A Nonlocal Cauchy Problem for Fractional Integrodifferential Equations

Abstract

This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential equations in a separable Banach space X. We establish an existence theorem for mild solutions to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point theorem for condensing maps. As an application, the existence of the mild solution to a nonlocal Cauchy problem for a concrete integrodifferential equation is obtained.

1. Introduction

Nonlocal Cauchy problem for equations is an initial problem for the corresponding equations with nonlocal initial data. Such a Cauchy problem has better effects than the normal Cauchy problem with the classical initial data when we deal with many concrete problem coming from engineering and physics (cf., e.g., [1–10] and references therein). Therefore, the study of this type of Cauchy problem is important and significant. Actually, as we have seen from the just mentioned literature, there have been many significant developments in this field.

On the other hand, fractional differential and integrodifferential equations arise from various real processes and phenomena appeared in physics, chemical technology, materials, earthquake analysis, robots, electric fractal network, statistical mechanics biotechnology, medicine, and economics. They have in recent years been an object of investigations with much increasing interest. For more information on this subject see for instance [9, 11–18] and references therein.

Throughout this paper, 𝑋 is a separable Banach space; 𝐿(𝑋) is the Banach space of all linear bounded operators on 𝑋; 𝐴 is the generator of an analytic and uniformly bounded semigroup {𝑇(𝑡)}_𝑡≥0 on 𝑋 with ‖𝑇(𝑡)‖_𝐿(𝑋)≤𝑀 for a constant 𝑀>0, and 𝐶([𝑎,𝑏],𝑋) is the space of all 𝑋-valued continuous functions on[𝑎,𝑏] with the supremum norm as follows:‖𝑥‖_[𝑎,𝑏]∶=max{‖𝑥(𝑡)‖∶𝑡∈[𝑎,𝑏]},forany𝑥∈𝐶([𝑎,𝑏],𝑋).(1.1)

Let 0<𝑞<1, 𝑇>0, Δ={(𝑡,𝑠)∈[0,𝑇]×[0,𝑇]∶𝑡≥𝑠}, 𝑓∶[0,𝑇]×𝐶([0,𝑇],𝑋)→𝑋, and ℎ∶Δ×𝐶([0,𝑇],𝑋)→𝑋. The nonlocal Cauchy problem for abstract fractional integrodifferential equations, with which we are concerned, is in the following form:^𝑐D^𝑞𝑥(𝑡)=𝐴𝑥(𝑡)+𝑓(𝑡,𝑥𝑥(𝑡()0))+=^𝑡0𝑔𝑘(𝑥(𝑡),+𝑠)𝑥ℎ₀(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈[0,𝑇],,(1.2) where 𝑘 and 𝑔 are given functions to be specified later and the fractional derivative is understood in the Caputo sense, this means that, the fractional derivative is understood in the following sense:^𝑐𝐷^𝑞𝑥(𝑡)∶=^𝐿𝐷^𝑞(𝑥(𝑡)−𝑥(0)),𝑡>0,0<𝑞<1,(1.3) and where^𝐿𝐷^𝑞𝑥(𝑡)∶=1Γ(1−𝑞)𝑑𝑑𝑡^𝑡0(𝑡−𝑠)^−𝑞𝑥(𝑠)𝑑𝑠,𝑡>0,0<𝑞<1(1.4) is the Riemann-Liouville derivative of order 𝑞 of 𝑥(𝑡), where Γ(⋅) is the Gamma function.

Our main purpose is to establish an existence theorem for the mild solutions to the nonlocal Cauchy problem based on a special measure of noncompactness under weak assumptions on the nonlinearity 𝑓 and the semigroup {𝑇(𝑡)}_𝑡≥0 generated by 𝐴.

2. Existence Result and Proof

As usual, we abbreviate ‖𝑢‖_𝐿^𝑝_{([0,𝑇],𝐑}⁺₎ with ‖𝑢‖_𝐿^𝑝, for any 𝑢∈𝐿^𝑝([0,𝑇],𝐑⁺).

As in [16, 17], we define the fractional integral of order 𝑞 with the lower limit zero for a function 𝑓∈𝐴𝐶[0,∞) as𝐼^𝑞𝑓(𝑡)=1Γ(𝑞)^𝑡0(𝑡−𝑠)^𝑞−1𝑓(𝑠)𝑑𝑠,𝑡>0,0<𝑞<1,(2.1) provided the right side is point-wise defined on [0,∞).

Now we recall some very basic concepts in the theory of measures of noncompactness and condensing maps (see, e.g., [19, 20]).

Definition 2.1. Let 𝐸 be a Banach space, 2^𝐸 the family of all nonempty subsets of 𝐸, (𝒜,≥) a partially ordered set, and 𝛼∶2^𝐸→𝒜. If for every Ω∈2^𝐸: 𝛼co(Ω)=𝛼(Ω),(2.2) then we say that 𝛼 is a measure of noncompactness in 𝐸.

Definition 2.2. Let 𝐸 be a Banach space, and 𝔉∶𝑌⊆𝐸→𝐸 is continuous. Let 𝛼 be a measure of noncompactness in 𝐸 such that(i)for any Ω₀,Ω₁∈2^𝐸 with Ω₀⊂Ω₁, 𝛼Ω₀≤𝛼Ω₁;(2.3)(ii) for every 𝑎₀∈𝐸, Ω∈2^𝐸, 𝛼𝑎₀∪Ω=𝛼(Ω).(2.4) If for every bounded set Ω⊆𝑌 which is not relatively compact, 𝛼(𝔉(Ω))<𝛼(Ω),(2.5) then we say that 𝔉 is condensing with respect to the measure of noncompactness 𝛼 (or 𝛼-condensing).

