We investigate the nonlocal fractional boundary value problem u00 D AcD˛uC f .t

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Vol. 19 (2018), No. 1, pp. 611–623 DOI: 10.18514/MMN.2018.1809

BOUNDARY VALUE PROBLEMS FOR BAGLEY–TORVIK FRACTIONAL DIFFERENTIAL EQUATIONS AT RESONANCE

SVATOSLAV STAN ˇEK Received 15 October, 2015

Abstract. We investigate the nonlocal fractional boundary value problem u⁰⁰ D A^cD^˛uC f .t; u;^cDu; u⁰/,u⁰.0/Du⁰.T /,.u/D0, at resonance. Here,˛2.1; 2/,2.0; 1/,f and WC Œ0; T !Rare continuous. We introduce a ”three-component” operatorS which first component is related to the fractional differential equation and remaining ones to the boundary conditions. Solutions of the problem are given by fixed points ofS. The existence of fixed points of Sis proved by the Leray–Schauder degree method.

2010Mathematics Subject Classification: 34A08; 26A33; 34B15

Keywords: fractional differential equation, boundary value problem at resonance, Caputo fractional derivative, Leray–Schauder degree, maximum principle

1. INTRODUCTION

LetT > 0be given andJ DŒ0; T . Denote byAthe set of (generally nonlinear) functionalsWC.J /!Rwhich are

(a) continuous,.0/D0,

(b) increasing, that is,x; y2C.J /,x.t / < y.t /onJ ).x/ < .y/.

Remark1. Let2Abe linear. Then it follows from property (b) of that takes bounded sets into bounded sets. Henceis a linear bounded functional.

Example 1. Let p2C.J / be positive, n2N, 0t0< t1< < tnT, and a_k> 0,kD0; 1; : : : ; n. Then the functionals

1.x/Dmaxfx.t /Wt2Jg; 2.x/Dminfx.t /Wt2Jg; 3.x/D

Z T 0

p.s/.x.s//^{2n 1}ds; 4.x/D

n

X

kD0

a_kx.t_k/ and their linear combinations with positive coefficients belong to the setA.

Supported by the grant No. 14-06958S of the Grant Agency of the Czech Republic.

c 2018 Miskolc University Press

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We discuss the fractional boundary value problem

u⁰⁰.t /DA^cD^˛u.t /Cf .t; u.t /;^cDu.t /; u⁰.t //; (1.1) u⁰.0/Du⁰.T /; .u/D0; 2A: (1.2) Here,^cDdenotes the Caputo fractional derivative,A2R,˛2.1; 2/,2.0; 1/, and the functionf satisfies the condition:

.H / there exists > 0such thatf 2C.DŒ ; /, where D DJŒ T; T Œ K; K/; KD T¹

.2 /; and

f .t; x; y; /0; f .t; x; y; /0 for.t; x; y/2D:

Equation1.1is the fractional differential equation of the Bagley-Torvik type. Its special case is the equationu⁰⁰DA^cD³⁼²uCauC'.t /. This equation with ^cD³⁼² replaced by the Riemann–Liouville fractional derivativeD³⁼²is called the Bagley–

Torvik equation. Torvik and Bagley [22] used this equation in modelling the motion of a rigid plate immersing in a Newtonian fluid. Analytical and numerical solutions of the problem

u⁰⁰DAD³⁼²uCauC'.t /; u.0/D0; u⁰.0/D0;

are given in [13,16,18], while for the problem

u⁰⁰DA^cD^˛uCauC'.t /; u.0/Du0; u⁰.0/Du1;

in [5,6,8,11,23]. The existence results for solutions of the generalized Bagley–Torvik equation (1.1) satisfying the boundary conditionsu⁰.0/D0,u.T /Cau⁰.T /D0are given in [20]. Here,f is a Carath´eodory function.

Definition 1. We say that u2C².J / is a solution of problem (1.1), (1.2) if u satisfies the boundary conditions (1.2) and (1.1) holds fort2J.

