Solvability for a nonlocal dispersal model governed by time and space integrals

1 Introduction and the main result

In this article, we are concerned with the following periodic problem of a nonlocal dispersal model governed by a Volterra type integral and two space integrals:

where ω ≥ 0 , f , k , a , K , W , φ , and b are given functions. The main assumptions imposed on the aforementioned symbols will be introduced later in this article. Here, the nonlocal dispersal model (1.1) is a model with a feedback control and two space integral kernel for describing biological invasion and disease spread. The feedback control W in the model (1.1) depends on the weighted values of u ( t , ⋅ ) over the whole habitat ∫ R φ ( η ) u ( t , η ) d η ; (for more introduction, see Malaguti and Rubbioni [1]). The time-continuous kernel k in the model (1.1) is involved in the Volterra type integral:

In particular, when k ( t , s ) = e − ( t − s ) ∕ T T , it accounts for a rapidly decreasing memory effect. The integral kernel K in the model (1.1) being involved in the space integral

accounts for long-distance interactions between individuals. We study the model that governed the time integral and the space integral at the same time. The model in this article is important in applications that introduce a time memory effect and a long-distance interaction concurrently. As far as we know, there are few papers discussing a model with Volterra-type integral and space integral simultaneously.

If the nonlinearity f in (1.1) is replaced by f ( t , u ( t , ξ ) ) , then (1.1) becomes a model only with one kernel K and the feedback control W (see [1, (4)]). In this case, it is also for many phenomena. We refer the reader to [1] and the references therein. In particular, for the case that without the feedback control W , several studies involving the space integral (1.3) and its analogous have recently been published. Alves et al. [2] obtained the existence of positive solution for a class of nonlocal problems. Benedetti et al. [3] studied a second-order partial differential equation with a space integral. They used an approximation solvability method, which combines a Schauder degree argument with an Hartman-type inequality, to prove existence of periodic solutions. Eigentler and Sherratt [4] considered the “nonlocal Klausmeier model,” i.e., a coupled reaction-advection-diffusion system. Hutson et al. [5], Jin and Zhao [6] and Wang et al. [7] all studied first-order differential equation with an integral diffusion term.

Equation (1.1) becomes a one without the feedback control W and the space integral (1.3) (see [8]) and is also a model in many applications, such as population dynamics, biology, and epidemiology. In recent years, the model involving the time integral (1.2) and its analogous has been studied in many papers. In [8], Bungardi et al. obtained an existence of mild solutions to a nonlocal semilinear integro-differential problem with a Volterra integral. They developed a vector valued measure of noncompactness involving a Volterra integral operator. It is worth mentioning that the nonlocal function there assumed to be a compact operator. Zhuang et al. [9] studied the periodic boundary value problems for integro-differential equations of Volterra type, concerning with a monotone method. Yu et al. [10] obtained the global solvability for a nonlinear Volterra delay evolution inclusion subjected to a nonlocal implicit initial condition.

Our article is inspired from the main results in [1,8]. Our aim is to prove an existence result for the periodic problem (1.1). The main methods we use in this article is that the one given in [1], i.e., the degree theory referring to condensing operators in Hilbert spaces. We combine it with a suitably assumed Hartman-type sign condition. We use a sufficiently regular measure of noncompactness (MNC) similarly defined as the one given in [8]. This case is important in applications and does not follow by a direct modification of the arguments in both [1,8]. Here, let us mention that the periodic condition is not a compactness assumption.

A function u : [ 0 , ω ] × R → R is said to be a solution of equation (1.1) if it satisfies the following conditions:

• u ( t , ⋅ ) ∈ L 2 ( R ) ;

• x : [ 0 , ω ] → L 2 ( R ) defined by x ( t ) = u ( t , ⋅ ) for all t ∈ [ 0 , ω ] ;

• x ∈ A C ( [ 0 , ω ] ; L 2 ( R ) ) ;

• x satisfies the equation in (1.1) for a.e. t ∈ [ 0 , ω ] .

