Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces

Jing Zhang; Haide Gou

doi:10.1515/math-2025-0165

Article Open Access

Existence of positive periodic solutions for evolution equations with delay in ordered Banach spaces

Jing Zhang and Haide Gou

Published/Copyright: June 23, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

The main focus of this study is to discuss the existence of positive ω -periodic mild solutions for evolution equation with delay in an ordered Banach space E . Under the ordered conditions of the growth exponent of the nonlinearity g with respect to the semigroup T ( t ) ( t ≥ 0 ) or the first eigenvalue of the operator A , the existence results of mild solutions and positive mild solutions for evolution equation are obtained by applying the Poincaré mapping and monotone iterative method, and adding appropriate conditions to the nonlinearity term g without assuming the existence of upper and lower solutions. Finally, we give an example to verify the applicability of our abstract results.

Keywords: monotone iterative method; periodic solutions; compact semigroup; Poincaré mapping; positive C 0-semigroup; evolution equations with delay

MSC 2010: 34G10; 34K45; 47D03; 34K30; 47H07; 47H08

1 Introduction

The theory of partial differential equations with delay has a broad physical background and realistic mathematical models, and has developed rapidly over the last 50 years. Such equations usually describe natural phenomena more realistically than those without delay [1,2]. The problem of periodic solutions and delays of partial differential equations is an important area of research because it can take into account seasonal fluctuations taking place in phenomena that occur in models and has been studied by some researchers in recent years. The existence and asymptotic stability of periodic solutions to evolution equations with delay have attracted much attention [3–19]. Li [20] discussed the time-periodic solution for the evolution equation with multiple delays in Hilbert space H

u ′ ( t ) + A u ( t ) = F ( t , u ( t ) , u ( t − τ 1 ) , … , u ( t − τ n ) ) , t ∈ R ,

where A : D ( A ) ⊂ H → H is a positive definite self-adjoint operator, having compact resolvent and the first eigenvalue λ 1 > 0 , F : R × H n + 1 → H is a nonlinear mapping, which is ω -periodic in t , and τ 1 , τ 2 , … , τ n are positive constants, which denote the time delays. The author has obtained the existence and uniqueness of time ω -periodic solutions and the unique time periodic solution is asymptotically stable by means of analytic semigroups theory and integral inequality with delays. However, due to the limitations of the research space and the particularity of the operator, the research results are not universal, and sometimes the conditions are not easy to verify in the application.

More recently, Liang et al. [21,22] also studied non-autonomous evolutionary equations with time delay and impulse. Under the nonlinearity of satisfying the continuous and Lipschitzian, Horn’s fixed point theorem or Sadovskii fixed point theorem proves the existence theorem for periodic temperature and the solution of nonautonomous delayed evolution equations. Wang and Zhu [23] used Horn’s fixed point theorem to obtain, under appropriate assumptions such as the boundability of the equation solutions, the existential theory of periodic solutions to abstract delayed evolutionary equations. In all these works, however, the key assumption or process of the priori boundedness of solutions is adopted.

On the other hand, in many practical models, such as the heat transfer equation, neutron transfer equation, reaction diffusion equation, etc., only the positive periodic solutions are significant. In [24], the existence and uniqueness of positive periodic mild solutions for the evolution equation without delay

u ′ ( t ) + A u ( t ) = F ( t , u ( t ) ) , t ∈ R ,

have been considered in an ordered Banach space E , where − A is the infinitesimal generator of a positive C 0 -semigroup, F : R × E → E is a continuous mapping, which is ω -periodic in t . In particular, Li and Li [25] obtained the existence and asymptotic stability of the positive periodic mild solutions for the abstract evolution equation with delay

u ′ ( t ) + A u ( t ) = F ( t , u ( t ) , u ( t − τ ) ) , t ∈ R ,

in an ordered Banach space E , by applying operator semigroup theory and some fixed point theorems. Furthermore, for the abstract evolution equation without delay, the periodic solutions have been discussed by many authors, see [26–30] and references therein.

Later, under the theory of semigroup of linear operators, Chen and Mu [31] discussed the existence and uniqueness of mild solutions to the initial value problem of impulsive integro-differential evolution in an abstract space. And then, under the conditions that nonlinear function and impulsive functions are mixed monotone, Chen and Li [32] established a mixed monotone iterative technique in the presence of a new concept of upper and lower solutions for the initial value problem of impulsive evolution equation

u ′ ( t ) + A u ( t ) = f ( t , u ( t ) , u ( t ) ) , t ∈ J , t ≠ t k , Δ u ∣ t = t k = I k ( u ( t k ) , u ( t k ) ) , k = 1 , 2 , … , m , u ( 0 ) = x 0 ,

where f ∈ C ( J × E × E , E ) , J = [ 0 , a ] , and a > 0 is a constant. Recently, Li and Gou [33] demonstrated the existence of mild solutions to the periodic boundary value problem of the first-order semilinear impulsive integro-differential evolution equation of Volterra type by using the monotone iterative technique in ordered Banach space.

However, the above results have all directly assumed that the equation has upper and lower solutions, but have not given or found specific upper and lower solutions. It is difficult to find upper and lower solutions of impulsive evolution equation. At present, Li et al. [34] discussed the positive S -asymptotically periodic solutions for the abstract fractional evolution equation

D t q c u ( t ) + A u ( t ) = f ( t , u ( t ) ) , t ≥ 0 , u ( 0 ) = u 0 ,

under order conditions and growth conditions. They chose the corresponding eigenfunction of smallest eigenvalue of operator A as a lower solution of fractional evolution equation, and then by using monotone iterative technique obtained the existence of positive mild solution.