We recall that the Riemann–Liouville fractional integral I of order > 0of a functionxWJ !Ris defined as [10,13,16]

Ix.t /D Z t

0

.t s/¹

. / x.s/ds;

and the Caputo fractional derivative ^cDx of order > 0, 62N, of a function xWJ !Ris given by the formula [10,13]

cDx.t /D dⁿ dtⁿ

Z t 0

.t s/^{n 1} .n / x.s/

n 1

X

kD0

x^.k/.0/

kŠ s^k

! ds;

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wherenDŒ C1,Œ means the integral part of and is the Euler gamma function. Ifx2Cⁿ.J /andn 1 < < n, then

cDx.t /D Z t

0

.t s/^{n 1}

.n / x^.n/.s/dsDIⁿx^.n/.t /:

In particular, ifx2C².J /and˛2.1; 2/,2.0; 1/, then

cD^˛x.t /D Z t

0

.t s/^{1 ˛}

.2 ˛/ x⁰⁰.s/ds; t 2J;

cD^˛x.t /D d dt

Z t 0

.t s/^{1 ˛}

.2 ˛/ .x⁰.s/ x⁰.0//dsD^cD^{˛ 1}x⁰.t /; t2J;

cDx.t /D d dt

Z t 0

.t s/

.1 /.x.s/ x.0//dsDI¹x⁰.t /; t 2J:

It is well known [10,13] that IWC.J /!C.J / for 2.0; 1/. Therefore, if x 2 C².J /, then^cD^˛x;^cDx2C.J /for˛2.1; 2/and2.0; 1/.

We will show that problem (1.1), (1.2) is at resonance. The linear functionx.t /D atCbis a solution of the problemu⁰⁰ A^cD^˛uD0,u⁰.0/Du⁰.T /, for eacha; b2R.

Let us consider the set of all functions atCb which are solutions of the equation .atCb/D0, whereis from (1.2).

Ifis linear, thenbD ^{a.t /}_.1/. Hence n

a

t _.1/^{.t /} Wa2R

o

is the set of solutions to problemu⁰⁰ A^cD^˛uD0, (1.2). This set is a one-dimensional linear subspace of C².J /.

Letbe nonlinear. IfaD0, thenbD0. Leta2Rn f0g. By our Lemma1(for D1), there existsa2J such thataaCbD0. HencebD aaand the equality .a.t a//D0 is true. a is determined uniquely. If this is not true, then there exists a 2J, a6Da, such that.a.t a//D0. Sincea.t a/6Da.t a/ for allt 2J, and therefore eithera.t a/ < a.t a/ora.t a/ > a.t a/on J, it follows from property (b) of that .a.t a//6D.a.t a//, which is impossible. Consequently,uD0andfa.t _a/Wa2Rn f0ggis the set of solutions to the problemu⁰⁰ A^cD^˛uD0, (1.2). In contrast to previous case, this set is not a one-dimensional linear subspace ofC².J /.

In order to show the solvability of problem (1.1), (1.2), we have to overcome troubles that derivatives are of fractional order, the problem is at resonance and finally thatin the boundary conditions (1.2) is generally a nonlinear functional. To this end, an auxiliary ”three-component” operatorS is introduced. Its first component is related to equation (1.1) and remaining ones to the boundary conditions (1.2).

Solutions of (1.1), (1.2) are given by fixed points ofS. The existence of fixed points ofS is proved by means of the Leray-Schauder degree method [7].

In the literature, see [1–4,12,14,19] and references therein, existence results for fractional boundary value problems at resonance are usually proved by using the the coincidence degree theory due to Mawhin [15].

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Our main result is as follows.

Theorem 1. Let.H / hold and letA > 0. Then problem.1:1/; .1:2/has at least one solution.

