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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions

Junfei Cao / Zaitang Huang
Published Online: 2018-07-17 | DOI: https://doi.org/10.1515/math-2018-0068

Abstract

In this paper we study a class of semilinear evolution equations with nonlocal initial conditions and give some new results on the existence of asymptotically periodic mild solutions. As one would expect, the results presented here would generalize and improve some results in this area.

Keywords: Semilinear evolution equation; Nonlocal initial condition; Asymptotic periodicity; Compact semigroup

MSC 2010: 35B10; 35K90

1 Introduction

It is well known that periodic oscillations are natural and important phenomena in the real world, which can be observed frequently in many fields, such as celestial mechanics, nonlinear vibration, electromagnetic theory, plasma physics, engineering, ecosphere and so on. However, real concrete systems usually exhibit internal variations and external perturbations which are only approximately periodic. Asymptotically periodic function is one of the concepts of approximately periodic function. From an applied perspective asymptotically periodic systems describe our world more realistically and more accurately than periodic ones, one can see [1, 2, 3] for more details.

In recent years, the theory of asymptotic periodicity and its various extensions have attracted a great deal of attention of many mathematicians due to both their mathematical interest and significance as well as applications in physics, mathematical biology, control theory and so forth. Some contributions have been made. For instance, Wei and Wang [2] investigated the asymptotically periodic Lotka-Volterra cooperative systems. Cushing [3] examined the forced asymptotically periodic solutions of predator-prey systems with or without hereditary effects. Wei and Wang [4] studied the existence and uniqueness of asymptotically periodic solution of some ecosystems with asymptotically periodic coefficients. Henriquez, Pierri and Táboas [5] gave a relationship between S-asymptotically ω-periodic functions and the class of asymptotically ω-periodic functions. Pierri [6] established some conditions under which an S-asymptotically ω-periodic function is asymptotically ω-periodic and discussed the existence of asymptotically ω-periodic solutions to an abstract integral equation. de Andrade and Cuevas [7] studied the existence and uniqueness of asymptotically ω-periodic solution for control systems and partial differential equations with linear part dominated by a Hille-Yosida operator with non-dense domain. Agarwal, Cuevas, Soto and El-Gebeily [8] examined the asymptotically ω-periodic solutions to an abstract neutral integro-differential equation with infinite delay etc. The study of asymptotically periodic solutions has been one of the most attracting topics in qualitative theory of various kinds of partial differential equations (see [9, 10, 11]), fractional differential equations (see [12, 13]), difference equations (see [14]), integro-differential equations (see [15, 16, 17]) and so forth. For more on these studies and related issues, we refer the reader to the references cited therein.

In this paper, we are concerned with the existence of asymptotically ω-periodic mild solutions for the following nonlocal problem

(P){x(t)=Ax(t)+F(t,x(t)),t>0,x(0)+g(x(t))=x0,(1) (2)

where A : D(A) ⊆ XX is a closed bounded linear operator on a Banach space X, F : ℝ+ × C(ℝ, X) → X is a given X-valued function, g : C(ℝ, X) → X is a given X-valued function and x0X.

The nonlocal problem (P) is motivated by physical problems. Indeed, the nonlocal initial conditions (2) can be applied in physics with better effect than the classical initial condition x(0) = x0. For example, they are used to determine the unknown physical parameters in some inverse heat conduction problems [18, 19]. The study of atomic reactors also gives rise to the nonlocal problem (P) [20, 21]. For this reason, the nonlocal problem (P) has got a considerable attention in recent years (see for instance [22, 23, 24, 25, 26, 27, 28, 29] and the references therein). See also [30, 31, 32, 33, 34] and the references cited there for recent generalizations of nonlocal problem (P) to various kinds of differential equations and differential inclusions.

