𝑡)[],𝑢(𝑡)𝜑(0)+=𝐽,𝑤=𝜓,for𝑡∈−𝑟,0=𝐽^𝑡₀𝑢^′(𝑠)𝐽𝑑𝑠,for𝑡∈𝐼,_∗(𝑢(𝑡((𝑡𝑤(𝑡)(0)+))∈𝐶𝑡,𝑤)),for𝑡∈𝐼,=𝜓^𝑡₀𝐽^∗𝑢(𝑠)𝑢𝑑𝑠,for𝑡∈𝐼,_′(𝑡)∈−𝑁_{𝐶(𝑡,𝑤(𝑡))}𝐽^∗𝑢(𝑡)+𝐹𝑡,𝜏(𝑡)𝑤,𝜏(𝑡)𝐽^∗𝑢,a.e.for𝑡∈𝐼, (2) where 𝜑,𝜓∈𝐶_𝑋([−𝑟,0]) are given with 𝜑(0)∈𝐶(0,𝜓(0)), 𝐽 is the duality mapping in 𝑋, and for each 𝑡∈[0,𝑇],𝜏(𝑡)∶𝐶_𝑋([−𝑟,𝑡])→𝐶_𝑋([−𝑟,0]),(𝜏(𝑡)𝑔)(𝑠)=𝑔(𝑠+𝑡), for all 𝑠∈[−𝑟,0].

In order to explain the mathematical motivation for this work, we mention some recent results in this domain. In a very recent paper, Aitalioubrahim [6] considered (2) when 𝑋 is a Hilbert spaces 𝐶 is a multifunction from 𝐼 to the family of nonempty closed nonconvex subsets of 𝑋, and 𝐹 is a multifunction defined on [0,𝑇]×𝐶_𝑋([−𝑟,0]) and with nonconvex noncompact values in 𝑋. Castaing et al. [7] considered a second order sweeping process without delay in a separable Hilbert space 𝐻 in the case when 𝐶 is a Lipschitz multifunction defined on 𝐼×𝐻, taking a closed 𝜌-proxy regular (𝜌>0) values in 𝐻, and 𝐹 is a convex weakly compact valued scalary uppersemicontinuous defined on 𝐼×𝐻×𝐻 and satisfying the growth condition 𝐹𝑡,𝑥,𝑦⊆1+‖𝑥‖+𝑦𝐵_𝐻, (3) for all (𝑡,𝑥,𝑦)∈[0,𝑇]×𝐻×𝐻. For other results concerning (2) with or without perturbation in Hilbert spaces, we refer to [8–13]. Bounkhel and Al-Yusof [14] initiated the extension of (1) from the Hilbert setting to the Banach spaces setting. In fact, Bounkhel [8] considered (2) when 𝐶 is a multifunction from 𝐼 to the family of nonempty closed convex subsets of 𝑋 and satisfyes a condition similar to Condition (𝐶₁) in the statement of our result (Theorem 18) and 𝐹 is an uppersemicontinuous multifunction defined on [0,𝑇]×𝑋 and with convex compact values in 𝑋^∗ such that 𝐹(𝑡,𝑥)⊆𝑍, for all (𝑡,𝑥)∈[0,𝑇]×𝑋, for some convex compact set 𝑍⊂𝑋^∗. More recently, Ibrahim and Aladsani [15] considered a second order sweeping process without delay in a separable 𝑝-uniformly convex and 𝑞-uniformly smooth Banach space 𝑋, and the perturbation 𝐹 is an upper semicontinuous defined from [0,𝑇]×𝑋×𝑋to the family of nonempty convex weakly compact sets of the topological dual space 𝑋^∗ of 𝑋 such that 𝐹𝑡,𝑥,𝑦⊆1+‖𝑥‖+𝑦𝐵_𝑋,∀𝑡,𝑥,𝑦∈[0,𝑇[×𝑋×𝑋. (4)

In this paper, our main purpose is to prove the existence of solutions of (2) in the case when the perturbation 𝐹 is nonconvex noncompact values. We will adapt the discretization technique used in [6].

The paper is organized as follows. Section 2 is devoted to some definitions and notations needed later. In Section 3, we prove in Theorem 18 the main result.

We note that if 𝑋 is a Hilbert space, then 𝐽 is equal to the identity operator on 𝑋, and 𝑋 is 2-uniformly convex and 2-uniformly smooth Banach space. Therefore, our work extends many problems from Hilbert spaces setting to Banach spaces setting. Moreover, our work improves many results in the literature concerning the existence of solutions for some evolution inclusions governed by sweeping process in Banach spaces, for example, [14, 15]. In addition our technique allows us to discuss some sweeping process problems with noncompact perturbation in Banach spaces.

2. Preliminaries and Notations

Let 𝐼=[0,𝑇], 𝑇>0, and let 𝑋 be a Banach space with topological dual space 𝑋^∗. Let 𝐵={𝑥∈𝑋∶‖𝑥‖≤1} and 𝐵_∗={𝑧∈𝑋^∗∶‖𝑧‖≤1}.

Definition 1 (see [1, Definition2.2.1]). A Banach space 𝑋 is said to be uniformly convex if for any 𝜀, 0<𝜀≤2, the inequalities ‖𝑥‖≤1,‖𝑦‖≤1, and ‖𝑥−𝑦‖≥𝜀 imply that there exists a 𝛿=𝛿(𝜀) such that ‖(1/2)(𝑥+𝑦)‖≤1−𝛿.

Lemma 2 (see [16, Theorems 2.2.4 and2.2.8]). (1)Every Hilbert space is uniformly convex.(2)Every uniformly convex space is strictly convex.(3)Every uniformly convex space is reflexive.

Definition 3 (see [1, 17, 18]). Let 𝑝>1 be a real number. A Banach space 𝑋 is said to be 𝑝-uniformly convex if there exists a constant 𝜆>0 such that 𝛿_𝑋(𝜀)≥𝜆𝜀^𝑝,∀𝜀∈(0,2]. (5)

Definition 4 (see [1, Definition2.4.1]). Let 𝑋^∗ be the topological dual of a Banach space 𝑋; then the multivalued mapping 𝐽∶𝑋⟶2^𝑋∗,𝐽(𝑥)=𝑦∈𝑋^∗∶𝑥,𝑦=‖𝑥‖²=𝑦² (6) is said to be the duality mapping (or duality mapping) in 𝑋, and the multivalued mapping 𝐽^∗∶𝑋^∗⟶2^𝑋,𝐽^∗𝑦=𝑥∈𝑋∶𝑦,𝑥=‖𝑥‖²=𝑦² (7) is called the normalized duality mapping (or duality mapping) in 𝑋^∗.

In the following lemma, we recall some properties of the duality mapping.