The main focus of this study is to discuss the existence of positive ω -periodic mild solutions for evolution equation with delay in an ordered Banach space E

(1.1) u ′ ( t ) + A u ( t ) = g ( t , u ( t ) , u ( t − η ) ) , t ∈ R ,

where A : D ( A ) ⊂ E → E is a closed linear operator, − A generates an exponentially stable positive compact C 0 -semigroup T ( t ) ( t ≥ 0 ) , and g : R + × E × E → E is a continuous mapping, which is ω -periodic in t . Under the ordered conditions of the growth exponent of the nonlinearity g with respect to the semigroup T ( t ) ( t ≥ 0 ) or the first eigenvalue of the operator A , the existence results of mild solutions and positive mild solutions for evolution equation are obtained by applying the Poincaré mapping and monotone iterative method, and adding appropriate conditions to the nonlinearity term g without assuming the existence of upper and lower solutions. Finally, we give an example to verify the applicability of our abstract results.

The primary contributions of this work are as follows:

We will discuss the existence of the positive ω -periodic mild solutions for equation (1.1) by using the Poincaré mapping and monotone iterative. To be more precise, the nonlinear term satisfies order conditions that relate to the growth exponent of the semigroup T ( t ) ( t ≥ 0 ) or the first eigenvalue of the operator A .
Without assuming upper and lower solutions, we first prove that corresponding linear periodic boundary value problem has a unique positive mild solution w 0 by using the Poincaré operator [28] and the contraction mapping principle.
By using the monotone iterative method in [ − w 0 , w 0 ] , the existence of mild solutions to equation (1.1) on E is obtained. Then, we establish an accurate estimate of spectral radius for the resolvent operator and obtain the uniqueness of mild solutions. Furthermore, choosing a function related to the eigenfunction of smallest eigenvalue of operator A as a lower solution, we establish the existence result of positive mild solutions to equation (1.1).
Further, we establish the existence results of positive mild solutions while − A generates an exponentially stable, positive, and compact C 0 -semigroup.

The structure of this study is as follows: Section 2 presents the preliminary details and in Section 3, the existence theorems for equation (1.1) are discussed. The final section illustrates the application of the obtained results through practical examples.

2 Preliminaries

We briefly present some basic results that are used in the rest of the study in this section. Let E be an ordered Banach space, whose positive cone K is a normal cone with normal constant N . Let I = [ 0 , + ∞ ) and r : I → E , consider the initial value problem of the linear evolution equation

(2.1) u ′ ( t ) + A u ( t ) = r ( t ) , t ∈ I , u ( 0 ) = x 0 .

We know that in [35, Chapter 4, Theorem 2.9], when x 0 ∈ D ( A ) and r ∈ C 1 ( I , E ) , the initial value problem (2.1) has a unique classical solution u ∈ C 1 ( I , E ) ∩ C ( I , E 1 ) expressed by

(2.2) u ( t ) = T ( t ) x 0 + ∫ 0 t T ( t − s ) r ( s ) d s ,

where E 1 = D ( A ) is Banach space, which have the graph norm ‖ ⋅ ‖ 1 = ‖ ⋅ ‖ + ‖ A ⋅ ‖ . In general, the function u given by (2.2) belongs to C ( I , E ) for x 0 ∈ E and r ∈ C ( I , E ) , and it is called a mild solution of the linear evolution equation (2.1).

Let C ω ( R , E ) be the Banach space { u ∈ C ( R , E ) ∣ u ( t + ω ) = u ( t ) , t ∈ R } endowed the maximum norm ‖ u ‖ C = max t ∈ I ‖ u ( t ) ‖ . Obviously, C ω ( R , E ) is also called an ordered Banach space with the partial order “ ≤ ” induced by the positive cone K C = { u ∈ C ω ( R , E ) ∣ u ( t ) ≥ θ , t ∈ R } and K C is also normal with the normal constant N .

Let A : D ( A ) ⊂ E → E be a linear operator and − A generates a C 0 -semigroup T ( t ) ( t ≥ 0 ) in E . Consistent with the exponential boundedness of C 0 -semigroup T ( t ) ( t ≥ 0 ) , there exist constants C ≥ 1 and ν ∈ R , such that

(2.3) ‖ T ( t ) ‖ ≤ C e ν t , t ≥ 0 .

The constant

ν 0 = inf { ν ∈ R ∣ there exists C ≥ 1 , such that ∣ ∣ T ( t ) ∣ ∣ ≤ C e ν t , t ≥ 0 }

is called growth index of the C 0 -semigroup T ( t ) ( t ≥ 0 ) . If ν 0 < 0 , then T ( t ) ( t ≥ 0 ) is called an exponentially stable C 0 -semigroup.

Definition 2.1

[35] A C 0 -semigroup T ( t ) ( t ≥ 0 ) on E is said to be positive, if T ( t ) x ≥ θ for each x ≥ θ , x ∈ E , and t ≥ 0 .

Definition 2.2

[35] A C 0 -semigroup T ( t ) ( t ≥ 0 ) on E is called compact for t > 0 , if T ( t ) is a compact operator for every t > 0 .

