On approximate solutions for a class of semilinear fractional-order

Kamenskii et al. Fixed Point Theory and Applications (2017) 2017:28 DOI 10.1186/s13663-017-0621-0

RESEARCH

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On approximate solutions for a class of semilinear fractional-order differential equations in Banach spaces Mikhail Kamenskii1 , Valeri Obukhovskii2,3 , Garik Petrosyan2 and Jen-Chih Yao4* *

Correspondence: [email protected] 4 Center for General Education, China Medical University, Taichung, 40402, Taiwan Full list of author information is available at the end of the article

Abstract We apply the topological degree theory for condensing maps to study approximation of solutions to a fractional-order semilinear differential equation in a Banach space. We assume that the linear part of the equation is a closed unbounded generator of a C0 -semigroup. We also suppose that the nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We justify the scheme of semidiscretization of the Cauchy problem for a differential equation of a given type and evaluate the topological index of the solution set. This makes it possible to obtain a result on the approximation of solutions to the problem. MSC: Primary 34A45; secondary 34A08; 34G20; 47H08; 47H10; 47H11 Keywords: fractional differential equation; semilinear differential equation; Cauchy problem; approximation; semidiscretization; index of the solution set; fixed point; condensing map; measure of noncompactness

1 Introduction The problem of approximation of solutions to semilinear differential equations in Banach spaces has attracted the attention of many researchers (see, e.g., [–], and the references therein). In these works, it was supposed that the linear part of the equation is a compact analytic semigroup or condensing C -semigroup. Recently this approach was extended to the case of a fractional-order semilinear equation in a Banach space. In particular, Liu, Li, and Piskarev [], using the general approximation scheme of Vainikko [], considered approximation of the Cauchy problem for the case where the linear part generates an analytic and compact resolution operator and the corresponding nonlinearity is sufficiently smooth. Our main goal in this paper is to apply the topological degree theory for condensing maps to study the problem of approximation of solutions to a Caputo fractional-order semilinear differential equation in a Banach space under the assumption that the linear part of the equation is a closed unbounded generator of a C -semigroup. It is also supposed that the nonlinearity is a continuous map satisfying a regularity condition expressed in terms of the Hausdorff measure of noncompactness. It is worth noting that the interest to the theory of differential equations of fractional order essentially strengthened in the recent years due to interesting applications in physics, © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Kamenskii et al. Fixed Point Theory and Applications (2017) 2017:28

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enginery, biology, economics, and other branches of natural sciences (see, e.g., monographs [–], and the references therein). Among a large amount of papers dedicated to fractional-order equations, let us mention works [–], where existence results of various types were obtained. Notice that results on the existence of solutions to the Cauchy and the periodic problems for semilinear differential inclusions in a Banach space under conditions similar to the above mentioned were obtained in the authors’ papers [, ]. In this paper, we justify the scheme of semidiscretization of the Cauchy problem for a differential equation of a given type and evaluate the topological index of the solution set. This makes it possible to present the main result of the paper (Theorem ) on the approximation of solutions to the above problem.

2 Differential equations of fractional order In this section, we recall some notions and definitions (details can be found, e.g., in [, , ]). Let E be a real Banach space. Definition  The Riemann-Liouville fractional derivative of order q ∈ (, ) of a continuous function g : [, a] → E is the function Dq g of the following form: Dq g(t) =

d  ( – q) dt

t

(t – s)–q g(s) ds, 

provided that the right-hand side of this equality is well defined. Here is the Euler gamma function

∞

(r) =

sr– e–s ds.



Definition  The Caputo fractional derivative of order q ∈ (, ) of a continuous function g : [, a] → E is the function C Dq g defined in the following way: C

Dq g(t) = Dq g(·) – g() (t),

provided that the right-hand side of this equality is well defined. Consider the Cauchy problem for a Caputo fractional-order semilinear differential equation of the form C

Dq x(t) = Ax(t) + f t, x(t) ,

t ∈ [, a],

(.)

with the initial condition x() = x ,

(.)

where  < q < , and a linear operator A satisfies the following condition: (A) A : D(A) ⊆ E → E is a linear closed (not necessarily bounded) operator generating a C -semigroup {U(t)}t≥ of bounded linear operators in E,