Lemma 5 (see [1, 19] Propositions2.4.5, 2.4.12, and2.4.15). (1)If 𝑋 is a Hilbert space, then 𝐽(𝑥)={𝑥}, for all 𝑥∈𝑋;(2)for each𝑥∈𝑋, 𝐽(𝑥) is nonempty closed convex and bounded subset of𝑋^∗;(3)𝐽(𝜆𝑥)=𝜆𝐽(𝑥), for all 𝑥∈𝑋 and for all 𝜆∈ℝ;(4)if 𝑋^∗ is strictly convex, 𝐽 is single valued;(5)if 𝑋 is strictly convex, 𝐽 is one to one; that is,𝑥≠𝑦⇒𝐽(𝑥)∩𝐽(𝑦)=𝜙;(6)if 𝑋^∗is uniformly convex, then 𝐽 is uniformly continuous on each bounded set in𝑋; that is, for all 𝜀>0 and 𝛼>0, there is a 𝛿=𝛿(𝜀,𝛼)>0 such that ‖𝑥‖𝑦𝐽(𝑥)𝑦≤𝛼,≤𝛼,𝑥−𝑦<𝛿⟹−𝐽<𝜀 (8) (note that if 𝑋^∗ is uniformly convex, then it is strictly convex, and hence 𝐽 is single-valued mapping);(7)if 𝑋 is reflexive, then 𝐽 is a mapping from 𝑋 onto𝑋^∗; that is, {𝐽(𝑥)∶𝑥∈𝑋}=𝑋^∗, (9)(8)if 𝑋 is reflexive strictly convex space with strictly convex conjugate space𝑋^∗, then 𝐽 and 𝐽^∗ are one-to-one, onto, and single-valued mappings and 𝐽⁻¹=𝐽^∗,𝐽𝐽^∗=𝐼_𝑋^∗,𝐽^∗𝐽=𝐼_𝑋, (10) where 𝐼_𝑋 is the identity mapping on 𝑋 and 𝐼_𝑋^∗ is the identity mapping, on 𝑋^∗.

For more properties of the duality mapping, we refer to [16, 19].

Definition 6 (see [1, Definition2.8.1]). The Banach space 𝑋 is said to be uniformly smooth if 𝜌^′_𝑋(0)=lim_𝑡→0𝜌_𝑋(𝑡)𝑡=0, (11) where 𝜌_𝑋 is the modulus of smoothness of 𝑋 (note that every Hilbert space is uniformly smooth).

Definition 7 (see [1, 17, 18]). Let 𝑞>1 be a real number. A Banach space 𝑋 is said to be 𝑞-uniformly smooth if there exist a constant 𝑐>0 such that 𝜌_𝑋(𝑡)≤𝑐𝑡^𝑞, for all 𝑡>0.

Clearly, every 𝑞-uniformly smooth is uniformly smooth (note that every Hilbert space is 2-uniformly smooth).

Lemma 8 (see [1, Theorems2.8.4,2.8.5, and2.8.6]). Let 𝑋 be a Banach space.(1)𝑋 is uniformly smooth if and only if 𝑋^∗is uniformly convex.(2)𝑋 is uniformly convex if and only if 𝑋^∗is uniformly smooth.(3)If 𝑋 is uniformly smooth, then 𝑋 is reflexive.

Lemma 9 (see [17, 18]). Let 𝑋 be a Banach space and 𝑝>1. (1)If 𝑋 is 𝑝-uniformly convex, then 𝑋^∗ is 𝑝^′-uniformly smooth where 𝑝^′=𝑝/(𝑝−1).(2)If 𝑋 is 𝑝-uniformly smooth, then 𝑋^∗  is 𝑝^′-uniformly convex where 𝑝^′=𝑝/(𝑝−1).

Now, let 𝑋 be a Banach space and 𝑋^∗  its topological dual. Let 𝑉∶𝑋^∗×𝑋⟶ℝ,𝑉_∗∶𝑋×𝑋^∗⟶ℝ (12) be two functions defined by 𝑉𝜑,𝑥=𝜑²−2𝜑,𝑥+‖𝑥‖²,𝑉_∗𝑥,𝜑=𝑉𝜑,𝑥. (13)

Based on the functional 𝑉, the generalized projection of a point 𝜑 in 𝑋^∗ onto a nonempty subset 𝐸 of 𝑋 can be defined as follows.

Definition 10 (see [16]). Let 𝑋 be a Banach space, 𝐸 a nonempty subset of 𝑋, and 𝜑∈𝑋^∗. If there exists a point 𝑧∈𝐸 satisfying 𝑉𝜑,𝑧=𝑑^𝑉_𝐸𝜑, (14) then 𝑧 is called a generalized projection of 𝜑 onto 𝐸; where 𝑑^𝑉_𝐸(𝜑)=inf_𝑥∈𝐸𝑉(𝜑,𝑥).
The set of all such points is denoted by 𝜋_𝐸(𝜑); that is, 𝜋_𝐸𝜑=𝑧∈𝐸∶𝑉𝜑,𝑧=𝑑^𝑉_𝐸𝜑⊆𝑋. (15)

Now, we list in the following theorem some properties of 𝑉 and 𝜋_𝐸(𝜑) (see [16]).

Lemma 11. Let 𝑋 be a Banach space and 𝑋^∗ its topological dual. (1)𝑉(𝐽(𝑥),𝑥)=0.(2)If𝑋 is uniformly convex or uniformly smooth, then 𝑉𝜑,𝑥=0⟺𝜑=𝐽(𝑥),∀𝑥∈𝑋,𝜑∈𝑋^∗. (16)(3)If 𝑋 is a Hilbert space, then 𝑉𝜑,𝑥=𝜑−𝑥². (17)(4)If 𝑋 is reflexive and 𝐸 is a nonempty closed and convex subset of 𝑋, then(a)𝜋_𝐸(𝜑)≠𝜙, for all 𝜑∈𝑋^∗;(b)𝑋 is strictly convex if and only if 𝜋_𝐸(𝜑) is singleton, for all 𝜑∈𝑋^∗;(c)if 𝑋 is also smooth, then, for any given 𝜑∈𝑋^∗, 𝑧∈𝜋_𝐸𝜑⟺⟨−𝐽(𝑧),𝑥−𝑧⟩≤0,∀𝑥∈𝐸. (18)

Definition 12 (see [4]). Let 𝑋 be a Banach space with topological dual 𝑋^∗, 𝐸 a nonempty closed convex subset of 𝑋, and 𝑧∈𝑋. The convex normal cone of 𝐸 at 𝑧 is defined by 𝑁_𝐸(𝑧)=𝜑∈𝑋^∗∶𝜑,𝑥−𝑧≤0,∀𝑥∈𝐸. (19)

For more details about the convex normal cone see [4, 20].

Lemma 13 (see [4]). Let 𝐸 be a nonempty, closed, and convex subset of Banach space 𝑋, and 𝑧∈𝑋. Then one has the following.(1)𝑁_𝐸(𝑧)∩𝐵_∗=𝜕𝑑_𝐸(𝑧), where 𝜕𝑑_𝐸(𝑧) is the subdifferential of the function 𝑧→𝑑_𝐸(𝑧) (𝑑_𝐸(𝑧) is the distance from 𝑧 to  𝐸).(2)If 𝑋 is reflexive and smooth, then 𝑧∈𝜋_𝐸𝜑⟺𝜑−𝐽(𝑧)∈𝑁_𝐸(𝑧),∀𝜑∈𝑋^∗,∀𝑧∈𝑋. (20)

Also, we need the following four Lemmas.

