Existence of Solutions in Some Interpolation Spaces for a Class of Semilinear Evolution Equations with Nonlocal Initial Conditions

Abstract

This paper is concerned with the existence of mild and strong solutions for a class of semilinear evolution equations with nonlocal initial conditions. The linear part is assumed to be a (not necessarily densely defined) sectorial operator in a Banach space 𝑋. Considering the equations in the norm of some interpolation spaces between 𝑋 and the domain of the linear part, we generalize the recent conclusions on this topic. The obtained results will be applied to a class of semilinear functional partial differential equations with nonlocal conditions.

1. Introduction

The purpose of this paper is to study the existence of mild and strong solutions to the following semilinear evolution problem with nonlocal initial conditions: 𝑢^′(𝑡)(𝑡)((𝑡[],𝑢(0)(𝑢)=𝐴𝑢+𝑓𝑡,𝑢)),𝑡∈0,𝑇+𝑔=𝑢₀∈𝑋,(1) in a Banach space (𝑋,‖⋅‖), where 𝑇>0, 𝑓 and 𝑔 are given 𝑋-valued functions which will be specified later, and 𝐴 is a linear closed operator in 𝑋. By letting (𝐹𝑢)(𝑡)∶=𝑓(𝑡,𝑢(𝑡)), we see that (1) can be considered as a special case of the following equation: 𝑢^′(𝑡)(𝑡)+(𝑡)[],𝑢(0)(𝑢)=𝐴𝑢𝐹𝑢)(,𝑡∈0,𝑇+𝑔=𝑢₀∈𝑋.(2) Since the nonlocal condition reflects physical phenomena more precisely than the classical initial condition 𝑢(0)=𝑢₀ does, this issue has caught an increasing interest in the past several years. For the importance of nonlocal Cauchy problems in applications, we refer to the papers [1, 2] and the references therein.

The existence and uniqueness of mild and classical solutions of (1) have been investigated by many authors, see for example [1–14]. A basic approach of this problem is to define the solution operator Φ∶𝐶([0,𝑇],𝑋)→𝐶([0,𝑇],𝑋) by (Φ𝑢)(𝑡)∶=𝑇(𝑡)𝑢₀−𝑔(𝑢)+^𝑡₀𝑇()𝑓((𝑠[],𝑡−𝑠𝑠,𝑢))𝑑𝑠,𝑡∈0,𝑇(3) and use various fixed-point theorems, including Schauder fixed-point theorem, Banach contraction principle, Leray-Schauder alternative, Sadovskii fixed-point theorem, and some others, to show that Φ has a fixed point, which is the mild solution of (1). To use fixed-point theorems, it is necessary that the semigroup 𝑇(𝑡) generated by the linear part of (1) is compact; that is, 𝑇(𝑡) is a compact operator for all 𝑡>0, so that the norm continuity of 𝑇(𝑡) for 𝑡>0 becomes a key point in the study of the existence of mild solution. In [13], the authors studied the existence of mild solution for (1) by using the Leray-Schauder alternative. However, as pointed out by Liu [11], the proof of the main result in [13] is invalid because of the absence of compactness of the solution operator at 𝑡=0. This gap can be filled by simply assuming complete continuity on the nonlocal term 𝑔, but it is too restrictive in applications. To take away this unsatisfactory condition, Liang et al. [7] observed that in many studies of nonlocal Cauchy problems, for example [1, 6, 9], the nonlocal condition 𝑔 is completely determined on [𝛿,𝑇], for some 𝛿>0; that is, such a 𝑔 ignores 𝑡=0; for instance, in [9] the function 𝑔(𝑢) is given by 𝑔(𝑢)=^𝑝_𝑖=0𝑐_𝑖𝑢𝑡_𝑖,(4) where the 𝑐_𝑖 are given constants, and in this case, we have measurements at 𝑡=0≤𝑡₀<𝑡₁<⋯<𝑡_𝑝≤𝑇 rather than just at 𝑡=0. Thus, by assuming that there is a 𝛿∈(0,𝑇) such that 𝑔𝜙=𝑔𝜓∀𝜙,𝜑∈𝑌_𝑟∶=𝜑∈𝐶([0,𝑇],𝑋);𝜑(𝑡)≤𝑟,∀𝑡∈[0,𝑇],(5) with 𝜙(𝑠)=𝜓(𝑠),𝑠∈[𝛿,𝑇],(6) the authors use their method of utilizing fixed-point theorem twice to deduce the existence of mild solution of (1) with the nonlinear term 𝐹 being Lipschitz continuous and the nonlocal term 𝑔 being continuous and mapping 𝑌_𝑟 into bounded sets. Afterward, Xiao and Liang [14] initiated the study of the nonautonomous case of (1) with nondensely defined sectorial linear part. More recently, Liu and Yuan [12] gave existence results by Schauder fixed-point theorem and a limiting process, under the following hypothesis: there is a𝛿∈(0,𝑇)such that𝐹𝜙=𝐹𝜓,𝑔𝜙=𝑔𝜓∀𝜙,𝜑∈𝑌_𝑟(7) with 𝜙(𝑠)=𝜓(𝑠),𝑠∈[𝛿,𝑇],(8) with the nonlinear term 𝐹 being bounded by an integrable function.

In the present paper, the linear part 𝐴 of (1) and (2) is assumed to be a (not necessarily densely defined) sectorial operator in 𝑋, which means that there exist constants 𝜔, 𝑀>0, and 𝜃∈(𝜋/2,𝜋) such that 𝜌(𝐴)⊃_𝜃,𝜔(,)𝑅(∶=𝜆∈ℂ∶𝜆≠𝜔,arg𝜆−𝜔)<𝜃‖(𝜆−𝜔𝜆,𝐴)‖≤𝑀,∀𝜆∈_𝜃,𝜔.(9) Existence results of solutions to (1) and (2) are discussed in some interpolation spaces 𝑋_𝛼,𝑝∶=(𝑋,𝐷(𝐴))_𝛼,𝑝, where 0<𝛼<1 and 1<𝑝≤∞ (see [15], Chapter 2). We establish some existence results of (1) and (2) without assuming Lipschitz condition on the nonlinear term and complete continuity on the nonlocal condition. Thus, the hypotheses in this note are less restricted than relevant works and hence the obtained results are more applicable than recent conclusions on this topic.

This work is organized as follows. In Section 2, we collect some known notions and results on the interpolation spaces between 𝑋 and 𝐷(𝐴) and give the basic hypotheses on (2). In Section 3, we study the existence of mild solutions to (2) and (1). Section 4 is devoted to the existence of strong solutions to (2) and (1), and in Section 5, we give an example to illustrate our abstract results.

2. Preliminaries

As usual, for a Banach space 𝑋, 𝐿(𝑋) is the space of linear operators on 𝑋 with operator norm denoted by ‖⋅‖_𝐿(𝑋). If 𝐴 satisfies (9), then it is known that the family of Dunford integrals 𝑇(𝑡)=12𝜋𝑖_{𝜔+𝛾𝑟,𝜂}𝑒^𝜆𝑡(𝜆−𝐴)⁻¹𝑑𝜆,(10) where 𝛾_𝑟,𝜂 is the curve {𝜆∈ℂ∶|arg𝜆|=𝜂,|𝜆|≥𝑟}∪{𝜆∈ℂ∶|arg𝜆|≤𝜂,|𝜆|=𝑟}, is an analytic semigroup satisfying the following properties (see [15], Proposition 2.1.1, p. 35). (A1)𝑇(𝑡)𝑥∈𝐷(𝐴^𝑘) for each 𝑡>0, 𝑥∈𝑋 and 𝑘∈ℕ. Moreover, 𝐴^𝑘𝑇(𝑡)𝑥=𝑇(𝑡)𝐴^𝑘𝑥 for all 𝑡≥0 and 𝑥∈𝐷(𝐴^𝑘). (A2)𝑇(𝑡)𝑇(𝑠)=𝑇(𝑡+𝑠) for all 𝑡,𝑠≥0. (A3) There are positive constants 𝑀₀,𝑀₁,𝑀₂,… such that ‖𝑇(𝑡)‖_𝐿(𝑋)≤𝑀₀𝑒^𝜔𝑡𝑡,𝑡>0,_𝑘(𝐴−𝜔𝐼)^𝑘𝑇(𝑡)_𝐿(𝑋)≤𝑀_𝑘𝑒^𝜔𝑡,𝑡>0.(11) (A4) The function 𝑡↦𝑇(𝑡) belongs to 𝐶^∞((0,∞),𝐿(𝑋)) and 𝑑^𝑘𝑑𝑡^𝑘𝑇(𝑡)=𝐴^𝑘𝑇(𝑡),𝑡>0.(12) Moreover, it has an analytic extension in the sector 𝜋𝜆∈ℂ∶𝜆≠0,arg𝜆<𝜃−2.(13) We see by (A3), that the family (𝑇(𝑡))_𝑡≥0 is uniformly bounded on the interval [0,𝑇]; that is, 𝑀∶=sup_{[]𝑡∈0,𝑇}‖𝑇(𝑡)‖<∞.(14) For 𝛼∈(0,1), 𝑝∈(1,∞], and 𝑡∈[𝑠,𝑇], we denote by 𝑋_𝛼,𝑝 the interpolation space (𝑋,𝐷)_𝛼,𝑝∶=𝑥∈𝑋∶themapping𝑡⟼𝑡^{1−𝛼−1/𝑝}𝐴𝑇(𝑡)𝑥belongsto𝐿^𝑝(0,1)(15) with norm |𝑥|_𝛼,𝑝=‖𝑥‖+𝜃_{1−𝛼−1/𝑝}𝐴𝑇(⋅)𝑥_𝑝,(16) where 𝜃_𝜆∶[0,𝑇]→ℝ is given by 𝜃_𝜆(𝑡)=𝑡^𝜆. Let 𝐸 be the Banach space 𝐶([0,𝑇],𝑋_𝛼,𝑝) endowed with the supnorm given by ‖𝑢‖_𝐸∶=sup_{0≤𝑡≤𝑇}|𝑢(𝑡)|_𝛼,𝑝for𝑥∈𝐸,(17) and for any 𝛿∈(0,𝑇), set 𝐸_𝛿∶=𝐶([𝛿,𝑇],𝑋_𝛼,𝑝). By (2.2.1) of [15] (p.44), we have 𝐶_𝑛∶=sup_{0<𝑡≤𝑇}𝑡^𝑛𝐴^𝑛𝑇(𝑡)_𝐿(𝑋)<∞,∀𝑛∈ℕ.(18) And hence the analytic semigroup (𝑇(𝑡))_𝑡≥0 also satisfies the following property. (A5) For every 𝛿>0𝐶_𝛿∶=sup_{𝑡∈[𝛿,𝑇]}‖𝐴𝑇(𝑡)‖_𝐿(𝑋)≤𝛿⁻¹𝐶₁<∞.(19) Moreover, by (2.2.20) of [15] (see the last line of p.51), there is a constant 𝐶>0 such that  (A6)‖𝐴^𝑛𝑇(𝑡)‖_{𝐿(𝑋,𝑋𝛼,𝑝)}≤𝐶𝑡^{−(𝑛+𝛼)},0<𝑡≤𝑇,𝑛∈ℕ∪{0}.