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and f : [, a] × E → E is a continuous map obeying the following properties: (f ) f maps bounded sets into bounded ones; (f ) there exists μ ∈ R+ such that, for every bounded set ⊂ E, we have: χ f (t, ) ≤ μχ() for t ∈ [, a], where χ is the Hausdorff measure of noncompactness in E: χ() = inf{ε > , for which has a finite ε-net in E}. Notice that we may assume, without loss of generality, that the semigroup {U (t)}t≥ is bounded: sup U (t) = M < ∞. t≥

Otherwise, we may replace the operator A with the operator Ax = Ax – ωx and, respec tively, the nonlinearity f with f (t, x) = f (t, x) + ωx, where ω >  is a constant greater than the order of growth of the semigroup. Definition  (cf. [, , ]) A mild solution of problem (.)-(.) is a function x ∈ C([, a]; E) that can be represented as x(t) = G (t)x +

t

(t – s)q– T (t – s)f s, x(s) ds,

t ∈ [, a],



where

G (t) =

∞

ξq (θ )U t q θ dθ ,



 ξq (θ ) = θ q

∞

T (t) = q

θ ξq (θ )U t q θ dθ ,

 –– q

q θ –/q ,

∞  (nq + ) q (θ ) = sin(nπq), (–)n– θ –qn– π n= n!

Remark  (See, e.g., []) ξq (θ ) ≥ ,

∞ 

ξq (θ ) dθ = ,

θ ∈ R+ . ∞ 

θ ξq (θ ) dθ =

 . (q+)

Lemma  (See [], Lemma .) The operator functions G and T possess the following properties: () For each t ∈ [, a], G (t) and T (t) are linear bounded operators; more precisely, for each x ∈ E, we have G (t)x ≤ M x E , E

(.)

T (t)x ≤ E

(.)

qM x E . ( + q)

() The operator functions G and T are strongly continuous, that is, the functions t ∈ [, a] → G (t)x and t ∈ [, a] → T (t)x are continuous for each x ∈ E.


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2.1 Measures of noncompactness and condensing maps Let E be a Banach space. Introduce the following notation: • Pb(E ) = {A ⊆ E : A = ∅ is bounded}; • Pv(E ) = {A ∈ Pb(E ) : A is convex}; • K(E ) = {A ∈ Pb(E ) : A is compact}; • Kv(E ) = Pv(E ) ∩ K(E ). Definition  (See, e.g., [, ]) Let (A, ≥) be a partially ordered set. A function β : Pb(E ) → A is called a measure of noncompactness (MNC) in E if for each ∈ Pb(E ), we have: β(co ) = β( ), where co denotes the closure of the convex hull of . A measure of noncompactness β is called: () monotone if for each  ,  ∈ Pb(E ),  ⊆  implies β(  ) ≤ β(  ); () nonsingular if for each a ∈ E and each ∈ Pb(E ), we have β({a} ∪ ) = β( ). If A is a cone in a Banach space, then the MNC β is called: () regular if β( ) =  is equivalent to the relative compactness of ∈ Pb(E ); () real if A is the set of all real numbers R with natural ordering; () algebraically semiadditive if β(  +  ) ≤ β(  ) + β(  ) for all  ,  ∈ Pb(E ). It should be mentioned that the Hausdorff MNC obeys all above properties. Other examples can be presented by the following measures of noncompactness defined on Pb(C([, a]; E)), where C([, a]; E) is the space of continuous functions with values in a Banach space E: (i) the modulus of fiber noncompactness ϕ( ) = sup χE (t) , t∈[,a]

where χE is the Hausdorff MNC in E, and (t) = {y(t) : y ∈ }; (ii) the modulus of equicontinuity defined as modC ( ) = lim sup max y(t ) – y(t ). δ→ y∈ |t –t |≤δ

Notice that these MNCs satisfy all above-mentioned properties except regularity. Nevertheless, notice that due to the Arzelà-Ascoli theorem, the relation ϕ( ) = modC ( ) =  provides the relative compactness of the set . Definition  A continuous map F : X ⊆ E → E is called condensing with respect to an MNC β (or β-condensing) if for every bounded set ⊆ X that is not relatively compact, we have β F ( ) β( ).