Lemma 14 (see [14, Proposition2.2]). Let Ω be an open subset in a normal vector space 𝑍, 𝑋 a Banach space, and 𝐶 a continuous set-valued mapping defined on Ω and with nonempty compact convex values in 𝑋. Let (𝜑_𝑛) be a sequence of 𝑋^∗ that converges weakly to 𝜑 in 𝑋^∗ and (𝑥_𝑛) a sequence in 𝑋 that converges strongly to 𝑥 in 𝑋 such that (𝑡_𝑛) a sequence in Ω that converges to 𝑡 in 𝐼. If 𝜑_𝑛∈𝜕𝑑_{𝐶(𝑡𝑛)}(𝑥_𝑛), then 𝜑∈𝜕𝑑_𝐶(𝑡)(𝑥).

Lemma 15 (see [14, Lemma 4.1 and Proposition4.2]). Let 𝑝,𝑞>1, 𝑋 be a 𝑝-uniformly convex and 𝑞-uniformly smooth Banach space, and let 𝐸 be a nonempty bounded subset of 𝑋; then there exist two constants 𝛼>0, 𝛽>0 such that 𝛼𝑥−𝑦^𝑝≤𝑉𝐽(𝑥),𝑦≤𝛽𝑥−𝑦^𝑞,∀𝑥,𝑦∈𝐸. (21)

Moreover, if 𝐸, in addition, is closed and 𝜑∈𝑋^∗, then 𝑑^𝑉_𝐸𝜑=0⟺𝐽^∗𝜑∈𝐸. (22)

Lemma 16 (see [21, Theorem2.2]). Let Ω be a subset of a Banach space 𝑋. Assume that 𝐹∶[𝑎,𝑏]×Ω→2^𝑥 is a multifunction such that(1)for every 𝑥∈Ω, the multifunction 𝐹(⋅,𝑥) is measurable on [𝑎,𝑏];(2)for every 𝑡∈[𝑎,𝑏], the multifunction 𝐹(𝑡,⋅) is continuous on Ω.
Then, for any measurable function 𝑥(⋅)∶[𝑎,𝑏]→Ω, the multifunction 𝐹(⋅,𝑥(⋅)) is measurable on [𝑎,𝑏].

Lemma 17 (see [21, Lemma2.3]). Let 𝑋 be a separable Banach space, 𝐺∶[𝑎,𝑏]→2^𝑋 a measurable multifunction, and 𝑧∶[𝑎,𝑏]→𝑋 a measurable function. Then for any positive measurable function 𝑟∶[𝑎,𝑏]→ℝ⁺, there exists a measurable selection 𝑔 of 𝐺 such that for almost 𝑡∈[𝑎,𝑏]𝑔(𝑡)−𝑧(𝑡)≤𝑑(𝑧(𝑡),𝐺(𝑡))+𝑟(𝑡). (23)

3. Existence of Solutions for Problem (2)

Theorem 18. Let 𝑝,𝑞>1, 𝑋 a separable 𝑝-uniformly convex and 𝑞-uniformly smooth Banach space, 𝐼=[0,𝑇], 𝑇>0, 𝐶 a multifunction defined from 𝐼×𝑋 to the family of nonempty closed convex subsets of 𝑋, and 𝐹 a multifunction defined from 𝐼×𝐶_𝑟×𝐶_𝑟 with nonempty closed values in 𝑋^∗. Assume that the following hypotheses hold: (𝐶₁) there is an absolutely continuous function 𝑣∶𝐼→ℝ and two positive real numbers 𝜆 and 𝛾 such that, for all 𝜉₁,𝜉₂∈𝑋^∗, all 𝑥,𝑦∈𝑋, and all 𝑡,𝑠∈𝐼, 𝑑^𝑉_{𝐶(𝑡,𝑥)}𝜉₁^1/𝑞′−𝑑^𝑉_{𝐶(𝑠,𝑦)}𝜉₂^1/𝑞′≤|𝑣(𝑡)−𝑣(𝑠)|+𝜆𝑥−𝑦+𝛾𝜉₁−𝜉₂, (24) where 𝑞^′=𝑞/(1−𝑞);(𝐶₂) there is a compact subset 𝐾 of 𝑋^∗ such that𝐽(𝐶(𝑡,𝑥))⊂𝐾,∀(𝑡,𝑥)∈𝐼×𝑋; (25)(𝐹₁) for each 𝜓,𝜑∈𝐶_𝑟, 𝑡→𝐹(𝑡,𝜓,𝜑) is a measurable;(𝐹₂) there is a function 𝑚∈𝐿¹(𝐼,ℝ^∗) such that, for all 𝑡∈𝐼, and for all 𝜓₁, 𝜑₁, 𝜓₂, 𝜑₂∈𝐶_𝑟, 𝐻𝐹𝑡,𝜓₁,𝜑₁,𝐹𝑡,𝜓₂,𝜑₂≤𝑚(𝑡)𝜓₁−𝜓₂+𝜑₁−𝜑₂; (26)(𝐹₃) there is a function 𝑝∈𝐿¹(𝐼,ℝ⁺) such that, for all 𝜓,𝜑∈𝐶_𝑟=𝐶([−𝑟,0],𝑋), 𝐹𝑡,𝜓,𝜑≤𝑝(𝑡)𝜓(0)+𝜑(0),𝑎.𝑒.𝑡∈𝐼. (27)
Then, for any 𝜑,𝜓∈𝐶_𝑟 with 𝜑(0)∈𝐶(0,𝜓(0)), there exist two continuous functions 𝑢∶[−𝑟,𝑇]→𝑋^∗ and 𝑤∶[−𝑟,𝑇]→𝑋 such that (2) is satisfied.

