Memoirs on Differential Equations and Mathematical Physics

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1 Memoirs on Differential Equations and Mathematical Physics Volume 41, 27, Robert Hakl and Sulkhan Mukhigulashvili ON A PERIODIC BOUNDARY VALUE PROBLEM FOR THIRD ORDER LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

2 Abstract. Unimprovable effective sufficient conditions are established for the unique solvability of the periodic problem 2 u t) = l i u i) )t) + qt) for t ω, i= u j) ) = u j) ω) j =, 1, 2), where ω >, l i : C [, ω] ) L [, ω] ) are linear bounded operators, and q L [, ω] ). 2 Mathematics Subject Classification. 34K6, 34K1. Key words and phrases. Linear functional differential equation, ordinary differential equation, periodic boundary value problem. u t) = 2 l i u i) )t) + qt) t ω, i= u j) ) = u j) ω) j =, 1, 2), ω > l i : C [, ω] ) L [, ω] )! " # $ % q L [, ω] )&

3 On a Periodic BVP for Third Order Linear FDE Introduction Consider the equation 2 u t) = l i u i) )t) + qt) for t ω 1.1) i= with the periodic boundary conditions u j) ) = u j) ω) j =, 1, 2), 1.2) where ω >, l i : C [, ω] ) L [, ω] ) are linear bounded operators, and q L [, ω] ). By a solution of the problem 1.1), 1.2) we understand a function u C 2 [, ω] ) which satisfies the equation 1.1) almost everywhere on [, ω] and fulfils the conditions 1.2). There are many works and interesting results on the existence and uniqueness of solution for the periodic boundary value problem for higher order ordinary differential equations see, e.g., [1], [2], [4] [6], [11] [15], [17], [18], [25], [26], [29], [3] and the references therein). But an analogous problem for functional differential equations, even in the case of linear equations, remains still little investigated. In 1972 H. H. Schaefer see [28, Theorem 4]) proved that there exists a linear bounded operator l : C [, ω] ) L [, ω] ) which is not strongly bounded, that is, it does not have the following property: there exists a summable function η : [, ω] [, + [ such that lx)t) ηt) x C for t ω, x C [, ω] ). It is well known see, e.g., [16]) that the general boundary value problem for linear functional differential equations with a strongly bounded linear operator has the Fredholm property, i.e., it is uniquely solvable iff the corresponding homogeneous problem has only the trivial solution. The same property Fredholmity) for functional differential equations with a nonstrongly bounded linear operator was not investigated till 2. The first step was made in [3], [8] for scalar first order functional differential equations. Those results were generalized for the n-th order functional differential systems in [1]. Thus, in the present paper, we study the problem 1.1), 1.2) under the assumptions that l is a strongly bounded operator and l i i = 1, 2) are bounded, not necessarily strongly bounded, operators. We establish new unimprovable integral conditions sufficient for the unique solvability of the problem 1.1), 1.2). Note that in [12], unlike the earlier known results, there are investigated, among others, the existence and uniqueness of an ω-periodic solution of the nonautonomous ordinary differential equation n 1 u n) t) = p k t)u k) t) + qt) for t ω k=

4 3 R. Hakl and S. Mukhigulashvili without the requirement on the function p to be of constant sign. In this paper we improve the result of [12] for n = 3 in a certain way see Corollary 2.3). Consequently, the obtained results are also new even if 1.1) is an ordinary differential equation u t) = 2 p i t)u i) t) + qt) for t ω. 1.3) i= For functional differential equations, one can name only a few papers devoted to the study of the periodic boundary value problem see, e.g., [7], [19] [24], [27]). All the results will be formulated for the differential equation with deviating arguments u t) = 2 p i t)u i) τ i t)) + qt) for t ω, 1.4) i= which is a particular case of the equation 1.1). Here p i, q L [, ω] ) and τ i : [, ω] [, ω] i =, 1, 2) are measurable functions. The method used for the investigation of the considered problem is based on the method developed in our previous papers see [7], [19] [21], [23]). In particular, the results presented in the paper generalize the results obtained in [7], [23]. The following notation is used throughout: N is the set of all natural numbers; R is the set of all real numbers, R + = [, + [ ; C [, ω] ) is the Banach space of continuous functions u : [, ω] R with the norm u C = max{ ut) : t ω}; C 2 [, ω] ) is the set of functions u : [, ω] R which are absolutely continuous together with their first and second derivatives; L [, ω] ) is the Banach space of Lebesgue integrable functions p : [, ω] R with the norm p L = ω ps) ds; If l : C [, ω] ) L [, ω] ) is a linear operator, then l = sup { lx) L : x C 1 }. Definition 1.1. We will say that a linear operator l : C [, ω] ) L [, ω] ) is nonnegative if for any nonnegative x C [, ω] ) the inequality lx)t) for t ω is fulfilled. We will say that a linear operator l : C [, ω] ) L [, ω] ) is monotone if either l or l is nonnegative.