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More generally, given a metric space of parameters, we say that a continuous map : × X → E is a condensing family with respect to an MNC β (or β-condensing family) if for every bounded set ⊆ X that is not relatively compact, we have β ( × ) β( ). Let V ⊂ E be a bounded open set, let K ⊆ E be a closed convex subset such that VK := V ∩ K = ∅, let β be a monotone nonsingular MNC in E , and let F : V K → K be a βcondensing map such that x = F (x) for all x ∈ ∂VK , where V K and ∂VK denote the closure and boundary of the set VK in the relative topology of K. In such a setting, the (relative) topological degree degK (i – F , V K ) of the corresponding vector field i – F satisfying the standard properties is defined (see, e.g., [, ]), where i is the identity map on E . In particular, the condition degK (i – F , V K ) =  implies that the fixed point set Fix F = {x : x = F (x)} is a nonempty subset of VK . To describe the next property, let us introduce the following notion. Definition  Suppose that β-condensing maps F , F : V K → K have no fixed points on the boundary ∂VK and there exists a β-condensing family : [, ] × V K → K such that: (i) x = (λ, x) for all (λ, x) ∈ [, ] × ∂VK ; (ii) (, ·) = F ; (, ·) = F . Then the vector fields  = i – F and  = i – F are called homotopic:  ∼  . The homotopy invariance property of the topological degree asserts that if  ∼  , then degK (i – F , V K ) = degK (i – F , V K ). Let us also mention the following properties of the topological degree. The normalization property. If F ≡ x ∈ K, then degK (i – F , V K ) =

 

if x ∈ VK , if x ∈/ V K .

Additive dependence property. Suppose that {VKj }m j= are disjoint open subsets of VK such

m that F has no fixed points on V K \ j= VKj . Then degK (i – F , V K ) =

m

degK (i – F , V Kj ).

j=

The map restriction property. Let F : V K → K be a β-condensing map without fixed points on the boundary ∂VK , and let L ⊂ K be a closed convex set such that F (V K ) ⊆ L.


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Then degK (i – F , V K ) = degL (i – F , V L ). Now, let N be an isolated component of the fixed point set Fix F , that is, there exists ε >  such that Wε (N) ∩ Fix F = N, where Wε (N) denotes the ε-neighborhood of a set N in K. From the additive dependence property of the topological degree it follows that the degree degK i – F , Wδ (N) does not depend on δ,  < δ < ε. This generic value of the topological degree is called the index of the set N with respect to F and is denoted as indF (N). In particular, if x is an isolated fixed point of F , then indF (x ) is called its index. We further need the following stability property of a nonzero index fixed point, which may be verified by using the property of continuous dependence of the fixed point set (see, e.g., [], Proposition ..). Proposition  Let {hn } be a sequence of positive numbers converging to zero, let H = {hn }, and let : H × V → E be a β-condensing family. Denoting Fn = (hn , ·), suppose that the map F = (, ·) has a unique fixed point x such that indF (x ) = . Then Fix Fn = ∅ for all sufficiently large n, and, moreover, xn → x for each sequence {xn } such that xn ∈ Fix Fn , n ≥ . We further also need the following statement. Proposition  (Corollary .. of []) Let E be a Banach space, and let {fi } ⊂ L ([, a]; E) be a semicompact sequence, i.e., it is integrably bounded and the set {fi (t)} is relatively compact in E for a.e. t ∈ [, a]. Then, for every δ > , there exist a measurable set mδ and a compact set Kδ ⊂ E such that meas mδ < δ and dist(fi (t), Kδ ) < δ for all i and t ∈ [, a] \ mδ .

3 Scheme of semidiscretization Along with equation (.), for a given sequence of positive numbers {hn } converging to zero, consider the equation Dq xh (t) = Ah xh (t) + fh t, xh (t) ,

t ∈ [, a],

(.)

where h ∈ H = {hn } is the semidiscretization parameter, Ah : D(Ah ) ⊂ Eh → Eh are closed linear operators in Banach spaces Eh generating C -semigroups {Uh (t)}t≥ . We assume


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that E = E, A = A, f = f , and continuous maps fh : [, a] × Eh → Eh satisfying conditions (f )-(f ) for each h ∈ H. We suppose that there exist linear operators Qh : Eh → E, h ∈ H, Q = I and projection operators Ph : E → Eh , P = I such that Ph Qh = Ih ,

(.)

where Ih is the identity on Eh , and Qh Ph x → x

(.)

as h →  for each x ∈ E. We suppose that the operators Ph and Qh are uniformly bounded in the sense that there exists a constant M such that Ph ≤ M ,

Qh ≤ M

(.)

for all h ∈ H. An initial condition for equation (.) is given by the equality xh () = Ph x .