Proof. At first, we note that since the Banach space 𝑋 is 𝑝-uniformly convex, then it is uniformly convex, and hence by Lemma 2 it is reflexive and strictly convex. Moreover, because 𝑋 is 𝑞-uniformly smooth, then, by Lemma 9, its conjugate 𝑋^∗ is 𝑞^′-uniformly convex, 𝑞^′=𝑞/(𝑞−1), and hence 𝑋^∗is strictly convex. Then, by Lemmas 5 and 8, the duality mapping 𝐽 is single valued, continuous, one-to-one, onto, and 𝐽𝐽^∗=𝐼_𝑋^∗ and 𝐽^∗𝐽=𝐼_𝑋. For notational convenience, we take 𝑇=1. Let 𝑘>0 such that 𝐾⊂𝑘𝐵^∗. Let 𝑛 be a fixed positive integer and 𝑧∶𝐼→𝑋^∗ a measurable function. In order to make it easier for the reader, we will divide the proof in the following steps.
Step 1. We will construct two continuous functions 𝑢_𝑛∶[−𝑟,1]→𝑋^∗ and 𝑤_𝑛∶[−𝑟,1]→𝑋, two functions 𝜃_𝑛, 𝛿_𝑛∶𝐼→𝐼 and 𝑓_𝑛∈𝐿¹(𝐼,𝑋^∗) such that 𝑢_𝑛 and 𝑤_𝑛  are absolutely continuous function on 𝐼=[0,1] and that (i)𝑢_𝑛(𝑡)=𝐽𝜑(𝑡),𝑤_𝑛(𝑡)(𝑡),[];()𝑓=𝜓for𝑡∈−𝑟,0ii_𝑛(𝑡)∈𝐹𝑡,𝜏𝛿_𝑛(𝑡)𝑤_𝑛,𝜏𝛿_𝑛(𝑡)𝐽^∗𝑢_𝑛,()𝐽for𝑡∈𝐼;iii^∗𝑢_𝑛𝜃_𝑛(𝑡)∈𝐶𝜃_𝑛(𝑡),𝑤_𝑛𝜃_𝑛(𝑡)∩𝐵^∗()()𝑤0,𝑘,for𝑡∈𝐼;iv_𝑛(𝑡)=𝑤_𝑛𝛿_𝑛(𝑡)+𝑡−𝛿_𝑛(𝑡)𝐽^∗𝑢_𝑛𝛿_𝑛(𝑡)𝑤for𝑡∈𝐼,_′𝑛(𝑡)=𝐽^∗𝑢_𝑛(𝑡)(v𝑓,a.e.for𝑡∈𝐼;)_𝑛(𝑡)(𝑡𝑧(𝑡)𝛿−𝑧)≤𝑑,𝐹𝑡,𝜏_𝑛(𝑡)𝑤_𝑛,𝜏𝛿_𝑛(𝑡)𝐽^∗𝑢_𝑛+1𝑛(𝑢,for𝑡∈𝐼;vi)^′_𝑛(𝑡)−𝑓_𝑛(𝑡≤1)𝛼^1/𝑞′𝑣^′(𝑡𝜓(0𝑝(𝑡,()𝑢)+𝛾)+𝑘)a.e.for𝑡∈𝐼;vii^′_𝑛(𝑡)−𝑓_𝑛(𝑡)∈−𝑁_{𝐶(𝜃𝑛(𝑡),𝑤𝑛(𝜃𝑛(𝑡))}𝐽^∗𝑢_𝑛𝜃_𝑛(𝑡),a.e.for𝑡∈𝐼. (28)
In view of Lemma 15, there is 𝛼>0 such that, for all 𝑧₁,𝑧₂∈𝜂𝐵^∗, we have 𝛼𝑧₁−𝑧₂^𝑞′≤𝑉_∗𝐽^∗𝑧₁,𝑧₂, (29) where 𝜂=𝑘+(𝑘+‖𝜓(0)‖)∫^𝑇₀𝑝(𝑠)𝑑𝑠.
For 𝑡∈[−𝑟,0], we define 𝑢_𝑛(𝑡)=𝐽(𝜑(𝑡)) and 𝑤_𝑛(𝑡)=𝜓(𝑡). In order to define  𝑢_𝑛  and 𝑤_𝑛  on 𝐼, we define a function 𝑒∶𝐼→ℝ⁺  by 𝑒(𝑡)1∶=𝛼^{1/𝑞′𝑡0}𝑣^′(𝑠)+𝛾𝜓(0)+𝑘𝑝(𝑠)𝑑𝑠+𝜆𝑘,∀𝑡∈𝐼. (30)
Let us consider a partition of 𝐼=[0,1] by the points 𝑡^𝑛_𝑖=𝑖/𝑛, 0≤𝑖≤𝑛. Let 𝐼^𝑛₀={𝑡^𝑛₀}={0}, 𝐼^𝑛_𝑖+1=]𝑡^𝑛_𝑖,𝑡^𝑛_𝑖+1], and 0≤𝑖≤𝑛−1. In view of Lemma 16, there exists a function 𝑓^𝑛₀∈𝐿¹([0,𝑡^𝑛₁],𝑋^∗)  such that 𝑓^𝑛₀(𝑡)∈𝐹(𝑡,𝜏(0)𝜓,𝜏(0)𝜑) and 𝑓^𝑛₀(𝑡)(𝑡(𝑧(𝑡),𝐹(0)(0)𝜑+1−𝑧)≤𝑑𝑡,𝜏𝜓,𝜏𝑛,∀𝑡∈0,𝑡^𝑛₁. (31)
We put 𝑥^𝑛₀=𝜑(0)∈𝐶(0,𝜓(0)) and for 𝑡∈𝐼^𝑛₁ we define 𝑤_𝑛(𝑡)∶=𝜓(0)+𝑡−𝑡^𝑛₀𝑥^𝑛₀,𝑢_𝑛(𝑡)∶=𝐽𝑥^𝑛₀+𝑒(𝑡)−𝑒𝑡^𝑛₀𝑒𝑡^𝑛₁−𝑒𝑡^𝑛₀×𝐽𝑥^𝑛₁−𝐽𝑥^𝑛₀−^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠+^𝑡_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠, (32)(33) where 𝑥^𝑛₁∶=𝜋_𝐶(𝑡^𝑛_{1,𝑤𝑛(𝑡}^𝑛₁₎₎𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠. (34)
According to Lemma 11, 𝑥^𝑛₁ is well defined. By recalling the definition of the generalized projection, we get 𝑥^𝑛₁∈𝐶(𝑡^𝑛₁,𝑤_𝑛(𝑡^𝑛₁)) and 𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠−𝐽𝑥^𝑛₁∈𝑁_𝐶(𝑡^𝑛_{1,𝑤𝑛(𝑡}^𝑛₁₎₎𝑥^𝑛₁. (35)