Definition 2.3. Let 𝜛_𝑞(𝜎)=1𝜋^∞_𝑛=1(−1)^𝑛−1𝜎^{−𝑞𝑛−1}Γ(𝑛𝑞+1)𝑛!sin(𝑛𝜋𝑞),𝜎∈(0,∞)(2.6) be a one-sided stable probability density, and 𝜉_𝑞(𝜎)=1𝑞𝜎^{−1−1/𝑞}𝜛_𝑞𝜎^−1/𝑞,𝜎∈(0,∞).(2.7) For any 𝑧∈𝑋, we define operators {𝑌(𝑡)}_𝑡≥0 and {𝑍(𝑡)}_𝑡≥0 by 𝑍𝑌((𝑡𝑡))𝑧𝑧==𝑞^∞0_∞0𝜉_𝑞(𝜎)𝑇(𝑡^𝑞𝜎)𝑧𝑑𝜎,𝜎𝑡^𝑞−1𝜉_𝑞(𝜎)𝑇(𝑡^𝑞𝜎)𝑧𝑑𝜎.(2.8) If a continuous function 𝑥∶[0,𝑇]→𝑋 satisfies 𝑥(𝑡)=𝑌(𝑡)𝑔(𝑥)+𝑥₀+^𝑡0𝑍(𝑡−𝑠)[𝑓(𝑠,𝑥(𝑠))+𝑎(𝑥)(𝑠)]𝑑𝑠,𝑡∈[0,𝑇],(2.9) then the function 𝑥 is called a mild solution of (1.2).
Our main result is as follows.

Theorem 2.4. Assume that (1)𝑓(⋅,𝑤) and ℎ(⋅,⋅,𝑤) are measurable for each 𝑤∈𝐶([0,𝑇],𝑋); 𝑘(𝑡,⋅) is measurable for each 𝑡∈[0,𝑇];(2)𝑓(𝑡,⋅) is continuous for a.e. 𝑡∈[0,𝑇]; 𝑔 is completely continuous; ℎ(𝑡,𝑠,⋅) is continuous for a.e. (𝑡,𝑠)∈Δ; the map 𝑡→𝑘_𝑡∶=𝑘(𝑡,⋅) is continuous from [0,𝑇] to 𝐿^∞([0,𝑇],𝐑);(3)there exist two positive functions 𝜇(⋅),𝜂(⋅)∈𝐿^𝑝(0,𝑇,𝐑⁺)(𝑝>1/𝑞>1) and two positive functions 𝑚(⋅,⋅) and 𝜁(⋅,⋅) on Δ with sup_{[]𝑡∈0,𝑇}^𝑡0𝑚(𝑡,𝑠)𝑑𝑠∶=𝑚^∗<∞,sup_{[]𝑡∈0,𝑇}^𝑡0𝜁(𝑡,𝑠)𝑑𝑠∶=𝜁^∗<∞,(2.10) such that ‖ℎ‖(𝑓𝑡,(𝑠𝑡,,𝑤𝑤))‖‖≤≤𝑚𝜇((𝑡𝑡),𝑠‖)𝑤‖‖𝑤‖(a.(ea..𝑡e.∈(𝑡,[0𝑠),𝑇∈])Δ,),(2.11) for all 𝑤∈𝐶([0,𝑇],𝑋), and 𝜒(𝜒ℎ((𝑓𝑡,𝑠(𝑡,,𝐷𝐷))))≤≤𝜁𝜂((𝑡𝑡,)𝑠𝜒)𝜒(𝐷(𝐷),),(a.(ea..𝑡e.∈(𝑡[,0𝑠,)T∈])Δ,),(2.12) for any bounded set 𝐷⊂𝐶([0,𝑇],𝑋), where 𝜒 is the Hausdorff measure of noncompactness: 𝜒(Ω)=inf{𝜀>0∶Ωhasafinite𝜀-net}.(2.13)(4)𝑔(⋅) satisfies ‖𝑔(𝑥)‖≤𝑏,∀𝑥∈𝐶([0,𝑇],𝑋),(2.14) for a positive constant 𝑏, and 𝑘(𝑡)∶=esssup||𝑘(𝑡,𝑠)||,0≤𝑠≤𝑡(2.15) is bounded on [0,𝑇].
Then the mild solutions set of problem (1.2) is a nonempty compact subset of the space 𝐶([0,𝑇],𝑋), in the case of 𝑞𝑀Γ(1+𝑞)𝑝𝑝𝑞−−11^{(𝑝−1)/𝑝}𝑇^{𝑞−1/𝑝}‖𝜇‖_𝐿^𝑝<1.(2.16)