If T ( t ) ( t ≥ 0 ) is continuous in the uniform operator topology for t > 0 , then ν 0 can also be expressed by spectral set σ ( A ) , i.e.,

ν 0 = − inf { Re λ : λ ∈ σ ( A ) } .

Moreover, if − A generates a positive and compact C 0 -semigroup T ( t ) ( t ≥ 0 ) , then for t > 0 , T ( t ) is continuous in the uniform operator topology. According to famous Krein-Rutman theorem, A has smallest eigenvalue λ 1 > 0 and the corresponding positive eigenfunction e 1 , and

λ 1 = inf { Re λ : λ ∈ σ ( A ) } ,

which implies that ν 0 = − λ 1 .

For any ν ∈ ( 0 , ∣ ν 0 ∣ ) , the equivalent norm ∣ ⋅ ∣ is defined by

∣ x ∣ = sup t ≥ 0 ∣ ∣ e ν t T ( t ) x ∣ ∣ ,

then ∣ ∣ x ∣ ∣ ≤ ∣ x ∣ ≤ C ∣ ∣ x ∣ ∣ . Denote ∣ T ( t ) ∣ the norm of T ( t ) in E , we have ∣ T ( t ) ∣ ≤ e − ν t .

To prove our main results, first, we consider the linear periodic bounded value problem and obtain the following lemma:

Lemma 2.1

Let A : D ( A ) ⊂ E → E be a linear operator and − A generates an exponentially stable positive C 0 -semigroup T ( t ) ( t ≥ 0 ) on E. For any r ∈ C ω ( I ω , E ) with r ≥ 0 , the linear periodic boundary value problem

(2.4) u ′ ( t ) + A u ( t ) = r ( t ) , t ∈ I ω = [ 0 , ω ] , u ( 0 ) = u ( ω )

has unique positive mild solution.

Proof

For any r ∈ C ω ( I ω , E ) , the solution of initial value problem (2.1) is given by (2.2). Thus, for initial value x 1 , x 2 ∈ E , and x 1 ≠ x 2 , we can define a Poincaré mapping

Π : x i ↦ u i ( ω , x i ) , i = 1 , 2 .

From calculation (2.2), for any ν ∈ ( 0 , ∣ ν 0 ∣ ) and t ∈ I ω , it follows that

∣ u 2 ( t ) − u 1 ( t ) ∣ ≤ ∣ T ( t ) x 2 − T ( t ) x 1 ∣ ≤ e − ν t ∣ x 2 − x 1 ∣ ,

and

∣ u 2 ( ω , x 2 ) − u 1 ( ω , x 1 ) ∣ = ∣ u 2 ( ω ) − u 1 ( ω ) ∣ ≤ ∣ T ( ω ) x 2 − T ( ω ) x 1 ∣ ≤ e − ν ω ∣ x 2 − x 1 ∣ .

Therefore, we have

∣ Π ( x 2 ) − Π ( x 1 ) ∣ ≤ e − ν ω ∣ x 2 − x 1 ∣ .

Obviously, Π is a contraction mapping. According to the positivity of operator r and semigroup T ( t ) , we can obtain that the periodic problem (2.6) has a unique positive mild solution u ∈ C ω ( I ω , E ) .□

Given r ∈ C ω ( R , E ) , we consider the existence of ω -periodic solution of the linear evolution equation

(2.5) u ′ ( t ) + A u ( t ) = r ( t ) , t ∈ R .

i.e., we consider the following linear periodic boundary value problem

(2.6) u ′ ( t ) + A u ( t ) = r ( t ) , t ∈ R , u ( t ) = u ( t + ω ) .

Lemma 2.2

Let A : D ( A ) ⊂ E → E be a linear operator and − A generates an exponentially stable positive C 0 -semigroup T ( t ) ( t ≥ 0 ) on E. For any r ∈ C ω ( R , E ) with r ≥ 0 , the linear evolution equation (2.6) has a unique positive periodic mild solution u given by

(2.7) u ( t ) = T ( t ) R ( r ) + ∫ 0 t T ( t − s ) r ( s ) d s ≔ Pr ( t ) ,

where R ( r ) = ( I − T ( ω ) ) − 1 [ ∫ 0 ω T ( ω − s ) r ( s ) d s ] and the solution operator P : C ω ( R , E ) → C ω ( R , E ) is a bounded linear operator with the spectral radius r ( P ) ≤ 1 ∣ v 0 ∣ = 1 λ 1 .

Proof

There exists M > 0 , such that for any ν ∈ ( 0 , ∣ ν 0 ∣ ) ,

‖ T ( t ) ‖ ≤ M e − ν t ≤ M , t ≥ 0 .

And we can define the equivalent norm ∣ ⋅ ∣ in E given by

∣ x ∣ = sup t ≥ 0 ‖ e ν t T ( t ) x ‖ ,

then we obtain that ‖ x ‖ ≤ ∣ x ∣ ≤ M ‖ x ‖ . Now, we denote the norm of T ( t ) in ( E , ∣ ⋅ ∣ ) , for x ∈ E and t ≥ 0 ,

∣ T ( t ) x ∣ = sup s ≥ 0 ‖ e ν s T ( s ) ⋅ T ( t ) x ‖ = sup s ≥ 0 ‖ e ν s T ( s + t ) x ‖ = e − ν t sup s ≥ 0 ‖ e ν ( s + t ) T ( s + t ) x ‖ ≤ e − ν t ∣ x ∣ ,

which means that ∣ T ( ω ) ∣ < e − ν ω < 1 . Hence, ( I − T ( ω ) ) has bounded inverse operator

( I − T ( ω ) ) − 1 = ∑ n = 0 ∞ T ( n ω ) ,

and its norm satisfies

∣ ( I − T ( ω ) ) − 1 ∣ ≤ 1 1 − ∣ T ( ω ) ∣ ≤ 1 1 − e − ν ω .