To begin with, let us consider the following conditions:

1. f : [ 0 , ω ] × R × R → R is a function such that

   1. f ( ⋅ , p ˜ , q ˜ ) is measurable for every p ˜ , q ˜ ∈ R ;

   2. f ( t , 0 , 0 ) = 0 for a.e. t ∈ [ 0 , ω ] ;

   3. f ( t , ⋅ , ⋅ ) is α ( t ) -Lipschitz continuous for a.e. t ∈ [ 0 , ω ] , i.e., there exists α ∈ L + 1 ( 0 , ω ) such that for each ( p ˜ 1 , q ˜ 1 ) , ( p ˜ 2 , q ˜ 2 ) ∈ R × R ,

      ∣ f ( t , p ˜ 1 , q ˜ 1 ) − f ( t , p ˜ 2 , q ˜ 2 ) ∣ ≤ α ( t ) ( ∣ p ˜ 1 − p ˜ 2 ∣ + ∣ q ˜ 1 − q ˜ 2 ∣ ) a.e. t ∈ [ 0 , ω ] .

2. k : Δ → R is a continuous function, where Δ ≔ { ( t , s ) ∈ [ 0 , ω ] × [ 0 , ω ] ; 0 ≤ s ≤ t ≤ ω } and φ ∈ L 2 ( R ) , ∣ ∣ φ ∣ ∣ L 2 ( R ) = 1 ;

3. a ∈ L + 1 ( 0 , ω ) and K : R × R → R satisfies

   K ∈ L 2 ( R × R ) , ∣ ∣ K ∣ ∣ L 2 ( R × R ) = 1 .

4. W : R × R → R is a function such that f 1 ( ξ , r ) ≤ W ( ξ , r ) ≤ f 2 ( ξ , r ) for every ( ξ , r ) ∈ R × R with

   1. W ( ⋅ , r ) ∈ L 2 ( R ) ∩ A C loc ( R ) for every r ∈ R and satisfying

      ∂ W ( ξ , r ) ∂ ξ ≤ l ( ξ ) for a.e. ξ ∈ R and every r ∈ R ,

      where l ∈ L 1 ( R ) ;

   2. f i : R × R → R are functions such that for i = 1 , 2 ,

      f i ( ⋅ , r ) ∈ L 2 ( R ) ∩ A C loc ( R ) for every r ∈ R

      and

      ∂ f i ( ξ , r ) ∂ ξ ≤ l ( ξ ) for a.e. ξ ∈ R and every r ∈ R ;

   3. there exists ψ : R × R + 0 → R such that ψ ( ⋅ , M ) ∈ L 2 ( R ) for every M > 0 and

      ∣ f i ( ξ , r ) ∣ ≤ ψ ( ξ , M ) for every ( ξ , r ) ∈ R × [ − M , M ] ;

   4. f 1 ( ξ , r ) ≤ f 2 ( ξ , r ) for every ( ξ , r ) ∈ R × R and

      f 1 ( ξ , r 0 ) ≥ lim sup r → r 0 f 1 ( ξ , r ) , f 2 ( ξ , r 0 ) ≤ lim inf r → r 0 f 2 ( ξ , r )

      for every ξ ∈ R and r 0 ∈ R .

5. b : [ 0 , ω ] × R → R is a measurable function and satisfies that there exist s 1 , s 2 ∈ L + 1 ( 0 , ω ) such that

   0 < s 1 ( t ) ≤ b ( t , ξ ) ≤ s 2 ( t ) for every ( t , ξ ) ∈ [ 0 , ω ] × R .

We now proceed to the statement of the main result in this article.

2 Reformulation of the model

Let X be a Banach space. Denote by ‖ ⋅ ‖ X the norms of X , ‖ ⋅ ‖ ℒ ( X ) the operator norm of linear and bounded operators on X . We reformulate equation (1.1) as an ordinary differential equation in the Hilbert space L 2 ( R ) . To this aim, we give the following definitions, and for these well-defined properties, see Section 3.

