# Existence of Mild Solutions for a Semilinear Integrodifferential Equation with Nonlocal Initial Conditions

Authors: Carlos Lizama; Juan C. Pozo
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Using Hausdorff measure of noncompactness and a fixed-point argument we prove the existence of mild solutions for the semilinear integrodifferential equation subject to nonlocal initial conditionsu′(t)=Au(t)+∫0tB(t-s)u(s)ds+f(t,u(t)), t∈[0,1], u(0)=g(u), where A:D(A)⊆X→X,  and for every t∈[0,1] the maps B(t):D(B(t))⊆X→X are linear closed operators defined in a Banach space X. We assume further that D(A)⊆D(B(t)) for every t∈[0,1], and the functions f:[0,1]×X→X and g:C([0,1];X)→X are X-valued functions which satisfy appropriate conditions.


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## Body

## 1. Introduction

The concept ofnonlocal initial condition has been introduced to extend the study of classical initial value problems. This notion is more precise for describing nature phenomena than the classical notion because additional information is taken into account. For the importance of nonlocal conditions in different fields, the reader is referred to [1–3] and the references cited therein.The earliest works related with problems submitted to nonlocal initial conditions were made by Byszewski [4–7]. In these works, using methods of semigroup theory and the Banach fixed point theorem the author has proved the existence of mild and strong solutions for the first order Cauchy problem
(1.1)u′(t)=Au(t)+f(t,u(t)),t∈[0,1],u(0)=g(u),
where A is an operator defined in a Banach space X which generates a semigroup {T(t)}t⩾0, and the maps f and g are suitable X-valued functions.Henceforth, (1.1) has been extensively studied by many authors. We just mention a few of these works. Byszewski and Lakshmikantham [8] have studied the existence and uniqueness of mild solutions whenever f and g satisfy Lipschitz-type conditions. Ntouyas and Tsamatos [9, 10] have studied this problem under conditions of compactness for the semigroup generated by A and the function g. Recently, Zhu et al. [11], have investigated this problem without conditions of compactness on the semigroup generated by A, or the function f.On the other hand, the study of abstract integrodifferential equations has been an active topic of research in recent years because it has many applications in different areas. In consequence, there exists an extensive literature about integrodifferential equations with nonlocal initial conditions, (cf., e.g., [12–25]). Our work is a contribution to this theory. Indeed, this paper is devoted to study the existence of mild solutions for the following semilinear integrodifferential evolution equation:
(1.2)u′(t)=Au(t)+∫0t‍B(t-s)u(s)ds+f(t,u(t)),t∈[0,1],u(0)=g(u),
where A:D(A)⊆X→X and for every t∈[0,1] the mappings B(t):D(B(t))⊆X→X are linear closed operators defined in a Banach space X. We assume further that D(A)⊆D(B(t)) for every t∈[0,1], and the functions f:[0,1]×X→X and g:C([0,1];X)→X are X-valued functions that satisfy appropriate conditions which we will describe later. In order to abbreviate the text of this paper, henceforth we will denote by I the interval [0,1], and C(I;X) is the space of all continuous functions from I to X endowed with the uniform convergence norm.The classical initial value version of (1.2), that is, u(0)=u0 for some u0∈X, has been extensively studied by many researchers because it has many important applications in different fields of natural sciences such as thermodynamics, electrodynamics, heat conduction in materials with memory, continuum mechanics and population biology, among others. For more information, see [26–28]. For this reason the study of existence and other properties of the solutions for (1.2) is a very important problem. However, to the best of our knowledge, the existence of mild solutions for the nonlocal initial value problem (1.2) has not been addressed in the existing literature. Most of the authors obtain the existence of solutions and well-posedness for (1.2) by establishing the existence of a resolvent operator {R(t)}t∈I and a variation of parameters formula (see, [29, 30]). Using adaptation of the methods described in [11], we are able to prove the existence of mild solutions of (1.2) under conditions of compactness of the function g and continuity of the function t↦R(t) for t>0. Furthermore, in the particular case B(t)=b(t)A for all t∈[0,1], where the operator A is the infinitesimal generator of a C0-semigroup defined in a Hilbert space H, and the kernel b is a scalar map which satisfies appropriate hypotheses, we are able to give sufficient conditions for the existence of mild solutions only in terms of spectral properties of the operator A and regularity properties of the kernel b. We show that our abstract results can be applied to concrete situations. Indeed, we consider an example with a particular choice of the function b and the operator A is defined by
(1.3)(Aw)(t,ξ)=a1(ξ)∂2∂ξ2w(t,ξ)+b1(ξ)∂∂ξw(t,ξ)+c¯(ξ)w(t,ξ),
where the given coefficients a1, b1, c¯ satisfy the usual uniform ellipticity conditions.