Let

(2.8) x 0 = ( ( I − T ( ω ) ) − 1 ) ∫ 0 ω T ( t − s ) r ( s ) d s ≔ R ( r ) ,

then the mild solution u ( t ) of the linear initial value problem (2.1) given by (2.2) satisfies the periodic boundary condition u ( 0 ) = u ( ω ) = x 0 . For t ∈ R , by (2.2) and the properties of the semigroup T ( t ) ( t ≥ 0 ) , we have

u ( t + ω ) = T ( t + ω ) u ( 0 ) + ∫ 0 t + ω T ( t + ω − s ) r ( s ) d s = T ( t ) ( T ( ω u ( 0 ) + ∫ 0 ω T ( ω − s ) r ( s ) d s ) ) + ∫ 0 t T ( t − s ) r ( s − ω ) d s = T ( t ) u ( 0 ) + ∫ 0 t T ( t − s ) r ( s ) d s = u ( t ) .

Therefore, the ω -periodic extension of u on R is a unique ω -periodic mild solution of equation (2.5). By (2.2), the ω -periodic mild solution can be shown by

u ( t ) = T ( t ) R ( r ) + ∫ 0 t T ( t − s ) r ( s ) d s ≔ Pr ( t ) .

Clearly, in view of the positivity of semigroup T ( t ) ( t ≥ 0 ) , we could obtain that P : C ω ( R , E ) → C ω ( R , E ) is a positive bounded linear operator. Thus, we obtain

‖ ( P r ) ( t ) ‖ ≤ ‖ T ( t ) ‖ ‖ R ( r ) ‖ + ∫ 0 t ‖ T ( t − s ) ‖ ‖ r ( s ) ‖ d s ≤ e − ν t ‖ R ( r ) ‖ + ∫ 0 t e − ν ( t − s ) d s ⋅ ‖ r ‖ C = e − ν t ‖ R ( r ) ‖ + 1 − e − ν t ν ‖ r ‖ C .

Since

‖ ( I − T ( ω ) ) − 1 ‖ = ∑ n = 0 ∞ T n ( ω ) ≤ ∑ n = 0 ∞ e − n ν ω = 1 1 − e − ν ω ,

by (2.6), we obtain that

‖ R ( r ) ‖ ≤ ‖ ( I − T ( ω ) ) − 1 ‖ ⋅ ∫ 0 ω T ( ω − s ) r ( s ) d s ≤ 1 1 − e − ν ω ∫ 0 ω ‖ T ( ω − s ) ‖ ‖ r ( s ) ‖ d s ≤ 1 1 − e − ν ω ∫ 0 ω e − ν ( ω − s ) d s ⋅ ‖ r ‖ C = 1 ν ‖ r ‖ C .

Thus, we have

‖ Pr ( t ) ‖ ≤ e − ν t ‖ R ( r ) ‖ + 1 − e − ν t ν ‖ r ‖ C ≤ 1 ν ‖ r ‖ C , t ∈ I ,

which shows that ‖ P ‖ ≤ 1 ν . Hence, r ( P ) ≤ ‖ P ‖ ≤ 1 ν . Therefore, by reason of the arbitrary of ν ∈ ( 0 , ∣ ν 0 ∣ ) , we can obtain r ( P ) ≤ 1 ∣ ν 0 ∣ = 1 λ 1 . The proof of Lemma 2.2 has been completed.□

3 Main results

The main results are given in this section. Obviously, for the partial order “ ≤ ,” C ω ( R , E ) is also an ordered Banach space induced by the positive cone K C = { u ∈ C ω ( R , E ) : u ( t ) ≥ 0 , t ∈ R } . Also, K C is normal with the normal constant N . We use [ v , w ] to denote the order interval { u ∈ C ω ( R , E ) : v ≤ u ≤ w } in C ω ( R , E ) , for v , w ∈ C ω ( R , E ) with v ≤ w , and [ v ( t ) , w ( t ) ] to denote the order interval { x ∈ E : v ( t ) ≤ x ( t ) ≤ w ( t ) , t ∈ R } in E .

Theorem 3.1

Let A : D ( A ) ⊂ E → E be a linear operator and − A generate an exponentially stable, positive, and compact C 0 -semigroup T ( t ) ( t ≥ 0 ) in E . If g : R × K × K → K is a continuous mapping, which is ω -periodic in t, and the following conditions are satisfied:

There exist a constant 0 < a + b < ∣ ν 0 ∣ and a function r ∈ C ω ( R , K ) , such that for any t ∈ R , x , y ∈ K ,
(3.1) g ( t , x , y ) ≤ a x + b y + r ( t ) , g ( t , − x , − y ) ≥ − a x − b y − r ( t ) ;
There exists a constant M ≥ 0 , for any t ∈ R + and v 0 ( t ) ≤ x 1 ≤ x 2 ≤ y 2 ≤ w 0 ( t ) , v 0 ( t ) ≤ y 1 ≤ y 2 ≤ x 2 ≤ w 0 ( t ) , such that
g ( t , x 2 , y 2 ) − g ( t , x 1 , y 2 ) ≥ − M ( x 2 − x 1 ) .