The paper is organized as follows. In Section 2 we state the results which are used in the next sections. Section 3 is devoted to auxiliary boundary value problems. To this end operatorsQ;S;KandHare introduced and their properties are given. In Section 4 Theorem1is proved. An example demonstrates our results.

Throughout the paper˛2.1; 2/,2.0; 1/,KD ^T_{.2 /}¹ andkxk Dmaxfjx.t /jWt2 Jgis the norm inC.J /.

2. PRELIMINARIES

This section contains the results that we will need in the next sections.

Lemma 1. Let2Aand let the equality

.x/C. 1/. x/D0

hold for somex2C.J /and2Œ0; 1. Then there exists2J such thatx./D0.

Proof. Assume that the statement is not true. Then eitherx > 0orx < 0onJ. If x > 0onJ, then.x/ > 0,. x/ < 0, and therefore .x/C. 1/. x/ > 0, which is impossible. Similarly,x < 0onJ leads to a contradiction.

The following maximal principle follows immediately from [17, Lemma 2.1] and [9, Lemma 2.7] and its proof.

Lemma 2(Maximum principle). Lett02.0; T , x2C¹Œ0; t0,x.t /x.t0/for t2Œ0; t0,x.0/ < x.t0/andx⁰.t0/D0. Let 2.0; 1/. Then

cDx.t /j^tDt0> 0:

Corollary 1. Lett02.0; T ,x2C¹Œ0; t0,x.t /x.t0/fort2Œ0; t0,x.0/ > x.t0/ andx⁰.t0/D0. Let2.0; 1/. Then

cDx.t /j^tDt₀< 0:

Lemma 3([21]). Letr2C.J /and 2.0; 1/. Then the initial value problem x⁰.t /DA^cDx.t /Cr.t /; x.0/Da; A; a2R;

has the unique solution x.t /DaC

Z t 0

r.s/dsCA Z t

0

Z s 0

.s / E1 ;1 A.s /¹ r./d

ds;

where

E1 ;1 .´/D

1

X

kD0

´^k

..kC1/.1 //; ´2R;

is the classical Mittag-Leffler function.

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Lemma 4 ([24, Lemma 2.2]). Let2.0; 1/ and letE; be the Mittag-Leffler function. Then

E;.´/ > 0; E_;⁰ .´/ > 0 for´2R:

We also need the following result.

Lemma 5. Leth2C.J /andA; c1; c22R. Then the initial value problem u⁰⁰.t /DA^cD^˛u.t /Ch.t /; u.0/Dc2; u⁰.0/Dc1; (2.1) has the unique solution

u.t /Dc1tCc2C Z t

0

.t s/h.s/ds CA

Z t 0

.t s/

Z s 0

.s /^{1 ˛}E2 ˛;2 ˛ A.s /^{2 ˛} h./d

ds:

(2.2)

Proof. Since ^cD^˛x.t /D^cD^{˛ 1}x⁰.t /for t 2J andx2C².J /, the equation of (2.1) can be written as

u⁰⁰.t /DA^cD^{˛ 1}u⁰.t /Ch.t /: (2.3) Hence, by Lemma3(forrDhand withxand replaced byu⁰and˛ 1),

u⁰.t /Dc1C Z t

0

h.s/ds CA

Z t 0

Z s 0

.s /^{1 ˛}E2 ˛;2 ˛ A.s /^{2 ˛} h./d

ds;

where u⁰.0/Dc1. Consequently, u.t /Dc2CRt

0u⁰.s/ds is the unique solution of

problem (2.1) and (2.2) follows.

3. OPERATORS

In this section auxiliary operators are introduced and their properties are proved.

The most important of these operators is an operator S by which the solvability of problem (1.1), (1.2) is proved in Section 4.

