ANTIPERIODIC BOUNDARY VALUE PROBLEM FOR A SEMILINEAR DIFFERENTIAL EQUATION OF FRACTIONAL ORDER Текст научной статьи по специальности «Математика»

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Серия «Математика»

2020. Т. 34. С. 51-66

УДК 517.929

MSC 34K09; 34K37; 47H04; 47H08; 47H10 DOI https://doi.org/10.26516/1997-7670.2020.34.51

Antiperiodic Boundary Value Problem for a Semilinear Differential Equation of Fractional Order*

G. G. Petrosyan

Voronezh State University of Engineering Technologies, Voronezh, Russian Federation

Abstract. The present paper is concerned with an antiperiodic boundary value problem for a semilinear differential equation with Caputo fractional derivative of order q £ (1, 2) considered in a separable Banach space. To prove the existence of a solution to our problem, we construct the Green's function corresponding to the problem employing the theory of fractional analysis and properties of the Mittag-Leffler function . Then, we reduce the original problem to the problem on existence of fixed points of a resolving integral operator. To prove the existence of fixed points of this operator we investigate its properties based on topological degree theory for condensing mappings and use a generalized B.N. Sadovskii-type fixed point theorem.

Keywords: Caputo fractional derivative, semilinear differential equation, boundary value problem, fixed point, condensing mapping, measure of noncompactness.

Recent years have seen a wide spread of the theory of fractional analysis and differential equations of fractional order in modern mathematics. The increasing interest in this subject is explained by numerous applications of the theory in various branches of applied mathematics, physics, engineering, biology, economics ant others (cf., e.g., the monographs [8; 9; 13; 15; 17]). Several different approaches for solving differential equations and their

1. Introduction

* The research was supported by RFBR grant no. 19-31-60011.

boundary value problems in the case of fractional order q € (0,1) have been introduced in the literature (see [1; 11; 12] and references therein). Differential equations of the fractional order q > 1 have received a particular attention in the past few years.

At the same time, along with periodic problems, antiperiodic boundary value problems are being presently studied intensively in view of their applications in physics and interpolation problems (see, e.g., [5], [6], [16] and references therein). Next, we shortly describe some of the existing results in this research direction. In the work [3], invoking Leray-Schauder degree theory and Green's functions method, the authors prove the existence of solutions for the antiperiodic boundary value problem

CDqx(t) = f (t,x(t)), t € [0,T],

x(0) = -x(T), x'(0) = -x'(T),

where CDq stands for the Caputo fractional derivative of order q € (1,2) and f : [0, T] x R ^ R is a continuous function. In the paper [2], applying Green's functions method and Krasnoselskii-Krein fixed point theorem the authors solve the following boundary value problem

CDqx(t) = f(t,x(t)), t € [0,T],

x(0) = -x(T), x'(0) = -x'(T), x''(0) = -x''(T), x'''(0) = -x'''(T),

for the case of fractional order q € (3,4) and a continuous function f : [0,T] x R ^ R.

Our problem is considered in the same vein. Namely, in the present work we investigate the solvability of the following boundary value problem for semilinear differential equation of fractional order

CDqx(t) = Ax(t) + f (t,x(t)), t € [0,T], (1.1)

with the antiperiodic boundary condition

x(0) = -x(T), x'(0) = -x'(T) (1.2)

in a separable Banach space E, where CDq is the Caputo fractional derivative of order q € (1, 2), A > 0, f: [0, T] x E ^ E is a nonlinear mapping.

The equation (1.1) has important applications in theoretical physics. For instance, when the space has a finite dimension it generalizes the Ginzburg-Landau equation of fractional order (see [17]):

C Dq ^(t) = a0(t) + b||^(t)||2 m (1.3)

where a, b are some constants. The equation (1.3) is used to describe the behaviour of a superconductor in media with dispersion in the absence of

an external magnetic field as well as in the study of the phenomena of superfluidity and propagation of nonlinear waves.

2. Preliminaries 2.1. Fractional integral and fractional derivative.

First, we introduce some notions and notation from fractional mathematical analysis necessary for our study (cf. the monographs [13; 15]).

Definition 1. A factional integral of order q > 0 of a function g : [0, T] ^ R is the function Iqg of the form:

1

W) = ±jJo(t-s)<-i9(s)ds,

where r is the Euler Gamma function.

We note that the following property holds for the Euler Gamma function (see, e.g., [15]):

-i-=0 for 9 = 0,-1,-2,.... (2.1)

r(q)

Definition 2. A factional derivative of order q > 0 of a function g € Cn([0,T]) is the function CDqg of the form:

1

q) Jo

1 r

CD«g{t) = Jo (t - s)n-q-lg^\s) ds, n = [q] + 1,

provided that the right-hand side is correctly defined. Definition 3. The function of the form

^ zn

is called the Mittag-Leffler function.

