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

Definition 2.1

([5]). A functionU(t) ∈ C_b(ℝ⁺, X) is said to be asymptoticallyω-periodic if it can be decomposed asU(t) = V(t) + W(t), where

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

Lemma 2.2

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

Remark 2.3

TakeU(t) ∈ AP_ω(ℝ⁺, X). Let us note that the decompositionU(t) = V(t)+W(t) is unique.

Definition 2.4

([5]). A jointly continuous functionF(t, x) from ℝ × XtoXis said to beω-periodic if

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

Definition 2.5

([5]). A functionF : ℝ⁺ × X → Xis said to be asymptoticallyω-periodic if it can be decomposed asF(t, x) = G(t, x) + Φ(t, x), where

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

Lemma 2.6

([42]). A setD ⊂ C₀(ℝ⁺, X) is relatively compact if

1. Dis equicontinuous;

2. limt→+∞x(t) = 0 uniformly forx ∈ D;

3. the setD(t) := {x(t): x ∈ D} is relatively compact inXfor everyt ∈ ℝ⁺.

Lemma 2.7

([43]). LetBbe a bounded closed and convex subset ofX, andJ₁, J₂be maps ofBintoXsuch thatJ₁x + J₂y ∈ Bfor every pairx, y ∈ B. IfJ₁is a contraction andJ₂is completely continuous, then the equationJ₁x + J₂x = xhas a solution onB.

Remark 3.1

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

Lemma 3.2

GivenF(t, x) = F₁(t, x) + F₂(t, x) ∈ AP_ω(ℝ⁺ × X, X) with

Then it yields that

Proof

To show our result, it suffices to verify that

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

It is clear that

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

Since F₁(t, x) ∈ C_ω(ℝ × X, X), then for t₀ + nω ≥ T,

which contradicts (11), completing the proof. □

Remark 3.3

In Lemma 3.2, (10) implies that whenF(t, x) meets the Lipschitz continuity with respect toxwith the Lipschitz constantL₁, thenF₁(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].

Lemma 3.4

Giveny ∈ X, U(t) ∈ C_ω(ℝ, X) andV(t) ∈ C₀(ℝ⁺, X). Let

ThenΥ(t) ∈ C₀(ℝ⁺, X).

Proof

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

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

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

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

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

We thus gain from the arguments above that

which implies Υ(t) ∈ C₀(ℝ⁺, X). □

Definition 3.5

A continuous functionx : ℝ⁺ → Xis called a mild solution to nonlocal problem (P) on ℝ⁺ifxsatisfies the integral equation of the form

Theorem 3.6

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

Proof

The proof is divided into the following five steps.

1. Define a mapping Λ on C_ω(ℝ, X) by

   (Λv)(t)=∫−∞tT(t−s)F1(s,v(s))ds,v(s)∈Cω(R,X),t∈R,

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

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

   ‖∫−∞tT(t−s)F1(s,v(s))ds‖≤M∫−∞te−δ(t−s)‖F1(s,v(s))‖ds≤Mδ‖F1(⋅,v(⋅))‖∞,

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

   (Λv)(t+ω)=∫−∞t+ωT(t+ω−s)F1(s,v(s))ds=∫−∞tT(t−s)F1(s+ω,v(s+ω))ds=∫−∞tT(t−s)F1(s+ω,v(s))ds=∫−∞tT(t−s)F1(s,v(s))ds=(Λv)(t),

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

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

   ‖(Λv1)(t)−(Λv2)(t)‖≤ML1∫−∞te−δ(t−s)‖v1(s)−v2(s)‖ds≤ML1δ‖v1−v2‖∞.

   As a result, one has

   ‖(Λv1)−(Λv2)‖∞≤ML1δ‖v1−v2‖∞.