## 2. Preliminaries

Most of the notations used throughout this paper are standard. So,ℕ, ℤ, ℝ, and ℂ denote the set of natural integers and real and complex numbers, respectively, ℕ0=ℕ∪{0}, ℝ+=(0,∞) and ℝ0+=[0,∞).In this workX and Y always are complex Banach spaces with norms ∥·∥X and ∥·∥Y; the subscript will be dropped when there is no danger of confusion. We denote the space of all bounded linear operators from X to Y by ℒ(X,Y). In the case X=Y, we will write briefly ℒ(X). Let A be an operator defined in X. We will denote its domain by D(A), its domain endowed with the graph norm by [D(A)], its resolvent set by ρ(A), and its spectrum by σ(A)=ℂ∖ρ(A).As we have already mentionedC(I;X) is the vector space of all continuous functions f:I→X. This space is a Banach space endowed with the norm
(2.1)∥f∥∞=supt∈I∥f(t)∥X.In the same manner, forn∈ℕ we write Cn(I;X) for denoting the space of all functions from I to X which are n-times differentiable. Further, C∞(I;X) represents the space of all infinitely differentiable functions from I to X.We denote byL1(I;X) the space of all (equivalent classes of) Bochner-measurable functions f:I↦X such that ∥f(t)∥X is integrable for t∈I. It is well known that this space is a Banach space with the norm
(2.2)∥f∥L1(I;X)=∫I‍∥f(s)∥Xds.We next include some preliminaries concerning the theory of resolvent operator{R(t)}t∈I for (1.2).Definition 2.1.
LetX be a complex Banach space. A family {R(t)}t∈I of bounded linear operators defined in X is called a resolvent operator for (1.2) if the following conditions are fulfilled. (R1)For each x∈X, R(0)x=x and R(·)x∈C(I;X).(R2) The map R:I→ℒ([D(A)]) is strongly continuous.(R3) For each y∈D(A), the function t↦R(t)y is continuously differentiable and
(2.3)ddtR(t)y=AR(t)y+∫0t‍B(t-s)R(s)yds=R(t)Ay+∫0t‍R(t-s)B(s)yds,t∈I.In what follows we assume that there exists a resolvent operator{R(t)}t∈I for (1.2) satisfying the following property.(P) The function t↦R(t) is continuous from (0,1] to ℒ(X) endowed with the uniform operator norm ∥·∥ℒ(X).Note that property(P) is also named in different ways in the existing literature on the subject, mainly the theory of C0-semigroups, namely, norm continuity for t>0, eventually norm continuity, or equicontinuity.The existence of solutions of thelinearproblem
(2.4)u′(t)=Au(t)+∫0t‍B(t-s)u(s)ds+f(t),t⩾0,u(0)=u0∈X
has been studied by many authors. Assuming that f:[0,+∞)→X is locally integrable, it follows from [29] that the function u given by
(2.5)u(t)=R(t)u0+∫0t‍R(t-s)f(s)ds,fort⩾0,
is a mild solution of the problem (2.4). Motivated by this result, we adopt the following concept of solution.Definition 2.2.
A continuous functionu∈C(I;X) is called a mild solution of (1.2) if the equation
(2.6)u(t)=R(t)g(u)+∫0t‍R(t-s)f(s,u(s))ds,t∈I,
is verified.The main results of this paper are based on the concept of measure of noncompactness. For general information the reader can see [31]. In this paper, we use the notion of Hausdorff measure of noncompactness. For this reason we recall a few properties related with this concept.Definition 2.3.
LetS be a bounded subset of a normed space Y. The Hausdorff measure of noncompactness of S is defined by
(2.7)η(S)=inf{ε>0:Shasafinitecoverbyballsofradiusε}.Remark 2.4.