As a rule, the function Eq,i is simply denoted by Eq. The Mittag-Leffler function is of great importance in fractional calculus. Consider the following Cauchy problem for the scalar differential equation of fractional order

CDqx(t) = Ax(t) + f (t), t € [0,T], 1 < q < 2, (2.2)

x(0) = ci,x'(0)= C2, (2.3)

where A € R, f : [0, T] ^ R a function for which there exists the fractional integral of order q. It is known (cf. [13]) that the unique solution to this problem is the function

x(t) = C1 Eq(Atq)+ C2iEq,2(Aiq) + /'(t - s)q-1Eq,q(A(t - s)q)f (s) dS. (2.4)

Jo

In what follows, we will use the following relations and statements (see [8])

V"1^,/»^)) = tl3~n~lEq^_n(\tq), (2.6)

d \ t)

f te-1E,>/3 (Atq )dt = Eq;e+1(Azq), (2.7)

J0

Lemma 1. For a function f € C([0, T]; E) and 1 < q < 2 we have

( f (i-s)q-1Eq,q(A(t-s)q)f (s) ds) t = f (i-s)q-2Eq;q-i(A(i-s)q)f (s) ds. 0 4 0 (2.8)

In order to establish a similar result for a function f € L^([0, T]; E) we need the following results.

Lemma 2. For any function f € L^([0,T];E) there exists a sequence {fn} C C([0,T];E) such that fn(t) ^ f(t) at all Lebesgue points of the function f from [0,T] and ^Jc^t];®) ^ llf ([0,T];E) •

Such a sequence can be constructed using the Steklov projector as follows

\ o,t € [0,t],

f(t) = / f(t),t€ [0,T]; f (t) = \ 0,t € [0,T].

Lemma 3. (see [4]) For any function f € L^([0,T];E) the set of its Lebesgue points is a set of full measure for [0, T].

Lemma 4. (see [7]) Let all the functions {fn} be differentiable on the interval [0, T] and the sequence of derivatives {f^} converge uniformly with respect to t € [0,T] in the entire interval. If the sequence {fn} converges at least at one point from [0,T], then the sequence {fn} converges uniformly in the entire interval and the limit function f is differentiable with f'(t) =

lim„^ fn(t).

The following statement is a consequence of the last three lemmas and Lebesgue's theorem on the passage to the limit under the integral sign.

Lemma 5. For a function f € L^([0, T]; E) and 1 < q < 2 we have

( f (ts)q-1Eqq(X(t-8)q)f (S) ds) t = f (t-s)q-2Eqq-1(X(t-s)q)f (s) ds. 0 4 0 (2.9)

2.2. Measures of noncompactness and condensing mappings.

Let E be a Banach space. We introduce the following notation:

- P (E ) = {A CE : A = 0} ;

- Pb(E) = {A € P(E) : A is bounded} ;

- Pv(E) = {A € P(E) : A is convex} ;

- K(E) = {A € Pb(E) : A is compact} ;

- Kv(E) = Pv(E) n K(E).

Definition 4. (see, e.g., [10; 14]). Let (A, >) be a partially ordered set. A function ft : Pb(E) ^ A is called a measure of noncompactness (MNC) in E, if for every Q € Pb(E) we have /?(coQ) = (3(Q), where coQ stands for the closure of the convex hull of Q.

The Hausdorff MNC x(Q) = inf {e > 0, for which Q has a finite e-net in E } is an example of a real MNC which is monotone, nonsingular, regular, algebraically semiadditive, and semi-homogeneous (cf., e.g., [10]).

The following notion and statement can be found in the monographs [10; 14].

Definition 5. Let X be a closed subset of E and ft be a MNK in E. A mapping f : X ^ E is called condensing relative to ft (or ft-condensing), if for every not relatively compact Q € Pb(X) we have ft(f (Q)) > ft(Q).

Theorem 1. (See [10]). Let M be a convex bounded closed subset of E and f : M ^ M be a continuous ft-condensing mapping, where ft is a nonsingular MNK in E. Then, the set of fixed points Fix f = {x : x = f (x)} is nonempty.

3. Construction of Green's function

In a separable Banach space E consider the boundary value problem (2.2)-(2.3):

C Dq x(t) = Ax(t) + f (t), t € [0, T], 1 <q< 2, x(0) = c\, x'(0) = c2,

where f : [0, T] ^ E.