Let 1.x/D

8 ˆ<

ˆ:

T forx > T ; x forjxj T ;

T forx < T ;

2.y/D 8 ˆ<

ˆ:

K fory > K;

y forjyj K;

K fory < K;

whereandKare from.H /. Let

f .t; x; y; ´/Q Df .t; 1.x/; 2.y/; ´/ for.t; x; y; ´/2JR²Œ ;

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and

f.t; x; y; ´/D 8 ˆˆ ˆˆ ˆ<

ˆˆ ˆˆ ˆ:

f .t; x; y; /Q C´

´ if´ > ; f .t; x; y; ´/Q ifj´j ; f .t; x; y; /Q ´C

´ if´ < : Under condition.H /,f2C.JR³/,

f.t; x; y; /0; f.t; x; y; /0 for.t; x; y/2JR²; f.t; x; y; ´/ < 0 for.t; x; y; ´/2JR². 1; /;

f.t; x; y; ´/ > 0 for.t; x; y; ´/2JR².;1/;

)

(3.1) and

jf.t; x; y; ´/j E for.t; x; y; ´/2JR³; (3.2) where

ED1Cmax n

jf .t; x; y; ´/jW.t; x; y/2D; ´2Œ ; o : Consider the fractional differential equation

u⁰⁰.t /DA^cD^˛u.t /Cf.t; u.t /;^cDu.t /; u⁰.t // (3.3) associated to equation (1.1). Keeping in mind Lemma5define operatorsQWC¹.J /! C.J /andSWC¹.J /R²!C¹.J /R²by the formulae

.Qx/.t /Df.t; x.t /;^cDx.t /; x⁰.t //

CA Z t

0

.t s/^{1 ˛}E2 ˛;2 ˛ A.t s/^{2 ˛}

f.s; x.s/;^cDx.t /; x⁰.s//ds;

S.x; c1; c2/D c1tCc2C Z t

0

.t s/.Qx/.s/ds; c1C Z T

0

.Qx/.s/ds; c2C.x/

!

; whereis from (1.2).

Lemma 6. Let.H /hold. If.x; c1; c2/is a fixed point of the operatorS, thenxis a solution of problem.3:3/; .1:2/andx⁰.0/Dc1,x.0/Dc2.

Proof. Let .x; c1; c2/ be a fixed point of the operatorS, that is, S.x; c1; c2/D .x; c1; c2/. Then

x.t /Dc1tCc2C Z t

0

.t s/.Qx/.s/ds; t 2J; (3.4) Z T

0

.Qx/.s/dsD0; (3.5)

.x/D0: (3.6)

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It follows from (3.4) and Lemma 5 (for h.t /Df.t; x.t /;^cDx.t /; x⁰.t //) that x.0/Dc2,x⁰.0/Dc1andxis a solution of (3.3).

Since (cf. (3.4))

x⁰.t /Dc1C Z t

0

.Qx/.s/ds; t 2J;

we conclude from (3.5) thatx⁰.T /Dc1. Hence x⁰.0/Dx⁰.T /. The last equality together with (3.6) give thatxsatisfies the boundary conditions (1.2). Consequently, xis a solution of problem (3.3), (1.2) andx⁰.0/Dc1,x.0/Dc2. In order to prove that the operatorS admits a fixed point, for2Œ0; 1, we first introduce an operatorKWC¹.J /R²!C¹.J /R²,

K.x; c1; c2/

D c1tCc2; c1C.1 /x⁰.0/C Z T

0

.Qx/.s/ds; c2C.x/C. 1/. x/

! : Let

˝Dn

.x; c1; c2/2C¹.J /R²

W kxk< TC1;kx⁰k< C1;jc1j< C1;jc2j< TC1o

;

(3.7) whereis from.H /.

Lemma 7. Let.H /hold and letA > 0. Then deg

I K1; ˝; 0

6D0; (3.8)

where ”deg” stands for the Leray-Schauder degree andIis the identity operator on C¹.J /R².

Proof. LetMWŒ0; 1C¹.J /R!C¹.J /R,M.; x; c1; c2/DK.x; c1; c2/.

Sincef2C.JR³/, we conclude from Lemma4thatQis a continuous operator.