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

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

   (Γ1ω)(t)=T(t)[x0−g(v|R++ω)]+∫0tT(t−s)[F1(s,v(s)+ω(s))−F1(s,v(s))]ds,(Γ2ω)(t)=−∫−∞0T(t−s)F1(s,v(s))ds+∫0tT(t−s)F2(s,v(s)+ω(s))ds

   for all ω(s) ∈ C₀(ℝ⁺, X), t ∈ ℝ⁺, and prove that Γ maps Ω_k0 into itself, where k₀ is a given constant.

   Firstly, from (6) it follows that

   ‖F1(s,v(s)+ω(s))−F1(s,v(s))‖≤L1‖ω(s)‖for alls∈R,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+sups∈R+‖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+Mδsups∈R‖F1(s,v(s))‖+MΦ(k0+‖v‖∞)ρ3≤k0.

   This enables us to conclude that for any t ∈ ℝ⁺ and ω₁(t), ω₂(t) ∈ Ω_k0,

   ‖(Γ1ω1)(t)+(Γ2ω2)(t)‖≤‖T(t)[x0−g(v|R++ω1)]‖+‖∫0tT(t−s)[F1(s,v(s)+ω1(s))−F1(s,v(s))]ds‖+‖∫−∞0T(t−s)F1(s,v(s))ds‖+‖∫0tT(t−s)F2(s,v(s)+ω2(s))ds‖≤M(‖x0‖+‖g(v|R++ω1)‖)+M∫0te−δ(t−s)‖F1(s,v(s)+ω1(s))−F1(s,v(s))‖ds+M∫−∞0e−δ(t−s)‖F1(s,v(s))‖ds+M∫0te−δ(t−s)‖F2(s,v(s)+ω2(s))‖ds≤M[‖x0‖+Ψ(k0+‖v‖∞)]+ML1k0δ+Mδsups∈R‖F1(s,v(s))‖+MΦ(k0+‖v‖∞)ρ3≤k0,

   which implies that

   (Γ1ω1)(t)+(Γ2ω2)(t)∈Ωk0.

   Thus Γ maps Ω_k0 into itself.

3. Show that Γ¹ is a contraction on Ω_k0.

   In fact, for any ω₁(t), ω₂(t) ∈ Ω_k0 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))]‖≤L1‖ω1(s)−ω2(s)‖for alls∈R+,ω1(s),ω2(s)∈X.

   Thus

   ‖(Γ1ω1)(t)−(Γ1ω2)(t)‖≤‖T(t)[x0−g(v|R++ω1)]−T(t)[x0−g(v|R++ω2)]‖+‖∫0tT(t−s)[F1(s,v(s)+ω1(s))−F1(s,v(s))]−[F1(s,v(s)+ω2(s))−F1(s,v(s))]ds‖≤ML2‖ω1−ω2‖0+ML1∫0te−δ(t−s)‖ω1(s)−ω2(s)‖ds≤[ML2+ML1δ]‖ω1−ω2‖∞.

   As a result, one has

   ‖Γ1ω1−Γ1ω2‖∞≤[ML2+ML1δ]‖ω1−ω2‖∞.

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

4. Show that Γ² is completely continuous on Ω_k0.

   Given ε > 0. Let {ωk}k=1+∞ ⊂ Ω_k0 with ω_k → ω₀ in C₀(ℝ⁺, X) as k → +∞. Since β(t) ∈ C₀(ℝ⁺), one may choose a t₁ > 0 big enough such that for all t ≥ t₁,

   3MΦ(k0+‖v‖∞)β(t)<δε.

   Also, in view of (H₁), we have F₂(s, v(s) + ω_k(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(t−s)F2(s,v(s)+ωk(s))ds−∫0tT(t−s)F2(s,v(s)+ω0(s))ds‖≤M∫0t1e−δ(t−s)‖F2(s,v(s)+ωk(s))−F2(s,v(s)+ω0(s))‖ds+2MΦ(k0+‖v‖∞)∫t1max{t,t1}e−δ(t−s)β(s)ds≤ε3+2ε3=ε

   whenever k ≥ N. Accordingly, Γ² is continuous on Ω_k0.