Proof. First of all, let us prove our definition of the mild solution to problem (1.2) is well defined and reasonable. Actually, the proof is basic. We present it here for the completeness of the proof as well as the convenience of reading.
Write 𝑎(𝑥𝑥)̂((𝜆𝑡))==^𝑡0_∞0𝑘(𝑡,𝑠)ℎ(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,𝑓(𝜆)=_∞0𝑒^−𝜆𝑡𝑥(𝑡)𝑑𝑡,𝑎̂(𝜆)=_∞0𝑒^−𝜆𝑡𝑓(𝑡,𝑥(𝑡))𝑑𝑡,𝑒^−𝜆𝑡𝑎(𝑥)(𝑡)𝑑𝑡.(2.17) Clearly, the nonlocal Cauchy problem (1.2) can be written as the following equivalent integral equation: 𝑥(𝑡)=𝑔(𝑥)+𝑥₀+1Γ(𝑞)^𝑡0(𝑡−𝑠)^𝑞−1[𝐴𝑥(𝑠)+𝑓(𝑠,𝑥(𝑠))+𝑎(𝑥)(𝑠)]𝑑𝑠,𝑡∈[0,𝑇],(2.18) provided that the integral in (2.18) exists. Formally taking the Laplace transform to (2.18), we have 𝑥̂(𝜆)=1𝜆𝑔(𝑥)+𝑥⁰+𝜆1_𝑞𝐴𝑥̂(𝜆)+1𝜆^𝑞𝑓(𝜆)+𝑎̂(𝜆).(2.19) Therefore, if the related integrals exist, then we obtain 𝑥̂(𝜆)==𝜆𝜆^𝑞−1_𝑞−1(𝜆^𝑞−𝐴)⁻¹𝑔(𝑥)+𝑥₀+(𝜆^𝑞−𝐴)⁻¹𝑓(𝜆)+𝑎̂(𝜆)^∞0=𝑞_∞0𝑒^{−𝜆𝑞𝑠}𝑇(𝑠)𝑔(𝑥)+𝑥₀𝑑𝑠+^∞0𝑒^{−𝜆𝑞𝑠}𝑇(𝑠)𝑓(𝜆)+𝑎̂(𝜆)𝑑𝑠+(𝜆𝑡)^𝑞−1𝑒^{−(𝜆𝑡)𝑞}𝑇(𝑡^𝑞)𝑔(𝑥)+𝑥₀𝑑𝑡_∞0=−1𝑒^{−(𝜆𝜏)𝑞}𝑞𝜏^𝑞−1𝑇(𝜏^𝑞)^∞0𝑒^−𝜆𝑡(𝑓(𝑡,𝑥(𝑡))+𝑎(𝑥)(𝑡))𝑑𝑡𝑑𝜏𝜆^∞0𝑑𝑑𝑡𝑒^{−(𝜆𝑡)𝑞}𝑇(𝑡^𝑞+)𝑔(𝑥)+𝑥₀𝑑𝑡_∞0^∞0=𝑒^{−𝜆𝜏𝜎}𝑞𝜏^𝑞−1𝜛_𝑞(𝜎)𝑇(𝜏^𝑞)^∞0𝑒^−𝜆𝑡(𝑓(𝑡,𝑥(𝑡))+𝑎(𝑥)(𝑡))𝑑𝑡𝑑𝜎𝑑𝜏_∞0^∞0+𝑞_∞0𝑒^{−𝜆𝑡𝜎}𝜎𝜛_𝑞(𝜎)𝑇(𝑡^𝑞)𝑔(𝑥)+𝑥₀𝑑𝜎𝑑𝑡^∞0𝑒^−𝜆𝜃𝜃^𝑞−1𝜎^𝑞𝜛_𝑞(𝜎)𝑇𝜃^𝑞𝜎^𝑞^∞0=𝑒^−𝜆𝑡(𝑓(𝑡,𝑥(𝑡))+𝑎(𝑥)(𝑡))𝑑𝑡𝑑𝜎𝑑𝜃_∞0𝑒^−𝜆𝑡^∞0𝜛_𝑞(𝜎)𝑇𝑡^𝑞+𝑞_∞0𝜎^𝑞𝑔(𝑥)+𝑥⁰𝑑𝜎𝑑𝑡^∞0^∞𝑡𝑒^−𝜆𝜏(𝜏−𝑡)^𝑞−1𝜎^𝑞𝜛_𝑞(𝜎)𝑇(𝜏−𝑡)^𝑞=𝜎^𝑞(𝑓(𝑡,𝑥(𝑡))+𝑎(𝑥)(𝑡))𝑑𝜏𝑑𝑡𝑑𝜎_∞0𝑒^−𝜆𝑡^∞0𝜛_𝑞(𝜎)𝑇𝑡^𝑞+𝑞_∞0𝜎^𝑞𝑔(𝑥)+𝑥⁰𝑑𝜎𝑑𝑡^∞0^𝜏0𝑒^−𝜆𝜏(𝜏−𝑡)^𝑞−1𝜎^𝑞𝜛_𝑞(𝜎)𝑇(𝜏−𝑡)^𝑞=𝜎^𝑞(𝑓(𝑡,𝑥(𝑡))+𝑎(𝑥)(𝑡))𝑑𝑡𝑑𝜏𝑑𝜎_∞0𝑒^−𝜆𝑡^∞0𝜛_𝑞(𝜎)𝑇𝑡^𝑞+𝜎^𝑞𝑔(𝑥)+𝑥⁰𝑑𝜎𝑑𝑡_∞0𝑒^−𝜆𝑡𝑞^𝑡0^∞0(𝑡−𝑠)^𝑞−1𝜎^𝑞𝜛_𝑞(𝜎)𝑇(𝑡−𝑠)^𝑞𝜎^𝑞(𝑓(𝑠,𝑥(𝑠))+𝑎(𝑥)(𝑠))𝑑𝜎𝑑𝑠𝑑𝑡.(2.20) Now using the uniqueness of the Laplace transform (cf., e.g., [21, Theorem  1.1.6]), we deduce that 𝑥(𝑡)=^∞0𝜛_𝑞(𝜎)𝑇𝑡^𝑞+𝑞_𝑡0𝜎^𝑞𝑔(𝑥)+𝑥⁰𝑑𝜎^∞0(𝑡−𝑠)^𝑞−1𝜎^𝑞𝜛_𝑞(𝜎)𝑇(𝑡−𝑠)^𝑞+𝑞_𝑡0𝜎^𝑞𝑓(𝑠,𝑥(𝑠))𝑑𝜎𝑑𝑠^∞0(𝑡−𝑠)^𝑞−1𝜎^𝑞𝜛_𝑞(𝜎)𝑇(𝑡−𝑠)^𝑞=𝜎^𝑞𝑎(𝑥)(𝑠)𝑑𝜎𝑑𝑠_∞0+𝑞𝜉_𝑞𝑡0(𝜎)𝑇(𝑡^𝑞𝜎)𝑔(𝑥)+𝑥₀𝑑𝜎^∞0+𝑞_𝑡0𝜎(𝑡−𝑠)^𝑞−1𝜉_𝑞(𝜎)𝑇((𝑡−𝑠)^𝑞𝜎)𝑓(𝑠,𝑥(𝑠))𝑑𝜎𝑑𝑠^∞0𝜎(𝑡−𝑠)^𝑞−1𝜉_𝑞(𝜎)𝑇((𝑡−𝑠)^𝑞𝜎)𝑎(𝑥)(𝑠)𝑑𝜎𝑑𝑠.(2.21) Consequently, we see that the mild solution to problem (1.2) given by Definition 2.3 is well defined.
Next, we define the operator ℱ∶𝐶([0,𝑇],𝑋)→𝐶([0,𝑇],𝑋) as follows: (ℱ𝑥)(𝑡)=𝑌(𝑡)𝑔(𝑥)+𝑥₀+^𝑡0𝑍(𝑡−𝑠)[𝑓(𝑠,𝑥(𝑠))+𝑎(𝑥)(𝑠)]𝑑𝑠,𝑡∈[0,𝑇].(2.22) It is clear that the operator ℱ is well defined.