(.)

Notice that a mild solution xh ∈ C([, a]; Eh ) to problem (.), (.) is defined by the equality xh (t) = Gh (t)Ph x +

t

(t – s)q– Th (t – s)fh s, xh (s) ds,

t ∈ [, a],



where the operator functions Gh and Th are defined analogously to Definition :

Gh (t) =

∞

ξq (θ )Uh t q θ dθ ,



Th (t) = q

∞

θ ξq (θ )Uh t q θ dθ .



We assume that (H) for each x ∈ E, Qh Uh (t)Ph x → U(t)x as h →  uniformly in t ∈ [, a]. Notice that conditions of validity of hypothesis (H) in terms of the strong convergence of the resolvents Qh (Ah + λI)– Ph x → (A + λI)– x being an analog of the Trotter-Kato theorem are given, for example, in [], Chapter IX, Theorem .; see also [], Theorem .. We also consider the map g : H × [, a] × E → E, g(h, t, x) = Qh fh (t, Ph x),


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for which we suppose the following: (g) g is continuous and bounded on bounded sets; (g) there exists k >  such that χE g(H, t, ) ≤ kχ( ) for each t ∈ [, a] and bounded ⊂ E. Consider the operator F : H × C([, a]; E) → C([, a]; E) defined by the equality

t

F(h, x)(t) = Qh Gh (t)Ph x +

(t – s)q– Qh Th (t – s)fh s, Ph x(s) ds.

(.)



Taking into account (.), we conclude that the operator F may be written in the following form:

t

F(h, x)(t) = Qh Gh (t)Ph x +

(t – s)q– Qh Th (t – s)Ph g h, s, Ph x(s) ds.

(.)



Notice that solutions xh of problem (.), (.) and fixed points of F(h, ·) are connected in the following way: if xh is a solution of (.), (.), then xh = Qh xh is a fixed point of F(h, ·), and, conversely, if xh is a fixed point of F(h, ·), then Ph xh is a solution of problem (.), (.). The continuity of the operator F follows from property (g) and the next assertion. Lemma  For each x ∈ E, we have the relations Qh Gh (t)Ph x → G (t)x

(.)

Qh Th (t)Ph x → T (t)x

(.)

and

as h →  uniformly in t ∈ [, a]. Proof Let us prove (.). Since the operator Qh is bounded, by Remark  and condition (H), for each x ∈ E, we have: Qh Gh (t)Ph x – G (t)x = E



∞

ξq (θ )Qh Uh t q θ Ph x dθ –

∞ 

ξq (θ )U t q θ x dθ

∞ q q = ξq (θ ) Qh Uh t θ Ph x – U t θ x dθ  E ∞ q q ≤ ξq (θ )Qh Uh t θ Ph x – U t θ xE dθ → . 

E


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In a similar way, we can get relation (.): Qh Th (t)Ph x – T (t)x = E

∞

θ ξq (θ )Qh Uh t θ Ph x dθ –

∞

q





q θ ξq (θ )U t θ x dθ

∞ q q = θ ξ (θ ) Q U θ P x – U t θ x dθ t q h h h  E ∞ q q ≤ θ ξq (θ )Qh Uh t θ Ph x – U t θ xE dθ → . 

E

Introduce in C([, a]; E) the measure of noncompactness ν : P(C([, a]; E)) → R+ with values in the cone R+ endowed with the natural order: ν( ) = max ψ(D), modC (D) , D ∈( )

(.)

where ( ) denotes the collection of all denumerable subsets of , ψ(D) = sup e–pt χE D(t) , t∈[,a]

and a constant p >  is chosen in the following way. Fix a constant d >  satisfying the relation qMM k dq  < . ( + q) q 

(.)

Then p is taken so that the following estimate holds:  qMM k  < . ( + q) pd–q 

(.)

The second component of the MNC ν is the modulus of equicontinuity defined in Section .. It is clear the the MNC ν is regular. Theorem  The operator F is a ν-condensing family. Proof Let ⊂ C([, a]; E) be a nonempty bounded subset such that ν F(H × ) ≥ ν( ).