The following conditions are basic assumptions of this paper. (H1) The operator family (𝑇(𝑡))_𝑡≥0 is compact; that is, 𝑇(𝑡) maps bounded subsets of 𝑋 into relatively compact subsets of 𝑋 for all 𝑡>0. (H2) The function 𝐹∶𝐸→𝐿¹([0,𝑇],𝑋) is continuous and for each 𝑘∈ℕ, there exists a positive function ℎ_𝑘∶[0,𝑇]→ℝ⁺ such that (1)𝑠↦ℎ_𝑘(𝑠)/(𝑡−𝑠)^𝛼 is integrable on [0,𝑡] for each 𝑡∈[0,𝑇],(2)sup_{‖𝑢‖𝐸<𝑘}‖(𝐹𝑢)(𝑡)‖≤ℎ_𝑘(𝑡) for almost all 𝑡∈[0,𝑇],(3)liminf_𝑘→∞(1/𝑘)(sup_{𝑡∈[0,𝑇]}∫^𝑡₀(ℎ_𝑘(𝑠)/(𝑡−𝑠)^𝛼)𝑑𝑠)=𝛾, (4)lim_𝜀→0(sup_{𝑡∈[0,𝑇]}∫^𝑡_𝑡−𝜀(ℎ_𝑘(𝑠)/(𝑡−𝑠)^𝛼)𝑑𝑠)=0.  (H3) The function 𝑔∶𝐿¹([0,𝑇],𝑋_𝛼,𝑝)→𝑋_𝛼,𝑝 is continuous and there exist constants 𝐿>0 and 𝛽∈[0,1] such that 𝑔(𝑢)_𝛼,𝑝≤𝐿‖𝑢‖_𝐸+1^𝛽(20) for 𝑢∈𝐸. Moreover, let 𝜉 be the number defined by 𝜉∶=lim_{‖𝑢‖𝐸→∞}𝐿‖𝑢‖_𝐸+1^𝛽‖𝑢‖_𝐸;(21) that is, 𝜉=𝐿 if 𝛽=1 and 𝜉=0 if 𝛽<1.

We will argue that the hypothesis (H2) is reasonable. In fact, consider a function 𝑓∶[0,𝑇]×𝑋_𝛼,𝑝→𝑋 satisfying that there exist ℒ>0 and 𝜗∈(0,1) such that 𝑓()||𝑡,𝑥−𝑓𝑠,𝑦≤ℒ𝑡−𝑠^𝜗+𝑥−𝑦_𝛼,𝑝,∀𝑡,𝑠∈[0,𝑇],𝑥,𝑦∈𝑋_𝛼,𝑝,(22) and define (𝐹𝑢)(𝑡)∶=𝑓(𝑡,𝑢(𝑡)). It is easy to see that 𝐹 satisfies (H2) by letting ℎ_𝑘(𝑡)∶=𝑘sup_{‖𝑢‖𝐸<𝑘}𝑓(⋅,𝑢(⋅))_{𝐿∞([0,𝑡],𝑋)}.(23) Moreover, one can choose the functions ℎ_𝑘 to be in 𝐿^∞(0,𝑇) and find easily that there is a lot more functions satisfying (H2).

3. Mild Solutions

Definition 1. Let 𝑢∶[0,𝑇]→𝑋 be a continuous function.(i)𝑢 is called a mild solution of (1) on [0,𝑇] if 𝑢 satisfies 𝑢(𝑡)=𝑇(𝑡)𝑢₀−𝑔(𝑢)+^𝑡₀𝑇(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠(24) for 𝑡∈[0,𝑇]. (ii)𝑢 is called a mild solution of (2) on [0,𝑇] if 𝑢 satisfies 𝑢(𝑡)=𝑇(𝑡)𝑢₀−𝑔(𝑢)+^𝑡₀𝑇(𝑡−𝑠)(𝐹𝑢)(𝑠)𝑑𝑠(25) for 𝑡∈[0,𝑇].

The part 𝐴_𝛼,𝑝 of 𝐴 in 𝑋_𝛼,𝑝 is defined by 𝐷𝐴_𝛼,𝑝=𝑥∈𝐷(𝐴)∶𝐴𝑥∈𝑋_𝛼,𝑝,𝐴_𝛼,𝑝𝑥=𝐴𝑥,∀𝑥∈𝐷𝐴_𝛼,𝑝.(26) By Proposition 2.2.7 of [15], we see that 𝐴_𝛼,𝑝 is sectorial in 𝑋_𝛼,𝑝. We denote by (𝑇_𝛼,𝑝(𝑡))_𝑡≥0 the analytic semigroup generated by 𝐴_𝛼,𝑝 on 𝑋_𝛼,𝑝. It is clear that 𝑇_𝛼,𝑝(𝑡)𝑥=𝑇(𝑡)𝑥 for all 𝑥∈𝑋_𝛼,𝑝 and 𝑇(𝑡)𝑥∈𝐷(𝐴)⊂𝑋_𝛼,𝑝 for all 𝑥∈𝑋 and 𝑡>0.

Remark 2. It is known that 𝑋_𝛼,𝑝 is contained in 𝐷(𝐴) ([15], Corollary 2.2.3 (ii), p.47) and that 𝑡↦𝑇(𝑡)𝑥 is continuous at 0 if 𝑥∈𝐷(𝐴) ([15], Proposition 2.1.4 (i), p.38).

Before stating and proving the main results, we need some lemmas.

Lemma 3. If (H1) is satisfied, then the following conclusions hold. (i)(𝑇_𝛼,𝑝(𝑡))_𝑡≥0 is norm-continuous for 𝑡>0.(ii)If 1−𝛼−1/𝑝≥0, then (𝑇_𝛼,𝑝(𝑡))_𝑡≥0 is compact.(iii)If {𝑥_𝑛}^∞_𝑛=1 is a bounded sequence in 𝑋 and 1−𝛼−1/𝑝≥0, then for any 𝑡>0, {𝑇_𝛼,𝑝(𝑡)𝑥_𝑛}^∞_𝑛=1 has a convergent subsequence in 𝑋_𝛼,𝑝.