The operator ℱ can be written in the form ℱ=ℱ₁+ℱ₂, where the operators ℱ_𝑖,𝑖=1,2 are defined as follows: ℱℱ₁𝑥(𝑡)=𝑌(𝑡)𝑔(𝑥)+𝑥₀,𝑡∈[0,𝑇₂],𝑥(𝑡)=^𝑡0𝑍(𝑡−𝑠)[𝑓(𝑠,𝑥(𝑠))+𝑎(𝑥)(𝑠)]𝑑𝑠,𝑡∈[0,𝑇].(2.23) The following facts will be used in the proof. (1)^∞0𝜉_𝑞(𝜎)𝑑𝜎=1,(2.24) which implies that ‖𝑌(𝑡)‖≤Const;(2.25)(2)^∞0𝜎^𝜈𝜉_𝑞(𝜎)𝑑𝜎=^∞01𝜎^𝑞𝜈𝜛_𝑞(𝜎)𝑑𝜎=Γ(1+𝜈)Γ(1+𝑞𝜈),𝜈∈(0,1],(2.26) which implies that‖𝑍(𝑡)‖≤𝑞𝑀Γ(1+𝑞)𝑡^q−1,𝑡>0.(2.27) Let {𝑥_𝑛}_𝑛∈𝐍⊂𝐶([0,𝑇],𝑋) such that lim_𝑛→∞‖‖𝑥_𝑛−𝑥‖‖_[0,𝑇]=0,(2.28) for an 𝑥∈𝐶([0,𝑇],𝑋). Then by the assumptions, we know that for almost every 𝑡∈[0,𝑇] and (𝑡,𝑠)∈Δ: lliimm_{𝑛→∞𝑛→∞}𝑓𝑡,𝑥_𝑛(𝑡)=𝑓(𝑡,𝑥(𝑡)),ℎ𝑡,𝑠,𝑥_𝑛(𝑠)=ℎ(𝑡,𝑠,𝑥(𝑠)).(2.29) Therefore, for sufficiently large 𝑛, we have ‖‖ℎ‖‖𝑡𝑓,𝑠,𝑡𝑥,_𝑛𝑥_𝑛(𝑡)−𝑓(𝑡,𝑥(𝑡))‖‖≤𝜇(𝑡)1+2‖𝑥‖_[0,𝑇],‖‖‖_𝑡0(𝑠)−ℎ(𝑡,𝑠,𝑥(𝑠))‖‖≤𝑚(𝑡,𝑠)1+2‖𝑥‖_[0,𝑇],𝑘(𝑡,𝑠)ℎ𝑡,𝑠,𝑥_𝑛(𝑠)𝑑𝑠−^𝑡0𝑘(𝑡,𝑠)ℎ(𝑡,𝑠,𝑥(𝑠))𝑑𝑠‖‖‖≤𝑚^∗𝑘^∗1+2‖𝑥‖_[0,𝑇],(2.30) where 𝑘^∗∶=sup_{[]𝑡∈0,𝑇}𝑘(𝑡).(2.31) Hence, lim_𝑛→∞‖‖‖^𝑡0𝑘(𝑡,𝑠)ℎ𝑡,𝑠,𝑥_𝑛(𝑠)𝑑𝑠−^𝑡0𝑘(𝑡,𝑠)ℎ(𝑡,𝑠,𝑥(𝑠))𝑑𝑠‖‖‖=0.(2.32) Thus, ‖‖ℱ₂𝑥_𝑛−ℱ₂𝑥‖‖_[0,𝑇]⟶0,as𝑛⟶∞,(2.33) since (2.27) implies that ‖‖‖^𝑡0𝑍(𝑡−𝑠)−𝑓𝑓(𝑠𝑠,,𝑥𝑥_𝑛(𝑠()𝑠))++_𝑠0^𝑠0𝑘(𝑠,𝜏)ℎ𝑠,𝜏,𝑥_𝑛(𝜏)𝑑𝜏≤𝑞𝑀𝑘(𝑠,𝜏)ℎ(𝑠,𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠‖‖‖Γ(1+𝑞)^𝑡0(𝑡−𝑠)^𝑞−1‖‖𝑓𝑠,𝑥^𝑛+‖‖‖(𝑠)−𝑓(𝑠,𝑥(𝑠))_𝑠0‖‖𝑘(𝑠,𝜏)ℎ𝑠,𝜏,𝑥_𝑛(𝜏)−ℎ(𝑠,𝜏,𝑥(𝜏))𝑑𝜏‖‖‖𝑑𝑠.(2.34) By (2.33) and our assumptions, we see that ℱ is continuous.
Since 𝜒 is the Hausdorff measure of noncompactness in 𝑋, we know that 𝜒 is monotone, nonsingular, invariant with respect to union with compact sets, algebraically semiadditive, and regular. This means that(i)for any Ω₀,Ω₁∈2^𝐸 with Ω₀⊂Ω₁, 𝜒Ω₀≤𝜒Ω₁;(2.35)(ii)for every 𝑎₀∈𝐸, Ω∈2^𝐸, 𝜒𝑎₀∪Ω=𝜒(Ω);(2.36)(iii)for every relatively compact set 𝐷⊂𝐸, Ω∈2^𝐸, 𝜒({𝐷}∪Ω)=𝜒(Ω);(2.37)(iv)for each Ω₀, Ω₁∈2^𝐸, 𝜒Ω₀+Ω₁≤𝜒Ω₀+𝜒Ω₁;(2.38)(v)𝜒(Ω)=0 is equivalent to the relative compactness of Ω.Noting that for any 𝜓∈𝐿¹([0,𝑇],𝑋), we have lim_𝐿→+∞sup_{[]𝑡∈0,𝑇}^𝑡0𝑒^{−𝐿(𝑡−𝑠)}𝜓(𝑠)𝑑𝑠=0.(2.39) So, there exists a positive constant 𝐿 such that 𝑞𝑀Γ(1+𝑞)sup_{[]𝑡∈0,𝑇}^𝑡0(𝑡−𝑠)^𝑞−1𝜂(𝑠)𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠=𝐿₁<1𝑞𝑀𝑘_∗3,𝜁^∗Γ(1+𝑞)sup_{[]𝑡∈0,𝑇}^𝑡0(𝑡−𝑠)^𝑞−1𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠=𝐿₂<1𝑞𝑀3,Γ(1+𝑞)sup_{[]𝑡∈0,𝑇}^𝑡0(𝑡−𝑠)^𝑞−1𝜇(𝑠)+𝑚^∗𝑘^∗𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠=𝐿₃<13.