5 On a Periodic BVP for Third Order Linear FDE Main Results Theorem 2.1. Let nonnegative operators l 1, l 2 : C [, ω] ) L [, ω] ) and bounded operators l 1, l 2 be such that and where ω 2 l 1 1)t) l 2 1)t), 2.1) l 1 1)s) ds < β 1, 2.2) ω l 1 1)s) ds ω β 1 ω2 l 1 1)s) ds l 2 1)s) ds 64β 1 ω ω2 β 1 l 2 1)s) ds, 2.3) β 1 = 1 ω 4 l 1 l 2. Then the problem 1.1), 1.2) with both has a unique solution. l = l 1 l 2 and l = l 2 l 1 ) l 1 1)s) ds, 2.4) Theorem 2.2. Let l i1, l i2 : C [, ω] ) L [, ω] ) i =, 1, 2) be nonnegative operators, l 1, l 2 admit the representations and let l 1 = l 11 l 12, l 2 = l 21 l 22, β 1 = 1 ω 4 max { l 11, l 12 } max { l 21, l 22 }. 2.5) Let, moreover, the conditions 2.1) 2.4) be fulfilled. 1.1), 1.2) with both Then the problem has a unique solution. l = l 1 l 2 and l = l 2 l 1 If the operator l is monotone, then from Theorem 2.2 we get the following assertion which is nonimprovable in a certain sense see Example 2.1). Corollary 2.1. Let a monotone operator l and strongly bounded operators l 1, l 2 be such that l 1)t), l i = l i1 l i2 i = 1, 2),

6 R. Hakl and S. Mukhigulashvili where l i1, l i2 : C [, ω] ) L [, ω] ) are nonnegative operators, and let ω 4 max { l 11, l 12 } + max { l 21, l 22 } < 1. Let, moreover, l 1)s) ds 128β 1 ω 2, 2.6) where β 1 is given by 2.5). Then the problem 1.1), 1.2) has a unique solution. Now consider the equation with deviating arguments 1.4), where q, p i L [, ω] ) and τ i : [, ω] [, ω] are measurable functions. Corollary 2.2. Put β 2 = 1 ω { ω 4 max [p 1 s)] + ds, max { ω [p 2 s)] + ds, } [p 1 s)] ds } [p 2 s)] ds. Let, moreover, p 1, p 2 L [, ω] ) be nonnegative functions such that ω 2 2.7) p 1 t) p 2 t), 2.8) p 1 s) ds < β 2, 2.9) ω p 1 s) ds ω p 2 s) ds, 2.1) β 2 ω2 p 1 s) ds p 2 s) ds 64β 2 ω ) ω ω2 p 1 s) ds. 2.11) β 2 Then the problem 1.4), 1.2) with both p t) = p 1 t) p 2 t) and p t) = p 2 t) p 1 t) for t ω has a unique solution. If τ i t) = t i =, 1, 2), then we get the following assertion. Corollary 2.3. Let p 1, p 2, p 1, p 2 L [, ω] ) be such that all the assumptions of Corollary 2.2 are fulfilled. Let, moreover, either p 1 t) + p 2 t), 2.12)