1. The Volterra integral operator k ˆ : C ( [ 0 , ω ] ; L 2 ( R ) ) → C ( [ 0 , ω ] ; L 2 ( R ) ) , which is defined by

   k ˆ x ( t ) = ∫ 0 t k ( t , s ) x ( s ) d s for all t ∈ [ 0 , ω ] and x ∈ C ( [ 0 , ω ] ; L 2 ( R ) ) .

2. The map f ˆ : [ 0 , ω ] × L 2 ( R ) × L 2 ( R ) → L 2 ( R ) , which is defined, for every ( t , x ˜ , y ˜ ) ∈ [ 0 , ω ] × L 2 ( R ) × L 2 ( R ) , by

   f ˆ ( t , x ˜ , y ˜ ) ( ξ ) = f ( t , x ˜ ( ξ ) , y ˜ ( ξ ) ) for ξ ∈ R .

   Particularly, we see that f ˆ ( t , x ˜ , y ˜ ) = f ( t , x ˜ ( ⋅ ) , y ˜ ( ⋅ ) ) for every ( t , x ˜ , y ˜ ) ∈ [ 0 , ω ] × L 2 ( R ) × L 2 ( R ) .

3. Define K ˆ : [ 0 , ω ] × L 2 ( R ) → L 2 ( R ) as follows:

   K ˆ ( t , x ˜ ) ( ξ ) = a ( t ) ∫ R K ( ξ , η ) x ˜ ( η ) d η for ξ ∈ R

   for every ( t , x ˜ ) ∈ [ 0 , ω ] × L 2 ( R ) .

4. Define the multimap W ˆ : L 2 ( R ) ⇝ L 2 ( R ) as

   W ˆ ( x ˜ ) = y ˜ ∈ L 2 ( R ) ∩ A C loc ( R ) : ∣ y ˜ ′ ( ξ ) ∣ ≤ l ( ξ ) for a.e. ξ ∈ R and f 1 ( ξ , ∫ R φ ( η ) x ˜ ( η ) d η ) ≤ y ˜ ( ξ ) ≤ f 2 ( ξ , ∫ R φ ( η ) x ˜ ( η ) d η ) for all ξ ∈ R

   for every x ˜ ∈ L 2 ( R ) .

5. The linear operator A ( t ) : L 2 ( R ) → L 2 ( R ) , for every t ∈ [ 0 , ω ] is defined, for every x ˜ ∈ L 2 ( R ) , by

   A ( t ) x ˜ ( ξ ) = − b ( t , ξ ) x ˜ ( ξ ) for every ξ ∈ R .

Now, we can write the reformulation of equation (1.1) as the following semilinear evolution inclusion in the Hilbert space L 2 ( R ) :

Obviously, the solutions for (2.1) with x ( 0 ) = x ( ω ) give rise to the one for (1.1).

3 Preliminaries

For the Volterra integral operator k ˆ presented in Section 2, we investigate firstly its main properties.

Then we can introduce a MNC υ in C ( [ 0 , ω ] ; L 2 ( R ) ) defined by, for every bounded Ω ⊂ C ( [ 0 , ω ] ; L 2 ( R ) ) ,

where

where mod C is the modulus of equicontinuity in C ( [ 0 , ω ] ; L 2 ( R ) ) , which has the following form:

for each bounded Ω ⊂ C ( [ 0 , ω ] ; L 2 ( R ) ) . Obviously, mod C ( Ω ) = 0 implies that Ω ( ⋅ ) is equicontinuous. The range for υ is the cone R + 2 and the maximum is taken in the sense of the partial ordering induced by this cone.

To prove our main result, we state the following version for Hausdorff measures of noncompactness in [12, Corollary 4.2.5, p. 113].

We next describe the main properties of the nonlinear term f ˆ .