In this paper we are interested in the case that A generates a compact C0-semigroup. In [24] the Leray-Schauder alternative was used to study the existence of solutions for the nonlocal problem (P). However, as was shown in [25], the proof of the main results in [24] does not work because the most important place at t = 0 was neglected when checking the compactness of the solution operator. To fill this gap, some authors added conditions on the compactness of g, see e.g. [25, 26, 35, 36, 37] and the references therein. However, in application to physics, these conditions are too strong. For example, as presented by Deng [27], the nonlocal problem (P) with the mapping g given by

g(x)=i=1pcix(ti),(3)

where 0 < t1 < t2 < ⋯ < tp < +∞, c1, c2, ⋯, cp are given constants, is used to describe the diffusion phenomenon of a small amount of gas in a transparent tube, and in many references on nonlocal Cauchy problems (see e.g. [27, 28, 29, 38]), the mapping g is also given by (3). Obviously, compactness condition is not valid for g in this case. Without assumptions on the compactness of g, Liang, Liu and Xiao [29] developed a method to deal with the case that F(t, x) is Lipschitz continuous in x. By Schauder’s fixed point theorem, the authors obtained the existence of mild solutions for the nonlocal problem (P).

To the best of our knowledge, much less is known about the existence of asymptotically periodic solutions to the nonlocal problem (P) when the nonlinearity F(t, x) as a whole loses the Lipschitz continuity with respect to x. In this work we will discuss the existence of asymptotically periodic mild solutions for the nonlocal problem (P). Some new existence theorems of asymptotically periodic mild solutions are established. In our results, the nonlinearity F(t, x) does not have to satisfy a (locally) Lipschitz condition with respect to x (see Remark 3.1). However, in many papers (for instance [5, 6, 9, 12, 13, 39, 40, 41]) on asymptotic periodicity, to be able to apply the well known Banach contraction principle, a (locally) Lipschitz condition for the nonlinearity of corresponding differential equations is needed. As can be seen, the hypotheses in our results are reasonably weak (see Remark 3.3), and our results generalize those as well as related research and have more broad applications.

The rest of this paper is organized as follows. In Section 2, some concepts, the related notations and some useful lemmas are introduced. In Section 3, we present some criteria ensuring the existence of asymptotically periodic mild solutions. An example is given to illustrate our results in Section 4.

2 Preliminaries

This section is concerned with some notations, definitions, lemmas and preliminary facts which are used in what follows.

From now on, ℝ and ℝ+ stand for the set of real numbers and nonnegative real numbers respectively. Denote by X a Banach space with norm ∥⋅∥. C(ℝ, X) stands for the Banach space of all continuous functions from ℝ to X. Cb(ℝ+, X) stands for the Banach space of all bounded and continuous functions x from ℝ+ to X. Furthermore, let C0(ℝ+, X) and Cω(ℝ, X) be the spaces of functions

C0(R+,X):={xCb(R+,X):limt+x(t)=0},Cω(R,X):={xC(R,X):x(t)isω-periodic}.

It is easy to see that C0(ℝ+, X) and Cω(ℝ, X) endowed with the norm

x=sup{x(t):tR}

are both Banach spaces.

We abbreviate C0(ℝ+, X) to C0(ℝ+) when X = ℝ+.

C(ℝ+ × X, X) stands for the set of all jointly continuous functions F(t, x) from ℝ+ × X to X, and let the notation C0(ℝ+ × X, X) be the set of functions

C0(R+×X,X):={FC(R+×X,X):limt+F(t,x)=0uniformly forxin any bounded subset ofX}.

For some r > 0, write

Ωr:={xC0(R+,X):xr}.

Now, we recall some basic definitions and results on asymptotically periodic functions.

Definition 2.1

([5]). A function U(t) ∈ Cb(ℝ+, X) is said to be asymptotically ω-periodic if it can be decomposed as U(t) = V(t) + W(t), where

V(t)Cω(R,X),W(t)C0(R+,X).

Denote by APω(ℝ+, X) the set of all such functions.

Lemma 2.2

([5]). APω(ℝ+, X) turns out to be a Banach space with the norm ∥⋅∥.

For some r > 0, let

Sr:={xAPω(R+,X):x0r}.

Remark 2.3

Take U(t) ∈ APω(ℝ+, X). Let us note that the decomposition U(t) = V(t)+W(t) is unique.