7 On a Periodic BVP for Third Order Linear FDE 33 or Then the problem 1.3), 1.2) with both p 1 t) and p 2 t). 2.13) p t) = p 1 t) p 2 t) and p t) = p 2 t) p 1 t) for t ω has a unique solution. Remark 2.1. In the paper [13] see Proposition 1.1 therein), there was proved that if p does not change its sign, then the problem u = p t)u + qt), u j) ) = u j) ω) j =, 1, 2) is uniquely solvable iff p t). Hence it follows that if the conditions 2.12) and 2.13) in Corollary 2.3 are violated, i.e., if the function p = p 1 p 2 does not change its sign and p i t) i = 1, 2), then other conditions dealing with the smallness of p in the integral sense are not important. If l 1 and l 2, then from Theorem 2.1 we get the following assertion which has been published in [7]. Corollary 2.4. Let nonnegative operators l 1, l 2 : C [, ω] ) L [, ω] ) be such that and l 1 1)t) l 2 1)t), l 1 1)s) ds < ω 2, ω l 1 1)s) ds ω 1 ω2 l 1 1)s) ds l 2 1)s) ds 64 ω Then the problem 1.1), 1.2) with both has a unique solution. 1 ω2 l 2 1)s) ds, l = l 1 l 2 and l = l 2 l 1 ) l 1 1)s) ds. If the operator l is monotone, then from Theorem 2.1 we get the following assertion which has been published in [23]. Corollary 2.5. Let a monotone operator l and bounded operators l 1, l 2 be such that l 1)t) and ω 4 l 1 + l 2 < 1.

8 34 R. Hakl and S. Mukhigulashvili Let, moreover, l 1)s) ds 128 ω 2 1 ω ) 4 l 1 l ) Then the problem 1.1), 1.2) has a unique solution. It is shown in [23] that Corollary 2.5 is unimprovable in a certain sense. We present an appropriate example here for the sake of completeness. Example 2.1. The example below shows that the conditions 2.6) and 2.14) in Corollaries 2.1 and 2.5, respectively, are optimal and they cannot be weakened. Let ω = 1, α k = π 2 k k, β 2 k = 1 8k 1 4, k N, and let the function u C 2 [, 1] ) be defined by the equality { ũt) for t 1/2 u t) = ũt 1/2) for 1/2 < t 1, where 1 t2 for t 1 2α k 4 1 8k ũt) = 1+ β k t+ β2 k 1 sin πk 1 4t) α k 2α k 16π 2 k 2α for 1 k 4 1 <t 1 8k 4 + 1, 8k t1 t) for 1 2α k 8α k < t 1 8k 2 and k N is such that ε)α k < ) Then it is clear that u j) ) = uj) 1) j =, 1, 2), and there exist constants λ 1 >, λ 2 > such that λ1 4 + λ 2 ) 1 u s) ds = ε)α k. 2.16) Now, let the measurable function τ : [, 1] [, 1] and the linear operators l i : C[, 1]) L[, 1]), i =, 1, 2) be given by the equalities { for u t) > τt) = 1/2 for u t), i 1 ) l x)t) = u t) xτt)), l ix)t) = λ i u t) x i = 1, 2). 4

9 On a Periodic BVP for Third Order Linear FDE 35 From 2.15) and 2.16) it follows that and l λ1 1 l 2 = λ 2 1 l 1)s) ds = 1 ) 1 u s) ds = u s) ds = 4 α k = ε)α k 4 4ε = < ) ε)α k ε)α k 4 l 1 l 2 + ε. Thus, all the assumptions of Corollaries 2.1 and 2.5 are satisfied except 2.6), resp. 2.14), instead of which the condition l 1)s) ds 128 ω 2 1 ω ) 4 l 1 l 2 + ε is fulfilled. On the other hand, from the definition of the functions u, τ and the operators l i it follows that u t) = u t) sgn u t) = u t) u τt)) = l u )t), l 1 u )t) + l 2 u )t) = λ 1 u ) + λ 2 u 1/4) ) u t) =, that is, u and u 1 t) are different solutions of the problem 1.1), 1.2) with ω = 1, qt). 3. Proofs Let v C 2 [, ω] ). Then for j =, 1, 2 put M j = max { v j) t) : t [, ω] }, m j = max { v j) t) : t [, ω] }. 3.1) The following lemma is a consequence of a more general result obtained in [9] see Theorem 1.1 and Remark 1.1 therein). Lemma 3.1. Let k {, 1}, v C 2 [, ω] ), and Then the estimate vt) const, v j) ) = v j) ω) j =, 1, 2). M k + m k < ω2 k d 2 k M 2 + m 2 ) holds, where d 1 = 4 and d 2 =. Lemma 3.2. Let l : C [, ω] ) L [, ω] ) be a nonnegative linear operator. Then for an arbitrary v C [, ω] ) the inequalities m l1)t) lv)t) M l1)t) for t ω are fulfilled, where M and m are defined by 3.1).