LetS1, S2 be bounded sets of a normed space Y. The Hausdorff measure of noncompactness has the following properties. (i)
IfS1⊆S2, then η(S1)⩽η(S2).(ii)
η(S1)=η(S1¯), where S1¯ denotes the closure of A.(iii)
η(S1)=0 if and only if S1 is totally bounded.(iv)
η(λS1)=|λ|η(S1) with λ∈ℝ.(v)
η(S1∪S2)=max{η(S1),η(S2)}.(vi)
η(S1+S2)⩽η(S1)+η(S2), where S1+S2={s1+s2:s1∈S1,s2∈S2}.(vii)
η(S1)=η(co¯(S1)), where co¯(S1) is the closed convex hull of S1.We next collect some specific properties of the Hausdorff measure of noncompactness which are needed to establish our results. Henceforth, when we need to compare the measures of noncompactness inX and C(I;X), we will use ζ to denote the Hausdorff measure of noncompactness defined in X and γ to denote the Hausdorff measure of noncompactness on C(I;X). Moreover, we will use η for the Hausdorff measure of noncompactness for general Banach spaces Y.Lemma 2.5.
LetW⊆C(I;X) be a subset of continuous functions. If W is bounded and equicontinuous, then the set co¯(W) is also bounded and equicontinuous.For the rest of the paper we will use the following notation. LetW be a set of functions from I to X and t∈I fixed, and we denote W(t)={w(t):w∈W}. The proof of Lemma 2.6 can be found in [31].Lemma 2.6.
LetW⊆C(I;X) be a bounded set. Then ζ(W(t))⩽γ(W) for all t∈I. Furthermore, if W is equicontinuous on I, then ζ(W(t)) is continuous on I, and
(2.8)γ(W)=sup{ζ(W(t)):t∈I}.A set of functionsW⊆L1(I;X) is said to be uniformly integrable if there exists a positive function κ∈L1(I;ℝ+) such that ∥w(t)∥⩽κ(t) a.e. for all w∈W.The next property has been studied by several authors; the reader can see [32] for more details.Lemma 2.7.
If{un}n∈ℕ⊆L1(I;X) is uniformly integrable, then for each n∈ℕ the function t↦ζ({un(t)}n∈ℕ) is measurable and
(2.9)ζ({∫0t‍un(s)ds}n=1∞)⩽2∫0t‍ζ({un(s)}n=1∞)ds.The next result is crucial for our work, the reader can see its proof in [33, Theorem 2].Lemma 2.8.
LetY be a Banach space. If W⊆Y is a bounded subset, then for each ε>0, there exists a sequence {un}n∈ℕ⊆W such that
(2.10)η(W)⩽2η({un}n=1∞)+ε.The following lemma is essential for the proof of Theorem3.2, which is the main result of this paper. For more details of its proof, see [34, Theorem 3.1].Lemma 2.9.
For all0⩽m⩽n, denote Cmn=(nm). If 0<ϵ<1 and h>0 and let
(2.11)Sn=ϵn+C1nϵn-1h+C2nϵn-2h22!+⋯+hnn!,n∈ℕ,
then limn→∞Sn=0.Clearly, a manner for proving the existence of mild solutions for (1.2) is using fixed-point arguments. The fixed-point theorem which we will apply has been established in [34, Lemma 2.4].Lemma 2.10.
LetS be a closed and convex subset of a complex Banach space Y, and let F:S→S be a continuous operator such that F(S) is a bounded set. Define
(2.12)F1(S)=F(S),Fn(S)=F(co¯(Fn-1(S))),n=2,3,….
If there exist a constant 0⩽r<1 and n0∈ℕ such that
(2.13)η(Fn0(S))⩽rη(S),
then F has a fixed point in the set S.


## 3. Main Results

In this section we will present our main results. Henceforth, we assume that the following assertions hold.(H1) There exists a resolvent operator {R(t)}t∈I for (1.2) having the property (P).(H2) The function g:C(I;X)→X is a compact map.(H3) The function f:I×X→X satisfies the Carathéodory type conditions; that is, f(·,x) is measurable for all x∈X and f(t,·) is continuous for almost all t∈I.(H4) There exist a function m∈L1(I;ℝ+) and a nondecreasing continuous function Φ:ℝ+→ℝ+ such that
(3.1)∥f(t,x)∥⩽m(t)Φ(∥x∥)
for all x∈X and almost all t∈I.(H5) There exists a function H∈L1(I;ℝ+) such that for any bounded S⊆X(3.2)ζ(f(t,S))⩽H(t)ζ(S)
for almost all t∈I.Remark 3.1.