(2.40) For every bounded subset Ω⊂𝐶([0,𝑇],𝑋), we define modΨ_𝑐((ΩΩ))∶∶==sliump_{𝛿→0[]𝑡∈0,𝑇}sup_𝑣∈Ωmax_{|𝑡1−𝑡2|≤𝛿}‖‖𝑣𝑡₁−𝑣𝑡₂‖‖,𝛼(Ω)∶=Ψ(Ω𝑒),^−𝐿𝑡m𝜒o(dΩ_𝑐(𝑡)),(Ω).(2.41) Then mod_𝑐(Ω) is the module of equicontinuity of Ω, and 𝛼 is a measure of noncompactness in the space 𝐶([0,𝑇],𝑋) with values in the cone 𝐑²⁺.
Let Ω⊂𝐶([0,𝑇],𝑋) be a nonempty, bounded set such that 𝛼(ℱ(Ω))≥𝛼(Ω).(2.42) By the assumptions and the continuity of 𝑇(𝑡) in the uniform operator topology for 𝑡>0, we get mod_𝑐ℱ₁Ω=0.(2.43) Clearly, ‖𝑓(𝑠,𝑥(𝑠))‖+‖𝑎(𝑥)(𝑠)‖≤𝜇(𝑠)+𝑚^∗𝑘^∗‖𝑥‖_[0,𝑇].(2.44) Let 𝛿>0, 𝑡₁,𝑡₂∈(0,𝑇] such that 0<𝑡₁−𝑡₂≤𝛿 and 𝑥∈Ω. Then ‖‖‖^𝑡1₀≤𝑍‖𝑥𝑡‖₁−𝑠_[0,𝑇][𝑓(𝑠,𝑥(𝑠))+𝑎(𝑥)(𝑠)]𝑑𝑠−^𝑡2₀𝑍𝑡₂−𝑠[𝑓(𝑠,𝑥(𝑠))+𝑎(𝑥)(𝑠)]𝑑𝑠‖‖‖^𝑡2₀≤𝑞‖𝑥‖_[0,𝑇]‖‖𝑍𝑡₁−𝑠−𝑍𝑡₂−𝑠‖‖𝜇(𝑠)+𝑚^∗𝑘^∗𝑑𝑠+^𝑡1_𝑡2‖‖𝑍𝑡₁−𝑠‖‖𝜇(𝑠)+𝑚^∗𝑘^∗𝑑𝑠^𝑡2₀^∞0|||𝑡₁−𝑠^𝑞−1−𝑡₂−𝑠^𝑞−1|||𝜎𝜉_𝑞+(𝜎)‖‖𝑇𝑡₁−𝑠^𝑞𝜎‖‖𝜇(𝑠)+𝑚^∗𝑘^∗𝑑𝜎𝑑𝑠_𝑡20^∞0𝑡₂×𝜇(𝑠)+𝑚−_∗𝑠^𝑞−1𝜎𝜉_𝑞(𝜎)‖‖𝑇𝑡₁−𝑠^𝑞𝜎−𝑇𝑡₂−𝑠^𝑞𝜎‖‖𝑘^∗𝑑𝜎𝑑𝑠+𝑞𝑀‖𝑥‖^[0,𝑇]Γ(1+𝑞)^𝑡1_𝑡2𝑡₁≤𝑞𝑀‖𝑥‖_[0,𝑇]−𝑠^𝑞−1𝜇(𝑠)+𝑚^∗𝑘^∗𝑑𝑠Γ(1+𝑞)^𝑡2₀|||𝑡₁−𝑠^𝑞−1−𝑡₂−𝑠^𝑞−1|||𝜇(𝑠)+𝑚_∗+𝑘^∗𝑑𝜎𝑑𝑠_𝑡1𝑡2+𝑡₁−𝑠^𝑞−1𝜇(𝑠)+𝑚^∗𝑘^∗𝑑𝑠_𝑡20^∞0𝑡₂−𝑠^𝑞−1𝜎𝜉_𝑞(𝜎)‖‖𝑇𝑡₁−𝑠^𝑞𝜎−𝑇𝑡₂−𝑠^𝑞𝜎‖‖𝜇(𝑠)+𝑚^∗𝑘^∗𝑑𝜎𝑑𝑠.(2.45) It is not hard to see that the right-hand side of (2.45) tend to 0 as 𝑡₂→𝑡₁. Thus, the set {(ℱ₂𝑥)(⋅)∶𝑥∈Ω} is equicontinuous, then mod_𝑐(ℱ₂Ω)=0. Combining with (2.43), we have mod_𝑐(ℱΩ)=0, which implies mod_𝑐(Ω)=0 from (2.42). Next, we show that Ψ(Ω)=0.
It is easy to see that Ψℱ₁Ω=0.(2.46) For any 𝑡∈[0,𝑇], we define ℱ₂(Ω)(𝑡)∶=^𝑡0𝑍(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠))𝑑𝑠∶𝑥∈Ω.(2.47) We consider the multifunction 𝑠∈[0,𝑡]⊸𝐺(𝑠): 𝐺(𝑠)={𝑍(𝑡−𝑠)𝑓(𝑠,𝑥(𝑠))∶𝑥∈Ω}.(2.48) Obviously, 𝐺 is integrable, that is, 𝐺 admits a Bochner integrable selection 𝔤∶[0,ℎ]→𝐸, and 𝔤(𝑡)∈𝐺(𝑡),fora.e.𝑡∈[0,ℎ].(2.49) From (2.27) and our assumptions, it follows that 𝐺 is integrably bounded, that is, there exists a function 𝜚∈𝐿¹([0,ℎ],𝐸) such that ‖𝐺(𝑡)‖∶=sup{‖𝔤‖∶𝔤∈𝐺(𝑡)}≤𝜚(𝑡),a.e.𝑡∈[0,ℎ].(2.50) Moreover, we have the following estimate for a.e. 