10 36 R. Hakl and S. Mukhigulashvili Proof. It is clear that vt) M, vt) + m for t ω. Then from nonnegativity of l we get lv M )t), lv + m )t) for t ω, whence follows the validity of our lemma. The next lemma on the Fredholm property, in the case where l 1 and l 2 are bounded operators not necessarily strongly bounded), immediately follows from [1, Theorem 1.1]. Lemma 3.3. The problem 1.1), 1.2) is uniquely solvable iff the corresponding homogeneous problem 2 v t) = l i v i) )t), 3.2) has only the trivial solution. i= v j) ) = v j) ω) j =, 1, 2) 3.3) Lemma 3.4. Let there exist α 1, α 2 R + such that for every x C [, ω] ) assuming both positive and negative values, the inequality l j x)s) ds α j max { xs 1 ) xs 2 ) : s 1, s 2 ω } j = 1, 2) 3.4) E holds, where E [, ω] is an arbitrary measurable set. Let, moreover, l 1, l 2 : C [, ω] ) L [, ω] ) be nonnegative operators such that β = 1 ωα 1 /4 α 2, and l 1 1)t) l 2 1)t), 3.5) ω 2 l 1 1)s) ds < β, 3.6) ω l 1 1)s) ds ω l 2 1)s) ds, 3.7) β ω2 l 1 1)s) ds l 2 1)s) ds 64β ω ) ω ω2 l 1 1)s) ds. 3.8) β Then the problem 3.2), 3.3) with both has only the trivial solution. l = l 1 l 2 and l = l 2 l 1

11 On a Periodic BVP for Third Order Linear FDE 37 Proof. First suppose that l = l 1 l 2 and assume on the contrary that the problem 3.2), 3.3) has a nontrivial solution v. Let M j and m j j =, 1, 2) be defined by 3.1) and t 1, t 2 [, ω[ be such that v t 1 ) = M 2, v t 2 ) = m ) By virtue of 3.5) we have that vt) const. Therefore, in view of 3.3), it is clear that v and v assume both positive and negative values, and thus M j >, m j > j = 1, 2). 3.1) According to 3.1), Lemma 3.1 with k = 1, and 3.4) we have l 1 v )s) + l 2 v )s) ds 1 β)m 2 + m 2 ) 3.11) E for an arbitrary measurable set E [, ω]. Assume that t 1 < t 2 and let I = [, t 1 ] [t 2, ω]. Then, in view of 3.9) and 3.3), it is clear that I v s) ds = M 2 + m 2, t 2 t 1 v s) ds = M 2 + m 2 ). Hence, on account of 3.2) and 3.11), we get βm 2 + m 2 ) l1 v)s) l 2 v)s) ) ds, 3.12) βm 2 + m 2 ) I t 2 t 1 l2 v)s) l 1 v)s) ) ds. 3.13) From 3.6) it follows that β >, and, in view of the fact that vt) const, we have βm + m ) >. Now the inequalities 3.12) and 3.13), according to Lemma 3.1 with k =, yield < βm + m ) < ω2 < βm + m ) < ω2 I t 2 t 1 l1 v)s) l 2 v)s) ) ds, 3.14) l2 v)s) l 1 v)s) ) ds, 3.15) respectively. On the other hand, integration of 3.2) from to ω, on account of 3.3) and 3.11), results in 1) k 1 where k = 1, 2. l1 v)s) l 2 v)s) ) ds + 1 β)m 2 + m 2 ), 3.16 k )