Definition 6. A solution of the boundary value problem (2.2)-(2.3) is a function x € C([0, T]; E) which satisfies

x(t) = ClEq(Atq) + C2iEq,2(Aiq) + / (t - s)q-1Eq,q(A(t - s)q)f (s) ds.

Jo

Lemma 6. Let f € C([0,T]; E) and

(1 + Eq(ATq))2 - Eq,o(ATq)Eq,2(ATq) = 0. (3.1)

Then, the boundary value problem (2.2), (1.2) has a unique .solution

x(t) = I G(t,s)f(s)ds, o

where Green's function G(t, s) has the form

r(f , = f-a + Eq{\Tq)){T - s)q~lEqA(\(T - s)q) V (l+Eq(ATq))2-Eq)0(ATq)Eq)2(ATq) , TEq,2(XTq)(T - s)q~2EqA_i(\(T - sf)\

{l + Eq{\Ti)y-Eqfi{\Ti)Eq,2{\T<i) ) qy n -(1 + Eq{\Tq)){T - s)q~2EqA_i(\(T - s)q) (1 + Eg(ATq))2 - Eqfl(ATq) Eq;2(ATq) T~lEqfl(\Tq))(T - s)q~lEqA(\(T - s)q) \

+ (1 , F iX^aWI-F f^^rr f^^ tEq^\tq) +

(1 + Eq(ATq))2 - Eq,o(ATq)Eq,2(ATq)

+ (t - s)q-1Eq,q(A(t - S)q),

if 0 < s <

, = ( ~(1 + Eq{\Tq)){T - s)q~lEq>q(\(T - s)q) ('J V (l+Eq(ATq))2-Eq;o(ATq)Eq)2(ATq) , TEg,2(ATq)(T - s^E^XjT - s)q)\

(l+Eq(ATq))2-Eq;o(ATq)Eq)2(ATq) J q{ П

-(1 + Eq(XTq))(T - s)q_2Eq;q_i(A(T - s)q) +

(1 + Eq(ATq))2 - Eq;c(ATq)Eq,2(AT^q) T~lEqfl(\Tq))(T - s)q~1Eq)q(A(T - S)q) \ (l + Eq(ATq))2-Eq;o(ATq)Eq)2(ATq) ,/ ^

if 0 < t < s < T.

Proof. A solution of the boundary value problem (2.2), (2.3) in the Banach space E has the following form

x(t) = ClEq(Atq) + C2tEq,2(Aiq) + ^(t - s)q-1Eq,q(A(t - s)q)f (s) ds.

o

Applying the formula (2.6) and Lemma 1, we can find the derivative

x'(t) = c1t-1Eq>o(At9) + C2EqA(Xtq) + f\t - s)q-2Eqq-l(\(t - s)q)f (s) ds.

J 0

Note that in view of the property p^jy = 0, for the function Eqfl(\tq) we have

F o.tq) V{uq)n 1 I V{xtq)n V{uq)n consequently

^ \ntqn-1

t~lEqfi{\tq) =

1 r(qn) •

n= 1

By virtue of the last formula we obtain x(0) = c1, x'(0) = c2. Now, using (1.2) we have the system

f-ci = ciEq(ATq)+c2TEq2(\Tq)+fQT(T - s)q-1Eqq(\(T - s)q)f (s) ds, \-C2 = ciT-1Eqo(ATq)+c2EQii(ATq)+/QT(T-s)q-2Eq,q-i(A(T-s)q)f (s) ds.

Solving the last system by Cramer's rule we have

_ -(1 + Eq{\Tq)) £(T - s)q~l EqA(\(T - s)q)f{s) ds Cl (1 + Eq(\Tq))2 - Eq>0(\Tq)Eq>2(\Tq)

TEq,2(XTq) - s)q~2 Eq^q_i(\(T - s)q)f{s) ds (1 + Eq{\Tq))2 - Eqfl{\Tq)Eqi2{\Tq)

-(1 + Eq{\Tq)) - s^E^XjT - s)q)f(s) ds C2 (1 + Eq(\Tq))2 - Eq>o(\Tq)Eq>2(\Tq)

T~lEqfi(\Tq) /0T(T - s)q~l Eq^q(\(T - s)q)f{s) ds (1 + Eq{\Tq))2 - Eqfl{\Tq)Eqi2{\Tq)

Inserting the coefficients that we have found into the solution's formula we obtain

-(1 + Eq{\Tq)) Jq(T - s)q~lEq,q(\(T - s)q)f{s) ds [ ) (1 + Eq{\Tq))2 - Eqfi{\Tq)Eq,2{\Tq) ) +

TEq>2{\Tq) f0T(T - s)q~2Eq>q_1(A(T - s)q)f(s) ds (1 + Eq(XTq))2 - Eqfl{\Tq)Eqß{\Tq)