Proof. (i) Since the operator family (𝑇(𝑡))_𝑡≥0 is, by (H1), compact for 𝑡>0, then (𝑇(𝑡))_𝑡>0 is norm-continuous for 𝑡>0. Note that 𝑇_𝛼,𝑝(𝑡)𝑥−𝑇_𝛼,𝑝(𝑠)𝑥_𝛼,𝑝=𝑇(𝑡)(𝑠𝑥‖+𝜃‖[−𝑇)]_{1−𝛼−1/𝑝}𝐴𝑇(⋅)[𝑇(𝑡)−𝑇(𝑠)]𝑥_𝑝≤‖𝑇(𝑡)−𝑇(𝑠)‖_𝐿(𝑋)‖𝑥‖+‖𝑇(𝑡)−𝑇(𝑠)‖_𝐿(𝑋)𝜃_{1−𝛼−1/𝑝}𝐴𝑇(⋅)𝑥_𝑝=‖𝑇(𝑡)−𝑇(𝑠)‖_𝐿(𝑋)|𝑥|_𝛼,𝑝,∀𝑡,𝑠∈(0,∞),𝑥∈𝑋_𝛼,𝑝,(27) which shows the norm continuity of (𝑇_𝛼,𝑝(𝑡))_𝑡≥0 for 𝑡>0. A similar argument also shows that 𝑇_𝛼,𝑝(𝑡)_{𝐿(𝑋𝛼,𝑝)}≤‖𝑇(𝑡)‖_𝐿(𝑋)(28) for all 𝑡≥0.
(ii) For the compactness of (𝑇_𝛼,𝑝(𝑡))_𝑡≥0 for 𝑡>0, it is sufficient to show that the sequence {𝑇_𝛼,𝑝(𝑡)𝑥_𝑛} has a convergent subsequence in 𝑋_𝛼,𝑝 for each sequence {𝑥_𝑛} in the unit ball of 𝑋_𝛼,𝑝 and 𝑡>0. Since 𝑥_𝑛≤𝑥_{𝑛𝛼,𝑝}=𝑥_𝑛+‖𝜃_{1−𝛼−1/𝑝}𝐴𝑇(⋅)𝑥‖_𝑝≤1,(29) then there is, by the compactness of 𝑇(𝑡/2), a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} and an 𝑥₀∈𝑋 so that 𝑇𝑡2𝑥_𝑛𝑘−𝑥₀⟶0as𝑛_𝑘⟶∞.(30) Thus, the boundedness of 𝐴𝑇(𝑡/2) implies that 𝐴𝑇(𝑡)𝑥_𝑛𝑘𝑡−𝐴𝑇2𝑥₀=𝑡𝐴𝑇2𝑇𝑡2𝑥_𝑛𝑘−𝑥₀⟶0as𝑛_𝑘⟶∞.(31) If 𝑝∈(1,∞) and 1−𝛼−1/𝑝≥0, then (30), (31), and the Lebesgue dominated convergence theorem insure that 𝑇_𝛼,𝑝(𝑡)𝑥_𝑛𝑘𝑡−𝑇2𝑥_0𝛼,𝑝=𝑇(𝑡)𝑥_𝑛𝑘𝑡−𝑇2𝑥₀+¹₀𝑠^1−𝛼𝐴𝑇(𝑠)𝑇(𝑡)𝑥_𝑛𝑘𝑡−𝑇2𝑥₀^𝑝𝑑𝑠𝑠^1/𝑝=𝑇𝑡2𝑇𝑡2𝑥_𝑛𝑘−𝑥₀+¹₀𝑠^{1−𝛼−1/𝑝}𝑇(𝑠)𝐴𝑇(𝑡)𝑥_𝑛𝑘𝑡−𝐴𝑇2𝑥₀^𝑝𝑑𝑠^1/𝑝⟶0(32) as 𝑛_𝑘→∞. On the other hand, using (14) and (31), we have 𝑇_𝛼,𝑝(𝑡)𝑥_𝑛𝑘𝑡−𝑇2𝑥_0𝛼,∞=𝑇(𝑡)𝑥_𝑛𝑘𝑡−𝑇2𝑥₀+𝜃_1−𝛼𝐴𝑇(⋅)𝑇(𝑡)𝑥_𝑛𝑘𝑡−𝑇2𝑥_0∞=𝑇𝑡2𝑇𝑡2𝑥_𝑛𝑘−𝑥₀+sup_{𝑠∈[0,𝑇]}𝜃_1−𝛼𝐴𝑇(𝑠)𝑇(𝑡)𝑥_𝑛𝑘𝑡−𝑇2𝑥₀≤𝑇𝑡2𝑇𝑡2𝑥_𝑛𝑘−𝑥₀+𝑀𝑇^1−𝛼𝑡𝐴𝑇2𝑇𝑡2𝑥_𝑛𝑘−𝑥₀⟶0(33) as 𝑛_𝑘→∞. Since 𝑇(𝑡/2)𝑥∈𝐷(𝐴)⊂𝑋_𝛼,𝑝, the assertion follows.
(iii) This follows immediately from the proof of (ii).

In the sequel, we assume that 𝛼∈(0,1) and 𝑝∈(1,∞] satisfy 1−𝛼−1/𝑝≥0. To see the existence of mild solution of nonlocal problem (2), we define the mapping Φ on 𝐸 by (Φ𝑢)(𝑡)∶=𝑇(𝑡)𝑢₀−𝑔(𝑢)+^𝑡₀𝑇(𝑠)[],𝑡−𝑠)(𝐹𝑢)(𝑑𝑠,𝑡∈0,𝑇(34) and for each 𝑘∈ℕ, let 𝐵_𝑘=𝑢∈𝐸∶|𝑢(𝑡)|_𝛼,𝑝≤𝑘,𝑡∈[0,𝑇].(35) We have the following results.

Lemma 4. Assume that the conditions (H1)–(H3) are satisfied. If 𝑢₀∈𝑋_𝛼,𝑝 and, in addition,(H4)𝑀𝜉+𝐶𝛾<1(36) also holds, then Φ𝐵_𝑘⊂𝐵_𝑘 for some 𝑘∈ℕ.

Proof. If this were not the case, then for each 𝑘∈ℕ, there would exist 𝑢_𝑘∈𝐵_𝑘 and 𝑡_𝑘∈[0,𝑇] such that |(Φ𝑢_𝑘)(𝑡_𝑘)|_𝛼,𝑝>𝑘 and hence we see by (14), (28), (H2), (H3), and (A6) that 𝑘<Φ𝑢_𝑘𝑡_{𝑘𝛼,𝑝}≤𝑇(𝑡)𝑢₀−𝑔𝑢_{𝑘𝛼,𝑝}+^𝑡𝑘₀𝑇𝑡_𝑘−𝑠𝐹𝑢_𝑘(𝑠)_𝛼,𝑝≤‖‖𝑑𝑠𝑇(𝑡)_{𝐿(𝑋𝛼,𝑝)}𝑢₀−𝑔𝑢_{𝑘𝛼,𝑝}+^𝑡𝑘₀𝑇(𝑡_𝑘−𝑠)_{𝐿(𝑋,𝑋𝛼,𝑝)}𝐹𝑢_𝑘(𝑠𝑢)𝑑𝑠≤𝑀_0𝛼,𝑝+𝐿𝑢_𝑘𝐸+1^𝛽+𝐶^𝑡𝑘₀ℎ_𝑘(𝑠)𝑡_𝑘−𝑠^𝛼𝑑𝑠.(37) Dividing on both sides by 𝑘 and taking the lower limit as ‖𝑢_𝑘‖_𝐸→∞ we obtain 𝑀𝜉+𝐶𝛾≥1(38) which contradicts (H4). Hence Φ𝐵_𝑘⊂𝐵_𝑘 for some 𝑘∈ℕ.

The following lemma is useful in the proof of our main result in this section.

Lemma 5. Assume that the conditions (H1)–(H4) are satisfied. If 𝑢₀∈𝑋_𝛼,𝑝 and, in addition, (H5) there exists a 𝛿∈(0,𝑇), such that 𝐹(𝑢)=𝐹(𝑣), 𝑔(𝑢)=𝑔(𝑣) for any 𝑢,𝑣∈𝐵_𝑘 with 𝑢(𝑠)=𝑣(𝑠), 𝑠∈[𝛿,𝑇],
then the nonlocal problem (2) has at least one mild solution in 𝐵_𝑘.