(.)

We will show that the set is relatively compact. Let the maximum in the left-hand side of relation (.) be achieved on a denumerable ∞ set D = {ym }. Then there exist sequences {xm }∞ m= ⊂ and {hnm }m= ⊂ H (which, for simplicity, will be denoted {hm }) such that ym (t) = Qhm Ghm (t)Phm x + 

for each m ≥ .

t

(t – s)q– Qhm Thm (t – s)Phm g hm , s, Phm xm (s) ds


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From Lemma  and the properties of algebraic semiadditivity and regularity of the MNC ν it follows that we can substitute the sequence {ym } with the sequence { ym }, where ym (t) =

t

(t – s)q– Qhm Thm (t – s)Phm g hm , s, Ph xm (s) ds.



From inequality (.) it follows that ψ { ym } ≥ ψ {xm }

(.)

modC { ym } ≥ modC {xm } .

(.)

and

By using property (g) we get the following estimates:

χE g H, s, xm (s) ≤ kχE xm (s) = eps ke–ps χE xm (s) ≤ eps k sup e–pξ χE xm (ξ ) = eps kψ {xm } . ξ ∈[,a]

The last inequality yields: qMM k e–pt χE ym (t) ≤ e–pt ( + q) ≤

qMM k

t

(t – s)q– eps ψ {xm } ds



ψ {xm }

( + q) × e–pt

t–d

(t – s)

q– ps

e ds + e



t

–pt

(t – s)

q– ps

e ds

t–d

–pt  ep(t–d) –  dq qMM k ψ {xm } e + ≤ ( + q) d–q p q  –pd q  e d qMM k ψ {xm } + ≤ ( + q) d–q p q dq qMM k  + ψ {xm } . ≤ –q ( + q) pd q Now, by using inequalities (.) and (.) we have  sup e–pt χE ym (t) ≤ ψ {xm } ,  t∈[,a]  ψ ym (t) ≤ ψ {xm } .  Taking into account inequality (.) together with the last one, we get:  ψ {xm } ≤ ψ {xm } , 


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and hence ψ {xm } = , implying χE xm (t) ≡ 

on [, a].

(.)

From (.) and conditions (H) and (g) it follows that F maps bounded sets into

bounded ones, and so the set t∈[,a] g(H, t, {xm (t)}) is bounded. Applying condition (g) and (.), we get that the set {g(hm , t, xm (t))} is relatively compact for all t ∈ [, a]. Then the sequence of functions g(hm , ·, xm (·)) is semicompact, and from Proposition  it follows that, for each δ > , there exist a compact set Kδ ⊂ E, a set mδ ⊂ [, a] of Lebesgue measure gm } ⊂ L ([, a]; E) such that meas mδ < δ, and a sequence of functions {

gm (t) ⊂ Kδ for t ∈ [, a] \ mδ ,

gm (t) =  for t ∈ mδ , and g hm , t, xm (t) – gm (t) < δ

for t ∈ [, a] \ mδ .

The set of functions vm (t) = Qhm Ghm (t)Phm x +

(t – s)q– Qhm Thm (t – s) gm (s) ds,

m ≥ ,

[t,]\mδ

is relatively compact in C([, a]; E) and forms a γδ -net of the set {ym }, where γδ →  as δ → , and we have the following estimate: modC ( ) ≤ modC {ym } = . Therefore ν( ) = (, ), which concludes the proof.

4 Index of the solution set The operator G : C([, a]; E) → C([, a]; E), Gx(t) = G (t)x +

t

(t – s)q– T (t – s)f s, x(s) ds,



is the solution operator of problem (.), (.) in the sense that the set of its fixed points Fix G coincides with the set of all mild solutions of this problem. From Theorem  it follows that G = F(, ·) is ν-condensing. It opens the possibility to introduce the following notion. Suppose that the set Fix G is bounded in C([, a]; E). Then its index indG (Fix G) is well defined.