-(1 + Eq{\Tq)) f0T(T - s)q~2Eq>q_1(A(T - s)q)f(s) ds

(1 + Eq{\Tq))2 - Eq>0{\Tq)Eq>2{\Tq) j +

58 G. G. PETROSYAN

T-1Eq,o(ATq) /0T(T - (A(T - s)q)f (s) ds

iEÇ;2(Aiq ) +

(1 + Eq(ATq))2 - Eq,o(ATq)Eq,2(ATq)

T(i - s)q-1Eq,q(A(i - s)q)f (s) ds = Г G(t, s)f (s)ds. oo

□

Reasoning as in deriving Lemma 5, we can construct Green's function for the boundary value problem (2.2), (1.2) having the same form as in the last lemma, but under the assumption that f € L^([0, T]; E).

4. Existence of a solution

We assume that the function f : [0,T] x E ^ E from the problem (1.1) - (1.2) has the following properties:

(f 1) for all x € E the function f (-,x) : [0,T] ^ E is measurable; (f2) for a.e. t € [0,T] the function f (t, ■) : E ^ E is continuous; (f3) for every r > 0 there exists a function wr € L^°([0,T]) such that for any x € E with ||x||E < r we have: ||f(t,x)||E < wr(t);

(f4) there exists a function ^ € L^°([0, T]) such that for any bounded set Q C E we have: x(f (t, Q)) < ^(t)x(Q), for a.e. t € [0,T], where x is the Hausdorff MNC in E.

The example of a function f satisfying the above properties in case of the infinite dimensional space E = L2[0, T] is given next:

f : [0,T] x L2[0,T] ^ L2[0,T],

f (t,x(t)) = f (t,6) = h( £ et2(s)d^et+g(t),

where h : R+ ^ R+ is a continuous bounded function, and g € C[0,T]. Since h and g are continuous bounded functions, the conditions (f 1) — (f 3) hold for the mapping f. We show that the condition (f4) is valid as well. Let Q C L2[0, T] be a bounded set and for e > 0 the set K = {yi, y2,..., yn} be a finite e-net for Q. Then, for any £ € Q there is y^ € K such that ||£ — y^U2 < e. Using the last inequality, for a.e. t € [0,T] we have:

||f(t,£) — f(t,yi)||L2 < — yi||L2 < ^e,

where ^ = supseK+ h(s). From this estimate it follows that for a.e. t € [0, T] the relatively compact set U™=1f (t,y») is a ^e-set for f (t, Q), hence

XL2 (f (t, Q)) < ^XL2 (Q).

Consider the operator F defined as follows:

Fx(t) = I G(i,s)/(s,x(s))ds. Jo

From the conditions (f 1)-(f4) it follows that for a function x € C([0, T]; E) the function f (-,x(-)) € L^([0,T]; E). In this case, from the definition of Green's function we infer that for any t € [0, T] and 1 < q < 2 : G(-, s) € Lp([0,T]),p > 1, and Green's function has a singularity only at the point s = T, hence F : C([0, T]; E) ^ C([0, T]; E). It is obvious that if a function x € C([0, T]; E) is a solution to the problem (1.1) - (1.2), then it is a fixed point of the operator F. Therefore, in what follows, we prove the existence of fixed points of the operator F. To this aim, we consider the operator S : L~([0,T]; E) ^ C([0,T]; E) of the form

S(f )(t) = At - s)q-1Eq,q(A(t - S)q)f (S) ds. Jo

We have the following statement (see [11]).

Lemma 7. For every compact set K C E and a bounded sequence {nn} C ([0,T];E) such that {nn(t)} C K for a.e. t € [0,T], the weak convergence ^ no in L1([0,T];E) implies the convergence S(nn) ^ S(no) in C([0,T]; E).

To prove that the operator F is condensing, consider the cone R+ = {Z = (Z1 , Z2) : Z1 > 0,(2 > 0} with the natural partial order, and introduce in the space C([0, T]; E) the vector measure of noncompactness v : P(C([0, T]; E)) ^ R+ defined as v(Q) = (^(Q),modc(Q)), where p(Q) is the modulus of fiber noncompactness

p(Q) = sup x({y(t) : y € Q}), ie[o,T ]

and the second component is the modulus of equicontinuity

modC(Q) = lim sup max ||y(t1) — y(t2)||. <^0 y€n |ti-t2|<<5

Theorem 2. Let the conditions (f 1) — (f4), (3.1) and

LMoo < (41)

A

hold, where

Eg(\Tq) - 1 + 3Eg,0(ATq)Eg(ATq)Eg,2(ATq) L |(1 + Eq(\Tq))2 — Eqfi(XTq)Eqt2(XTq)\ }

is the function from (f 4), then the operator F is v-condensing.