Then, in C ω ( R , K ) , equation (1.1) has at least one positive ω -periodic mild solution.

Proof

For r ( t ) ∈ C ω ( R , K ) in (H1), we investigate the following linear evolution equation:

(3.2) u ′ ( t ) + ( A − a I ) u ( t ) = r ( t ) + b u ( t − η ) , t ∈ R ,

in E . Since a + b < ∣ ν 0 ∣ , there is no doubt that − ( A + a I ) generates an exponentially stable, positive, and compact C 0 -semigroup S ( t ) ( t ≥ 0 ) = e a t T ( t ) and ‖ S ( t ) ‖ ≤ C e − ( ∣ ν 0 ∣ − a ) t . For any u ∈ K , by reason of Lemma 2.2, we can deduce that equation (3.2) has unique positive mild solution w 0 ( t ) ∈ C ω ( R , K ) and

w 0 ′ ( t ) + A w 0 ( t ) = r ( t ) + a w 0 ( t ) + b w 0 ( t − η ) ≥ g ( t , w 0 ( t ) , w 0 ( t − η ) ) .

Choosing w 0 as an upper solution of equation (1.1), obviously, let v 0 ( t ) = − w 0 ( t ) , then v 0 is a lower solution of equation (1.1). Now, let M > a be a constant in (H2), for r 1 ( t ) = g ( t , u ( t ) , u ( t − η ) ) + M u ( t ) ∈ C ω ( R , K ) , we consider the following evolution equation:

(3.3) u ′ ( t ) + A u ( t ) + M u ( t ) = r 1 ( t ) , t ∈ R .

In light of characteristics of T ( t ) ( t ≥ 0 ) , it is simplicity to know that − A + M I generates a positive compact C 0 -semigroup T 1 ( t ) ( t ≥ 0 ) = e − M t T ( t ) and ∣ ∣ T 1 ( t ) ∣ ∣ ≤ e − ( M + ∣ ν 0 ∣ ) t . For t ∈ R , from Lemma 2.2, the solution of equation (3.3) is given by

u ( t ) = T 1 ( t ) R ( r ) + ∫ 0 t T 1 ( t − s ) r 1 ( s ) d s ,

where R ( r ) = ( I − T 1 ( ω ) ) − 1 ∫ 0 ω T 1 ( ω − s ) r 1 ( s ) d s .

Now, we use the monotone iterative method to obtain the existence of mild solutions of equation (1.1) in [ v 0 , w 0 ] . For any u ∈ [ v 0 , w 0 ] , we define the mapping Q : [ v 0 , w 0 ] → C ω ( R , E ) by

(3.4) Q u ( t ) = T 1 ( t ) R ( r ) + ∫ 0 t T 1 ( t − s ) ( g ( s , u ( s ) , u ( s − η ) ) + M u ( s ) ) d s ,

where R ( r ) = ( I − T 1 ( ω ) ) − 1 ∫ 0 ω T 1 ( ω − s ) ( g ( s , u ( s ) , u ( s − η ) ) + M u ( s ) ) d s . Apparently, Q : [ v 0 , w 0 ] → C ω ( R , E ) is continuous. By Lemma 2.2, the mild solution of equation (1.1) is equivalent to the fixed point of the operator Q . By the assumptions (H2), Q is increasing in [ v 0 , w 0 ] .

We first show v 0 ≤ Q v 0 , Q w 0 ≤ w 0 . Let r 1 ( t ) = w 0 ′ ( t ) + A w 0 ( t ) + M w 0 ( t ) . By Lemma 2.2, we know that

w 0 ( t ) = T 1 ( t ) w 0 ( 0 ) + ∫ 0 t T 1 ( t − s ) r 1 ( s ) d s ≥ T 1 ( t ) w 0 ( 0 ) + ∫ 0 t T 1 ( t − s ) ( g ( s , w 0 ( s ) , w 0 ( s − η ) ) + M w 0 ( s ) ) d s .

Especially,

w 0 ( ω ) = T 1 ( ω ) w 0 ( 0 ) + ∫ 0 ω T 1 ( ω − s ) r 1 ( s ) d s .

Combining this inequality with periodic condition w 0 ( 0 ) = w 0 ( ω ) , we can obtain that

w 0 ( 0 ) = ( I − T 1 ( ω ) ) − 1 ∫ 0 ω T 1 ( ω − s ) r 1 ( s ) d s ≥ ( I − T 1 ( ω ) ) − 1 ∫ 0 ω T 1 ( ω − s ) ( g ( s , w 0 ( s ) , w 0 ( s − η ) ) + M w 0 ( s ) ) d s = R ( w 0 ) .

Moreover, for t ∈ R , we obtain that

Q ( w 0 ) ( t ) = T 1 ( t ) R ( w 0 ) + ∫ 0 t T 1 ( t − s ) ( g ( s , w 0 ( s ) , w 0 ( s − η ) ) + M w 0 ( s ) ) d s .

Hence, for all t ∈ R , w 0 ( t ) − Q ( w 0 ) ( t ) ≥ T 1 ( t ) ( w 0 ( 0 ) − R ( w 0 ) ) ≥ θ . It implies that Q w 0 ≤ w 0 . Similarly, it can be shown that v 0 ≤ Q v 0 . Therefore, Q : [ v 0 , w 0 ] → [ v 0 , w 0 ] is a continuously increasing operator.