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Definition  The value indG (Fix G) is called the index ind() of a bounded solution set of problem (.), (.). We further assume that the nonlinearity f satisfies the following condition: (f  ) χE (f ([, a] × )) ≤ μχE () for each bounded ⊂ E, where μ ≥ . Notice that this condition is fulfilled if, for example, f satisfies condition (f ) and for each bounded set ⊂ E, the function f (·, x) : [, a] → E is uniformly continuous w.r.t. x ∈ . Theorem  If is a bounded solution set of (.), (.) then ind() = . Proof Take an arbitrary open bounded set V ⊂ C([, a]; E) containing the solution set and let

L = x ∈ C [, a]; E : x() = x . From the map restriction property of the topological degree it follows that ind() = degL (i – G, V L ). Take an arbitrary x∗ ∈ and for n = , , . . . , consider the operators Gn : L → L defined as

t

Gn x(t) =

(t + /n – s)q– T (t + /n – s)f s, x(s – /n) ds + x∗ (t)



–

t

(t + /n – s)q– T (t + /n – s)f s, x∗ (s – /n) ds,



where x ∈ C([–, a]; E) denotes the extension of a function x ∈ L defined as x on [–, ]. Let us show that, for a sufficiently large n, the operators Gn are ν-condensing (the choice of the coefficient p is described later). Let ⊂ L be a nonempty bounded set, and let ν Gn ( ) ≥ ν( ).

(.)

Let us show that the set is relatively compact. Let the maximum in the left-hand part of inequality (.) is achieved on a denumerable set D = {yk }. Then there exists a sequence {xk } ⊂ such that

t

yk (t) =

(t + /n – s)q– T (t + /n – s)f s, xk (s – /n) ds + x∗ (t)



t

– 

(t + /n – s)q– T (t + /n – s)f s, x∗ (s – /n) ds.


Page 13 of 20

It is clear that it is sufficient to consider the sequence { yk } defined as yk (t) =

t

(t + /n – s)q– T (t + /n – s)f s, xk (s – /n) ds.



From (.) it follows that ψ { yk } ≥ ψ {xk } , modC { yk }∞ n= ≥ modC {xk } .

(.) (.)

The following estimates hold: χE f s, xk (s – /n) ≤ μχE xk (s – /n)

≤ ep(s–/n) μ sup e–p(ξ –/n) χE xk (ξ – /n) ξ ∈[,a]

≤ ep(s–/n) μψ {xk } .

(.)

Now, take d such that, for all n ≥ ,  Mμ (d + /n)q – (/n)q < ( + q) 

(.)

and then choose p >  such that  qMμ  · –q < . ( + q) pd 

(.)

Now, using estimate (.), we get that, for each t ∈ [, a], t qMμ yk (t) ≤ e–pt (t + /n – s)q– ep(s–/n) ψ {xk } ds e–pt χE ( + q)  qMμ ψ {xk } e–p/n ≤ ( + q) t–d t –pt q– ps –pt q– ps × e (t + /n – s) e ds + e (t + /n – s) e ds 

t–d

 ep(t–d) –  (d + /n)q – (/n)q qMμ ψ {xk } e–pt + ≤ ( + q) (d + /n)–q p q  e–pd (d + /n)q – (/n)q qMμ ψ {xk } + ≤ ( + q) d–q p q q  (d + /n) – (/n)q qMμ ψ {xk } . + ≤ ( + q) pd–q q Now, using inequalities (.) and (.), from the last estimate we have:  sup e–pt χE yk (t) ≤ ψ {xk } ,  t∈[–,a]  ψ yk (t) ≤ ψ {xk } . 


Page 14 of 20

Combining the last inequality with (.), we get:  ψ {xk } ≤ ψ {xk } ,  implying ψ {xk } = .

(.)

Since the operator Gn maps bounded sets into bounded ones, the sequence of functions {fk } defined by fk (t) = f (t, xk (t – /n)) is bounded. From relation (.) and condition (f ) it follows that the set {fk (t)} is relatively compact for each t ∈ [, a], and hence the sequence {fk } is weakly compact in L ([, a]; E) (see, e.g., Proposition .. in []). From condition (.) it follows that, for each δ > , there exist a compact set Kδ ⊂ E and a set mδ ⊂ [–, a] with the Lebesgue measure meas(mδ ) < δ such that {fk (t)} ⊂ Kδ for t ∈ [, a]\mδ . Therefore the sequence of functions (t + /n – s)q– T (t + /n – s)fk (s) ds

vk (t) = [,t]\mδ

is relatively compact in C([, a]; E) and forms a γδ -net of the set {yk }, where γδ →  as δ → , and hence the sequence { yk } is relatively compact. By (.) we get modC ( ) ≤ modC {yk } = . Therefore ν( ) = (, ). Let us prove that, for all sufficiently large n, we have the equality degL (i – Gn , V L ) = degL (i – G, V L ).