Indeed, if there exist

V1(t),V2(t)Cω(R,X),W1(t),W2(t)C0(R+,X)

such that

U(t)=V1(t)+W1(t)=V2(t)+W2(t),

then one can find that for fixed t ∈ ℝ,

V1(t)V2(t)=V1(t+nω)V2(t+nω)=W2(t+nω)W1(t+nω),nωt.

Taking the limit as n → +∞, one has that V1(t) = V2(t), t ∈ ℝ as required.

Definition 2.4

([5]). A jointly continuous function F(t, x) from ℝ × X to X is said to be ω-periodic if

F(t+ω,x)=F(t,x)foralltRandxX.

The set of such functions will be denoted by Cω(ℝ × X, X).

Definition 2.5

([5]). A function F : ℝ+ × XX is said to be asymptotically ω-periodic if it can be decomposed as F(t, x) = G(t, x) + Φ(t, x), where

G(t,x)Cω(R×X,X),Φ(t,x)C0(R+×X,X).

Denote by APω(ℝ+ × X, X) the set of all such functions.

In the following, we present the following compactness criterion, which is a special case of the general compactness result of Theorem 2.1 in [42].

Lemma 2.6

([42]). A set DC0(ℝ+, X) is relatively compact if

  1. D is equicontinuous;

  2. limt+ x(t) = 0 uniformly for xD;

  3. the set D(t) := {x(t): xD} is relatively compact in X for every t ∈ ℝ+.

The following Krasnoselskii’s fixed point theorem plays a key role in the proofs of our main results, which can be found in many books.

Lemma 2.7

([43]). Let B be a bounded closed and convex subset of X, and J1, J2 be maps of B into X such that J1x + J2yB for every pair x, yB. If J1 is a contraction and J2 is completely continuous, then the equation J1x + J2x = x has a solution on B.

3 Main results

In this section, we study the existence of asymptotically periodic mild solutions for the following nonlocal problem

(P){x(t)=Ax(t)+F(t,x(t)),t>0,x(0)+g(x(t))=x0(4) (5)

in the Banach space X. Here, x0X, the operator A : D(A) ⊂ XX is the infinitesimal generator of a compact and uniformly exponentially stable C0-semigroup {T(t)}t≥0, i.e. there exist two constants M, δ > 0 such that

T(t)Meδtfor allt>0.

F : ℝ+ × C(ℝ, X) → X is a given function satisfying the following assumption:

(H1) F(t, x) = F1(t, x) + F2(t, x) ∈ APω(ℝ+ × X, X) with

F1(t,x)Cω(R×X,X),F2(t,x)C0(R+×X,X),

and there exists a constant L1 > 0 such that

F1(t,x)F1(t,y)L1xyfor alltR,x,yX.(6)

Moreover, there exist a function β(t) ∈ C0(ℝ+) and a nondecreasing function Φ : ℝ+ → ℝ+ such that for all t ∈ ℝ+ and xX with ∥x∥ ≤ r,

F2(t,x)β(t)Φ(r)andlim infr+Φ(r)r=ρ1.(7)

The function g : APω(ℝ+, X) → X satisfies the following assumption:

(H2) There exists a constant L2 > 0 such that

g(x)g(y)L2xyfor allx,yX.(8)

Moreover, there exists a nondecreasing function Ψ: ℝ+ → ℝ+ such that for all xSr,

g(x)Ψ(r)andlim infr+Ψ(r)r=ρ2.(9)

Remark 3.1

Assume that F(t, x) satisfies the assumption (H1), it is noted that F(t, x) does not have to meet the Lipschitz continuity with respect to x. Such class of asymptotically ω-periodic functions F(t, x) are more complicated than those satisfying Lipschitz continuity and little is known about them.

Lemma 3.2

Given F(t, x) = F1(t, x) + F2(t, x) ∈ APω(ℝ+ × X, X) with

F1(t,x)Cω(R×X,X),F2(t,x)C0(R+×X,X).

Then it yields that

suptRF1(t,x)F1(t,y)suptR+F(t,x)F(t,y),x,yX.(10)

Proof

To show our result, it suffices to verify that

{F1(t,x)F1(t,y):tR}{F(t,x)F(t,y):tR+}¯,x,yX.