Proof. Let Q C C([0, T]; E) be a nonempty bounded set such that

v(F(Q)) > v(Q). (4.3)

We prove that Q is a relatively compact set. From the inequality (4.3) it follows that

V(F(Q)) > p(Q).

(4.4)

Using the condition (f 4) along with the monotonicity, algebraically semi-additivity, and semi-homogeneity of the Hausdorff MNK for t € [0, T], we obtain the following estimates

x (f mt)) <

(1 + Eq{XTq)) ¡J (T - s)q~l Eq>q(X(T - syMïljs)) ds |(1 + Eq{XTq))2 - Eq>0(XTq)Eq>2(XTq)\

TEq,2(XTq) Jq (T - s)q~2 Eq^q_\(X(T - s)q)X(n(s)) ds |(1 + Eq{XTq))2 - Eqfl{XTq)Eq,2{XTq)\

(1 + Eq{\Tq)) Jq(T - s)q~2Eqa_i(\(T - s)qMm) ds |(1 + Eq{XTq))2 - Eq>0(XTq)Eq>2(XTq)\

T~lEqfi(\Tq) Jq (T - s)q~lEq,q{X{T - s)q)x(n(s)) ds |(1 + Eq{XTq))2 - Eqfl{XTq)Eq,2{XTq)\

те Eq(Xtq )+

те Eq (Xtq ) + те tEq,2(\tq) + те tEq,2(Xtq) +

At - S)q-1Eq,q(A(t - S)q)Х(ОД) ds < J 0

(1 + Eq{\Tq)) /0г(г - s)q~l Eq^q(X(T - s)q) ds |(1 + Eq(XTq))2 - Eqfl{XTq)Eq,2{XTq)\

TEqt2(\Tq) - g)q-2^,g_i(A(T - s)q) ds |(1 + Eq(XTq))2 - Eqfi{XTq)Eq,2{XTq)\

(1 + Eq{XTq)) J0T(T - s)q~2EqA_i(X(T - s)q) ds |(1 + Eq{XTq))2 - Eq>0(XTq)Eq>2(XTq)\

T~lEqfl{XTq) /0г(Г - s)q~lEq!q(X(T - s)q) ds |(1 + Eq(XTq))2 - Eqfi{XTq)Eq,2{XTq)\

Eq (Xtq M^)+

те ^ q

те Eq(Xtq)^(Q) + те tEq,2 (Xtq)^(Q) +

те tEq,2(XtqMQ) +

,f(Q) f \t - s)q-lEqq(X(t - s)q) ds. J 0

To further estimate x (F(Q)(t)) we calculate the integrals in the last expression with the help of the formula (2.7):

IT (T - s)q-1Eq,q (X(T - S)q )ds = - [T (T - s)q-1Eq,q (X(T - s)q )d(T - s) = 00

зо

cT

/ yq-1Eq,q (Ay9 )dy = T9 Eq,q+1(ATq ).

Jo

Similarly, we have

IT(T — S)9-2Eq,q-1 (A(T — S)9) ds = Tq-1Eq,q(AT9), Jo

At — s)9-1Eg,g (A(t — s)9 )ds = t9 Eg,g+1 (At9). Jo

Now, we note that if we take P = 1 in the formula (2.5), then we obtain ^(AT<0 = f{T) + XTqE^XTq) = 1 + AT«WAn,

Eq(\tq) = ^ + Ai«Si>i+i(Ai«) = 1 + Ai^+^Ai9).

Making use of the property (2.1), if we take P = 0 in the formula (2.5), then we have

Eq,o{\Tq) = + \TqEq>q{\Tq) = \TqEqyq(\Tq).

Therefore, we obtain the following equalities

rT 1 1

Jo {T-s)q-lEq>q{\{T-s)q)ds = Tq— (Eq(\Tq) - 1) = - (Eq(\Tq) - 1),

i'T 1

J (T - s)q~2Eq^q_i(\(T - s)q) ds = —Eq,0(XTq),

ft 1

J (t - s)q~lEq^\{t - s)q)ds = - (Eq(Xtq) - 1).