Assuming that the functiong satisfies the hypothesis (H2), it is clear that g takes bounded set into bounded sets. For this reason, for each R⩾0 we will denote by gR the number gR=sup{∥g(u)∥:∥u∥∞⩽R}.The following theorem is the main result of this paper.Theorem 3.2.
If the hypotheses(H1)–(H5) are satisfied and there exists a constant R⩾0 such that
(3.3)KgR+KΦ(R)∫01‍m(s)ds⩽R,
where K=sup{∥R(t)∥:t∈I}, then the problem (1.2) has at least one mild solution.Proof.
DefineF:C(I;X)→C(I;X) by
(3.4)(Fu)(t)=R(t)g(u)+∫0t‍R(t-s)f(s,u(s))ds,t∈I,
for all u∈C(I;X).
We begin showing thatF is a continuous map. Let {un}n∈ℕ⊆C(I;X) such that un→u as n→∞ (in the norm of C(I;X)). Note that
(3.5)∥F(un)-F(u)∥⩽K∥g(un)-g(u)∥+K∫01‍∥f(s,un(s))-f(s,u(s))∥ds,
by hypotheses (H2) and (H3) and by the dominated convergence theorem we have that ∥F(un)-F(u)∥→0 when n→∞.
LetR⩾0 and denote BR={u∈C(I;X):∥u(t)∥⩽R∀t∈I} and note that for any u∈BR we have
(3.6)∥(Fu)(t)∥⩽∥R(t)g(u)∥+∥∫0t‍R(t-s)f(s,u(s))ds∥⩽KgR+KΦ(R)∫01‍m(s)ds⩽R.
ThereforeF:BR→BR and F(BR) is a bounded set. Moreover, by continuity of the function t↦R(t) on (0,1], we have that the set F(BR) is an equicontinuous set of functions.
Defineℬ=co¯(F(BR)). It follows from Lemma 2.5 that the set ℬ is equicontinuous. In addition, the operator F:ℬ→ℬ is continuous and F(ℬ) is a bounded set of functions.
Letε>0. Since the function g is a compact map, by Lemma 2.8 there exists a sequence {vn}n∈ℕ⊂F(ℬ) such that
(3.7)ζ(F(ℬ)(t))⩽2ζ({vn(t)}n=1∞)+ε⩽2ζ(∫0t‍{R(t-s)f(s,un(s))}n=1∞ds)+ε.
By the hypothesis(H4), for each t∈I we have ∥R(t-s)f(s,un(s))∥⩽KΦ(R)m(s). Therefore, by the condition (H5) we have
(3.8)ζ(F(ℬ)(t))⩽4K∫0t‍ζ({f(s,un(s))}n=1∞ds+ε⩽4K∫0t‍H(s)ζ({un(s)}n∈ℕ)ds+ε⩽4Kγ(ℬ)∫0t‍H(s)ds+ε.
Since the function H∈L1(I;ℝ+), for α<1/4K there exists φ∈C(I;ℝ+) satisfying ∫01‍|H(s)-φ(s)|ds<α. Hence,
(3.9)ζ(F(ℬ)(t))⩽4Kγ(ℬ)[∫0t‍|H(s)-φ(s)|ds+∫0t‍φ(s)ds]+ε⩽4Kγ(ℬ)[α+Nt]+ε,
where N=∥φ∥∞. Since ε>0 is arbitrary, we have
(3.10)ζ(F(ℬ)(t))⩽(a+bt)γ(ℬ),wherea=4αK,b=4KN.
Let ε>0. Since the function g is a compact map and applying the Lemma 2.8 there exists a sequence {wn}n∈ℕ⊆co¯(F(ℬ)) such that
(3.11)ζ(F2(ℬ)(t))⩽2ζ(∫0t‍{R(t-s)f(s,wn(s))}n=1∞ds)+ε⩽4K∫0t‍ζ{f(s,wn(s))}n=1∞ds+ε⩽4K∫0t‍H(s)ζ(co¯(F1(ℬ)(s)))+ε=4K∫0t‍H(s)ζ(F1(ℬ)(s))+ε.