In fact, if this is not the case, then for fixed x, yX, there exist some t0 ∈ ℝ and ε > 0 such that

(F1(t0,x)F1(t0,y))(F(t,x)F(t,y))εfor alltR+.

It is clear that

limt+F2(t,x)F2(t,y)=0,

which implies that there exists a positive number T such that for all tT,

F2(t,x)F2(t,y)<ε.(11)

Since F1(t, x) ∈ Cω(ℝ × X, X), then for t0 + nωT,

F2(t0+nω,x)F2(t0+nω,y)F(t0+nω,x)F(t0+nω,y)F1(t0,x)+F1(t0,y)F1(t0+nω,x)F1(t0,x)F1(t0+nω,y)F1(t0,y)=F(t0+nω,x)F(t0+nω,y)F1(t0,x)+F1(t0,y)ε,

which contradicts (11), completing the proof. □

Remark 3.3

In Lemma 3.2, (10) implies that when F(t, x) meets the Lipschitz continuity with respect to x with the Lipschitz constant L1, then F1(t, x) satisfies (6). Note that in many papers (for instance [5, 6, 9, 12, 13, 39, 40, 41]) on asymptotic periodicity, to be able to apply the well known Banach contraction principle, a (locally) Lipschitz condition for the nonlinearities of corresponding differential equations is needed. Thus our conditions in the assumption (H1) are weaker than those of [5, 6, 9, 12, 13, 39, 40, 41].

In the proofs of our results, we need the following auxiliary result.

Lemma 3.4

Given yX, U(t) ∈ Cω(ℝ, X) and V(t) ∈ C0(ℝ+, X). Let

Υ(t):=T(t)y0T(ts)U(s)ds+0tT(ts)V(s)ds,tR+.

Then Υ(t) ∈ C0(ℝ+, X).

Proof

From the exponential stability of {T(t)}t≥0 it is clear that for all t ∈ ℝ+,

T(t)y0T(ts)U(s)ds+0tT(ts)V(s)dsMeδty+M0eδ(ts)U(s)ds+M0teδ(ts)V(s)dsMy+Mδ[U+V],

which implies that Υ(t) is well-defined and continuous on ℝ+. Furthermore

T(t)y0T(ts)U(s)dsMeδty+M0eδ(ts)U(s)dsMeδty+MeδtδU0ast+.

On the other hand, since V(t) ∈ C0(ℝ+, X), given ε > 0, one can choose a T > 0 such that

V(t)<εfor allt>T.

This, together with the exponential stability of {T(t)}t≥0, enables us to conclude that for all tT,

TtT(ts)V(s)dsMTteδ(ts)V(s)dsMδε.

Also, we derive, by the exponential stability of {T(t)}t≥0

0TT(ts)V(s)dsM0Teδ(ts)V(s)dsMeδtδV0ast+.

We thus gain from the arguments above that

Υ(t)T(t)y0T(ts)U(s)ds+0tT(ts)V(s)dsT(t)y0T(ts)U(s)ds+0TT(ts)V(s)ds+TtT(ts)V(s)ds0ast+,

which implies Υ(t) ∈ C0(ℝ+, X). □

We give the following definition of mild solution to nonlocal problem (P).

Definition 3.5

A continuous function x : ℝ+X is called a mild solution to nonlocal problem (P) on+ if x satisfies the integral equation of the form

x(t)=T(t)[x0g(x)]+0tT(ts)F(s,x(s))dsforallt0.

Let β(t) be the function involved in the assumption (H1). Define

σ(t):=0teδ(ts)β(s)ds,tR+.

Then σ(t) ∈ C0(ℝ+). Put ρ3 := suptR+ σ(t).

Now we are in a position to present our results.

Theorem 3.6

Under the assumptions (H1) and (H2), the nonlocal problem (P) has at least one asymptotically ω-periodic mild solution provided that

Mmax{L2,ρ2}+ML1δ1+Mρ1ρ3<1.(12)

Proof

The proof is divided into the following five steps.