Next we establish two sequences { v n } and { w n } in [ v 0 , w 0 ] by the iterative schemes

(3.5) v n = Q v n − 1 , w n = Q w n − 1 , n = 1 , 2 , … ,

hence in view of the monotonicity of Q , which shows that

(3.6) v 0 ≤ v 1 ≤ v 2 ≤ … ≤ v n ≤ … ≤ w n ≤ … ≤ w 2 ≤ w 1 ≤ w 0 .

Next we prove that { v n } and { w n } are uniformly convergent. Let B = { w n ∣ n ∈ N } , and B 0 = { w n − 1 ∣ n ∈ N } , then B 0 = { w 0 } ∪ B and B = Q B 0 . For any w n − 1 ∈ B 0 , let

( W w n − 1 ) ( t ) = ∫ 0 t T 1 ( t − s ) ( g ( s , w n − 1 ( s ) , w n − 1 ( s − η ) ) + M w n − 1 ( s ) ) d s , ( P w n − 1 ) ( r ) = ( I − T 1 ( ω ) ) − 1 ( ( W w n − 1 ) ( ω ) ) ,

thus, Q ( w n − 1 ) ( t ) = ( P w n − 1 ) ( r ) + ( W w n − 1 ) ( t ) . We prove that for any 0 < t ≤ ω , W ( t ) = { ( W w n − 1 ) ( t ) : w n − 1 ∈ B 0 } is precompact in E . For 0 < ε < t and w n − 1 ∈ B 0 ,

(3.7) ( W ε w n − 1 ) ( t ) = ∫ 0 t − ε T 1 ( t − s ) ( g ( s , w n − 1 ( s ) , w n − 1 ( s − η ) ) + M w n − 1 ( s ) ) d s = T 1 ( ε ) ∫ 0 t − ε T 1 ( t − s − ε ) ( g ( s , w n − 1 ( s ) , w n − 1 ( s − η ) ) + M w n − 1 ( s ) ) d s .

With the assumption (H2), for x ∈ [ v 0 ( t ) , w 0 ( t ) ] , we obtain

g ( s , v 0 ( s ) , v 0 ( s − η ) ) + M v 0 ( s ) ≤ g ( s , x ( s ) , x ( s − η ) ) + M x ( s ) ≤ g ( s , w 0 ( s ) , w 0 ( s − η ) ) + M w 0 ( s ) .

As the cone K is normal, there exists ℳ > 0 , such that

‖ g ( s , w n − 1 ( s ) , w n − 1 ( s − η ) ) + M w n − 1 ( s ) ‖ ≤ ℳ , w n − 1 ∈ B 0 .

By the compactness of T 1 ( ε ) , W ε ( t ) = { ( W ε w n − 1 ) ( t ) : w n − 1 ∈ B 0 } is precompact in E . Let M ¯ = sup t ∈ I ‖ T 1 ( t ) ‖ , since

‖ ( W w n − 1 ) ( t ) − ( W ε w n − 1 ) ( t ) ‖ ≤ ∫ t − ε t ‖ T 1 ( t − s ) ‖ ⋅ ∥ ( g ( s , w n − 1 ( s ) , w n − 1 ( s − η ) ) + M w n − 1 ( s ) ) ∥ d s ≤ ε M ¯ ℳ ,

the set W ( t ) is totally bounded in E . By the completeness of E , we know that W ( t ) is precompact in E . Using the same method, we can obtain that V is precompact, it implies that ( P w n − 1 ) ( r ) is precompact.

For 0 ≤ t ≤ ω , since { Q w n − 1 ( t ) : w n − 1 ∈ B 0 } = { ( P w n − 1 ) ( r ) + ( W w n − 1 ) ( t ) : w n − 1 ∈ B 0 } , and Q w n − 1 ( 0 ) = R ( r ) = w n − 1 ( ω ) are precompact in E . Therefore, { w n ( t ) } = { Q w n − 1 ( t ) : w n − 1 ∈ B 0 } is precompact in C ω ( R , E ) , by the monotonicity of { w n } , we can prove that { w n } itself is uniformly convergent. Let { w n ( t ) } → u ¯ ( t ) for t ∈ R . Similarly, we have { v n ( t ) } → u ̲ ( t ) for t ∈ R .

Clearly, u ̲ ( t ) and u ¯ ( t ) are bounded integrable in R , for any t ∈ R , v n ( t ) = Q ( v n − 1 ) ( t ) , w n ( t ) = Q ( w n − 1 ) ( t ) , letting n → ∞ , we have u ̲ ( t ) = Q u ̲ ( t ) , u ¯ ( t ) = Q u ¯ ( t ) by Lebesgue dominated convergence theorem. Therefore, equation (1.1) has at least one positive ω -periodic mild solution in C ω ( R , K ) .□

Next we assume that the positive cone K is a regeneration cone. By reason of the characteristic of positive semigroup [36], we have that λ 0 I + A has positive bounded inverse operator ( λ 0 I + A ) − 1 for sufficiently large λ 0 > − inf { Re λ ∣ λ ∈ σ ( A ) } . Since σ ( A ) ≠ ϕ , the spectral radius r ω ( λ 0 I + A ) − 1 = 1 dist ( − λ 0 , σ ( A ) ) > 0 . By reason of the famous Krein-Rutman theorem, we know that A has the first eigenvalue λ 1 , which has a positive eigenfunction e 1 , and

λ 1 = inf { Re λ ∣ λ ∈ σ ( A ) } ,

which implies that ν 0 = − λ 1 . Hence, in the light of Theorem 3.1, the following corollary is obtained.