Using the last equalities, we can continue to estimate x (F(Q)(t)) for t € [0, T] as follows

X (F(Q)(t)) <

{l+Eq{\Tq)) {Eq{\Tq)-l)+TEqt2{\Tq)±Eqfi{\Tq) ll^l^

|(1 + Eg (AT9 ))2 — Eg;o(AT9 )E,,2 (AT9 )|

Eq (At9 )p(Q)+

(l+Eq(\Tq))±Eq,0(\Tq)+T-iEq,0(\Tq) (Eq(XTq)-l) mioo

|(1 + Eq(\Tq))2 - Eqfl{\Tq)Eq>2{\Tq)\ A ttjq'2{Xt MU)

E2q(XTq)-l + Eq,2(XTq)Eq,0(XTq) \{l + Eq{\Tq)Y - Eqfi{\Tq)Eq,2{\Tq)\ A mU)+

62 G. G. PETROSYAN

2T~l Eqfl(\Tq)Eq(\Tq) ll,,,^2(AT>(Q) +

|(1 + Eq(ATq))2 - Eq;c(ATq)Eq;2(ATq)| A

(E2q{\Tq) - 1 + 3Eq,o(XTq)Eq(XTq)Eq,2(XTq) _ x

I \{l + Eq{\Tq)Y-Eq>0{\Tq)Eq>2{\Tq)\ ^ j Г

^(Q) = L-^-pV(Q).

A rv ' A

From the last estimate we see that supt€[0,T] X (E(Q)(t)) < ip(Q),

or, which is the same, that ip(F(Q)) < L ip(il). Taking into account the conditions (4.1) and (4.4) together with the last inequality, we obtain <p(Q) = 0.

In the work [11] it was proved that the set of functions

M = {S(f)(t) = J*(t - s)q-1Eq,q(A(t - s)q)f (s, x(s))ds : x € q|

is equicontinuous, consequently modC (Q) = 0, hence v(Q) = (0,0), which proves the relative compactness of the set Q. □

Now, we are in a position to prove the main result of our work.

Theorem 3. Let the conditions (f 1), (f2), (f4), (3.1) hold. In addition, assume that instead of the condition (f 3) we have (f3') : there exists a function a € L°([0,T]) such that ||f(t,£)||E < a(t)(1 +1|£||E). If Lk/A < 1, where k = max {||a||^, ||^||^} , ^ is the function from the condition (f4), L is the constant defined by the formula (4.2), then the problem (1.1)—(1.2) has a solution.

Proof. Take an arbitrary x € C = C([0,T]; E), then for t € [0,T] we have the following estimate:

||Fx(t)||E <

(1 + Eq(\Tq)) Jq(T — s)q~lEqtq(\(T — s)q) ds x\\r)E(Xti)+

|(1 + Eq(\Tq))2 - Eqfi{\Tq)Eq,2{\Tq)\ l|a|l°° 11 + WxWc)^M )+

TEq,2(XTq) fQT(T - s)g~2Eg>g_i(A(T - 8)*) ds

|(1 + Eq(\Tq))2 - Eq,0(\Tq)Eq,2(\Tq)\ l|a|l~ 11 + WWW* )+

(1+Eq(ATq)) /0T(T-s)q-2Eq,q-l(A(T - s)q) ds

'gV^'JJJO °r / — II II (л ,|| || (\4-q\i

\{l+Eq{\Tq)Y-Eqfi{\Tq)Eq,2{\Tq)\ c)tbq^M

T~1Eqft(\Tq) J0T(T - s)q~l Eq>q(\(T - s)q) ds П,|ЫП№ (Xfq),

|(1 + Eq(\Tq))2 - Eqfi{\Tq)Eq,2{\Tq)\ 11 + IWIcJWWt )+

IML (1 + IMIc) /V - s)q-1Eq,q(X(t - s)q) ds = J 0

(1 + Eq(XTq)) (Eq(XTq) - 1) + TEq,2(\Tq)±Eq,o(\Tq)

|(1 + Eq (XTq ))2 - Eqfi(XTq )Eq,2(XTq )|

Hall

A -(1 + \\x\\c)Eq(Xtq)+

(1 + Eq{XTq))±Eqfl{XTq) + T~lEqfl(XTq) (Eq{XTq) - 1) |(1 + Eq{XTq)f - Eqfl{XTq)Eq,2{XTq)\ X

llall Hall

+ \\x\\c)tEq>2{Xtq) + 1-^(1 + \\x\\c) (Eq{Xtq) - 1) <

E2(XTq) - 1 + Eq 2(XTq)Eq o(XTq) Hall gV ; ; q,o\ ; ll«lloc(1 + \\x\\c)Eq{XTq) +

|(1 + Eq (XTq ))2 - Eqo(XTq )Eq,2 (XTq )| X

2T^EqAXTq)Eq(XTq) "a"°°(l + \\x\\c)TEq>2(XTq) +

|(1 + Eq (XTq ))2 - Eqo(XTq )Eq,2(XTq )| X a

l-Lj^(l + \\x\\c)(Eq(XTq)-l) =

(E2q{XTq) - 1 + 3EqAXTq)Eq(XTq)EqAXTq) _ x

I |(1 + Eq{XTq))2 — Eq>o(XTq)Eq>2(XTq)\ ^ j JX

a k

11 Moo-(l + ll^llc) <LT(1 + Nlc).

Now, if we take R > ; then the inequality ||a;||c < R implies

that ||Fx||C < R. Hence, the operator F maps the closed ball BR(0) c C into itself. Therefore, the operator F satisfies all the assumptions of the theorem 1, and thus F has fixed points, and the problem (1.1)-(1.2) has a solution. □