As a result of Condition (𝐶₂), ‖𝜑(0)‖=‖𝐽(𝜑(0))‖=‖𝐽(𝑥^𝑛₀)‖≤𝑘, ‖𝐽(𝑥^𝑛₁)‖≤𝑘. Moreover, by Condition (𝐹₃)^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)≤𝑑𝑠_{𝑡𝑛1𝑡𝑛0}𝑝(𝑠𝜏(0)𝜓(0+𝜏(0)𝜑(0≤𝜓(0+𝜑(0)))𝑑𝑠)^𝑡𝑛1_𝑡𝑛0𝑝(𝑠)≤𝜓(0𝑑𝑠)+𝑘^𝑡𝑛1_𝑡𝑛0𝑝(𝑠)𝑑𝑠. (36)
Therefore, 𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝜓(0𝑑𝑠≤𝑘+)+𝑘^𝑡𝑛1_𝑡𝑛0𝑝(𝑠)𝑑𝑠=𝜂. (37)
Consequently, by (29) we obtain 𝛼𝐽𝑥^𝑛₁−𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠^𝑞′≤𝑉_∗𝐽^∗𝐽𝑥^𝑛₁,𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)=𝑥𝑑𝑠_𝑛12−2𝑥^𝑛₁,𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)+𝐽𝑥𝑑𝑠_𝑛0+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠²=𝑉𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠,𝑥^𝑛₁=𝑑^𝑉_𝐶(𝑡^𝑛_{1,𝑤𝑛(𝑡}^𝑛₁₎₎𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠. (38)
We note that, from Lemma 15 and the choice 𝑥^𝑛₀,𝑑^𝑉_𝐶(𝑡^𝑛_{0,𝑤𝑛(𝑡}^𝑛₀₎₎(𝐽(𝑥^𝑛₀))=0. Then, by (𝐶₁) and (36), the last inequality gives us 𝛼^1/𝑞′𝐽𝑥^𝑛₁−𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)≤𝑑𝑑𝑠_{𝑉𝐶(𝑡𝑛1,𝑤𝑛(𝑡𝑛1))}𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠^1/𝑞′=𝑑^𝑉_𝐶(𝑡^𝑛_{1,𝑤𝑛(𝑡}^𝑛₁₎₎𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠^1/𝑞′−𝑑^𝑉_𝐶(𝑡^𝑛_{0,𝑤𝑛(𝑡}^𝑛₀₎₎𝐽𝑥^𝑛₀^1/𝑞′≤𝑣𝑡^𝑛₁−𝑣𝑡^𝑛₀+𝛾^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑤𝑑𝑠+𝜆_𝑛𝑡^𝑛₁−𝑤_𝑛𝑡^𝑛₀≤^𝑡𝑛1_𝑡𝑛0𝑣^′(𝑠)𝑑𝑠+𝛾𝜓(0)+𝑘^𝑡𝑛1_𝑡𝑛0𝑝(𝑠)𝑑𝑠+𝜆𝑘. (39)
This inequality with (30) leads to 𝐽𝑥^𝑛₁−𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)≤1𝑑𝑠𝛼^{1/𝑞′𝑡𝑛1𝑡𝑛}₀𝑣^′(𝑠𝜓(0𝑝(𝑠=𝑒𝑡)+𝛾)+𝑘)𝑑𝑠+𝜆𝑘_𝑛1−𝑒𝑡^𝑛₀. (40)
Observe that from (33), for 𝑡∈]𝑡^𝑛₀,𝑡^𝑛₁[, we get 𝑢^′_𝑛(𝑡)=𝑒^′(𝑡)𝑒𝑡^𝑛₁−𝑒𝑡^𝑛₀×𝐽𝑥^𝑛₁−𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠+𝑓^𝑛₀(𝑡). (41) From this equality with (40), we get 𝑢^′_𝑛(𝑡)−𝑓^𝑛₀(𝑡≤𝑒_′)(𝑡=1)𝛼^1/𝑞′𝑣^′(𝑡)+𝛾𝜓(0)+𝑘𝑝(𝑡),∀𝑡∈𝑡^𝑛₀,𝑡^𝑛₁. (42)
Note that, from (33), we have 𝐽𝑥^𝑛₀+^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)𝑑𝑠−𝐽𝑥^𝑛₁∈𝑁_𝐶(𝑡^𝑛_{1,𝑤𝑛(𝑡}^𝑛₁₎₎𝑥^𝑛₁. (43)
Therefore, by (41), 𝑢^′_𝑛(𝑡)−𝑓^𝑛₀(𝑡)∈−𝑁_𝐶(𝑡^𝑛_{1,𝑤𝑛(𝑡}^𝑛₁₎₎𝐽^∗𝑢_𝑛𝑡^𝑛₁,fora.e.𝑡∈𝐼^𝑛₁. (44)
Moreover, by (30), (33), (36), and (40), for 𝑡,𝑠∈[0,𝑡^𝑛₁](𝑠<𝑡)   we have 𝑢_𝑛(𝑡)−𝑢_𝑛(𝑠≤𝑒(𝑡)(𝑠))−𝑒𝑒𝑡^𝑛₁−𝑒𝑡^𝑛₀𝐽𝑥^𝑛₁−𝐽𝑥^𝑛₀−^𝑡𝑛1_𝑡𝑛0𝑓^𝑛₀(𝑠)+𝑑𝑠_𝑡𝑠𝑓^𝑛₀𝛽≤|𝑒(𝑡)(𝑠+𝜓(0𝑑𝛽−𝑒)|)+𝑘^𝑡_𝑠𝑝𝛽=1𝑑𝛽𝛼^{1/𝑞′𝑡𝑠}𝑣^′𝛽𝜓(0𝑝𝛽+𝜓(0+𝛾)+𝑘𝑑𝛽+𝜆𝑘)+𝑘^𝑡_𝑠𝑝𝛽=1𝑑𝛽𝛼^{1/𝑞′𝑡𝑠}𝑣^′𝛽+𝛾𝜓(0)+𝑘𝑝𝛽+𝛼^1/𝑞′𝜓(0𝑝𝛽=1)+𝑘𝑑𝛽+𝜆𝑘𝛼^{1/𝑞′𝑡𝑠}𝑣^′𝛽+𝜓(0×)+𝑘𝛾+𝛼^1/𝑞′𝑝𝛽𝑑𝛽+𝜆𝑘, (45) this shows that 𝑢_𝑛 is absolutely continuous on [0,𝑡^𝑛₁].