12 38 R. Hakl and S. Mukhigulashvili Now we will show that v assumes both positive and negative values. Assume on the contrary that v does not change its sign. It is sufficient to discuss the following two cases: t 1 < t 2, 1) k 1 vt) for t ω k = 1, 2) k ) If ) is satisfied, then from 3.12), 3.14), and ), in view of Lemma 3.2 since l 1 and l 2 are nonnegative operators), we obtain m βm 2 + m 2 ) M M β ω2 l 2 1)s) ds M From 3.18) and 3.2) we get m β l 1 1)s) ds, 3.18) ) l 1 1)s) ds < m β, 3.19) l 1 1)s) ds + 1 β)m 2 + m 2 ). 3.2) l 2 1)s) ds M l 1 1)s) ds, which, together with 3.19), contradicts 3.7). If ) is satisfied, from 3.13), 3.15), and ) we can obtain a contradiction to 3.7) in an analogous way. Consequently, the contradictions obtained guarantee that v changes its sign and thus M >, m >. 3.21) Without loss of generality we can assume that t 1 < t 2, and therefore the inequalities 3.12) 3.16 k ) hold. Now, from 3.14) and 3.15), according to 3.21) and Lemma 3.2, we obtain ω 2 ω 2 βm + βm < M l 1 1)s) ds + m l 2 1)s) ds, 3.22) βm + βm < M ω 2 I t 2 I ω 2 t 2 l 2 1)s) ds + m l 1 1)s) ds. 3.23) t 1 t 1 The inequalities 3.22) and 3.23), by virtue of 3.6) and 3.21), yield ) < M β ω2 ω 2 ) l 1 1)s) ds < m l 2 1)s) ds β, < m β ω2 I t 2 t 1 ) ω 2 l 1 1)s) ds < M I t 2 t 1 ) l 2 1)s) ds β.

13 On a Periodic BVP for Third Order Linear FDE 39 Multiplying the last two inequalities, on account of the inequality 4AB A + B) 2, we get β 2 β ω2 l 1 1)s) ds < 1 4 ω 2 l 2 1)s) ds 2β) 2, which contradicts 3.8). Therefore our assumption is invalid and the problem 3.2), 3.3) with l = l 1 l 2 has only the trivial solution. Now let l = l 2 l 1. Put for every x : [, ω] R and ϑx)t) def = xω t) for t ω, l i x)t) def = ϑ l i ϑx)) ) t), li x)t) def = 1) i 1 ϑ l i ϑx)) ) t) i=1, 2). Then, obviously, the operators l i i = 1, 2) are also nonnegative and l i = l i, l i = l i i = 1, 2). 3.24) On the other hand, v 1 is a solution of 3.2), 3.3) iff is a solution of the problem v 2 t) def = ϑv 1 )t) with 2 v t) = l i v i) )t), v j) ) = v j) ω) j =, 1, 2) 3.25) i= l = l 1 l 2. From 3.24) it follows that all the assumptions of the lemma are fulfilled for the problem 3.25). However, 3.25) has only the trivial solution, as was shown in the first part of this proof. Consequently, the problem 3.2), 3.3) has also only the trivial solution. Proof of Theorem 2.1. Put α 1 = l 1, α 2 = l 2. Then all the assumptions of Lemma 3.4 are fulfilled, and the conclusion of theorem follows from Lemma 3.3. Proof of Theorem 2.2. Put α 1 = max { l 11, l 12 }, α 2 = max { l 21, l 22 }. Then all the assumptions of Lemma 3.4 are fulfilled, and the conclusion of theorem follows from Lemma 3.3. Corollary 2.1 follows immediately from Theorem 2.2 with l 1.

14 4 R. Hakl and S. Mukhigulashvili Proof of Corollary 2.2. Let Then l i x)t) def = p i t)xτ i t)) i =, 1, 2), l 1 x)t) def = p 1 t)xτ t)), l 2 x)t) def = p 2 t)xτ t)), l i1 x)t) def = [p i t)] + xτ i t)), l i2 x)t) def = [p i t)] xτ i t)) i = 1, 2). l i1 = [p i t)] + dt, l i2 = [p i t)] dt i = 1, 2) and the conditions 2.7) 2.11) yield the conditions 2.1) 2.4) with β 1 defined by 2.5). Consequently, the assumptions of Theorem 2.2 are fulfilled. Acknowledgement The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AVZ11953 and the Grant No. 21/6/254 of the Grant Agency of the Czech Republic. References 1. P. Amster, P. De Nápoli, and M. C. Mariani, Periodic solutions of a resonant third-order equation. Nonlinear Anal. 625), No. 3, S. R. Baslandze and I. T. Kiguradze, On the unique solvability of a periodic boundary value problem for third-order linear differential equations. Russian) Differ. Uravn. 4226), No. 2, ; English transl.: Differ. Equ. 4226), No. 2, E. Bravyi, A note on the Fredholm property of boundary value problems for linear functional differential equations. Mem. Differential Equations Math. Phys. 22), A. Cabada, The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems. J. Math. Anal. Appl ), No. 2,. 5. A. Cabada, The method of lower and upper solutions for third-order periodic boundary value problems. J. Math. Anal. Appl ), No. 2, G. T. Gegelia, On boundary value problems of periodic type for ordinary odd order differential equations. Arch. Math. Brno) 21984), No. 4, R. Hakl, Periodic boundary value problem for third order linear functional differential equations. Nonlinear Oscil. in press). 8. R. Hakl, A. Lomtatidze, and I. P. Stavroulakis, On a boundary value problem for scalar linear functional differential equations. Abstr. Appl. Anal., 24, No. 1, R. Hakl and S. Mukhigulashvili, On one estimate for periodic functions. Georgian Math. J. 1225), No. 1, R. Hakl and S. Mukhigulashvili, On a boundary value problem for n-th order linear functional differential systems. Georgian Math. J. 1225), No. 2, I. Kiguradze, Bounded and periodic solutions of higher-order linear differential equations. Russian) Mat. Zametki ), No. 1,