5. Conclusion

In the paper, the existence of a solution to an antiperiodic boundary value problem for a semilinear differential equation of fractional order q € (1,2) was considered in a separable Banach space. The original problem was reduced to the problem on existence of fixed points of the corresponding resolving integral operator. Using the topological degree theory for condensing mappings and a generalized B.N. Sadovskii-type fixed point theorem, conditions which guarantee the existence of fixed points for the resolving operator were obtained.

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G. G. PETROSYAN References

Afanasova M., Petrosyan G. On the boundary value problem for functional-differential inclusion of fractional order with general initial condition in a Banach space. Russian Mathematics, 2019, vol. 63, no. 9, pp. 1-12. https://doi.org/10.3103/S1066369X19090019

Agarwal R.P., Ahmad B. Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Computers and Mathematics with Applications, 2011, vol. 62, pp. 1200-1214. https://doi.org/10.1016/j.camwa.2011.03.001

Ahmad B., Nieto J.J. Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topological Methods in Nonlinear Analysis, 2010, vol. 35, pp. 295-304. Bogdan V.M. Generalized vectorial Lebesgue and Bochner integration theory. arXiv:1006.3881v1 [math.FA], 2010, 86 p.

Chen Y., Nieto J.J., O'Regan D. Antiperiodic solutions for fully nonlinear first-order differential equations. Math. Comput. Modelling, 2007, vol. 46, pp. 1183-1190. https://doi.org/10.1016/j.mcm.2006.12.006

Delvos F. J., Knoche L. Lacunary interpolation by antiperiodic trigonometric polynomials. BIT, 1999, vol. 39, pp. 439-450. https://doi.org/10.1023/A:1022314518264

Fichtenholz G.M. Course in Differential and Integral Calculus. Moscow, Fizmatlit Publ., 2006, vol. 1, 607 p. (in Russian)

Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V. Mittag-Leffler Functions, Related Topics and Applications. Berlin, Heidelberg, Springer-Verlag, 2014, 443 p. Hilfer R. Applications of Fractional Calculus in Physics. Singapore, World Scientific, 2000, 472 p.

Kamenskii M., Obukhovskii V., Zecca P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Berlin, New-York, de Gruyter Series in Nonlinear Analysis and Applications, 7, Walter de Gruyter, 2001, 231 p. Kamenskii M., Obukhovskii V., Petrosyan G., Yao J.C. On semilinear fractional order differential inclusions in Banach spaces. Fixed Point Theory 2017, vol. 18, no. 1, pp. 269-292. https://doi.org/10.24193/fpt-ro.2017.L22 Kamenskii M., Obukhovskii V., Petrosyan G., Yao J.C. On approximate solutions for a class of semilinear fractional-order differential equations in Banach spaces. Fixed Point Theory and Applications, 2017, vol. 28, no. 4, pp. 1-28. https://doi.org/10.1186/s13663-017-0621-0

Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of Fractional Differential Equations. Amsterdam, Elsevier Science B.V., North-Holland Mathematics Studies, 2006, 523 p.

Obukhovskii V.V., Gelman B.D. Multivalued Maps and Differential Inclusions. Elements of Theory and Applications. Singapore, World Scientific, 2020, 220 p. Podlubny I. Fractional Differential Equations. San Diego, Academic Press, 1999, 340 p.

Shao J. Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays. Phys. Lett. A, 2008, vol. 372, pp. 5011-5016. https://doi.org/10.1016/j.physleta.2008.05.064

Tarasov V.E. Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. London, New York, Springer, 2010, 504 p.

Garik Petrosyan, Candidate of Sciences (Physics and Mathematics), Leading Researcher, Research Center, Voronezh State University of Engineering Technologies, 19, Revolutsii Prospect, Voronezh, 394036, Russian Federation, tel.: (3952)242210, e-mail: garikpetrosyan@yandex.ru, ORCID iD https://orcid.org/0000-0001-8154-6299.