Asis continuous and takes bounded sets into bounded sets, it is easy to prove that M is a completely continuous operator.

Due to

K0. x; c1; c2/D K0.x; c1; c2/ for.x; c1; c2/2C¹.J /R²; K0is an odd operator.

Assume thatM.0; x; c1; c2/D.x; c1; c2/for some.x; c1; c2/2C¹.J /R²and 02Œ0; 1. Then

x.t /Dc1tCc2; t 2J; (3.9) .1 0/x⁰.0/C0

Z T 0

.Qx/.s/dsD0; (3.10)

.x/C.0 1/. x/D0: (3.11)

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Lemma 1 together with (3.11) give x./D0 for some 2J. Hence (cf. (3.9)) c1Cc2D0, and thereforex.t /Dc1.t /onJ.

We now prove that

jc1j : (3.12)

Letc1> . Thenx⁰Dc1> onJ, and thereforef.t; x.t /;^cDx.t /; x⁰.t // > 0 fort 2J by (3.1). This fact together withA > 0and Lemma4imply.Qx/.t / > 0 onJ. Hence.1 0/c1C0RT

0 .Qx/.s/ds > 0, which contradicts (3.10). Therefore c1 . Similarly, if c1< , then we have f.t; x.t /;^cDx.t /; x⁰.t // < 0 and .Qx/.t / < 0fort2J, which again contradicts (3.10). Hence (3.12) is true.

Consequently, jx.t /j D jc1.t /j T, jx⁰.t /j D jc1j , j^cDx.t /j D jI¹x⁰.t /j KonJ andjc2j D jx.0/j T. As a result,

M.; x; c1; c2/6D.x; c1; c2/ for.x; c1; c2/2@˝and2Œ0; 1:

Hence, by the Borsuk antipodal theorem and the homotopy property, the relations deg

I K0; ˝; 0 6D0;

deg

I K0; ˝; 0 Ddeg

I K1; ˝; 0

hold. Combining these relations we obtain (3.8).

Finally, let for2Œ0; 1an operatorHWC¹.J /R²!C¹.J /R²be defined as

H.x; c1; c2/D c1tCc2C Z t

0

.t s/.Qx/.s/ds; c1C Z T

0

.Qx/.s/ds; c2C.x/

! : Then, for.x; c1; c2/2C¹.J /R²,

H0.x; c1; c2/DK1.x; c1; c2/; (3.13) H1.x; c1; c2/DS.x; c1; c2/: (3.14) Lemma 8. Let .H / hold. Let VWŒ0; 1C¹.J /R ! C¹.J /R and V .; x; c₁; c₂/DH.x; c₁; c₂/. ThenV is a completely continuous operator.

Proof. We first prove thatV is continuous. To this end letfxng C¹.J /,fcn;ig R,iD1; 2,fng Œ0; 1be convergent sequences and let limn!1xnDxinC¹.J /, limn!1cn;iDci, limn!1nDinR, wherex2C¹.J /,ci; 2R,iD1; 2. Then limn!1f.t; xn.t /;^cDxn.t /; x⁰_n.t //Df.t; x.t /;^cDx.t /; x⁰.t // uniformly on J. This together with Lemma4imply that limn!1.Qxn/.t /D.Qx/.t /uniformly onJ. Hence

nlim!1

cn;1tCcn;2Cn

Z t 0

.t s/.Qxn/.s/ds

Dc1tCc2C Z t

0

.t s/.Qx/.s/ds;

nlim!1

cn;1Cn

Z t 0

.Qxn/.s/ds

Dc1C Z t

0

.Qx/.s/ds

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uniformly onJ. Besides,

nlim!1 cn;1C Z T

0

.Qxn/.s/ds

!

Dc1C Z T

0

.Qx/.s/ds;

nlim!1 cn;2C.xn/

Dc2C.x/:

Consequently,V is a continuous operator.