We reiterate this process for constructing two sequences (𝑓^𝑛_𝑖),(𝑥^𝑛_𝑖),𝑖=0,1,…,𝑛−1, and two functions 𝑢_𝑛∶[−𝑟,1]→𝑋^∗, 𝑤_𝑛∶[−𝑟,1]→𝑋such that, for all 𝑖=0,1,…,𝑛−1, the following properties hold: 𝑓^𝑛_𝑖(𝑡)∈𝐹𝑡,𝜏𝑡^𝑛_𝑖𝑤_𝑛,𝜏𝑡^𝑛_𝑖𝐽^∗𝑢_𝑛,𝑡∈𝐼^𝑛_𝑖+1,𝑓^𝑛_𝑖(𝑡)(𝑡𝑧(𝑡)𝑡−𝑧)≤𝑑,𝐹𝑡,𝜏^𝑛_𝑖𝑤_𝑛,𝜏𝑡^𝑛_𝑖𝐽^∗𝑢_𝑛+1𝑛,𝑡∈𝐼^𝑛_𝑖+1,𝑤_𝑛(𝑡)=𝑤_𝑛𝑡^𝑛_𝑖+𝑡−𝑡^𝑛_𝑖𝐽^∗𝑢_𝑛𝑡^𝑛_𝑖,𝑡∈𝐼^𝑛_𝑖+1,𝑥^𝑛_𝑖+1=𝜋_𝐶(𝑡^𝑛_{𝑖+1,𝑤𝑛(𝑡}^𝑛_𝑖+1))𝐽𝑥^𝑛_𝑖+^{𝑡𝑛𝑖+1}_𝑡𝑛𝑖𝑓^𝑛_𝑖(𝑠)𝑑𝑠,𝑡∈𝐼^𝑛_𝑖+1,𝑢_𝑛(𝑡)=𝐽𝑥^𝑛_𝑖+𝑒(𝑡)−𝑒𝑡^𝑛_𝑖𝑒𝑡^𝑛_𝑖+1−𝑒𝑡^𝑛_𝑖×𝐽𝑥^𝑛_𝑖+1−𝐽𝑥^𝑛_𝑖−^{𝑡𝑛𝑖+1}_𝑡𝑛𝑖𝑓^𝑛_𝑖(𝑠)+𝑑𝑠_{𝑡𝑡𝑛𝑖}𝑓^𝑛_𝑖(𝑠)𝑑𝑠,𝑡∈𝐼^𝑛_𝑖+1,𝐽𝑥^𝑛_𝑖+1−𝐽𝑥^𝑛_𝑖+^{𝑡𝑛𝑖+1}_𝑡𝑛𝑖𝑓^𝑛_𝑖(𝑠)𝑑𝑠≤𝑒𝑡^𝑛_𝑖+1−𝑒𝑡^𝑛_𝑖,𝑢^′_𝑛(𝑡)−𝑓^𝑛_𝑖(𝑡≤𝑒_′)(𝑡=1)𝛼^1/𝑞′𝑣^′(𝑡)+𝛾𝜓(0)+𝑘𝑝(𝑡),fora.e.𝑡∈𝐼^𝑛_𝑖+1,𝑢^′_𝑛(𝑡)−𝑓^𝑛_𝑖(𝑡)∈−𝑁_𝐶(𝑡^𝑛_{𝑖+1,𝑤𝑛(𝑡}^𝑛_𝑖+1)𝐽^∗𝑢_𝑛𝑡^𝑛_𝑖,fora.e.𝑡∈𝐼^𝑛_𝑖+1,𝑥^𝑛_𝑖+1∈𝐶𝑡^𝑛_𝑖+1,𝑤_𝑛𝑡^𝑛_𝑖+1∩𝐵^∗(0,𝑘), (46)(47)(48)(49)(50)(51)(52)(53)(54) where 𝐽^∗(𝑢_𝑛)(𝑡)=𝐽^∗(𝑢_𝑛(𝑡)).
Note that, by (48), for every 𝑖=0,1,2,…,𝑛−1, 𝑤_𝑛𝑡^𝑛_𝑖+1≤𝑤_𝑛𝑡^𝑛_𝑖+𝑡^𝑛_𝑖+1−𝑡^𝑛_𝑖𝐽^∗𝑢_𝑛𝑡^𝑛_𝑖≤𝑤_𝑛𝑡^𝑛_𝑖+𝑡^𝑛_𝑖+1−𝑡^𝑛_𝑖𝑘. (55)
Then, from this recurrence relation, we can get 𝑤_𝑛𝑡^𝑛_𝑖+1≤𝜓(0)+𝑘,forevery𝑖=0,1,2,…,𝑛−1. (56)
This relation with (46), (54), and Condition (𝐹₃)gives us, for 𝑡∈𝐼^𝑛_𝑖+1, 𝑓^𝑛_𝑖(𝑡)≤𝑝(𝑡)𝜏𝑡^𝑛_𝑖𝑤_𝑛(0)+𝜏𝑡^𝑛_𝑖𝐽^∗𝑢_𝑛0(𝑡)𝑤()≤𝑝_𝑛𝑡^𝑛_𝑖+𝐽^∗𝑢_𝑛𝑡^𝑛_𝑖(𝑡)𝜓(0(𝑡)𝜓(0.≤𝑝)+𝑘+𝑘=𝑝)+2𝑘 (57)
Therefore, by (34), (50), (51), and (57), for 𝑡,𝑠∈𝐼^𝑛_𝑖+1(𝑠<𝑡) we have 𝑢_𝑛(𝑡)−𝑢_𝑛(𝑠≤𝑒(𝑡)(𝑠))−𝑒𝑒𝑡^𝑛_𝑖+1−𝑒𝑡^𝑛_𝑖𝐽𝑥^𝑛_𝑖+1−𝐽𝑥^𝑛_𝑖−^{𝑡𝑛𝑖+1}_𝑡𝑛𝑖𝑓^𝑛_𝑖(𝑠)+𝑑𝑠_𝑡𝑠𝑓^𝑛_𝑖𝛽≤|𝑒(𝑡)(𝑠+𝜓(0𝑑𝛽−𝑒)|)+2𝑘^𝑡_𝑠𝑝𝛽=1𝑑𝛽𝛼^{1/𝑞′𝑡𝑠}𝑣^′𝛽𝜓(0×𝑑𝛽+𝛾)+𝑘_𝑡𝑠𝑝𝛽+𝜓(0𝑑𝛽+𝜆𝑘)+2𝑘^𝑡_𝑠𝑝𝛽=1𝑑𝛽𝛼^{1/𝑞′𝑡𝑠}𝑣^′𝛽𝜓(0𝑝𝛽+𝜓(0+𝛾)+𝑘𝑑𝛽+𝜆𝑘)+2𝑘^𝑡_𝑠𝑝𝛽𝑑𝛽. (58)
By addition, the last inequality holds for 𝑡,𝑠∈𝐼, with 𝑠<𝑡; hence, 𝑢_𝑛 is absolutely continuous on 𝐼.
Now, for each positive integer 𝑛 we define the functions 𝜃_𝑛, 𝛿_𝑛∶𝐼→𝐼 and 𝑓_𝑛∈𝐿¹(𝐼,𝑋^∗) by setting 𝜃_𝑛(0)=𝑡^𝑛₁,𝛿_𝑛(0)𝑓=0,_𝑛(0)=𝑓^𝑛_𝑖(0),𝜃_𝑛(𝑡)=𝑡^𝑛_𝑖+1,𝛿_𝑛(𝑡)=𝑡^𝑛_𝑖,𝑓_𝑛(𝑡)=𝑓^𝑛_𝑖(𝑡)for𝑡∈𝐼^𝑛_𝑖+1. (59)
Then obviously, the relations (46), (47), (48), (52), (53), and (54) imply (28)(ii)→(vii), respectively.
Step 2. The sequences (𝑢_𝑛) and (𝑤_𝑛)  converge uniformly to continuous functions 𝑢∶[−𝑟,1]→𝑋^∗, 𝑤∶[−𝑟,1]→𝑋, respectively, with 𝑢(𝑡)=𝐽(𝜑(𝑡)) and 𝑤(𝑡)=𝜓(𝑡) for 𝑡∈[−𝑟,0] and 𝐽^∗(𝑢(𝑡))∈𝐶(𝑡,𝑤(𝑡)) for 𝑡∈𝐼.
Indeed, by (58) for 𝑡∈𝐼 we have lim_𝑛→∞‖𝑢_𝑛(𝜃_𝑛(𝑡))−𝑢_𝑛(𝑡)‖=0. Moreover, by (28)(iii), for any 𝑛≥1 and any 𝑡∈𝐼, we have 𝑢_𝑛(𝜃_𝑛(𝑡))∈𝐾. Then, for any 𝑡∈𝐼, the set {𝑢_𝑛(𝑡)∶𝑛≥1} is relatively compact subset in 𝑋^∗. In addition, from (28)(vi), Condition (𝐹₃), and (57) we get for a.e. 𝑡∈𝐼𝑢^′_𝑛(𝑡≤1)𝛼^1/𝑞′𝑣^′(𝑡𝜓(0𝑝(𝑡+𝑓)+𝛾)+𝑘)_𝑛(𝑡≤1)𝛼^1/𝑞′𝑣^′(𝑡𝜓(0𝑝(𝑡(𝑡)𝑤)+𝛾)+𝑘)+𝑝_𝑛𝛿_𝑛(𝑡)+𝐽^∗𝑢_𝑛𝛿_𝑛(𝑡)≤1𝛼^1/𝑞′𝑣^′(𝑡𝜓(0𝑝(𝑡(𝑡𝜓(0.)+𝛾)+𝑘)+𝑝))+2𝑘 (60)