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

   t→∫−∞0T(t−s)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(t−s)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)=∫0t−ε0T(t−ε0−s)F2(s,v(s)+ω(s))ds

   is uniformly bounded for ω(t) ∈ Ω_k0. This, together with the compactness of T(ε₀), yields that the set {T(ε₀)(Π_ε0ω)(t) : ω(t) ∈ Ω_k0} is relatively compact in X.

   On the other hand

   ‖(Πω)(t)−T(ε0)(Πε0ω)(t)‖≤‖∫t−ε0tT(t−s)F2(s,v(s)+ω(s))ds‖≤M∫t−ε0te−δ(t−s)‖F2(s,v(s)+ω(s))‖ds→0asε0→0+,

   this, together with the total boundedness, yields that the set {(Πω)(t) : ω(t) ∈ Ω_k0} is relatively compact in X for each t ∈ ℝ⁺.

   Next, we verify the equicontinuity of the set {(Πω)(t): ω(t) ∈ Ω_k0}.

   Let k > 0 be small enough and t₁, t₂ ∈ ℝ⁺, ω(t) ∈ Ω_k0. Then by (6) we have that for the case when 0 < t₁ < t₂,

   ‖(Πω)(t2)−(Πω)(t1)‖=‖∫0t2T(t2−s)F2(s,v(s)+ω(s))ds−∫0t1T(t1−s)F2(s,v(s)+ω(s))ds‖≤∫t1t2‖T(t2−s)F2(s,v(s)+ω(s))‖ds+∫0t1−k‖[T(t2−s)−T(t1−s)]F2(s,v(s)+ω(s))‖ds+∫t1−kt1‖[T(t2−s)−T(t1−s)]F2(s,v(s)+ω(s))‖ds≤MΦ(k0+‖v‖∞)∫t1t2e−δ(t2−s)β(s)ds+Φ(k0+‖v‖∞)supt∈[0,t1−k]‖T(t2−s)−T(t1−s)‖∫0t1−kβ(s)ds+MΦ(k0+‖v‖∞)∫t1−kt1(e−δ(t2−s)+e−δ(t1−s))β(s)ds→0ast2−t1→0,k→0,

   and for the case when 0 = t₁ < t₂,

   ‖(Πω)(t2)−(Πω)(t1)‖≤MΦ(k0+‖v‖∞)∫0t2e−δ(t2−s)β(s)ds→0ast2→0,

   which verifies that the result follows.

   Finally, as

   ‖(Πω)(t)‖≤MΦ(k0+‖v‖∞)∫0te−δ(t−s)β(s)ds→0ast→+∞

   uniformly for ω(t) ∈ Ω_k0, in view of σ(t) ∈ C₀(ℝ⁺), we conclude that (Πω)(t) vanishes at infinity uniformly for ω(t) ∈ Ω_k0.

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

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) ∈ Ω_k0, furthermore ω(t) ∈ C₀(ℝ⁺, X).

   Then, consider the following coupled system of integral equations

   {v(t)=∫−∞tT(t−s)F1(s,v(s))ds,t∈R+,ω(t)=T(t)[x0−g(v|R++ω)]+∫0tT(t−s)F(s,v(s)+ω(s))ds−∫−∞tT(t−s)F1(s,v(s))ds,t∈R+.(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 ofF₁(t, x) being locally Lipschitz continuous:

for allt ∈ ℝ⁺andx, y ∈ Xsatisfying ∥x∥, ∥y∥ ≤ r.

Corollary 3.8

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

Theorem 3.9

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

(H2′)The functiong : AP_ω(ℝ⁺, X) → Xis completely continuous, and there exists a nondecreasing functionΘ : ℝ⁺ → ℝ⁺such that for allx ∈ S_r,

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

Proof

Let the operator Λ be defined in the same way as in Theorem 3.6 and v(t) ∈ C_ω(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