  • Step 1

    Define a mapping Λ on Cω(ℝ, X) by

    (Λv)(t)=tT(ts)F1(s,v(s))ds,v(s)Cω(R,X),tR,

    and prove Λ has a unique fixed point v(t) ∈ Cω(ℝ, X).

    Firstly, from the exponential stability of {T(t)}t≥0 it is clear that

    tT(ts)F1(s,v(s))dsMteδ(ts)F1(s,v(s))dsMδF1(,v()),

    which implies that Λ is well-defined and continuous on ℝ+. Moreover, one easily calculates, by v(t) ∈ Cω(ℝ, X) and F1(t, x) ∈ Cω(ℝ × X, X),

    (Λv)(t+ω)=t+ωT(t+ωs)F1(s,v(s))ds=tT(ts)F1(s+ω,v(s+ω))ds=tT(ts)F1(s+ω,v(s))ds=tT(ts)F1(s,v(s))ds=(Λv)(t),

    this implies that Λ maps Cω(ℝ, X) into itself.

    On the other hand, for any v1(t), v2(t) ∈ Cω(ℝ, X), by (6) one has

    (Λv1)(t)(Λv2)(t)ML1teδ(ts)v1(s)v2(s)dsML1δv1v2.

    As a result, one has

    (Λv1)(Λv2)ML1δv1v2.

    This, together with (12), proves that Λ is a contraction on Cω(ℝ, X). Thus, the Banach’s fixed point theorem implies that Λ has a unique fixed point v(t) ∈ Cω(ℝ, X).

  • Step 2

    For the above v(t), define a mapping Γ := Γ1 + Γ2 as

    (Γ1ω)(t)=T(t)[x0g(v|R++ω)]+0tT(ts)[F1(s,v(s)+ω(s))F1(s,v(s))]ds,(Γ2ω)(t)=0T(ts)F1(s,v(s))ds+0tT(ts)F2(s,v(s)+ω(s))ds

    for all ω(s) ∈ C0(ℝ+, X), t ∈ ℝ+, and prove that Γ maps Ωk0 into itself, where k0 is a given constant.

    Firstly, from (6) it follows that

    F1(s,v(s)+ω(s))F1(s,v(s))L1ω(s)for allsR,ω(s)X,

    which implies that

    F1(,v()+ω())F1(,v())C0(R+,X)for eachω()C0(R+,X).

    By (7), one has for all s ∈ ℝ+ and ω(s) ∈ X with ∥ω(s)∥ ≤ r,

    F2(s,v(s)+ω(s))β(s)Φ(r+supsR+v(s)),(13)

    which implies that

    F2(,v()+ω())C0(R+,X)asβ(s)C0(R+).

    Those, together with Lemma 3.4, yield that Γ is well-defined and maps C0(ℝ+, X) into itself.

    On the other hand, in view of (7), (9) and (12) it is not difficult to see that there exists a constant k0 > 0 such that

    M[x0+Ψ(k0+v)]+ML1k0δ+MδsupsRF1(s,v(s))+MΦ(k0+v)ρ3k0.

    This enables us to conclude that for any t ∈ ℝ+ and ω1(t), ω2(t) ∈ Ωk0,

    (Γ1ω1)(t)+(Γ2ω2)(t)T(t)[x0g(v|R++ω1)]+0tT(ts)[F1(s,v(s)+ω1(s))F1(s,v(s))]ds+0T(ts)F1(s,v(s))ds+0tT(ts)F2(s,v(s)+ω2(s))dsM(x0+g(v|R++ω1))+M0teδ(ts)F1(s,v(s)+ω1(s))F1(s,v(s))ds+M0eδ(ts)F1(s,v(s))ds+M0teδ(ts)F2(s,v(s)+ω2(s))dsM[x0+Ψ(k0+v)]+ML1k0δ+MδsupsRF1(s,v(s))+MΦ(k0+v)ρ3k0,

    which implies that

    (Γ1ω1)(t)+(Γ2ω2)(t)Ωk0.

    Thus Γ maps Ωk0 into itself.