Using the inequality (3.10) we have that
(3.12)ζ(F2(ℬ)(t))⩽4K∫0t‍[|H(s)-φ(s)|+|φ(s)|](a+bs)γ(ℬ)ds+ε⩽4K(a+bt)γ(ℬ)∫0t‍|H(s)-φ(s)|ds+4KNγ(ℬ)(at+bt22)+ε⩽a(a+bt)+b(at+bt22)+ε⩽(a2+2bt+(bt)22)γ(ℬ)+ε.
Sinceε>0 is arbitrary, we have
(3.13)ζ(F2(ℬ)(t))⩽(a2+2bt+(bt)22)γ(ℬ).
By an inductive process, for alln∈ℕ, it holds
(3.14)ζ(Fn(ℬ)(t))⩽(an+C1nan-1bt+C2nan-2(bt)22!+⋯+(bt)nn!)γ(ℬ),
where, for 0⩽m⩽n, the symbol Cmn denotes the binomial coefficient (nm).
In addition, for alln∈ℕ the set Fn(ℬ) is an equicontinuous set of functions. Therefore, using the Lemma 2.6 we conclude that
(3.15)γ(Fn(ℬ))⩽(an+C1nan-1b+C2nan-2b22!+⋯+bnn!)γ(ℬ).
Since0⩽a<1 and b>0, it follows from Lemma 2.7 that there exists n0∈ℕ such that
(3.16)(an0+C1n0an0-1b+C2n0an0-2b22!+⋯+bn0n0!)=r<1.
Consequently,γ(Fn0(ℬ))⩽rγ(ℬ). It follows from Lemma 2.9 that F has a fixed point in ℬ, and this fixed point is a mild solution of (1.2).Our next result is related with a particular case of (1.2). Consider the following Volterra equation of convolution type:
(3.17)u′(t)=Au(t)+∫0t‍b(t-s)Au(s)ds+f(t,u(t)),t∈I,u(0)=g(u),
where A is a closed linear operator defined on a Hilbert space H, the kernel b∈Lloc1(ℝ+;ℝ), and the function f is an appropriate H-valued map.Since (3.17) is a convolution type equation, it is natural to employ the Laplace transform for its study.LetX be a Banach space and a∈Lloc1(ℝ+;ℝ). We say that the function a is Laplace transformable if there is ω∈ℝ such that ∫0∞‍e-ωt|a(t)|dt<∞. In addition, we denote by a^(λ)=∫0∞‍e-λta(t)dt, for Reλ>ω, the Laplace transform of the function a.We need the following definitions for proving the existence of a resolvent operator for (3.17). These concepts have been introduced by Prüss in [28].Definition 3.3.
Leta∈Lloc1(ℝ+;ℝ) be Laplace transformable and k∈ℕ. We say that the function a is k-regular if there exists a constant C>0 such that
(3.18)|λna^(n)(λ)|⩽C|a^(λ)|
for all Reλ⩾ω and 0<n⩽k.Convolutions ofk-regular functions are again k-regular. Moreover, integration and differentiation are operations which preserve k-regularity as well. See [28, page 70].Definition 3.4.
Letf∈C∞(ℝ+;ℝ). We say that f is a completely monotone function if and only if (-1)nf(n)(λ)⩾0 for all λ>0 and n∈ℕ.Definition 3.5.
Leta∈Lloc1(ℝ+;ℝ) such that a is Laplace transformable. We say that a is completely positive function if and only if
(3.19)1λa^(λ),-a^′(λ)(a^(λ))2
are completely monotone functions.Finally, we recall that a one-parameter family{T(t)}t⩾0 of bounded and linear operators is said to be exponentially bounded of type (M,ω) if there are constants M⩾1 and ω∈ℝ such that
(3.20)∥T(t)∥⩽Meωt,∀t⩾0.The next proposition guarantees the existence of a resolvent operator for (3.17) satisfying the property (P). With this purpose we will introduce the conditions (C1) and (C2). (C1) The kernel a defined by a(t)=1+∫0t‍b(s)ds, for all t⩾0, is 2-regular and completely positive.(C2) The operator A is the generator of a semigroup of type (M,ω) and there exists μ0>ω such that
(3.21)lim|μ|→∞∥1b^(μ0+iμ)+1(μ0+iμb^(μ0+iμ)+1-A)-1∥=0.Proposition 3.6.