Proof. Let 𝐵_𝑘(𝛿)∶={𝑢∈𝐶([𝛿,𝑇],𝑋_𝛼,𝑝)∶|𝑢(𝑡)|_𝛼,𝑝≤𝑘,forall𝑡∈[𝛿,𝑇]} where 𝛿 is the number defined in (H5). For 𝑢∈𝐵_𝑘(𝛿), find 𝑣∈𝐵_𝑘 such that 𝑢(𝑡)=𝑣(𝑡) for all 𝑡∈[𝛿,𝑇] and define 𝔉∶𝐵_𝑘(𝛿)→𝐿([0,𝑇],𝑋) and 𝔤∶𝐵_𝑘(𝛿)→𝑋_𝛼,𝑝 by (𝔉𝑢)(𝑡)=(𝐹𝑣)(𝑡) for 𝑡∈[0,𝑇] and 𝔤(𝑢)=𝑔(𝑣), respectively. By (H2), (H3), and (H5), we see that 𝔉 and 𝔤 are well-defined, continuous and satisfy ‖(𝔉𝑢)(𝑡)‖≤ℎ_𝑘(𝑡),𝑎.𝑒.𝑡∈[0,𝑇],∀𝑢∈𝐵_𝑘(𝛿),sup_{𝑢∈𝐵𝑘(𝛿)}‖𝔤(𝑢)‖=sup_{𝑣∈𝐵𝑘}𝑔(𝑣)_𝛼,𝑝≤𝐿(𝑘+1)^𝛽<∞.(39) Now let us define a mapping Φ_𝛿 on 𝐵_𝑘(𝛿) as follows: Φ_𝛿𝑢(𝑡)∶=𝑇(𝑡)𝑢₀−𝔤(𝑢)+^𝑡₀𝑇(𝑠)[].𝑡−𝑠)(𝔉𝑢)(𝑑𝑠,𝑡∈𝛿,𝑇(40) We will show that Φ_𝛿 has a fixed point in 𝐵_𝑘(𝛿). To see this, note first that Φ_𝛿 is continuous by the continuity of 𝔉 and 𝔤. Then, in view of Schauder’s fixed point theorem, it suffices to show that the set {Φ_𝛿𝑢∶𝑢∈𝐵_𝑘(𝛿)} is relatively compact in 𝐶([𝛿,𝑇],𝑋_𝛼,𝑝). Thus, we will prove that for each 𝑡∈[𝛿,𝑇], both sets 𝑇_𝛼,𝑝(𝑡)𝑢₀−𝔤(𝑢);𝑢∈𝐵_𝑘(𝛿,𝑉(𝑡))∶=^𝑡₀𝑇(𝑡−𝑠)(𝔉𝑢)(𝑠)𝑑𝑠;𝑢∈𝐵_𝑘(𝛿)(41) are relatively compact in 𝑋_𝛼,𝑝, and that both 𝑇_𝛼,𝑝(⋅)𝑢₀−𝔤(𝑢)∶𝑢∈𝐵_𝑘(𝛿)_⋅0,𝑇_𝛼,𝑝(⋅−𝑠)(𝔉𝑢)(𝑠)𝑑𝑠∶𝑢∈𝐵_𝑘(𝛿)(42) are equicontinuous families of functions on [𝛿,𝑇].
In fact, from the compactness of 𝑇_𝛼,𝑝(𝑡) for 𝑡∈[𝛿,𝑇] it follows that {𝑇_𝛼,𝑝(𝑡)(𝑢₀−𝔤(𝑢));𝑢∈𝐵_𝑘(𝛿)} is relatively compact in 𝑋_𝛼,𝑝 for 𝑡∈[𝛿,𝑇]. Moreover, for each 𝑡∈[𝛿,𝑇], 𝜀∈(0,𝑡), and 𝑢∈𝐵_𝑘(𝛿), consider the mapping defined by 𝑃_𝜀𝑢(𝑡)=𝑇_𝛼,𝑝(𝜀)₀^𝑡−𝜀𝑇_𝛼,𝑝(𝑡−𝜀−𝑠)(𝔉𝑢)(𝑠)𝑑𝑠.(43) Since 𝑉_𝜀(𝑡)={(𝑃_𝜀𝑢)(𝑡);𝑢∈𝐵_𝑘(𝛿)} is relatively compact in 𝑋_𝛼,𝑝 by the compactness of 𝑇_𝛼,𝑝(𝜀) and since ^𝑡₀𝑇(𝑡−𝑠)(𝔉𝑢)(𝑠)𝑑𝑠−(𝑃_𝜀𝑢)(𝑡)_𝛼,𝑝≤^𝑡_𝑡−𝜀‖𝑇(𝑡−𝑠)‖_{𝐿(𝑋,𝑋𝛼,𝑝)}‖(𝔉𝑢)(𝑠)‖𝑑𝑠≤𝐶^𝑡_𝑡−𝜀ℎ_𝑘(𝑠)(𝑡−𝑠)^𝛼𝑑𝑠,(44) then we see by (H2) that there are relative compact sets arbitrarily close to 𝑉(𝑡) and hence 𝑉(𝑡) is also relatively compact in 𝑋_𝛼,𝑝. Therefore, the set {(Φ_𝛿𝑢)(𝑡)∶𝑢∈𝐵_𝑘(𝛿)} is relatively compact in 𝑋_𝛼,𝑝 for each 𝑡∈[𝛿,𝑇].
Now, by the norm continuity of 𝑇_𝛼,𝑝(𝑡) for 𝑡>0, we see that 𝑇_𝛼,𝑝(𝑡+ℎ)−𝑇_𝛼,𝑝(𝑡)(𝑢₀−𝔤(𝑢))_𝛼,𝑝⟶0,asℎ⟶0(45) independently of 𝑢∈𝐵_𝑘, and hence {𝑇_𝛼,𝑝(⋅)(𝑢₀−𝔤(𝑢))∶𝑢∈𝐵_𝑘(𝛿)} is an equicontinuous family of functions on [𝛿,𝑇]. Finally, let 𝜖>0 be arbitrarily small and we see by (A6) with 𝑛=0 that ₀^𝑡+ℎ𝑇(𝑡+ℎ−𝑠)(𝔉𝑢)(𝑠)𝑑𝑠−^𝑡₀𝑇(𝑡−𝑠)(𝔉𝑢)(𝑠)𝑑𝑠_𝛼,𝑝≤₀^𝑡−𝜖|[𝑇(𝑡+ℎ−𝑠)−𝑇(𝑡−𝑠)](𝔉𝑢)(𝑠)|_𝛼,𝑝𝑑𝑠+^𝑡_𝑡−𝜖|[𝑇(𝑡+ℎ−𝑠)−𝑇(𝑡−𝑠)](𝔉𝑢)(𝑠)|_𝛼,𝑝+𝑑𝑠_𝑡𝑡+ℎ|𝑇(𝑡+ℎ−𝑠)(𝔉𝑢)(𝑠)|_𝛼,𝑝≤𝑑𝑠_0𝑡−𝜖𝑇𝑡−𝑠2+ℎ−𝑇𝑡−𝑠2_{𝐿(𝑋𝛼,𝑝)}×𝑇𝑡−𝑠2_{𝐿(𝑋,𝑋𝛼,𝑝)}𝑠+‖(𝔉𝑢)()‖𝑑𝑠_{𝑡𝑡−𝜖}‖𝑇(𝑡+ℎ−𝑠)−𝑇(𝑡−𝑠)‖_{𝐿(𝑋,𝑋𝛼,𝑝)}𝑠+‖(𝔉𝑢)()‖𝑑𝑠_𝑡𝑡+ℎ‖𝑇(𝑡+ℎ−𝑠)‖_{𝐿(𝑋,𝑋𝛼,𝑝)}‖(𝔉𝑢)(𝑠)‖𝑑𝑠≤2^𝛼𝐶₀^𝑡−𝜖𝑇𝑡−𝑠2+ℎ−𝑇𝑡−𝑠2_{𝐿(𝑋𝛼,𝑝)}ℎ_𝑘(𝑠)(𝑡−𝑠)^𝛼𝑑𝑠+𝐶^𝑡_𝑡−𝜖ℎ_𝑘(𝑠)(𝑡+ℎ−𝑠)^𝛼𝑑𝑠+𝐶^𝑡_𝑡−𝜖ℎ_𝑘(𝑠)(𝑡−𝑠)^𝛼𝑑𝑠+𝐶_𝑡^𝑡+ℎℎ_𝑘(𝑠)(𝑡+ℎ−𝑠)^𝛼𝑑𝑠≤2^𝛼𝐶₀^𝑡−𝜖𝑇𝑡−𝑠2+ℎ−𝑇𝑡−𝑠2_{𝐿(𝑋𝛼,𝑝)}ℎ_𝑘(𝑠)(𝑡−𝑠)^𝛼𝑑𝑠+𝐶^𝑡+ℎ_𝑡−𝜖ℎ_𝑘(𝑠)(𝑡+ℎ−𝑠)^𝛼𝑑𝑠−^𝑡_𝑡−𝜖ℎ_𝑘(𝑠)(𝑡−𝑠)^𝛼𝑑𝑠+2^𝑡_𝑡−𝜖ℎ_𝑘(𝑠)(𝑡−𝑠)^𝛼𝑑𝑠≤2^𝛼𝐶₀^𝑡−𝜖𝑇𝑡−𝑠2+ℎ−𝑇𝑡−𝑠2_{𝐿(𝑋𝛼,𝑝)}ℎ_𝑘(𝑠)(𝑡−𝑠)^𝛼𝜃𝑑𝑠+𝐶_−𝛼∗ℎ_𝑘(𝑡+ℎ)−𝜃_−𝛼∗ℎ_𝑘(𝑡+_0𝑡−𝜀)ℎ_𝑘(𝑠)(𝑡−𝑠)^𝛼−ℎ_𝑘(𝑠)(𝑡+ℎ−𝑠)^𝛼𝑑𝑠+2^𝑡_𝑡−𝜖ℎ_𝑘(𝑠)(𝑡−𝑠)^𝛼𝑑𝑠(46) which is, by the norm continuity of 𝑇_𝛼,𝑝(𝑡) for 𝑡>0 and dominated convergence theorem, arbitrarily small independently of 𝑢∈𝐵_𝑘 as ℎ→0. Therefore, ^⋅₀𝑇_𝛼,𝑝(⋅−𝑠)(𝔉𝑢)(𝑠)𝑑𝑠∶𝑢∈𝐵_𝑘(𝛿)(47) is an equicontinuous family of functions on [𝛿,𝑇].
Consequently, by Arzela-Ascoli’s theorem, it follows that {Φ_𝛿𝑢∶𝑢∈𝐵_𝑘(𝛿)} is relatively compact in 𝐸_𝛿. Thus there is a function 𝜑∈𝐵_𝑘(𝛿) such that Φ_𝛿𝜑=𝜑; that is, 𝜑(⋅)=𝑇(⋅)𝑢₀−𝔤𝜑+^⋅₀𝑇(⋅−𝑠)𝔉𝜑(𝑠)𝑑𝑠∈𝐵_𝑘(𝛿),(48) or 𝜑(𝑡)=𝑇(𝑡)𝑢₀−𝔤𝜑+^𝑡₀𝑇()𝑠)[].𝑡−𝑠𝔉𝜑(𝑑𝑠,∀𝑡∈𝛿,𝑇(49)
Now, define a function 𝜓 on [0,𝑇] by 𝜓(𝑡)=𝑇(𝑡)𝑢₀−𝔤𝜑+^𝑡₀𝑇()𝑠)[].𝑡−𝑠𝔉𝜑(𝑑𝑠,∀𝑡∈0,𝑇(50) Then 𝜓=𝜑 on [𝛿,𝑇] and we have 𝜓∈𝐵_𝑘. By the definition of 𝔉, 𝔤, and (H5), we see that 𝔉𝜑=𝐹𝜑=𝐹𝜓, 𝔤(𝜑)=𝑔(𝜑)=𝑔(𝜓) and consequently, 𝜓(𝑡)=𝑇(𝑡)𝑢₀−𝑔𝜓+^𝑡₀𝑇()𝑠)[];𝑡−𝑠𝐹𝜓(𝑑𝑠,∀𝑡∈0,𝑇(51) that is, 𝜓 is a mild solution of (2), and the proof is completed.