𝑠∈[0,𝑡]: 𝜒(𝐺(𝑠))≤𝑞𝑀Γ(1+𝑞)(𝑡−𝑠)^𝑞−1≤𝑞𝑀𝜒(𝑓(𝑠,Ω(𝑠)))Γ(1+𝑞)(𝑡−𝑠)^𝑞−1=𝑞𝑀𝜂(𝑠)𝜒(Ω(𝑠))Γ(1+𝑞)(𝑡−𝑠)^𝑞−1≤𝑞𝑀𝜂(𝑠)𝑒^𝐿𝑠𝑒^−𝐿𝑠𝜒(Ω(𝑠))Γ(1+𝑞)(𝑡−𝑠)^𝑞−1𝜂(𝑠)𝑒^𝐿𝑠Ψ(Ω).(2.51) Therefore, since 𝑋 is a separable Banach space, we know by [20, Theorem  4.2.3] that 𝜒ℱ₂(Ω)(𝑡)=𝜒^𝑡0𝐺(𝑠)𝑑𝑠≤𝑞𝑀Γ(1+𝑞)^𝑡0(𝑡−𝑠)^𝑞−1𝜂(𝑠)𝑒^𝐿𝑠𝑑𝑠⋅Ψ(Ω).(2.52) So sup_{[]𝑡∈0,𝑇}𝑒^−𝐿𝑡𝜒ℱ₂(Ω)(𝑡)≤𝑞𝑀Γ(1+𝑞)sup_{[]𝑡∈0,𝑇}^𝑡0=𝐿₁(t−𝑠)^𝑞−1𝜂(𝑠)𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠⋅Ψ(Ω)Ψ(Ω).(2.53) Similarly, if we set ℱ₂(Ω)(𝑡)=^𝑡0𝑍(𝑡−𝑠)𝑎(𝑥)(𝑠)𝑑𝑠∶𝑥∈Ω,(2.54) then we see that the multifunction 𝑠∈[0,𝑡]⊸𝐺(𝑠), 𝐺(𝑠)={𝑍(𝑡−𝑠)𝑎(𝑥)(𝑠)∶𝑥∈Ω}(2.55) is integrable and integrably bounded. Thus, we obtain the following estimate for a.e. 𝑠∈[0,𝑡]: 𝜒𝐺(𝑠)≤𝑞𝑀𝑘^∗𝜁^∗Γ(1+𝑞)(𝑡−𝑠)^𝑞−1sup_{[]𝑡∈0,𝑇}𝑒^𝐿𝑠Ψ(Ω),𝑒^−𝐿𝑡𝜒ℱ₂(Ω)(𝑡)≤𝑞𝑀𝑘^∗𝜁^∗Γ(1+𝑞)sup_{[]𝑡∈0,𝑇}^𝑡0=𝐿₂(𝑡−𝑠)^𝑞−1𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠⋅Ψ(Ω)Ψ(Ω).(2.56) Now, from (2.53) and (2.56), it follows that Ψ(ℱ(Ω))≤Ψℱ₁(Ω)+Ψℱ₂(Ω)≤𝐿₁+𝐿₂Ψ(Ω)=𝐿Ψ(Ω),(2.57) where 0<𝐿<1. Then by (2.42), we get Ψ(Ω)=0. Hence 𝛼(Ω)=(0,0). Thus, Ω is relatively compact due to the regularity property of 𝛼. This means that ℱ is 𝛼-condensing.
Let us introduce in the space 𝐶([0,𝑇],𝑋) the equivalent norm defined as ‖𝑥‖_∗=sup_{[]𝑡∈0,𝑇}𝑒^−𝐿𝑡‖𝑥(𝑡)‖.(2.58) Consider the set 𝐵_𝑟=𝑥∈𝐶([0,𝑇],𝑋)∶‖𝑥‖_∗≤𝑟.(2.59) Next, we show that there exists some 𝑟>0 such that ℱ𝐵_𝑟⊂𝐵_𝑟. Suppose on the contrary that for each 𝑟>0 there exist 𝑥_𝑟(⋅)∈𝐵_𝑟, and some 𝑡∈[0,𝑇] such that ‖(ℱ𝑥_𝑟)(𝑡)‖_∗>𝑟.
From the assumptions, we have ‖‖ℱ₁𝑥_𝑟(𝑡)‖‖_∗≤𝑀𝑏+‖‖𝑥₀‖‖.(2.60) Moreover, ‖‖ℱ₂𝑥_𝑟(𝑡)‖‖≤≤^𝑡0𝑞‖‖𝑀𝑍(t−𝑠)𝑓𝑠,𝑥_𝑟(𝑠)+𝑎𝑥_𝑟(𝑠)‖‖𝑑𝑠Γ(1+𝑞)^𝑡0=𝑞𝑀(𝑡−𝑠)^𝑞−1𝜇(𝑠)‖‖𝑥_𝑟(𝑠)‖‖+𝑚^∗𝑘^∗𝑒^𝐿𝑠‖‖𝑥_𝑟‖‖_∗𝑑𝑠Γ(1+𝑞)^𝑡0+𝑚(𝑡_∗−𝑠)^𝑞−1𝜇(𝑠)𝑒^𝐿𝑠𝑒^−𝐿𝑠‖‖𝑥_𝑟(𝑠)‖‖𝑑𝑠𝑘^∗^𝑡0(𝑡−𝑠)^𝑞−1𝑒^𝐿𝑠𝑑𝑠⋅‖‖𝑥_𝑟‖‖_∗≤𝑞𝑀𝑟Γ(1+𝑞)^𝑡0(𝑡−𝑠)^𝑞−1𝜇(𝑠)+𝑚^∗𝑘^∗𝑒^𝐿𝑠𝑑𝑠.(2.61) Therefore, 𝑟<≤𝑀sup𝑏_{𝑡∈[0,𝑇]}+𝑒‖‖𝑥^−𝐿𝑡₀‖‖ℱ𝑥_𝑟(𝑡)‖‖‖‖+𝑞𝑀𝑟Γ(1+𝑞)sup_{[]𝑡∈0,𝑇}^𝑡0(𝑡−𝑠)^𝑞−1𝜇(𝑠)+𝑚^∗𝑘^∗𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠.(2.62) Dividing both sides of (2.62) by 𝑟, and taking 𝑟→∞, we have 𝑞𝑀Γ(1+𝑞)sup_{[]𝑡∈0,𝑇}^𝑡0(𝑡−𝑠)^𝑞−1𝜇(𝑠)+𝑚^∗𝑘^∗𝑒^{−𝐿(𝑡−𝑠)}𝑑𝑠≥1.(2.63) This is a contradiction. Hence for some positive number 𝑟, ℱ𝐵_𝑟⊂𝐵_𝑟. According to the following known fact.