Suppose thatA is the generator of a C0-semigroup of type (M,ω) in a Hilbert space H. If the conditions (C1)-(C2)are satisfied, then there exists a resolvent operator {R(t)}t∈I for (3.17) having the property (P).Proof.
Integrating in time (3.17) we get
(3.22)u(t)=∫0t‍a(t-s)Au(s)ds+∫0t‍f(s,u(s))+g(u).
Since the scalar kernel a is completely positive and A generates a C0-semigroup, it follows from [28, Theorem 4.2] that there exists a family of operators {R(t)}t∈I strongly continuous, exponentially bounded which commutes with A, satisfying
(3.23)R(t)x=x+∫0t‍a(t-s)AR(s)xds,∀x∈D(A).
On the other hand, using the condition (C2) and since the scalar kernel a is 2-regular, it follows from [35, Theorem 2.2] that the function t↦R(t) is continuous for t>0. Further, since a∈C1(ℝ+;ℝ), it follows from (3.23) that for all x∈D(A) the map R(·)x is differentiable for all t⩾0 and satisfies
(3.24)ddtR(t)x=AR(t)x+∫0t‍b(t-s)AR(s)xds,t∈I.
From the quality (3.24), we conclude that {R(t)}t∈I is a resolvent operator for (3.17) having the property (P).Corollary 3.7.
Suppose thatA generates a C0-semigroup of type (M,ω) in a Hilbert space H. Assume further that the conditions (C1)-(C2) are fulfilled. If the hypotheses (H2)–(H5) are satisfied and there exists R⩾0 such that
(3.25)KgR+KΦ(R)∫01‍m(s)ds⩽R,whereK=sup{∥R(t)∥:t∈I},
then (3.17) has at least one mild solution.Proof.
It follows from Proposition3.6 that there exists a resolvent operator {R(t)}t∈I for the equation and this resolvent operator has the property (P). Since the hypotheses (H2)–(H5) are satisfied, we apply Theorem 3.2 and conclude that (3.17) has at least one mild solution.


## 4. Applications

In this section we apply the abstract results which we have obtained in the preceding section to study the existence of solutions for a partial differential equation submitted to nonlocal initial conditions. This type of equations arises in the study of heat conduction in materials with memory (see [26, 27]). Specifically, we will study the following problem:
(4.1)∂w(t,ξ)∂t=Aw(t,ξ)+∫0t‍βe-α(t-s)Aw(s,ξ)ds+p1(t)p2(w(t,ξ)),t∈I,w(t,0)=w(t,2π),fort∈I,w(0,ξ)=∫01‍∫0ξ‍qk(s,ξ)w(s,y)dsdy,0⩽ξ⩽2π,
where k:I×[0,2π]→ℝ+ is a continuous function such that k(t,2π)=0 for all t∈I, the constant q∈ℝ+ and the constants α,β satisfy the relation -α⩽β⩽0⩽α. The operator A is defined by
(4.2)(Aw)(t,ξ)=a1(ξ)∂2∂ξ2w(t,ξ)+b1(ξ)∂∂ξw(t,ξ)+c¯(ξ)w(t,ξ),
where the coefficients a1, b1, c¯ satisfy the usual uniformly ellipticity conditions, and D(A)={v∈L2([0,2π];ℝ):v′′L2([0,2π];ℝ)}. The functions p1:I→ℝ+ and p2:ℝ→ℝ satisfy appropriate conditions which will be specified later.Identifyingu(t)=w(t,·) we model this problem in the space X=L2(𝕋;ℝ), where the group 𝕋 is defined as the quotient ℝ/2πℤ. We will use the identification between functions on 𝕋 and 2π-periodic functions on ℝ. Specifically, in what follows we denote by L2(𝕋;ℝ) the space of 2π-periodic and square integrable functions from ℝ into ℝ. Consequently, (4.1) is rewritten as