This, with the absolutely continuity of 𝑢_𝑛, assures the equicontinuity of 𝑢_𝑛. Thus, by Arzela-Ascoli’s Theorem, we can select a subsequence of 𝑢_𝑛, again denoted by 𝑢_𝑛, which converges uniformly to an absolutely continuous function 𝑢 on 𝐼 such that, for any 𝑡∈𝐼, 𝑢(𝑡)=𝐽𝑥^𝑛₀+^𝑡₀𝑢^′(𝑠)𝑑𝑠. (61)
Moreover, (𝑢^′_𝑛) converges weakly to 𝑢^′∈𝐿¹(𝐼,𝑋^∗). We extend the definition of 𝑢 in such a way that 𝑢=𝐽(𝜑) on [−𝑟,0]. Obviously, (𝑢_𝑛) converges uniformly to 𝑢 on [−𝑟,1]. Let us define 𝑢^∗_𝑛(𝑡)=𝐽^∗(𝑢_𝑛(𝑡)), 𝑡∈𝐼. Since 𝐽^∗  is continuous on the compact set 𝐾 and 𝑢_𝑛(𝑡)∈𝐾 for 𝑡∈𝐼, then the sequence (𝑢^∗_𝑛) converges uniformly to a function 𝑢^∗=𝐽^∗(𝑢). In addition to the fact that 𝑤^′_𝑛(𝑡)=𝐽^∗(𝑢_𝑛(𝑡)), a.e. for 𝑡∈𝐼, we confirm that the sequence (𝑤_𝑛) converges uniformly to a function 𝑤(𝑡)=𝜓(0)+∫^𝑡₀𝐽^∗(𝑢(𝑠))𝑑𝑠,𝑡∈𝐼. Let us show that 𝐽^∗(𝑢(𝑡))∈𝐶(𝑡,𝑤(𝑡)) for 𝑡∈𝐼. Note that lim_𝑛→∞𝑢_𝑛𝜃_𝑛(𝑡)−𝑢(𝑡)≤lim_𝑛→∞𝑢_𝑛𝜃_𝑛(𝑡)−𝑢_𝑛(𝑡)+lim_𝑛→∞𝑢_𝑛(𝑡)−𝑢(𝑡)=0. (62)
Similarly, lim_𝑛→∞𝑤_𝑛𝜃_𝑛(𝑡)−𝑤(𝑡)=0. (63)
Moreover, as 𝐽^∗(𝑢_𝑛(𝜃_𝑛(𝑡)))=𝐽^∗(𝑢_𝑛(𝑡^𝑛_𝑖+1))=𝐽^∗(𝐽(𝑥^𝑛_𝑖+1))=𝑥^𝑛_𝑖+1∈𝐶(𝜃_𝑛(𝑡),𝑤_𝑛(𝜃_𝑛(𝑡))), it follows that 𝑑^𝑉_{𝐶𝜃𝑛(𝑡),𝑤𝑛(𝜃𝑛(𝑡))}𝑢_𝑛𝜃_𝑛(𝑡)=0,𝑡∈𝐼. (64) Then, from (62), (63), (64), Condition (𝐶₁) and one obtains, for 𝑡∈𝐼, 𝑑^𝑉_{𝐶((𝑡𝑡,𝑤))}(𝑢(𝑡))^1/𝑞′≤𝑑^𝑉_{𝐶((𝑡𝑡,𝑤))}(𝑢(𝑡))^1/𝑞′−𝑑^𝑉_{𝐶𝜃𝑛(𝑡),𝑤𝑛(𝜃𝑛(𝑡))}𝑢_𝑛𝜃_𝑛(𝑡)^1/𝑞′≤𝑣(𝑡)−𝑣𝜃_𝑛(𝑡)+𝛾𝑢_𝑛𝜃_𝑛(𝑡)−𝑢(𝑡)+𝜆𝑤_𝑛𝜃_𝑛(𝑡)−𝑤(𝑡). (65)
By taking the limit as 𝑛→∞, we conclude that 𝐽^∗(𝑢(𝑡))∈𝐶(𝑡,𝑤(𝑡)) for 𝑡∈𝐼.
Step 3. For any 𝑡∈𝐼, the sequence 𝜏(𝛿_𝑛(𝑡)𝑢^∗_𝑛) converges to 𝜏(𝑡)𝑢^∗, and the sequence 𝜏(𝛿_𝑛(𝑡)𝑤_𝑛) converges to 𝜏(𝑡)𝑤 in 𝐶_𝑟. Let 𝑡∈𝐼. We have 𝜏𝛿_𝑛(𝑡)𝑢^∗_𝑛−𝜏(𝑡)𝑢^∗≤𝜏𝛿_𝑛(𝑡)𝑢^∗_𝑛−𝜏(𝑡)𝑢^∗_𝑛+𝜏(𝑡)𝑢^∗_𝑛−𝜏(𝑡)𝑢^∗≤sup_{−𝑟≤𝑠≤0}𝑢^∗_𝑛𝛿_𝑛(𝑡)+𝑠−𝑢^∗_𝑛(+𝜏(𝑡)𝑢𝑡+𝑠)_∗𝑛−𝜏(𝑡)𝑢^∗≤sup_{−𝑟≤𝑠1≤𝑠2≤1|𝑠1−𝑠2|≤1/𝑛}𝑢^∗_𝑛𝑠₁−𝑢^∗_𝑛𝑠₂+𝜏(𝑡)𝑢^∗_𝑛−𝜏(𝑡)𝑢^∗≤sup_{−𝑟≤𝑠1,𝑠2≤0|𝑠1−𝑠2|≤1/𝑛}𝑢^∗_𝑛𝑠₁−𝑢^∗_𝑛𝑠₂+sup_{−𝑟≤𝑠1≤0≤𝑠2≤1|𝑠1−𝑠2|≤1/𝑛}𝑢^∗_𝑛𝑠₁−𝑢^∗_𝑛𝑠₂+sup_{0≤𝑠1≤𝑠2≤1}𝑢^∗_𝑛𝑠₁−𝑢^∗_𝑛𝑠₂+𝜏(𝑡)𝑢^∗_𝑛−𝜏(𝑡)𝑢^∗≤sup_{−𝑟≤𝑠1,𝑠2≤0|𝑠1−𝑠2|≤1/𝑛}𝜑𝑠₁−𝜑𝑠₂+sup_{−𝑟≤𝑠1≤0|𝑠1|≤1/𝑛}𝑢^∗_𝑛𝑠₁−𝑢^∗_𝑛(0)+sup_{0≤𝑠2≤1|𝑠2|≤1/𝑛}𝑢^∗_𝑛(0)−𝑢^∗_𝑛𝑠₂+sup_{0≤𝑠1≤𝑠2≤1|𝑠1−𝑠2|≤1/𝑛}𝑢^∗_𝑛𝑠₁−𝑢^∗_𝑛𝑠₂+𝜏(𝑡)𝑢^∗_𝑛−𝜏(𝑡)𝑢^∗≤2sup_{−𝑟≤𝑠1,𝑠2≤0|𝑠1−𝑠2|≤1/𝑛}𝜑𝑠₁−𝜑𝑠₂+2sup_{0≤𝑠1,𝑠2≤1|𝑠1−𝑠2|≤1/𝑛}𝑢^∗_𝑛𝑠₁−𝑢^∗_𝑛𝑠₂+𝜏(𝑡)𝑢^∗_𝑛−𝜏(𝑡)𝑢^∗. (66)
By the continuity of 𝜑, the uniform convergence of 𝑢^∗_𝑛 towards 𝑢^∗, and the preceding estimate, we get lim_𝑛→∞𝜏𝛿_𝑛(𝑡)𝑢^∗_𝑛−𝜏(𝑡)𝑢^∗=0. (67)
Similarly, we can show that lim_𝑛→∞‖𝜏(𝛿_𝑛(𝑡))𝑤_𝑛−𝜏(𝑡)𝑤‖=0.
Step 4. The sequence (𝑓_𝑛) converges pointwise to a function 𝑓∈𝐿¹(𝐼,𝑋^∗) satisfying 𝑓(𝑡)∈𝐹(𝑡,𝜏(𝑡)𝑤,𝜏(𝑡)𝑢^∗), 𝑡∈𝐼.