For our main result in this section, we introduce in the following a family of nonlocal Cauchy problems. Firstly, we define, for each 𝛿∈(0,𝑇), an operator ℬ_𝛿 on 𝐸 by ℬ_𝛿𝑢𝑡)𝑢(𝛿),𝑢(𝑡),(∶=0≤𝑡≤𝛿,𝛿≤𝑡≤𝑇,∀𝑢∈𝐸.(52) It is clear that ℬ_𝛿 is bounded on 𝐸 and ‖ℬ_𝛿‖_𝐿(𝐸)≤1, and hence ℬ_𝛿𝐵_𝑘⊂𝐵_𝑘. Now, for each 𝛿∈[0,𝑇], we define 𝑔_𝛿∶𝐸→𝑋_𝛼,𝑝 by 𝑔_𝛿(𝑢)=𝑔ℬ_𝛿𝑢,𝑢∈𝐸,(53) and 𝐹_𝛿∶𝐸→𝐿([0,𝑇],𝑋) by 𝐹_𝛿𝑢=𝐹ℬ_𝛿𝑢,𝑢∈𝐸,(54) and consider the nonlocal Cauchy problems: 𝑢^′(𝑡)=𝐴𝑢(𝑡)+𝐹_𝛿𝑢𝑡)[],𝑢(0)(,𝑡∈0,𝑇+𝑔_𝛿(𝑢)=𝑢₀∈𝑋_𝛼,𝑝.NCP_𝛿 In view of (52), (53), and (54), the following result is an immediate corollary of Lemma 5.

Lemma 6. Suppose that (H1)–(H4) are satisfied. Then for any 𝛿∈(0,𝑇], the problem NCP_𝛿 has at least one mild solution in 𝐵_𝑘.

Theorem 7. Suppose that the hypotheses (H1)–(H4) are satisfied. If 𝑢₀∈𝑋_𝛼,𝑝, then the problem (2) has at least one mild solution in 𝐵_𝑘 for some 𝑘∈ℕ.

Proof. Choose a decreasing sequence {𝛿_𝑛}_𝑛∈ℕ⊂(0,𝑇) so that lim_𝑛→∞𝛿_𝑛=0 and then by Lemma 6, we see that for each 𝑛∈ℕ, 𝑢_𝑛(𝑡)=𝑇(𝑡)𝑢₀−𝑔_𝛿𝑛𝑢_𝑛+^𝑡₀𝑇(𝑡−𝑠)𝐹_𝛿𝑛𝑢_𝑛𝑠)[],(𝑑𝑠,𝑡∈0,𝑇(55) or 𝑢_𝑛=𝑇(⋅)𝑢₀−𝑔ℬ_𝛿𝑛𝑢_𝑛+^⋅₀𝑇(⋅−𝑠)𝐹ℬ_𝛿𝑛𝑢_𝑛(𝑠)𝑑𝑠∈𝐵_𝑘(56) is a mild solution for the nonlocal Cauchy problem (NCP_𝛿𝑛).
Since the sequence {ℬ_𝛿𝑛𝑢_𝑛}_𝑛∈ℕ lies in 𝐵_𝑘, then a similar argument as in the proof of Lemma 5 (see (44)–(46) shows that [],𝑢forany𝑡∈0,𝑇theset_𝑛(𝑡)∶𝑛∈ℕ,𝑡∈[0,𝑇]⊆𝑋_𝛼,𝑝,(57) that foreach𝑡∈[0,𝑇],theset_𝑡0𝑇(𝑡−𝑠)𝐹(ℬ_𝛿𝑛𝑢_𝑛)(𝑠)𝑑𝑠_𝑛∈ℕisrelativelycompactin𝑋_𝛼,𝑝,(58) and that ^⋅₀𝑇(⋅−𝑠)𝐹(ℬ_𝛿𝑛𝑢_𝑛)(𝑠)𝑑𝑠_𝑛∈ℕisanequicontinuoussequenceoffunctionson[0,𝑇].(59) Hence it follows by Ascoli-Arzela theorem that ^⋅₀𝑇(⋅−𝑠)𝐹(ℬ_𝛿𝑛𝑢_𝑛)(𝑠)𝑑𝑠_𝑛∈ℕisrelativelycompactin𝐸.(60) Now, let {𝜖_𝑛}_𝑛∈ℕ⊂(0,𝑇) be a decreasing sequence such that lim_𝑛→∞𝜖_𝑛=0 and let {𝑢_𝑛0}_𝑛0∈ℕ be a subsequence of {𝑢_𝑛}_𝑛∈ℕ. Then, a similar argument as in the proof of Lemma 5 insures that {𝑇_𝛼,𝑝(⋅)(𝑢₀−𝑔(ℬ_𝛿𝑛0𝑢_𝑛0))}_𝑛0∈ℕ is an equicontinuous sequence of functions on [𝜖₁,𝑇]. Thus, Ascoli-Arzela theorem guarantees that the sequence 𝑇_𝛼,𝑝(⋅)(𝑢₀−𝑔(ℬ_𝛿𝑛0𝑢_𝑛0))_𝑛0∈ℕisrelativelycompactin𝐶𝜖₁,𝑇,𝑋_𝛼,𝑝.(61) Thus by (60) and (61), we see that {𝑢_𝑛0}_𝑛0∈ℕ is relatively compact in 𝐶([𝜖₁,𝑇],𝑋_𝛼,𝑝), and hence we can select a subsequence of {𝑢_𝑛0}_𝑛0∈ℕ, denoted by {𝑢_𝑛1}_𝑛1∈ℕ which is a Cauchy sequence in 𝐶([𝜖₁,𝑇],𝑋_𝛼,𝑝). By a similar process, we can select a subsequence of {𝑢_𝑛1}_𝑛1∈ℕ, denoted by {𝑢_𝑛2}_𝑛2∈ℕ which is a Cauchy sequence in 𝐶([𝜖₂,𝑇],𝑋_𝛼,𝑝). Repeat the above argument and use a diagonal argument to obtain a subsequence of {𝑢_𝑛0}_𝑛0∈ℕ, denoted by {𝑣_𝑛}_𝑛∈ℕ. Then for every 𝑡∈(0,𝑇], {𝑣_𝑛(𝑡)}_𝑛∈ℕ is a Cauchy sequence in 𝑋_𝛼,𝑝 and so, we can define the function 𝑣_∞ by 𝑣_∞(𝑡)=0,𝑡=0;lim_𝑛→∞𝑣_𝑛(𝑡),0<𝑡≤𝑇.(62) It is clear that 𝑣_∞ is strongly measurable, 𝑣_∞∈𝐵_𝑘 and ^𝑇₀𝑣_𝑛(𝑡)−𝑣_∞(𝑡)_𝛼,𝑝𝑑𝑡≤2𝑇𝑘.(63) It therefore follows by Lebesgue’s dominated convergence theorem that there is a subsequence {𝜏_𝑛}_𝑛∈ℕ of {𝛿_𝑛}_𝑛∈ℕ such that ^𝑇₀ℬ_𝜏𝑛𝑣_𝑛(𝑡)−𝑣_∞(𝑡)_𝛼,𝑝≤𝑑𝑡_𝑇0ℬ_𝜏𝑛𝑣_𝑛(𝑡)−𝑣_𝑛(𝑡)_𝛼,𝑝+𝑑𝑡_𝑇0𝑣_𝑛(𝑡)−𝑣_∞(𝑡)_𝛼,𝑝𝑑𝑡⟶0(64) as 𝑛→∞. This shows that the sequence 𝑇(⋅)(𝑢₀−𝑔(ℬ_𝛿𝑛𝑢_𝑛))_𝑛∈ℕisrelativelycompactin𝐸.(65)
By (60) and (65), we see the relative compactness of {𝑢_𝑛}_𝑛∈ℕ in 𝐸. Thus, there is a subsequence of {𝑢_𝑛}_𝑛∈ℕ, denoted by {𝑢_𝜈}_𝜈∈ℕ and a function 𝑢_∞∈𝐸 such that lim_𝜈→∞𝑢_𝜈−𝑢_∞𝐸=0.(66) It is clear that 𝑢_∞∈𝐵_𝑘. Since ℬ_𝛿𝜈𝑢_𝜈−𝑢_∞𝐸=sup_{0≤𝑡≤𝑇}ℬ_𝛿𝜈𝑢_𝜈(𝑡)−𝑢_∞(𝑡)_𝛼,𝑝≤sup_{0≤𝑡≤𝛿𝜈}𝑢_𝜈𝛿_𝜈−𝑢_∞(𝑡)_𝛼,𝑝+sup_{𝛿𝜈≤𝑡≤𝑇}𝑢_𝜈(𝑡)−𝑢_∞(𝑡)_𝛼,𝑝≤𝑢_𝜈𝛿_𝜈−𝑢_∞𝛿_{𝜈𝛼,𝑝}+sup_{0≤𝑡≤𝛿𝜈}𝑢_∞𝛿_𝜈−𝑢_∞(𝑡)_𝛼,𝑝+𝑢_𝜈−𝑢_∞𝐸≤2𝑢_𝜈−𝑢_∞𝐸+sup_{0≤𝑡≤𝛿𝜈}𝑢_∞𝛿_𝜈−𝑢_∞(𝑡)_𝛼,𝑝,(67) then (66) and the uniform continuity of 𝑢_∞ imply that lim_𝜈→∞‖ℬ_𝛿𝜈𝑢_𝜈−𝑢_∞‖_𝐸=0. By taking limits in (56), we see that 𝑢_∞ is a mild solution of (2) and this completes the proof.