  • Step 3

    Show that Γ1 is a contraction on Ωk0.

    In fact, for any ω1(t), ω2(t) ∈ Ωk0 and t ∈ ℝ+, from (6) it follows that

    [F1(s,v(s)+ω1(s))F1(s,v(s))][F1(s,v(s)+ω2(s))F1(s,v(s))]L1ω1(s)ω2(s)for allsR+,ω1(s),ω2(s)X.

    Thus

    (Γ1ω1)(t)(Γ1ω2)(t)T(t)[x0g(v|R++ω1)]T(t)[x0g(v|R++ω2)]+0tT(ts)[F1(s,v(s)+ω1(s))F1(s,v(s))][F1(s,v(s)+ω2(s))F1(s,v(s))]dsML2ω1ω20+ML10teδ(ts)ω1(s)ω2(s)ds[ML2+ML1δ]ω1ω2.

    As a result, one has

    Γ1ω1Γ1ω2[ML2+ML1δ]ω1ω2.

    Thus, in view of (12), one obtains the conclusion.

  • Step 4

    Show that Γ2 is completely continuous on Ωk0.

    Given ε > 0. Let {ωk}k=1+Ωk0 with ωkω0 in C0(ℝ+, X) as k → +∞. Since β(t) ∈ C0(ℝ+), one may choose a t1 > 0 big enough such that for all tt1,

    3MΦ(k0+v)β(t)<δε.

    Also, in view of (H1), we have F2(s, v(s) + ωk(s)) → F2(s, v(s) + ω0(s)) for all s ∈ [0, t1] as k → +∞, and

    F2(,v()+ωk())F2(,v()+ω0())2Φ(k0+v)β()L1(0,t1).

    Hence, by the Lebesgue dominated convergence theorem we deduce that there exists an N > 0 such that for any t ∈ ℝ+,

    (Γ2ωk)(t)(Γ2ω0)(t)=0tT(ts)F2(s,v(s)+ωk(s))ds0tT(ts)F2(s,v(s)+ω0(s))dsM0t1eδ(ts)F2(s,v(s)+ωk(s))F2(s,v(s)+ω0(s))ds+2MΦ(k0+v)t1max{t,t1}eδ(ts)β(s)dsε3+2ε3=ε

    whenever kN. Accordingly, Γ2 is continuous on Ωk0.

    In the sequel, we consider the compactness of Γ2. Since the function

    t0T(ts)F2(s,v(s))ds

    belongs to C0(ℝ+, X) due to Lemma 2.6 and is independent of ω, it suffices to show that the mapping

    (Πω)(t)=0tT(ts)F2(s,v(s)+ω(s))ds,ω(s)C0(R+,X)

    is compact.

    Let t ∈ ℝ be fixed. For given ε0 > 0, from (13) it follows that

    (Πε0ω)(t)=0tε0T(tε0s)F2(s,v(s)+ω(s))ds

    is uniformly bounded for ω(t) ∈ Ωk0. This, together with the compactness of T(ε0), yields that the set {T(ε0)(Πε0ω)(t) : ω(t) ∈ Ωk0} is relatively compact in X.

    On the other hand

    (Πω)(t)T(ε0)(Πε0ω)(t)tε0tT(ts)F2(s,v(s)+ω(s))dsMtε0teδ(ts)F2(s,v(s)+ω(s))ds0asε00+,

    this, together with the total boundedness, yields that the set {(Πω)(t) : ω(t) ∈ Ωk0} is relatively compact in X for each t ∈ ℝ+.

    Next, we verify the equicontinuity of the set {(Πω)(t): ω(t) ∈ Ωk0}.