Let˚ C¹.J /R²be bounded and letkxk L,kx⁰k L,jc1j L,jc2j L for.x; c1; c2/2˚, whereLis a positive constant. LetW DE2 ˛;2 ˛ jAjT^{2 ˛}

. Then, by (3.2) and Lemma4, the relation

j.Qx/.t /j EC jAjE Z t

0

.t s/^{1 ˛}E2 ˛;2 ˛ A.t s/^{2 ˛} ds EC jAjEW

Z t 0

.t s/^{1 ˛}dsEC jAjEWT^{2 ˛} 2 ˛ DH holds fort2J and.x; c1; c2/2˚. Hence

ˇ ˇ ˇ ˇ

c1tCc2C Z t

0

.t s/.Qx/.s/ds ˇ ˇ ˇ

ˇL.T C1/CH T² 2 ; ˇ

ˇ ˇ ˇ ˇ

c1C Z T

0

.Qx/.s/ds ˇ ˇ ˇ ˇ ˇ

LCH T;

jc2C.x/j LCmaxfj. L/j; .L/g

fort2J,.x; c₁; c₂/2˚and2Œ0; 1, and therefore the setV .Œ0; 1˚ /D fV .; x; c₁; c₂/W2 Œ0; 1; .x; c1; c2/2˚gis bounded inC¹.J /R². In view ofkQxk H we see that

the setn

c1CRt

0.Qx/.s/dsW.x; c1; c2/2˚; 2Œ0; 1o

is equicontinuous onJ. Hence the Arzel`a-Ascoli theorem and the Bolzano–Weierstrass compactness theorem inRguarantee that the setV .Œ0; 1˚ /is relatively compact inC¹.J /R².

Consequently,V is completely continuous.

4. THE PROOF OF THEOREM 1AND AN EXAMPLE

Proof. Suppose that .x; c1; c2/2C¹.J /R² is a fixed point of H for some 2Œ0; 1, that is,H.x; c1; c2/D.x; c1; c2/. IfD0, then it follows from the proof of Lemma7(cf. (3.13)) that.x; c1; c2/2˝, where˝is given in (3.7). Let2.0; 1.

Then

x.t /Dc1tCc2C Z t

0

.t s/.Qx/.s/ds; t 2J; (4.1) Z T

0

.Qx/.s/dsD0; (4.2)

.x/D0: (4.3)

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Hence

x⁰.t /Dc1C Z t

0

.Qx/.s/ds; t 2J; (4.4)

sox⁰.0/Dc1, and, by (4.2),x⁰.T /Dc1CRT

0 .Qx/.s/dsDc1. Consequently,

x⁰.0/Dx⁰.T /: (4.5)

Suppose thatc1> , whereis from.H /. Thenf.0; x.0/; 0; c1/ > 0by (3.1), and therefore f.t; x.t /;^cDx.t /; x⁰.t // > 0 on a right neighbourhood of t D0.

If there is some 2.0; T such that f.t; x.t /;^cDx.t /; x⁰.t // > 0 on Œ0; /and f.; x./;^cDx.t /jtD; x⁰.//D0, then .Qx/.t / > 0 on Œ0; because A > 0, which givesx⁰.t / > c1fort2.0; . Hencef.t; x.t /;^cDx.t /; x⁰.t // > 0onŒ0; , contrary tof.; x./;^cDx.t /jtD; x⁰.//D0. Consequently,

f.t; x.t /;^cDx.t /; x⁰.t // > 0; .Qx/.t / > 0; t2J:

Thusx⁰.T / > c1Dx⁰.0/, which contradicts (4.5). Hencec1. Similarly, we can prove thatc1 . To summarize,jc1j .