We will consider a case more generally; that is, the nonlocal condition 𝑔 is defined on 𝐸 rather than 𝐿¹([0,𝑇],𝑋).

Theorem 8. Suppose that the hypotheses (H1) and (H2) are satisfied and, in addition, that there hold the following conditions. (H6) The function 𝑔∶𝐸→𝑋_𝛼,𝑝 is continuous and there exist constants 𝐿>0 and 𝛽∈[0,1] such that 𝑔(𝑢)_𝛼,𝑝≤𝐿‖𝑢‖_𝐸+1^𝛽(68) for 𝑢∈𝐸. Moreover, the real number 𝜉 is defined by 𝜉∶=lim_{‖𝑢‖𝐸→∞}𝐿‖𝑢‖_𝐸+1^𝛽‖𝑢‖_𝐸.(69) (H7)lim_𝜖→0sup_{𝜙∈𝐵𝑘}‖𝑔(𝜙)−𝑔(𝜙^𝜖)‖_𝐸=0, where 𝜙^𝜖(𝑡)=^{𝜙(𝜖),0≤𝑡≤𝜖,}_{𝜙(𝑡),𝜖≤𝑡≤𝑇.}
If the inequality (H4) holds and 𝑢₀∈𝑋_𝛼,𝑝, then the problem (2) has at least one mild solution in 𝐵_𝑘 for some 𝑘∈ℕ.

Proof. Let {𝑢_𝑛}_𝑛∈ℕ be the sequence defined as in the proof of Theorem 7. By the same arguments as in the proofs of Lemma 4 and Theorem 7, we see that Φ𝐵_𝑘⊂𝐵_𝑘 for some 𝑘∈ℕ, (60) and (61) also hold and for every subsequence {𝑢_𝑛0}_𝑛0∈ℕ of {𝑢_𝑛}_𝑛∈ℕ, and there exist a subsequence {𝑣_𝑛}^∞_𝑛=1 and a function 𝑣_∞∶(0,𝑇]→𝑋_𝛼,𝑝 such that 𝑣_∞ is continuous on (0,𝑇] and for every 𝜖_𝑘lim_𝑛→∞max_{𝑡∈[𝜖𝑘,𝑇]}𝑣_𝑛(𝑡)−𝑣_∞(𝑡)_𝛼,𝑝=0,orlim_𝑛→∞ℬ_𝜖𝑘[𝑣_𝑛−𝑣_∞]_𝐸=0.(70) Let 𝜀>0 be given. It follows by (H7) and (70) that there is a 𝛿>0 such that 𝑔𝜙−𝑔𝜓_𝛼,𝑝<𝜀4whenever𝜙=𝜓on[𝛿,𝑇](71) and that for every 𝑘∈ℕ there is an 𝑁_𝑘 such that 𝑛>𝑁_𝑘 implies that 𝑔ℬ_𝜖𝑘𝑣_𝑛−𝑣_∞𝛼,𝑝<𝜀4.(72) Choose 𝐾 large enough so that 𝜖_𝐾<𝛿 and define 𝜑∶[0,𝑇]→𝑋_𝛼,𝑝 by 𝜑(𝑡)=𝑣_∞𝜖_𝐾,0≤𝑡≤𝜖_𝐾,𝑣_∞(𝑡),𝜖_𝐾≤𝑡≤𝑇.(73) Thus, (H7), (71), and (72) insures that 𝑔𝑣_𝑛−𝑔𝑣_{𝑚𝛼,𝑝}≤𝑔𝑣_𝑛−𝑔ℬ_𝜖𝐾𝑣_{𝑛𝛼,𝑝}+𝑔ℬ_𝜖𝐾𝑣_𝑛−𝑔𝜑_𝛼,𝑝+𝑔𝜑−𝑔𝐵_𝜖𝐾𝑣_{𝑚𝛼,𝑝}+𝑔ℬ_𝜖𝐾𝑣_𝑚−𝑔𝑣_{𝑚𝛼,𝑝}<𝜀.(74) And hence by the continuity of 𝑔 and compactness of 𝑇_𝛼,𝑝(𝑡) for 𝑡>0, (65) is also valid in this case. Therefore, the same argument as in the last paragraph of the proof of Theorem 7 shows the existence of mild solution for (2).

Now, we consider the nonlocal Cauchy problem (1). For this, we introduce the following hypothesis.(H8) The function 𝑓∶[0,𝑇]×𝑋_𝛼,𝑝→𝑋 satisfies the Carathéodory condition; that is, 𝑓(⋅,𝑥) is measurable for each 𝑥∈𝑋 and 𝑓(𝑡,⋅) is continuous for any 𝑡∈[0,𝑇]. Moreover, for each 𝑘∈ℕ, there exists a function ℎ_𝑘∶[0,𝑇]→ℝ⁺ such that (1)𝑠↦ℎ_𝑘(𝑠)/(𝑡−𝑠)^𝛼 is integrable on [0,𝑡] for each 𝑡∈[0,𝑇],(2)sup_{‖𝑢‖𝐸<𝑘}‖𝐹(𝑡,𝑢(𝑡))‖≤ℎ_𝑘(𝑡) for almost all 𝑡∈[0,𝑇],(3)liminf_𝑘→∞(1/𝑘)(sup_{𝑡∈[0,𝑇]}∫^𝑡₀(ℎ_𝑘(𝑠)/(𝑡−𝑠)^𝛼)𝑑𝑠)=𝛾,(4)lim_𝜀→0(sup_{𝑡∈[0,𝑇]}∫^𝑡_𝑡−𝜀(ℎ_𝑘(𝑠)/(𝑡−𝑠)^𝛼)𝑑𝑠)=0. The following existence results are immediate corollaries of Theorems 7 and 8.

Theorem 9. (i) Suppose that (H1), (H3), and (H8) are satisfied. If the inequality (H4) also holds and 𝑢₀∈𝑋_𝛼,𝑝, then (1) has at least one mild solution in 𝐵_𝑘 for some 𝑘∈ℕ.
(ii) Suppose that (H1) and (H6)–(H8) are satisfied. If the inequality (H4) also holds and 𝑢₀∈𝑋_𝛼,𝑝, then (1) has at least one mild solution in 𝐵_𝑘 for some 𝑘∈ℕ.

Proof. By letting (𝐹𝑢)(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈[0,𝑇](75) one sees that (H8) implies (H2), and hence we obtain the existence results of (1) by Theorems 7 and 8.

4. The Existence of Strong Solutions

Definition 10. A mild solution 𝑢 of (1) is called a strong solution if 𝑢 is continuously differentiable on (0,𝑇] and satisfies (1). The strong solution of (2) is defined in a similar way.

In the following, we establish existence of strong solutions for (2).