    Let k > 0 be small enough and t1, t2 ∈ ℝ+, ω(t) ∈ Ωk0. Then by (6) we have that for the case when 0 < t1 < t2,

    (Πω)(t2)(Πω)(t1)=0t2T(t2s)F2(s,v(s)+ω(s))ds0t1T(t1s)F2(s,v(s)+ω(s))dst1t2T(t2s)F2(s,v(s)+ω(s))ds+0t1k[T(t2s)T(t1s)]F2(s,v(s)+ω(s))ds+t1kt1[T(t2s)T(t1s)]F2(s,v(s)+ω(s))dsMΦ(k0+v)t1t2eδ(t2s)β(s)ds+Φ(k0+v)supt[0,t1k]T(t2s)T(t1s)0t1kβ(s)ds+MΦ(k0+v)t1kt1(eδ(t2s)+eδ(t1s))β(s)ds0ast2t10,k0,

    and for the case when 0 = t1 < t2,

    (Πω)(t2)(Πω)(t1)MΦ(k0+v)0t2eδ(t2s)β(s)ds0ast20,

    which verifies that the result follows.

    Finally, as

    (Πω)(t)MΦ(k0+v)0teδ(ts)β(s)ds0ast+

    uniformly for ω(t) ∈ Ωk0, in view of σ(t) ∈ C0(ℝ+), we conclude that (Πω)(t) vanishes at infinity uniformly for ω(t) ∈ Ωk0.

    Now an application of Lemma 2.6 justifies the compactness of Π, which together with the representation of Γ2 implies that Γ2 is compact.

  • Step 5

    Show that the nonlocal problem (P) has at least one asymptotically ω-periodic mild solution.

    Firstly, as shown in Step 3 and Step 4 respectively, Γ1 is a strict contraction and Γ2 is completely continuous. Accordingly, we deduce, thanks to Lemma 2.7, that Γ has at least one fixed point ω(t) ∈ Ωk0, furthermore ω(t) ∈ C0(ℝ+, X).

    Then, consider the following coupled system of integral equations

    {v(t)=tT(ts)F1(s,v(s))ds,tR+,ω(t)=T(t)[x0g(v|R++ω)]+0tT(ts)F(s,v(s)+ω(s))dstT(ts)F1(s,v(s))ds,tR+.(14)

    From the result of step 1, together with the above fixed point ω(t) ∈ C0(ℝ+, X), it follows that

    (v(t),ω(t))Cω(R,X)×C0(R+,X)

    is a solution to system (14). Thus

    x(t):=v(t)+ω(t)APω(R+,X),

    and it is an asymptotically ω-periodic mild solution to the nonlocal problem (P). □

Remark 3.7

Note that the condition (6) in (H1) of Theorem 3.6 can be easily extended to the case of F1(t, x) being locally Lipschitz continuous:

F1(t,x)F1(t,y)L(r)XY

for all t ∈ ℝ+ and x, yX satisfyingx∥, ∥y∥ ≤ r.

Corollary 3.8

Assume that the hypothesis (H1) holds and g(x) = 0. Then the problem (P) has at least one asymptotically ω-periodic mild solution provided that

ML1δ1+Mρ1ρ3<1.

In the following, we prove the existence of asymptotically ω-periodic mild solutions to the nonlocal problem (P) for the case of g being completely continuous.

Theorem 3.9

Let the hypothesis (H1) hold. Assume in addition that

(H2) The function g : APω(ℝ+, X) → X is completely continuous, and there exists a nondecreasing function Θ : ℝ+ → ℝ+ such that for all xSr,

g(x)Θ(r)andlim infr+Θ(r)r=ρ2.

Then the nonlocal problem (P) has at least one asymptotically ω-periodic mild solution provided that

Mρ2+ML1δ1+Mρ1ρ3<1.(15)

Proof

Let the operator Λ be defined in the same way as in Theorem 3.6 and v(t) ∈ Cω(ℝ, X) come from the Step 1 in the proof of Theorem 3.6, is a unique fixed point of Λ.

Consider a mapping ϒ = Υ1 + Υ2 defined by

(Υ1ω)(t)=0tT(ts)[F1(s,v(s)+ω(s))F1(s,v(s))]ds,ω(s)C0(R+,X),tR+,(Υ2ω)(t)=T(t)[x0g(v|R++ω)]0T(ts)F1(s,v(s))ds+0tT(ts)F2(s,v(s)+ω(s))ds,ω(s)C0(R+,X),tR+.