Corollary 3.1

Let A : D ( A ) ⊂ E → E be a linear operator and − A generate an exponentially stable, positive, and compact C 0 -semigroup T ( t ) ( t ≥ 0 ) in E. If g ∈ C ( I × E , E ) and the following conditions are satisfied:

There exist a constant 0 < a + b < λ 1 and a function r 0 ∈ C ω ( R , K ) , such that for any t ∈ R , x , y ∈ K ,
(3.8) g ( t , x , y ) ≤ a x + b y + r 0 ( t ) , g ( t , − x , − y ) ≥ − a x − b y − r 0 ( t ) ;
There exists a constant M ≥ 0 , for any t ∈ R + and v 0 ( t ) ≤ x 1 ≤ x 2 ≤ y 2 ≤ w 0 ( t ) , v 0 ( t ) ≤ y 1 ≤ y 2 ≤ x 2 ≤ w 0 ( t ) , such that
g ( t , x 2 , y 2 ) − g ( t , x 1 , y 2 ) ≥ − M ( x 2 − x 1 ) .

Then, equation (1.1) has at least one positive ω -periodic mild solution in C ω ( R , K ) .

Replacing conditions ( H 1 ) ′ by the following assumptions:

There exist a constant 0 < a + b < λ 1 , a function r ∈ C ω ( R , E ) , and a positive constant σ , such that
g ( t , x , y ) ≤ a x + b y + r ( t ) , x , y ∈ K , t ∈ R ,

g ( t , x , y ) ≥ λ 1 x , 0 ≤ x ≤ σ .

Consequently, the existence result of positive mild solution is obtained as follows.

Theorem 3.2

Let A : D ( A ) ⊂ E → E be a linear operator and − A generate an exponentially stable, positive, and compact C 0 -semigroup T ( t ) ( t ≥ 0 ) . If g : R × K × K → K is continuous and conditions ( H 1 ) ″ and (H2) hold, then equation (1.1) has at least one positive ω -periodic mild solution in C ω ( R , K ) .

Proof

According to the proof of Theorem 3.1 and conditions ( H 1 ) ″ , equation (1.1) has an upper solution w 0 . Further, in view of the compactness of C 0 -semigroup T ( t ) ( t ≥ 0 ) , we can deduce that v 0 = σ e 1 ( σ is a small enough nonnegative constant) is a lower solution of equation (1.1). From the monotone iterative technique, we only prove that v 0 ≤ w 0 . By Riesz-Schauder spectral theorem and eigenvalue problem, we can obtain that A e 1 = λ 1 e 1 . Choosing σ ∈ ( 0 , H e 1 ( λ 1 − ( a + b ) ) ) , H = min t ∈ R r ( t ) , for any t ≥ 0 , let u ( t ) = w 0 ( t ) − v 0 ( t ) , we have

u ′ ( t ) + A u ( t ) = ( w 0 ( t ) − v 0 ( t ) ) ′ + A ( w 0 ( t ) − v 0 ( t ) ) = a w 0 ( t ) + b w 0 ( t − η ) + r ( t ) − λ 1 σ e 1 = a u ( t ) + a σ e 1 + b u ( t − η ) + b σ e 1 + r ( t ) − λ 1 σ e 1 = a u ( t ) + b u ( t − η ) + r ( t ) + ( a + b − λ 1 ) σ e 1 ≥ a u ( t ) + b u ( t − η ) .

Moreover, we can apply the maximum principle to ensure that u ( t ) ≥ 0 . Therefore, we can deduce that equation (1.1) has at least one positive ω -periodic mild solution in C ω ( R , K ) .□

4 Example

An example is given to demonstrate the applicability of the abstract results in this section. Let Ω ⊂ R N ( N ≥ 1 ) be a bounded domain with a sufficiently smooth boundary ∂ Ω , f : Ω ¯ × R × R 2 → R be continuous. Consider the impulsive parabolic periodic boundary problem

(4.1) ∂ t u ( x , t ) − Δ u ( x , t ) = f ( x , t , u ( x , t ) , u ( x , t − τ ) ) , x ∈ Ω , t ∈ R , u ∣ ∂ Ω = 0 .

Let E = L 2 ( Ω ) , P = { u ∈ L 2 ( Ω ) : u ( x ) ≥ 0 , a.e. x ∈ Ω } , then P is a regular cone of E . And define the operator A in E as follows:

D ( A ) = { u ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) : u ∣ ∂ Ω = 0 } , A u = − Δ u ,

from [35], which implies that − A generates a positive, exponentially stable, and analytic C 0 -semigroup T ( t ) ( t ≥ 0 ) with growth index ν 0 = − λ 1 . According to analyticity of T ( t ) and compactness of resolvent of A , we can obtain that T ( t ) is also a compact semigroup in E . Hence, we have existence result as follows:

Theorem 4.1

Assume λ 1 > 0 is the first eigenvalue of Laplace operator − Δ with boundary condition u ∣ ∂ Ω = 0 . The function f ∈ C ( Ω ¯ × R × R 2 , R ) and f ( x , t , ξ , ψ ) has continuous partial derivatives f ξ ′ ( x , t , ξ , ψ ) and f ψ ′ ( x , t , ξ , ψ ) with f ( x , t , ξ , ψ ) > θ . If the following listed conditions are satisfied:

For any t ∈ R , x ∈ Ω ¯ , ψ ∈ R , ξ ∈ R with ξ ≥ 0 , there exists positive constants a, b satisfying 0 < a + b < λ 1 such that
0 < sup t ∈ R f ξ ′ ( x , t , ξ , ψ ) < a , 0 < sup t ∈ R f ψ ′ ( x , t , ξ , ψ ) < b .
There exists a constant M ≥ 0 , for any t ∈ R + , x ∈ Ω ¯ , and v 0 ( t ) ≤ ξ 1 ≤ ξ 2 ≤ w 0 ( t ) , v 0 ( t ) ≤ ψ 1 ≤ ψ 2 ≤ w 0 ( t ) , such that
f ( x , t , ξ 2 , ψ 2 ) − f ( x , t , ξ 1 , ψ 2 ) ≥ − M ( ξ 2 − ξ 1 ) .

Then, the delay parabolic boundary value problem (4.1) has at least one positive ω -periodic solution u ∈ C 2,1 ( Ω ¯ × R ) .

Proof

Let u ( t ) = u ( ⋅ , t ) , g ( t , u ( t ) , u ( t − η ) ) = f ( ⋅ , t , u ( ⋅ , t ) , u ( ⋅ , t − η ) ) , then the delay parabolic boundary value problem (4.1) can be transformed into the abstract evolution equation (1.1) in E , then g : R × E × E → E is continuous in v and partial derivative g u ′ ( t , u , v ) , g v ′ ( t , u , v ) is bounded. It is obvious that f satisfies the conditions ( H 1 ) ′ and (H2). From Corollary 3.1, we can obtain that the parabolic periodic boundary value problem (4.1) has at least one positive mild solution u ∈ L loc 2 ( R , H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) ∩ W loc 1 , 2 ( R , L 2 ( Ω ) ) . By the analytic of T ( t ) and regularization method used in [37], we can know that u ∈ C 2,1 ( Ω ¯ × R ) is a classical time ω -periodic solution to equation (4.1).□

Corollary 4.1

Let λ 1 > 0 be the first eigenvalue of Laplace operator − Δ with boundary condition u ∣ ∂ Ω = 0 . g ∈ C ( Ω ¯ × R × R 2 , R ) and f ( x , t , ξ , ψ ) have continuous partial derivatives f ξ ′ ( x , t , ξ , ψ ) and f ψ ′ ( x , t , ξ , ψ ) with f ( x , t , ξ , ψ ) > θ . If the following conditions hold:

For any t ∈ R , x ∈ Ω ¯ , ψ ∈ R , ξ ∈ R with ξ ≥ 0 ,
lim ¯ ξ → + ∞ sup t ∈ R f ξ ′ ( x , t , ξ , ψ ) < λ 1 , lim ¯ ξ → 0 + inf t ∈ R f ψ ′ ( x , t , ξ , ψ ) > λ 1 .
There exists a constant M ≥ 0 , such that
f ( x , t , ξ 2 , ψ 2 ) − f ( x , t , ξ 1 , ψ 2 ) ≥ − M ( ξ 2 − ξ 1 )
for any t ∈ R + , x ∈ Ω ¯ , and v 0 ( t ) ≤ ξ 1 ≤ ξ 2 ≤ w 0 ( t ) , v 0 ( t ) ≤ ψ 1 ≤ ψ 2 ≤ w 0 ( t ) .

Then, the delay parabolic boundary value problem (4.1) has at least one positive ω -periodic solution u ∈ C 2,1 ( Ω ¯ × R ) .

Proof

Let u ( t ) = u ( ⋅ , t ) , g ( t , u ( t ) , u ( t − η ) ) = f ( ⋅ , t , u ( ⋅ , t ) , u ( ⋅ , t − η ) ) , then the delay parabolic boundary value problem (4.1) can be transformed into the abstract evolution equation (1.1) in E , then g : R × E × E → E is continuous in v and partial derivatives g u ′ ( t , u , v ) and g v ′ ( t , u , v ) are bounded. It is obvious that f satisfies the conditions ( H 1 ) ″ and (H2). From Theorem 3.2, we can obtain that the parabolic periodic boundary value problem (4.1) has at least one positive mild solution u ∈ L loc 2 ( R , H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) ∩ W loc 1 , 2 ( R , L 2 ( Ω ) ) . By the analytic of T ( t ) and regularization method used in [37], we can know that u ∈ C 2,1 ( Ω ¯ × R ) is a classical time ω -periodic solution to equation (4.1).□

Acknowledgment

The authors would like to express their deep thanks to the referees for many helpful comments and suggestions.

Funding information: This work was supported by Science Research Project for Colleges and Universities of Gansu Province (No. 2022A-010) Project of NWNU-LKQN2023-02 and Natural Science Foundation of Gansu Province (No. 25JRRA362).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflicts of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2023-10-21

Revised: 2025-01-23

Accepted: 2025-05-10

Published Online: 2025-06-23

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2025-0165

Keywords for this article

monotone iterative method; periodic solutions; compact semigroup; Poincaré mapping; positive C 0-semigroup; evolution equations with delay

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