15 On a Periodic BVP for Third Order Linear FDE I. Kiguradze, On periodic solutions of nth order ordinary differential equations. Nonlinear Anal. 42), No. 1-8, Ser. A: Theory Methods, I. Kiguradze and T. Kusano, On periodic solutions of higher-order nonautonomous ordinary differential equations. Russian) Differ. Uravn ), No. 1, 72 78; English transl.: Differential Equations ), No. 1, I. Kiguradze and T. Kusano, On conditions for the existence and uniqueness of a periodic solution of nonautonomous differential equations. Russian) Differ. Uravn. 362), No. 1, ; English transl.: Differ. Equ. 362), No. 1, I. Kiguradze and T. Kusano, On periodic solutions of even-order ordinary differential equations. Ann. Mat. Pura Appl. 4) 1821), No. 3, I. Kiguradze and B. Půža, On boundary value problems for systems of linear functional-differential equations. Czechoslovak Math. J )1997), No. 2, A. Lasota and Z. Opial, Sur les solutions périodiques des équations différentielles ordinaires. Ann. Polon. Math ), A. Lasota and F. H. Szafraniec, Sur les solutions périodiques d une équation différentielle ordinaire d ordre n. Ann. Polon. Math ), A. Lomtatidze and S. Mukhigulashvili, On periodic solutions of second order functional differential equations. Mem. Differential Equations Math. Phys ), S. Mukhigulashvili, On a periodic boundary value problem for second-order linear functional differential equations. Bound. Value Probl., 25, No. 3, S. Mukhigulashvili, On periodic solutions of second order functional differential equations. Ital. J. Pure Appl. Math., 26, No. 2, S. Mukhigulashvili, On the solvability of the periodic problem for nonlinear secondorder function-differential equations. Differ. Uravn. 4226), No. 3, ; English transl.: Differ. Equ. 4226), No. 3, S. Mukhigulashvili, On a periodic boundary value problem for third order linear functional differential equations. Nonlinear Anal. 6627), No. 2, Ser. A: Theory Methods, S. Mukhigulashvili and B. Půža, On a periodic boundary value problem for third order linear functional differential equations. Funct. Differ. Equ. in press). 25. J. J. Nieto, Periodic solutions for third order ordinary differential equations. Comment. Math. Univ. Carolin. 1991), No. 3, P. Omari and M. Trombetta, Remarks on the lower and upper solutions method for second- and third-order periodic boundary value problems. Appl. Math. Comput ), No. 1, A. Ronto and A. Samoĭlenko, On the unique solvability of some boundary value problems for linear functional-differential equations in a Banach space. Russian) Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 22, No. 9, H. H. Schaefer, Normed tensor products of Banach lattices. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces Jerusalem, 1972), Israel J. Math ), G. Villari, Soluzioni periodiche di una classe di equazioni differenziali del terz ordine quasi lineari. Ann. Mat. Pura Appl. 4) ), M. Wypich, On the existence of a solution of nonlinear periodic boundary value problems. Univ. Iagel. Acta Math. 1991, No. 28, Received )

16 42 R. Hakl and S. Mukhigulashvili Authors addresses: R. Hakl Institute of Mathematics Academy of Sciences of the Czech Republic Žižkova 22, Brno Czech Republic S. Mukhigulashvili Current address: Mathematical Institute Academy of Sciences of the Czech Republic Žižkova 22, Brno Czech Republic Permanent address: A. Razmadze Mathematical Institute 1, M. Aleksidze St., Tbilisi 193 Georgia