From our assumptions it follows that ϒ is well defined and maps C0(ℝ+, X) into itself. Moreover, there exists a constant k0 > 0 such that

Υ1ω1+Υ2ω2Ωk0for every pairω1(t),ω2(t)Ωk0,

(see the Step 2 in the proof of Theorem 3.6 for more details). Thus, to be able to apply Lemma 2.7 to obtain a fixed point of ϒ, we need to prove that Υ1 is a strict contraction and Υ2 is completely continuous on Ωk0.

From (15) and the Step 3 in the proof of Theorem 3.6 it follows that Υ1 is a strict contraction. Also, since g : APω(ℝ+, X) → X is completely continuous, it follows from the Step 4 in the proof of Theorem 3.6 that Υ2 is completely continuous. Now, applying Lemma 2.7 we obtain that ϒ has a fixed point ω(t) ∈ C0(ℝ+, X), which gives rise to an asymptotically ω-periodic mild solution v(t) + ω(t). □

Corollary 3.10

The nonlocal problem (P) with g(x) given by (3) for xAPω(ℝ+, X) has at least one asymptotically ω-periodic mild solution provided that

Mi=1p|ci|+ML1δ+Mρ1ρ3<1.

4 Applications

In this section, an example is given to illustrate the practical usefulness of the theoretical results established in the preceding section.

Consider the partial differential equation with homogeneous Dirichlet boundary condition and nonlocal initial condition of the form

{u(t,ξ)t=2ξ2u(t,ξ)+14sintsinu(t,ξ)+14etu(t,ξ)cosu3(t,ξ),tR+,ξ[0,π],u(t,0)=u(t,π)=0,tR+,u(0,ξ)=u0(ξ)+i=14116u(ti,ξ),ξ[0,π],(16)

where 0 < t1 < t2 < t3 < t4 < +∞.

Take X = L2[0,π] with norm ∥⋅∥ and define A : D(A) ⊂ XX given by Ax=2x(ξ)ξ2 with the domain

D(A)={x()X:xX,xXis absolutely continuous on[0,π],x(0)=x(π)=0}.

It is well known that A is self-adjoint, with compact resolvent and is the infinitesimal generator of an analytic as well as compact semigroup {T(t)}t≥0 satisfying ∥T(t)∥ ≤ et for t > 0.

Now, let

x0=u0(ξ)X,F1(t,x(ξ))=14sintsinx(ξ)for alltRandxX,F2(t,x(ξ))=14etx(ξ)cosx3(ξ)for alltR+andxX,g(x(t))=i=14116x(ti)forxAP2π(R+,X).

Then it is easy to verify that F1(t, x) ∈ C2π(ℝ × X, X) and

F1(t,x)F1(t,y)14xyfor alltR,x,yX,

F2 : ℝ+ × XX is continuous and

F2(t,x)14etxfor alltR+,xX,

which implies F2(t, x) ∈ C0(ℝ+ × X, X), furthermore

F(t,x)=F1(t,x)+F2(t,x)AP2π(R+×X,X),

g : AP2π(ℝ+, X) → X satisfies

g(x)g(y)14uvfor allx,yAP2π(R+,X).

Thus, equation (16) can be reformulated as the abstract nonlocal problem (P) and the assumption (H1) as well as the assumption (H2) holds with

L1=14,L2=ρ2=14,Φ(r)=r,Ψ(r)=u0+r4,β(t)=14et,ρ1=1,ρ314,

and

Mi=1p|ci|+ML1δ+Mρ1ρ334<1.

Then from Corollary 3.10 it follows that equation (16) at least has one asymptotically 2π-periodic mild solution.

Acknowledgement

This research was supported by the NNSF of China (No.11561009), the Guangdong Province Natural Science Foundation (No.2015A030313896), the Characteristic Innovation Project (Natural Science) of Guangdong Province (No.2016KTSCX094), the Science and Technology Program Project of Guangzhou (No.201707010230).

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About the article

caojunfei@gdei.edu.cn


Received: 2017-10-15

Accepted: 2018-05-15

Published Online: 2018-07-17


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 792–805, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0068.

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© 2018 Cao and Huang, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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