Theorem 11. (i) Suppose that the hypotheses (H1) and (H3) are satisfied, and in addition, that there holds the following conditions.(H9) The function 𝐹∶𝐸→𝐿¹([0,𝑇],𝑋) is continuous and for each 𝑘∈ℕ, there exists a ℎ_𝑘∶[0,𝑇]→ℝ⁺ such that (1)ℎ_𝑘 is locally bounded function on (0,𝑇], (2)𝜃_−𝛼∗ℎ_𝑘∈𝐿¹([0,𝑇],ℝ⁺) is locally Hölder continuous on (0,𝑇], (3)sup_{‖𝑢‖𝐸<𝑘}‖(𝐹𝑢)(𝑡)‖≤ℎ_𝑘(𝑡) for almost all 𝑡∈[0,𝑇], (4)liminf_𝑘→∞(1/𝑘)(sup_{𝑡∈[0,𝑇]}∫^𝑡₀(ℎ_𝑘(𝑠)/(𝑡−𝑠)^𝛼)𝑑𝑠)=𝛾,(5)lim_𝜀→0(sup_{𝑡∈[0,𝑇]}∫^𝑡_𝑡−𝜀(ℎ_𝑘(𝑠)/(𝑡−𝑠)^𝛼)𝑑𝑠)=0. (H10) If 𝑢∈𝐸 is locally Hölder continuous on (0,𝑇], then 𝐹𝑢 is locally Hölder continuous on (0,𝑇].
If the inequality (H4) also holds and 𝑢₀∈𝑋_𝛼,𝑝, then (2) has a strong solution on [0,𝑇].
(ii) Suppose that the hypotheses (H1), (H6), (H7), (H9), and (H10) are satisfied. If the inequality (H4) also holds and 𝑢₀∈𝑋_𝛼,𝑝, then (2) has a strong solution on [0,𝑇].

Proof. It is clear that (H9) implies (H2). Thus, the conditions (H1)–(H4) hold by assumption and hence Theorem 7 shows that if 𝑢₀∈𝑋_𝛼,𝑝, then (2) has a mild solution on [0,𝑇]; that is, there is a 𝑢∈𝐸 such that 𝑢(𝑡)∶=𝑇(𝑡)𝑢₀−𝑔(𝑢)+^𝑡₀𝑇(𝑡−𝑠)(𝐹𝑢)(𝑠)𝑑𝑠(76) for 𝑡∈[0,𝑇]. We will show that this mild solution is a strong solution. In view of [16] (p.113, Corollary 4.3.3), it suffices to prove that 𝐹𝑢 is locally Hölder continuous on (0,𝑇].
From (76), for 0<𝑡₁<𝑡₂≤𝑇, one derives the following: 𝑢𝑡₂−𝑢𝑡_1𝛼,𝑝≤𝑇𝑡₂−𝑇𝑡₁𝑢₀−𝑔(𝑢)_𝛼,𝑝+^𝑡1₀𝑇𝑡₂−𝑠−𝑇𝑡₁−𝑠(𝐹𝑢)(𝑠)𝑑𝑠_𝛼,𝑝+^𝑡2_𝑡1𝑇(𝑡₂−𝑠)(𝐹𝑢)(𝑠)𝑑𝑠_𝛼,𝑝.(77) By (18), (A5), and (A6) we have 𝑇𝑡₂−𝑇𝑡₁𝑢₀−𝑔(𝑢)_𝛼,𝑝=^𝑡2_𝑡1𝐴𝑇(𝑠)(𝑢₀−𝑔(𝑢))𝑑𝑠_𝛼,𝑝≤^{𝑡2−𝑡1}₀𝐴𝑇(𝑡₁+𝑠)(𝑢₀−𝑔(𝑢))_𝛼,𝑝≤𝑑𝑠_{𝑡2−𝑡10}‖𝑇(𝑠)‖_{𝐿(𝑋,𝑋𝛼,𝑝)}𝐴𝑇𝑡₁𝑢₀(𝑢≤−𝑔)𝑑𝑠_{𝑡2−𝑡10}𝐶𝑠^−𝛼𝑡₁⁻¹𝐶₁𝑢₀(𝑢=−𝑔)𝑑𝑠𝐶𝐶₁𝑡₁()𝑢1−𝛼⁰−𝑔(𝑢)𝑡₂−𝑡₁^1−𝛼,(78) that ^𝑡1₀𝑇𝑡₂−𝑠−𝑇𝑡₁−𝑠(𝐹𝑢)(𝑠)𝑑𝑠_𝛼,𝑝≤^𝑡1₀^{𝑡2−𝑡1}₀𝐴𝑇𝜏+𝑡₁−𝑠(𝐹𝑢)(𝑠)_𝛼,𝑝≤𝑑𝜏𝑑𝑠_{𝑡10𝑡2−𝑡10}𝐴𝑇𝜏+𝑡₁−𝑠_{𝐿(𝑋,𝑋𝛼,𝑝)}‖(𝐹𝑢)(𝑠)‖𝑑𝜏𝑑𝑠≤𝐶^𝑡1₀^{𝑡2−𝑡1}₀ℎ_𝑘(𝑠)𝜏+𝑡₁−𝑠^1+𝛼𝐶𝑑𝜏𝑑𝑠≤−𝛼^𝑡1₀1𝜏+𝑡₁−𝑠^{𝛼𝑡2−𝑡1}₀ℎ_𝑘(𝑠)≤𝐶𝑑𝑠𝛼^𝑡1₀1𝑡₁−𝑠^𝛼−1𝑡₂−𝑠^𝛼ℎ_𝑘(𝑠)𝑑𝑠,(79) where we use (A6) with 𝑛=1 and that ^𝑡2_𝑡1𝑇(𝑡₂−𝑠)(𝐹𝑢)(𝑠)𝑑𝑠_𝛼,𝑝≤^𝑡2_𝑡1𝑇(𝑡₂−𝑠)(𝐹𝑢)(𝑠)_𝛼,𝑝≤𝑑𝑠_𝑡2𝑡1𝑇(𝑡₂−𝑠)_{𝐿(𝑋,𝑋𝛼,𝑝)}‖(𝐹𝑢)(𝑠)‖𝑑𝑠≤𝐶^𝑡2_𝑡1ℎ_𝑘(𝑠)𝑡₂−𝑠^𝛼𝑑𝑠.(80) Since ℎ_𝑘 is locally bounded and 𝜃_−𝛼∗ℎ_𝑘 is locally Hölder continuous on (0,𝑇] by (H9), then 𝐶𝛼^𝑡1₀1𝑡₁−𝑠^𝛼−1𝑡₂−𝑠^𝛼ℎ_𝑘(𝑠)𝑑𝑠+𝐶^𝑡2_𝑡1ℎ_𝑘(𝑠)𝑡₂−𝑠^𝛼=𝐶𝑑𝑠𝛼𝜃_−𝛼∗ℎ_𝑘𝑡₁−𝜃_−𝛼∗ℎ_𝑘𝑡₂1+𝐶𝛼+1^𝑡2_𝑡1ℎ_𝑘(𝑠)𝑡₂−𝑠^𝛼≤𝐶𝑑𝑠𝛼𝜃_−𝛼∗ℎ_𝑘𝑡₂−𝜃_−𝛼∗ℎ_𝑘𝑡₁1+𝐶𝛼+1^𝑡2_𝑡1𝐻_𝑘𝑡₂−𝑠^𝛼𝑑𝑠(81) for some 𝐻_𝑘>0. This shows that 𝑢 is locally Hölder continuous on (0,𝑇] and so, by (H10), 𝐹𝑢 is locally Hölder continuous on (0,𝑇]. This completes the proof of (i) and the assertion (ii) follows in a similar argument.

Now, we establish existence results for strong solutions of the nonlocal Cauchy problem (1) and for this purpose, we introduce the following hypotheses.(H11) The function 𝑓∶[0,𝑇]×𝑋_𝛼,𝑝→𝑋 satisfies the Carathéodory condition; that is, 𝑓(⋅,𝑥) is measurable for each 𝑥∈𝑋 and 𝑓(𝑡,⋅) is continuous for any 𝑡∈[0,𝑇]. Moreover, for each 𝑘∈ℕ, there exists a function ℎ_𝑘∶[0,𝑇]→ℝ⁺ such that (1)ℎ_𝑘 is locally bounded function on (0,𝑇], (2)𝜃_−𝛼∗ℎ_𝑘∈𝐿¹([0,𝑇],ℝ⁺) is locally Hölder continuous on (0,𝑇], (3)sup_{‖𝑢‖𝐸<𝑘}‖𝑓(𝑡,𝑢(𝑡))‖≤ℎ_𝑘(𝑡) for almost all 𝑡∈[0,𝑇], (4)liminf_𝑘→∞(1/𝑘)(sup_{𝑡∈[0,𝑇]}∫^𝑡₀(ℎ_𝑘(𝑠)/(𝑡−𝑠)^𝛼)𝑑𝑠)=𝛾, and (5)lim_𝜀→0(sup_{𝑡∈[0,𝑇]}∫^𝑡_𝑡−𝜀(ℎ_𝑘(𝑠)/(𝑡−𝑠)^𝛼)𝑑𝑠)=0. (H12) If 𝑢∈𝐸 is locally Hölder continuous on (0,𝑇], then 𝑓(𝑡,𝑢(𝑡)) is locally Hölder continuous on (0,𝑇].

Theorem 12. (i) Suppose that (H1), (H3), (H11), and (H12) are satisfied. If the inequality (H4) also holds and 𝑢₀∈𝑋_𝛼,𝑝, then (1) has at least one strong solution in 𝐵_𝑘 for some 𝑘∈ℕ.
(ii) Suppose that (H1), (H6), (H7), (H11), and (H12) are satisfied. If the inequality (H4) also holds and 𝑢₀∈𝑋_𝛼,𝑝, then (1) has at least one strong solution in 𝐵_𝑘 for some 𝑘∈ℕ.