Let 𝔐 be a bounded convex closed subset of 𝐸 and 𝔉∶𝔐→𝔐 a 𝛼-condensing map. Then Fix𝔉={𝑥∶𝑥=𝔉(𝑥)} is nonempty. we see that problem (1.2) has at least one mild solution.
Next, for 𝑐∈(0,1], we consider the following one-parameter family of maps: ℋ∶[0,(1𝑐],𝑥×)𝐶⟶([0ℋ,𝑇](𝑐,,𝑋𝑥))=⟶𝑐ℱ𝐶(𝑥([)0.,𝑇],𝑋)(2.64) We will demonstrate that the fixed point set of the family ℋ, Fixℋ={𝑥∈ℋ(𝑐,𝑥)forsome𝑐∈(0,1]}(2.65) is a priori bounded. Indeed, let 𝑥∈Fixℋ, for 𝑡∈[0,𝑇], we have ‖𝑥(𝑡)‖≤≤𝑀𝑀‖‖𝑏𝑔+(𝑥‖‖)𝑥+₀𝑥₀‖‖+^𝑡0‖𝑍(𝑡−𝑠)‖‖𝑓(𝑠,𝑥(𝑠))+𝑎(𝑥)(𝑠)‖𝑑𝑠‖‖+𝑞𝑀Γ(1+𝑞)^𝑡0+𝑚_∗(𝑡−𝑠)^𝑞−1𝜇(𝑠)‖𝑥(𝑠)‖𝑑𝑠𝑘^∗^𝑡0(𝑡−𝑠)^𝑞−1sup_{[]𝜏∈0,𝑠}‖𝑥(𝜏)‖𝑑𝑠.(2.66)
Noting that the Hölder inequality, we have ^𝑡0(𝑡−𝑠)^𝑞−1𝜇(𝑠)𝑑𝑠≤𝑝−1𝑝𝑞−1^{(𝑝−1)/𝑝}⋅𝑡^{(𝑝𝑞−1)/𝑝}⋅‖𝜇‖_𝐿^𝑝≤𝑝−1𝑝𝑞−1^{(𝑝−1)/𝑝}⋅𝑇^{𝑞−1/𝑝}⋅‖𝜇‖_𝐿^𝑝.(2.67) Therefore, from (2.66), we obtain ‖𝑥(𝑡)‖≤𝑀𝑏+‖‖𝑥₀‖‖+𝑞𝑀Γ(1+𝑞)𝑝𝑝𝑞−−11^{(𝑝−1)/𝑝}+𝑞𝑀𝑚_∗𝑇^{𝑞−1/𝑝}‖𝜇‖_𝐿^𝑝sup_{[]𝑠∈0,𝑡}‖𝑥(𝑠)‖𝑘^∗Γ(1+𝑞)^𝑡0(𝑡−𝑠)^𝑞−1sup_{[]𝜏∈0,𝑠}‖𝑥(𝜏)‖𝑑𝑠.(2.68) We denote 𝑦(𝑡)∶=sup_{𝑠∈[0,𝑡]}‖𝑥(𝑠)‖.(2.69) Let 𝑡̃∈[0,𝑡] such that 𝑦(𝑡)=‖𝑥(𝑡)̃‖. Then, by (2.68), we can see 𝑦(𝑡)≤𝑀𝑏+‖‖𝑥₀‖‖+𝑞𝑀Γ(1+𝑞)𝑝𝑝𝑞−−11^{(𝑝−1)/𝑝}+𝑞𝑀𝑚_∗𝑇^{𝑞−1/𝑝}‖𝜇‖_𝐿^𝑝𝑦(𝑡)𝑘^∗Γ(1+𝑞)^𝑡0(𝑡−𝑠)^𝑞−1𝑦(𝑠)𝑑𝑠.(2.70) By a generalization of Gronwall’s lemma for singular kernels ([22, Lemma  7.1.1]), we deduce that there exists a constant 𝜅=𝜅(𝑞) such that 𝑦(𝑡)≤𝑀𝑏+‖‖𝑥⁰‖‖1−(𝑞𝑀/Γ(1+𝑞))((𝑝−1)/(𝑝𝑞−1))^{(𝑝−1)/𝑝}𝑇^{𝑞−1/𝑝}‖𝜇‖_𝐿^𝑝+𝜅𝑀𝑏+‖‖𝑥⁰‖‖𝑞𝑀𝑚^∗𝑘^∗/Γ(1+𝑞)1−(𝑞𝑀/Γ(1+𝑞))((𝑝−1)/(𝑝𝑞−1))^{(𝑝−1)/𝑝}𝑇^{𝑞−1/𝑝}‖𝜇‖_𝐿^𝑝²^𝑡0≤𝑀𝑏+‖‖𝑥₀(𝑡−𝑠)^𝑞−1𝑑𝑠‖‖1−(𝑞𝑀/Γ(1+𝑞))((𝑝−1)/(𝑝𝑞−1))^{(𝑝−1)/𝑝}𝑇^{𝑞−1/𝑝}‖𝜇‖_𝐿^𝑝+𝜅𝑀𝑏+‖‖𝑥⁰‖‖𝑞𝑀𝑚^∗𝑘^∗/Γ(1+𝑞)𝑇^𝑞𝑞1−(𝑞𝑀/Γ(1+𝑞))((𝑝−1)/(𝑝𝑞−1))^{(𝑝−1)/𝑝}𝑇^{𝑞−1/𝑝}‖𝜇‖_𝐿^𝑝²∶=𝑤.(2.71) Hence, sup_{𝑡∈[0,𝑇]}‖𝑥(𝑡)‖≤𝑤.
Now we consider a closed ball: 𝐵_𝑅=𝑥∈𝐶([0,𝑇],𝑋)∶‖𝑥‖_[0,𝑇]≤𝑅⊂𝐶([0,𝑇],𝑋).(2.72) We take the radius 𝑅>0 large enough to contain the set Fixℋ inside itself. Moreover, from the proof above, ℱ∶𝐵_𝑅→𝐶([0,𝑇],𝑋) is 𝛼-condensing. Consequently, the following known fact implies our conclusion: Let 𝑉⊂𝐸 be a bounded open neighborhood of zero and 𝔉∶𝑉→𝐸 a 𝛼-condensing map satisfying the boundary condition: 𝑥≠𝜆𝔉(𝑥),(2.73)for all 𝑥∈𝜕𝑉 and 0<𝜆≤1. Then, Fix𝔉is nonempty compact.