(.)

We will show that equality (.) follows from the homotopy of the above fields, which is realized for sufficiently large n by the linear transfer : [, ] × V L → C([, a]; E) defined as (λ, x) = λGn x + ( – λ)Gx. From the properties of the MNC ν it easily follows that the family is ν-condensing. Let us demonstrate that x = (λ, x) for all (λ, x) ∈ [.] × ∂VL . Supposing the contrary, we will have the sequences {xm }, {nm }, and {λm } such that xm ∈ ∂VL , nm → ∞, λm ∈ [, ], λm → λ , and xm = λm Gnm xm + ( – λm )Gxm .

(.)


Page 15 of 20

Let us show that the sequence {xm } is relatively compact. From relations (.) it follows that xm (t) = λm 

t

(t + /nm – s)q– T (t + /nm – s)f s, xm (s – /nm ) ds + λm x∗ (t)

t

– λm

(t + /nm – s)q– T (t + /nm – s)f s, x∗ (s – /nm ) ds



+ ( – λm )Gxm (t),

t ∈ [, a].

Choose a constant p >  such that, for d >  satisfying the inequality  Mμdq < , ( + q) 

(.)

we have qMμ   < . –q ( + q) pd 

(.)

Notice that t–/nm   = λm xm t – (t – s)q– T (t – s)f s, xm (s – /nm ) ds + λm x∗ t – nm nm  t–/nm – λm (t – s)q– T (t – s)f s, x∗ (s – /nm ) ds 

 + ( – λm ) G t – x nm q– t–/nm   t– + –s T t– – s f s, xm (s) ds , nm nm   ,a . t∈ nm Let us make the substitution of variables s + /nm = ξ in the last integral and denote ξ by s again. Then we get t–/nm   = λm xm t – (t – s)q– T (t – s)f s, xm (s – /nm ) ds + λm x∗ t – nm nm  t–/nm – λm (t – s)q– T (t – s)f s, x∗ (s – /nm ) ds 

 x + ( – λm ) G t – nm t q– + (t – s) T (t – s)f s – /nm , xm (s – /nm ) ds , /nm

t∈

 ,a . nm


Page 16 of 20

Set f (s, x) =

f (s, x) for s ≥ , f (, x) for s < .

Notice that the sequences of functions λm

t

(t – s)

q–

T (t – s)f s, xm (s – /nm ) ds ,

t–/nm

λm

t–/nm

(t – s)



q–

T (t – s)f

s, x∗m (s – /nm )

ds ,

λm x∗ (t – /nm ) ,

( – λm )G (t – /nm )x , and

/nm

(t – s)q– T (t – s)f s – /nm , x(s – /nm ) ds



converge uniformly for t belonging to each interval [γ , a] with γ ∈ (, a). Let us estimate for t >  the following value: e–pt χE xm (t – /nm ) t –pt (t – s)q– T (t – s) = e χE 

× λ f s, xm (s – /nm ) + ( – λ )f s – /nm , xm (s – /nm ) ds

.

Since by (f  ) χE λ f s, xm (s – /nm ) + ( – λ )f s – /nm , xm (s – /nm ) ≤ KχE xm (s – /nm ) , we get qMμ e–pt χE xm (t – /nm ) ≤ sup e–ps χE xm (s – /nm ) ( + q) s∈[,a] t–d t (t – s)q– eps ds + e–pt (t – s)q– eps ds × e–pt 

t–d

qMμ ≤ sup e–ps χE xm (s – /nm ) ( + q) s∈[,a]  ep(t–d) –  dq + × e–pt –q (d) p q

 e–pd dq qMμ –ps sup e χE xm (s – /nm ) + ≤ ( + q) s∈[,a] d–q p q q  d qMμ sup e–ps χE xm (s – /nm ) . + ≤ –q ( + q) s∈[,a] pd q


Page 17 of 20

Now, using (.) and (.), we have  sup e–pt χE xm (t – /nm ) ≤ sup e–ps χE xm (s – /nm ) ,  s∈[,a] t∈[,a] implying sup e–pt χE xm (t – /nm ) = , t∈[,a]

and hence χE xm (t – /nm ) = ,

t ∈ [, a].