(4.3)u′(t)=Au(t)+∫0t‍b(t-s)Au(s)ds+f(t,u(t)),t∈I,u(0)=g(u),
where the function g:C(I;X)→X is defined by g(w)(ξ)=∫01‍∫0ξ‍qk(s,ξ)w(s,y)dsdy, and f(t,u(t))=p1(t)p2(u(t)), where p1 is integrable on I, and p2 is a bounded function satisfying a Lipschitz type condition with Lipschitz constant L.We will prove that there existsq>0 sufficiently small such that (4.3) has a mild solution on L2(𝕋;ℝ).With this purpose, we begin noting that∥g∥⩽q(2π)1/2(∫02π‍∫01‍k(s,ξ)2dsdξ)1/2. Moreover, it is a well-known fact that the g is a compact map.Further, the functionf satisfies ∥f(t,u(t))∥⩽p1(t)Φ(∥u(t)∥), with Φ(∥u(t)∥)≡∥p2∥ and ∥f(t,u1(t))-f(t,u2(t))∥⩽Lp1(t)∥u1-u2∥. Thus, the conditions (H2)–(H5) are fulfilled.Definea(t)=1+∫0t‍βe-αsds, for all t∈ℝ0+. Since the kernel b defined by b(t)=βe-αt is 2-regular, it follows that a is 2-regular. Furthermore, we claim that a is completely positive. In fact, we have
(4.4)a^(λ)=λ+α+βλ(λ+α).
Define the functions f1 and f2 by f1(λ)=1/(λa^(λ)) and f2(λ)=-a^′(λ)/[a^(λ)]2, respectively. In other words
(4.5)f1(λ)=λ+αλ+α+β,f2(λ)=λ2+2(α+β)λ+αβ+α2(λ+α+β)2.
A direct calculation shows that
(4.6)f1(n)(λ)=(-1)n+1β(n+1)!(λ+α+β)n+1,f2(n)(λ)=(-1)n+1β(α+β)(n+1)!(λ+α+β)n+2forn∈ℕ.
Since -α⩽β⩽0⩽α, we have that f1 and f2 are completely monotone. Thus, the kernel a is completely positive.On the other hand, it follows from [36] that A generates an analytic, noncompact semigroup {T(t)}t⩾0 on L2(𝕋;ℝ). In addition, there exists a constant M>0 such that
(4.7)M=sup{∥T(t)∥:t⩾0}<+∞.
It follows from the preceding fact and the Hille-Yosida theorem that z∈ρ(A) for all z∈ℂ such that Re(z)>0. Let z=μ0+iμ. By direct computation we have
(4.8)Re(μ0+iμb^(μ0+iμ)+1)=μ03+μ02α+μ02(α+β)+μ0α(α+β)+μ0μ2-μ2β(α+β)2+2μ0(α+β)+μ02+μ2.
Hence, Re((μ0+iμ)/(b~(μ0+iμ)+1))>0 for all z=μ0+iμ, such that μ0>0. This implies that
(4.9)(μ0+iμb~(μ0+iμ)+1-A)-1∈ℒ(X),∀μ0>0.
Since the semigroup generated by A is an analytic semigroup we have
(4.10)∥1b^(μ0+iμ)+1(μ0+iμb^(μ0+iμ)+1-A)-1∥⩽∥Mμ0+iμ∥.
Therefore,
(4.11)lim|μ|→∞∥1b^(μ0+iμ)+1(μ0+iμb^(μ0+iμ)+1-A)-1∥=0.
It follows from Proposition 3.6 that (4.3) admits a resolvent operator {R(t)}t∈I satisfying property (P).LetK=sup{∥R(t)∥:t∈I} and c=(2π)1/2(∫02π‍∫01‍k(s,ξ)2dsdξ)1/2.A direct computation shows that for eachR⩾0 the number gR is equal to gR=qcR.Therefore the expression(KgR+KΦ(R)∫01‍m(s)ds) is equivalent to (qcKR+∥p1∥1LK).Since there existsq>0 such that qcK<1, then there exists R⩾0 such that
(4.12)qcKR+∥p1∥1LK⩽R.From Corollary3.7 we conclude that there exists a mild solution of (4.1).



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*Source: 647103-2012-12-31.xml*