Received 06.07.2020

Антипериодическая задача для полулинейного дифференциального уравнения дробного порядка

Г. Г. Петросян

Воронежский государственный университет инженерных технологий, Воронеж, Российская Федерация

Аннотация. Рассматривается антипериодическая краевая задача для полулинейного дифференциального уравнения с дробной производной Капуто порядка q £ (1, 2) в сепарабельном банаховом пространстве. Для разрешения поставленной задачи мы конструируем, используя теорию дробного анализа и свойства функции Миттаг-Леффлера, соответствующую задаче функцию Грина. Затем исходная задача сводится к задаче о существовании неподвижных точек разрешающего интегрального оператора. Для доказательства существования неподвижных точек разрешающего оператора мы исследуем его свойства на основе теории топологической степени для уплотняющих отображений и используем обобщенную теорему типа Б. Н. Садовского о неподвижной точке.

Ключевые слова: дробная производная Капуто, полулинейное дифференциальное уравнение, краевая задача, неподвижная точка, уплотняющее отображение, мера некомпактности.

Список литературы

1. Афанасова М. С., Петросян Г. Г. О краевой задаче для функционально-дифференциального включения дробного порядка с общим начальным условием в банаховом пространстве // Известия вузов. Математика. 2019. № 9. С. 3-15. https://doi.org/10.26907/0021-3446-2019-9-3-15

2. Agarwal R. P., Ahmad B. Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions // Computers and Mathematics with Applications. 2011. Vol. 62. P. 1200-1214. https://doi.org/10.1016/jxamwa.2011.03.001

3. Ahmad B., Nieto J. J. Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory // Topological Methods in Nonlinear Analysis. 2010. Vol. 35. P. 295-304.

4. Bogdan V. M. Generalized vectorial Lebesgue and Bochner integration theory // arXiv:1006.3881v1 [math.FA]. 2010. 86 p.

5. Chen Y., Nieto J. J., O'Regan D. Antiperiodic solutions for fully nonlinear first-order differential equations // Math. Comput. Modelling. 2007. Vol. 46. P. 1183-1190. https://doi.org/10.1016/j.mcm.2006.12.006

6. Delvos F. J., Knoche L. Lacunary interpolation by antiperiodic trigonometric polynomials // BIT. 1999. Vol. 39. P. 439-450. https://doi.org/10.1023/A:1022314518264

7. Фихтенгольц Г. М. Курс дифференциального и интегрального исчисления. М. : Физматлит, 2006. Т. 1. 607 с.

8. Mittag-Leffler Functions, Related Topics and Applications / R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin. Berlin Heidelberg : Springer-Verlag, 2014. 443 p.

9. Hilfer R. Applications of Fractional Calculus in Physics. Singapore : World Scientific, 2000. 472 p.

10. Kamenskii M., Obukhovskii V., Zecca P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces. Berlin ; New-York : Walter de Gruyter, 2001. 231 p. (de Gruyter Series in Nonlinear Analysis and Applications ; vol. 7).

11. On semilinear fractional order differential inclusions in Banach spaces / M. Kamenskii, V. Obukhovskii, G. Petrosyan, J. C. Yao // Fixed Point Theory. 2017. Vol. 18, N 1. P. 269-292. https://doi.org/10.24193/fpt-ro.2017.L22

12. On approximate solutions for a class of semilinear fractional-order differential equations in Banach spaces / M. Kamenskii, V. Obukhovskii, G. Petrosyan, J. C. Yao // Fixed Point Theory and Applications. 2017. Vol. 28, N 4. P. 1-28. https://doi.org/10.1186/s13663-017-0621-0

13. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. Amsterdam : Elsevier Science B.V., North-Holland Mathematics Studies, 2006. 523 p.

14. Obukhovskii V. V., Gelman B. D. Multivalued Maps and Differential Inclusions. Elements of Theory and Applications. Singapore : World Scientific, 2020. 220 p.

15. Podlubny I. Fractional Differential Equations. San Diego : Academic Press, 1999. 340 p.

16. Shao J. Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays // Phys. Lett. A. 2008. Vol. 372. P. 5011-5016. https://doi.org/10.1016/j.physleta.2008.05.064

17. Tarasov V.E. Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. London, New York : Springer, 2010. 504 p.

Гарик Гагикович Петросян, кандидат физико-математических наук, ведущий научный сотрудник, Научно-образовательный центр, Воронежский государственный университет инженерных технологий, Российская Федерация, 394036, г. Воронеж, пр. Революции, 19, тел.: 8-473255-38-75, e-mail: garikpetrosyan@yandex.ru, ORCID iD https:// orcid.org/0000-0001-8154-6299.

Поступила в 'редакцию 06.07.2020