The proof of the equicontinuity of the sequence {xm } is similar to the previous proof of the equicontinuity of the sequence { yk }. So the sequence {xm } is relatively compact, and we can assume, without loss of generality, that xm → x . Then x ∈ ∂VL , that is, x = x∗ . Passing to the limit in (.) as m → ∞, we get the contradiction Gx = x . Now, let us show that the map S : [, ] × V L → L given by the formula S(λ, x) = λGn x + x∗ – λGn x∗ is the homotopy connecting maps G (x) ≡ x∗ and Gn . In fact, for each λ ∈ [, ], the equation S(λ, x) = x

(.)

has a unique solution x = x∗ . For λ = , this is evident, and for λ = , if equation (.) has a solution y, then y() = x , and hence y(t – /n) = x∗ (t – /n), t ∈ [, /n], so y(t) = x∗ (t), t ∈ [, /n]. Then y(t – /n) = x∗ (t – /n), t ∈ [/n, /n]. Continuing further, we get the equality y(t) = x∗ (t), t ∈ [, a]. Then, using the map restriction and normalization properties of the topological degree, we have: degL (i – Gn , V L ) = degL (i – G , V L ) = , yielding degL (i – G, V L ) = , which concludes the proof of the theorem.

Gn : L → C([, a]; E) given by the Remark  On the set L, let us consider the operators formula Gn x(t) = G (t)x +

t 

(t + /n – s)q– T (t + /n – s)f s, x(s – /n) ds.


Page 18 of 20

Fixed points xn of the operators Gn may be found by the method of steps successively on ], k = , , . . . , [an]. Finding these fixed points is analogous to the Tonelli the intervals [ nk , k+ n procedure (see, e.g., [] or [, ], Chapter II, Theorem .) of solving problem (.)(.). If it is known that the sequence {xn } is bounded on the interval [, a], then, repeating the reasonings for {xm } given by formula (.) with λm = , we get the compactness of the sequence {xn }. If {xnk } is a convergent subsequence, then passing to the limit in the equalities xnk = Gnk xnk , we obtain that the limit of the subsequence {xnk } may be only a fixed point of the operator G, that is, a solution of problem (.)-(.). Let us mention also that, instead of condition (f  ), in this case, it is sufficient to assume condition (f ).

5 The main result Now we are in position to present the main result of this paper. Theorem  Under conditions (A), (f ), (f  ), (.), (.), (.), (g), (g), suppose that problem (.)-(.) has a unique solution x on the interval [, a]. Then, for a sufficiently small h > , problems (.), (.) have solutions xh on the interval [, a], and Qh x h → x as h → . Proof It is sufficient to apply Proposition  to the operator F given by formula (.).

In conclusion, let us present two examples of the construction of Ah and fh . Example  Let hn = /n, and let Ahn = An be the Yosida approximations (see, e.g., []), Ehn = E, Phn = Qhn = I, and fhn = f . Then condition (H) is fulfilled for the semigroups generated by the operators An . So, the transfer from a unbounded operator A in equation (.) to a bounded operator Ah in equation (.) can be justified. Example  Let f satisfy (f ). If we set fh (t, xh ) = Ph f (t, Qh xh ), then, for the operator g, condition (g) is fulfilled with the constant k = μ.

Acknowledgements The work of the first, second, and third authors is supported by the Ministry of Education and Science of the Russian Federation in the frameworks of the project part of the state work quota (Project No. 1.3464.2017/4.6). JCY was partially supported by the Grant MOST 106-2923-E-039-001-MY3. Competing interests The authors declare that they have no competing interests.


Page 19 of 20

Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Author details 1 Faculty of Mathematics, Voronezh State University, Voronezh, 394006, Russia. 2 Faculty of Physics and Mathematics, Voronezh State Pedagogical University, Voronezh, 394043, Russia. 3 RUDN University, 6 Miklukho-Maklaya st., Moscow, 117198, Russia. 4 Center for General Education, China Medical University, Taichung, 40402, Taiwan.

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