Let 𝑡∈𝐼 be fixed. In view of (28)(v) and Condition  (𝐹₂), we obtain for 𝑛≥1𝑓_𝑛+1(𝑡)−𝑓_𝑛(𝑡𝐹𝛿)≤𝐻𝑡,𝜏_𝑛(𝑡)𝑤_𝑛,𝜏𝛿_𝑛(𝑡)𝑢^∗_𝑛,𝐹𝑡,𝜏𝛿_𝑛+1(𝑡)𝑤_𝑛+1,𝜏𝛿_𝑛+1(𝑡)𝑢^∗_𝑛+1+1(𝑡)𝜏𝛿𝑛+1≤𝑚_𝑛(𝑡)𝑤_𝑛−𝜏𝛿_𝑛+1(𝑡)𝑤_𝑛+1+𝜏𝛿_𝑛(𝑡)𝑢^∗_𝑛−𝜏𝛿_𝑛+1(𝑡)𝑢^∗_𝑛+1+1.𝑛+1 (68)
From Step 3, the right-hand side of (68) tends to zero when 𝑛→∞. Hence, (𝑓_𝑛(𝑡)) is a Cauchy sequence in 𝑋^∗; then it converges pointwise to a function 𝑓∈𝐿¹(𝐼,𝑋^∗). Moreover, by (28)(v) and Condition (𝐹₂), we get 𝑑𝑓(𝑡),𝐹𝑡,𝜏(𝑡)𝑤,𝜏(𝑡)𝑢^∗≤𝑓(𝑡)−𝑓_𝑛(𝑡𝐹(𝑡)(𝑡)𝑢)+𝐻𝑡,𝜏𝑤,𝜏^∗,𝐹𝑡,𝜏𝛿_𝑛(𝑡)𝑤_𝑛,𝜏𝛿_𝑛(𝑡)𝑢^∗_𝑛≤𝑓(𝑡)−𝑓_𝑛(𝑡(𝑡)×𝜏(𝑡)𝛿)+𝑚𝑤−𝜏_𝑛(𝑡)𝑤_𝑛+𝜏(𝑡)𝑢^∗−𝜏𝛿_𝑛(𝑡)𝑢^∗_𝑛. (69)
Again, by Step 3, the right-hand side of this inequality tends to zero when 𝑛→∞. Hence, 𝑓(𝑡)∈𝐹(𝑡,𝜏(𝑡)𝑤,𝜏(𝑡)𝐽^∗(𝑢)), and 𝑡∈𝐼.
Step 5. For almost 𝑡∈𝐼, 𝑢^′(𝑡)−𝑓(𝑡)∈−𝑁_{𝐶(𝑡,𝑤(𝑡))}(𝐽^∗(𝑢(𝑡))).
We note the weak convergence of 𝑢^′_𝑛−𝑓_𝑛 towards 𝑢^′−𝑓 in 𝐿¹(𝐼,𝑋^∗), and Mazur’s Lemma gives us𝑢^′(𝑡)−𝑓(𝑡)∈_𝑛Co𝑢^′_𝑚(𝑡)−𝑓_𝑚(𝑡)∶𝑚≥𝑛,fora.e.𝑡∈𝐼. (70)
Fix any 𝑡 such that the preceding relation is satisfied, and consider 𝑥∈𝑋. The last relation above yields 𝑢^′(𝑡)−𝑓(𝑡),𝑥≤inf_𝑛Sup_𝑚≥𝑛𝑢^′_𝑚(𝑡)−𝑓_𝑚(𝑡),𝑥. (71)
Moreover, the relations (28)(vi) and (vii) tell us for a.e. 𝑡∈𝐼𝑢^′_𝑛(𝑡)−𝑓_𝑛(𝑡)∈−𝑁_{𝐶(𝜃𝑛(𝑡),𝑤𝑛(𝜃𝑛(𝑡)))}𝐽^∗𝑢_𝑛𝜃_𝑛(𝑡)∩𝐵^∗0,𝑒^′(𝑡), (72) where the function 𝑒  is defined by (34). Hence, by Lemma 13, for a.e. 𝑡∈𝐼,  we have 𝑢^′_𝑛(𝑡)−𝑓_𝑛(𝑡)∈−𝑒^′(𝑡)𝜕𝑑_{𝐶(𝜃𝑛(𝑡),𝑤𝑛(𝜃𝑛(𝑡)))}𝐽^∗𝑢_𝑛𝜃_𝑛(𝑡). (73)
In view of (63) and (64), for 𝑥∈𝑋 and a.e. 𝑡∈𝐼, one obtains 𝑢^′(𝑡)−𝑓(𝑡),𝑥≤lim_𝑛→∞Sup𝑢^′_𝑚(𝑡)−𝑓_𝑚(𝑡),𝑥≤𝑒^′(𝑡)lim_𝑛→∞Sup𝜎𝑥,−𝜕𝑑_{𝐶(𝜃𝑛(𝑡),𝑤𝑛(𝜃𝑛(𝑡)))}𝐽^∗𝑢_𝑛𝜃_𝑛(𝑡), (74) where 𝜎 is the support function. From the uppersemicontinuity property, Lemma 14 and the last relation above yield for 𝑥∈𝑋 and a.e. 𝑡∈𝐼𝑢^′(𝑡)−𝑓(𝑡),𝑥≤𝑒^′(𝑡)𝜎𝑥,−𝜕𝑑_{𝐶(𝑡,𝑤(𝑡))}𝐽^∗(𝑢(𝑡)). (75)
As the set 𝜕𝑑_{𝐶(𝑡,𝑤𝑛(𝑡))}(𝐽^∗(𝑢(𝑡))) is closed and convex, we get 𝑢^′(𝑡)−𝑓(𝑡)∈−𝑁_{𝐶(𝑡,𝑤(𝑡))}𝐽^∗(𝑢(𝑡)),fora.e.𝑡∈𝐼. (76) This proves the claim.
Finally, by Steps 4 and 5 we have for almost 𝑡∈𝐼𝑢^′(𝑡)∈−𝑁_{𝐶(𝑡,𝑤(𝑡))}𝐽^∗(𝑢(𝑡(𝑡)(𝑡)𝐽))+𝐹𝑡,𝜏𝑤,𝜏^∗(𝑢(𝑡)), (77) and the proof is complete.

4. Conclusion

In this paper, existence problem of solution of a functional sweeping process in Banach space with noncompact perturbation has been considered. Some sufficient conditions have been obtained. The importance of this work due to the fact that the values of the perturbation are not necessarily convex or compact and the considerable space is a Banach space.

Acknowledgment

The authors gratefully acknowledge the Deanship of Scientific Research, King Faisal University of Saudi Arabia, for its financial support concerning this research project.