Proof. By letting (𝐹𝑢)(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡∈[0,𝑇](82) one sees that (H11) implies (H2), and hence we obtain the existence results of (1) by using a similar argument as the proof of Theorem 11.

5. An Example

In this section, we consider the following system: 𝜕𝑢𝑢()=𝜕𝜕𝑡𝑡,𝑥²𝑢(𝑡,𝑥)𝜕𝑥²(),()+𝑓𝑡,𝑢𝑡,𝑥𝜕𝑢𝑡,𝑥,[],𝑢()()[],𝑢()+𝜕𝑥𝑡∈0,𝑇𝑡,0=𝑢𝑡,1=0,𝑡∈0,𝑇0,𝑥^𝑇₀𝑘₁(𝑡,𝑥)¹₀𝑘₂𝑢(),()𝑡,𝑟𝜕𝑢𝑡,𝑟𝜕𝑟𝑑𝑟𝑑𝑡=𝜙₀(𝑥),𝑥∈[0,1],(83) where 𝑢∶[0,𝑇]×[0,1]→ℝ, 𝑓∶[0,𝑇]×ℝ×ℝ→ℝ, 𝑘₁∶[0,𝑇]×[0,1]→ℝ, and 𝑘₂∶ℝ×ℝ→ℝ. This system will be solved by transforming it into (2). To do this, let 𝑋=𝐶([0,1],ℝ) and let 𝐷={𝜙∶𝜙∈𝐶²([0,1],ℝ);𝜙(0)=𝜙(1)=0}. Define 𝐴∶𝐷→𝑋 by 𝐴𝑢=𝑢^′′. By [15], it is well known that 𝐴 is a sectorial operator on 𝑋 and 𝐴 generates a semigroup (𝑇(𝑡))_𝑡≥0 on 𝑋. The nonautonomous cases have been studied by Xiao and Liang [14] and they have shown there that 𝑇(𝑡) is a compact operator for each 𝑡>0. Choose 𝛼∈(1/2,1) and again by [15] (P.107, (3.1.76)) one sees that 𝑋_𝛼,∞=𝐶₀^2𝛼([0,1],ℝ). Suppose that 𝑓, 𝑘₁, and 𝑘₂ satisfy the following conditions.

(1) 𝑓∶[0,𝑇]×ℝ×ℝ→ℝ is continuous and there is a function ℎ∈𝐿^∞([0,𝑇],ℝ) such that |𝑓(𝑡,𝑥,𝑦)|≤ℎ(𝑡) for 𝑡∈[0,𝑇] and 𝑥, 𝑦∈ℝ.

(2) 𝑘₁∶[0,𝑇]×[0,1]→ℝ is continuous and for each 𝑡∈[0,𝑇], the mapping 𝑠↦𝑘₁(𝑡,𝑠) belongs to 𝑋_𝛼,∞=𝐶₀^2𝛼([0,1],ℝ).

(3) 𝑘₂∶ℝ×ℝ→ℝ is continuous and uniformly bounded. Moreover, there is a constant 𝐿>0 such that |𝑘₂(𝑥₁,𝑦₁)−𝑘₂(𝑥₂,𝑦₂)|≤𝐿(|𝑥₁−𝑥₂|+|𝑦₁−𝑦₂|) for (𝑥_𝑖,𝑦_𝑖)∈ℝ² and 𝑖=1,2.

Let 𝐸 denote the space 𝐶([0,𝑇],𝑋_𝛼,∞) equipped the supnorm ‖⋅‖_𝐸 and for each 𝑢∈𝐸 or 𝐿¹([0,𝑇],𝑋_𝛼,∞), let 𝑢(⋅,⋅)∶[0,𝑇]×[0,1]→ℝ be given by 𝑢(𝑡,𝑠)∶=(𝑢(𝑡))(𝑠), for all 𝑡∈[0,𝑇] and 𝑠∈[0,1]. Consider the following two operators.(i)𝐹∶𝐸→𝐿¹([0,𝑇],𝑋) is defined by [(𝐹𝑢)(𝑡)](𝑠)=𝑓𝑡,𝑢(𝑡,𝑠),𝐷₂𝑢(),[][],𝑡,𝑠𝑡∈0,𝑇,𝑠∈0,1(84) where 𝐷₂ denotes the partial derivative with respect to the second variable.(ii)𝑔∶𝐿¹([0,𝑇],𝑋_𝛼,∞)→𝑋_𝛼,∞ is defined by 𝑔(𝑢)(𝑠)=^𝑇₀¹₀𝑘₁(𝑡,𝑠)𝑘₂𝑢(𝑡,𝑟),𝐷₂𝑢()[].𝑡,𝑟𝑑𝑟𝑑𝑡,𝑠∈0,1(85)From assumptions (1)–(3), it is easy to see that these operators are well defined.

Now, (83) can be transformed into 𝑑𝑢(𝑡)(𝑡)+(𝐹(𝑢𝑡)[],𝑢(0)(𝑢)𝑑𝑡=𝐴𝑢))(,𝑡∈0,𝑇+𝑔=𝜙₀.(86) We will show that 𝐹 and 𝑔 satisfy the hypotheses (H2) and (H3), respectively. In fact, it follows from assumption (1) that sup_{‖𝑢‖𝐸<𝑘}‖(𝐹𝑢)(𝑡)‖≤ℎ(𝑡),liminf_𝑘→01𝑘^𝑡₀ℎ(𝑠)(𝑡−𝑠)^𝛼𝑑𝑠=𝛾=0(87) for each 𝑡∈[0,𝑇] and 𝑘∈ℕ. From assumptions (2) and (3), it is easy to see that 𝑔 is continuous and satisfies (H3) with 𝜉=0. Hence (H4) also holds. So, Theorem 7 ensures the following.

Theorem 13. If the conditions (1)–(3) hold and 𝜙₀∈𝐶₀^2𝛼([0,1],ℝ), then (86) has a mild solution.

For the existence of strong solutions to (86), the following assumption is also needed.

(4) There are constants 𝐿^′>0 and 𝜂∈(0,1] such that 𝑓𝑡₁,𝑥₁,𝑦₁−𝑓𝑡₂,𝑥₂,𝑦₂≤𝐿^′𝑡₁−𝑡₂^𝜂+𝑥₁−𝑥₂^𝜂+𝑦₁−𝑦₂^𝜂(88) for (𝑡₁,𝑥₁,𝑦₁), (𝑡₂,𝑥₂,𝑦₂)∈[0,𝑇]×ℝ×ℝ.

Next, for 𝑡₂>𝑡₁>0, we have 𝜃_−𝛼∗ℎ𝑡₂−𝜃_−𝛼∗ℎ𝑡₁=^𝑡2₀ℎ(𝑠)𝑡₂−𝑠^𝛼𝑑𝑠−^𝑡1₀ℎ(𝑠)𝑡₁−𝑠^𝛼≤𝑑𝑠_𝑡10ℎ(𝑠)𝑡₂−𝑠^𝛼𝑑𝑠−^𝑡1₀ℎ(𝑠)𝑡₁−𝑠^𝛼+𝑑𝑠_𝑡2𝑡1ℎ(𝑠)𝑡₂−𝑠^𝛼≤𝑑𝑠_𝑡101𝑡₂−𝑠^𝛼−1𝑡₁−𝑠^𝛼ℎ(𝑠)+𝑑𝑠_𝑡2𝑡1ℎ(𝑠)𝑡₂−𝑠^𝛼≤‖ℎ‖𝑑𝑠_∞𝑡101𝑡₁−𝑠^𝛼−1𝑡₂−𝑠^𝛼+‖ℎ‖𝑑𝑠_{∞𝑡2𝑡1}1𝑡₂−𝑠^𝛼=‖ℎ‖𝑑𝑠_∞−𝑡1−𝛼¹−𝑠^1−𝛼|^𝑡1₀+𝑡₂−𝑠^1−𝛼|^𝑡1₀−𝑡₂−𝑠^1−𝛼|^𝑡2_𝑡1=‖ℎ‖_∞𝑡1−𝛼^11−𝛼+𝑡₂−𝑡₁^1−𝛼−𝑡₂^1−𝛼+𝑡₂−𝑡₁^1−𝛼.(89) By Mean Value Theorem, it is easy to see that 𝑡₂^1−𝛼−𝑡₁^1−𝛼=1−𝛼𝛿^𝛼𝑡₂−𝑡₁(90) for some 𝛿∈(𝑡₁,𝑡₂). So, it easy to see that 𝜃_−𝛼∗ℎ is locally Hölder continuous on (0,𝑇]. Hence, (H9) holds. Finally, from (4), it is easy to see that (H10) holds. So from Theorem 11, we have the following.

Theorem 14. If the conditions (1)–(4) hold and 𝜙₀∈𝐶₀^2𝛼([0,1],ℝ), then (86) has a strong solution.

Acknowledgment

This research is supported in part by the National Science Council of Taiwan.