3. Example

In this section, let 𝑋=𝐿²([0,𝜋]), we consider the following nonlocal Cauchy problem for an integrodifferential problem:𝜕^𝑞𝑡𝑢(𝑡,𝜉)=𝜕²𝜕𝜉²𝑢(𝑡,𝜉)+1𝑘√^𝑘𝑡⋅1+𝑢(𝑢𝑡,(𝜉𝑡,)𝜉)+^𝑡0(𝑡−𝑠)sin√𝑠⋅𝑢(𝑠,𝜉)𝑢(0,𝜉)𝑢=(𝑡,_𝑗0)=𝑢(𝑡,𝜋)=0,𝑡∈[0,1]𝑡𝑑𝑠,𝑡∈(0,1]_𝑖=0^𝜋0𝑐_𝑖(𝜉,𝑦)𝑢𝑡^𝑖,𝑦1+𝑢𝑡_𝑖,𝑦𝑑𝑦+𝑢⁰(𝜉),(3.1) where 𝜕^𝑞𝑡 is the Caputo fractional partial derivative of order 0<𝑞<1; 𝜉∈[0,𝜋]; 𝑘>0 is a constant to be specified later; 𝑢₀(𝜉)∈𝑋;𝑗∈𝐍⁺;0<𝑡₀<𝑡₁<⋯<𝑡_𝑗<1;(3.2)𝑐_𝑖(⋅,⋅)(𝑖=0,1,…,𝑗) are continuous functions and there exists a positive constant 𝑏 such that^𝑗_𝑖=0^𝜋0‖‖𝑐_𝑖(𝜉,𝑦)‖‖𝑑𝑦≤𝑏.(3.3) For 𝑡∈(0,1], 𝜉∈[0,𝜋], we set𝑔𝑥((𝑥𝑡))((𝜉𝜉))==𝑢_𝑗(𝑡,𝜉),_𝑖=0^𝜋0𝑐_𝑖(𝜉,𝑦)𝑥𝑡^𝑖(𝑦)ℎ(𝑡,𝑠,𝑥(𝑘𝑠)()𝑡,(𝜉𝑠))==𝑡si−n𝑠,√1+𝑥𝑡_𝑖(𝑦)𝑑𝑦,𝑠⋅𝑥(𝑠)(𝜉)𝑓(𝑡,𝑥(𝑡))(𝜉)=1𝑡,𝑘√^𝑘𝑡⋅1+𝑥(𝑥𝑡)((𝑡)𝜉()𝜉).(3.4) On the other hand, it is known that the operator 𝐴 (𝐴𝑢=𝑢^ with 𝐷(𝐴)=𝐻²([0,𝜋])∩𝐻¹⁰([0,𝜋])) generates an analytic semigroup and uniformly bounded semigroup {𝑇(𝑡)}_𝑡≥0 on 𝑋 with ‖𝑇(𝑡)‖_𝐿(𝑋)≤1. Therefore, (3.1) is a special case of (1.2).

Moreover, we have(1)for all 𝑡∈(0,1], ‖𝑓(𝑡,𝑥)‖≤1𝑘√^𝑘𝑡‖𝑥‖∶=𝜇(𝑡)‖𝑥‖;(3.5)(2)for any 𝑤,𝑤∈𝑋, ‖‖𝑓(𝑡,𝑤)−𝑓𝑡,𝑤‖‖≤1𝑘√^𝑘𝑡‖‖𝑤−𝑤‖‖,(3.6) that is, for any bounded set 𝐷⊂𝑋, 𝜒(𝑓(𝑡,𝐷))≤1𝑘√^𝑘𝑡𝜒(𝐷),(3.7) for a.e. 𝑡∈[0,1];(3)for almost all (𝑡,𝑠)∈Δ, ‖ℎ(𝑡,𝑠,𝑥)‖=‖‖‖‖sin√𝑠⋅𝑥(𝑠)(𝜉)𝑡‖‖‖‖≤𝑚(𝑡,𝑠)‖𝑥‖,(3.8) where 𝑚(𝑡,𝑠)∶=√𝑠/𝑡, and 𝑚^∗=sup_[]𝑡∈0,1^𝑡0𝑚(𝑡,𝑠)𝑑𝑠=sup_[]𝑡∈0,1^𝑡0√𝑠𝑡𝑑𝑠=32;(3.9)(4)‖‖ℎ(𝑡,𝑠,𝑤)−ℎ𝑡,𝑠,𝑤‖‖≤√𝑠𝑡‖‖𝑤−𝑤‖‖,(3.10) that is, for any bounded set 𝐷⊂𝑋, 𝜒(ℎ(𝑡,𝑠,𝐷))≤𝜁(𝑡,𝑠)𝜒(𝐷),(3.11) where 𝜁(𝑡,𝑠)∶=√𝑠/𝑡, and sup_[]𝑡∈0,1^𝑡0𝜁(𝑡,𝑠)𝑑𝑠=23.(3.12) Therefore, Theorem 2.4 implies that the problem (3.1) has at least a mild solution when𝑞⋅((𝑝−1)/(𝑝𝑞−1))^{(𝑝−1)/𝑝}Γ(1+𝑞)‖𝜇‖^𝐿𝑝<1.(3.13)

Acknowledgment

The work was supported partly by the Chinese Academy of Sciences, the NSF of Yunnan Province (2009ZC054M) and the NSF of China (11171210).