Suppose that maxfx⁰.t /Wt2Jg Dx⁰./ > . Then2.0; T /andx⁰./ x⁰.0/ >

0. By (4.4),x2C².J /andx⁰⁰DQx. Hencex⁰⁰./D0and by Lemma5and (2.3) (forh.t /Df.t; x.t /;^cDx.t /; x⁰.t //) the equality

x⁰⁰.t /DA^cD^{˛ 1}x⁰.t /Cf.t; x.t /;^cDx.t /; x⁰.t //; t2J;

holds. Lemma 2 (for t0 D , D ˛ 1 and x replaced by x⁰) shows that

cD^{˛ 1}x⁰.t /jtD> 0. Hence

x⁰⁰./DA^cD^{˛ 1}x⁰.t /jtDCf.; x./;^cDx.t /jtD; x⁰.// > 0;

which is impossible. Hencex⁰.t /fort2J. Similarly, by Corollary1, we can prove thatx⁰ onJ. Consequently,

jx⁰.t /j ; t2J:

Next, it follows from (4.3) and Lemma1thatx. /D0for some2J. Therefore jx.t /j Dˇ

ˇ ˇ

Rt

x⁰.s/dsˇ ˇ

ˇjt j T,j^cDx.t /j D jI¹x⁰.t /j K. Asc1D x⁰.0/andc2Dx.0/, we have proved

kxk T;k^cDxk K; kx⁰k ; jc1j T; jc2j ; (4.6) which impliesV .; x; c1; c2/6D.x; c1; c2/for.x; c1; c2/2@˝and2Œ0; 1, where V is from Lemma8. Combinig Lemma8with the homotopy property we have

deg

I H0; ˝; 0 Ddeg

I H1; ˝; 0 : This equality together with (3.8) and (3.13) give

deg

I H1; ˝; 0 6D0:

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Hence there exists a fixed point.x; c1; c2/ofH1. Lemma6and (3.14) guarantee that x is a fixed point of problem (3.3), (1.2) and c1Dx⁰.0/, c2Dx.0/. Due to (4.6), f.t; x.t /;^cDx.t /; x⁰.t //Df .t; x.t /;^cDx.t /; x⁰.t //fort2J, and thereforexis

a solution of problem (1.1), (1.2).

Example2. Letp2C.JR³/be bounded,a; b; c2C.J /,c > 0onJ, andn2N, ˇ; 2.0; 2n 1/. Then the function

f .t; x; y; ´/Dp.t; x; y; ´/Ca.t /jxj^{ˇ 1}xCb.t /jyjCc.t /´^{2n 1}

satisfies condition .H /. Really, let jp.t; x; y; ´/j Lfor .t; x; y; ´/2J R³ and cDminfc.t /Wt2Jg. Since

vlim!1

Lv^{1 2n}C kakT^ˇv^ˇ^C^{1 2n}C kbkKv^C^{1 2n}

D0; KD T¹ .2 /; there exists > 0such that

L^{1 2n}C kakT^ˇ^ˇ^C^{1 2n}C kbkK^C^{1 2n}c:

Hence LC kak.T /^ˇ C kbk.K/ c^{2n 1}, and therefore for .t; x; y/2D, whereD is from.H /, the inequalities

f .t; x; y; / L kak.T /^ˇ kbk.K/Cc^{2n 1}0;

f .t; x; y; /LC kak.T /^ˇC kbk.K/ c^{2n 1}0 hold. Theorem1gives that the equation

u⁰⁰DA^cD^˛uCp.t; u;^cDu; u⁰/

Ca.t /juj^{ˇ 1}uCb.t /j^cDujCc.t /.u⁰/^{2n 1}; A > 0; (4.7) has at least one solutionusatisfying the boundary conditions (1.2) andkuk T, k^cDuk K,ku⁰k .

In particular, there exists a solution of (4.7) satisfying the boundary conditions minfu.t /Wt2Jg D0; u⁰.0/Du⁰.T /;

that is,uis a nonnegative solution of the problem.

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Author’s address

Svatoslav Stanˇek

Department of Mathematical Analysis, Faculty of Science

Current address: Palack´y University, 17. listopadu 12, 771 46 Olomouc, Czech Republic E-mail address:svatoslav.stanek@upol.cz