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Research ArticleExistence and Uniqueness of Periodic Solutions forSecond Order Differential Equations
Yuanhong Wei
College of Mathematics Jilin University Changchun 130012 China
Correspondence should be addressed to Yuanhong Wei yhweiamssaccn
Received 12 June 2014 Accepted 16 August 2014 Published 27 August 2014
Academic Editor Mohamed Abdalla Darwish
Copyright copy 2014 Yuanhong Wei This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We study some second order ordinary differential equations We establish the existence and uniqueness in some appropriatefunction space By using Schauderrsquos fixed-point theorem new results on the existence and uniqueness of periodic solutions areobtained
1 Introduction
In this paper we are concerned with the existence anduniqueness of periodic solution for the nonlinear equation
where 119891 R times R times R rarr R is continuous periodic in 119905 withperiod 119879 and 1198621 with respect to (119909 1199091015840)
Because of wide interests in physics and engineeringperiodic solutions of second order differential equations havebeen investigated by many authors We refer the reader to [1ndash7] and the references cited therein
The purpose of this paper is to study the existence anduniqueness of periodic solution in some appropriate functionspace 1198621
119879 To be precise we first derive a result of existence
and uniqueness when the nonlinearity is a 1198621 functionwith respect to (119909 119909
1015840) Then a similar result concerning
the existence of periodic solutions is obtained when thenonlinearity is not a 1198621 function
Throughout this paper we use the following assumptions
(A1) There exist two continuous functions 119886(119905) and 119887(119905)such that for all 119905 isin [0 119879]
Theorem 1 Let assumptions (A1) and (A2) hold Then (1) hasa unique 119879-periodic solution
Remark 2 We point out that the condition 0 ≨ 119886(119905) ⩽ 119891119909
in (A1) is necessary Let us take the following equation forexample
11990910158401015840= 119862 (4)
where119862 is a constant and119862 = 0The equation has no periodicsolution
Remark 3 Consider the following example
11990910158401015840= sin 119905 sdot sin119909 sdot sin1199091015840 + 119909 (5)
We can easily check that assumptions (A1) and (A2) hold ByTheorem 1 the equation has a unique 2120587-periodic solution
We also consider the case when the right-hand side ofthe equation is only continuous In this case we establish theexistence of periodic solutions for the differential equation
By Theorem 4 the equation has at least one 2120587-periodicsolution
To establish the main results we introduce some appro-priate function space By using Schauderrsquos fixed-point theo-rem the existence and uniqueness of periodic solutions areobtained We know from the anonymous referee that theproof of this paper can be simplified by using the results in[8] The proof of this paper can be seen as an application ofSchauderrsquos fixed-point theorem
2 Preliminary
In this sectionWefirst introduce the function space inwhichwe will obtain the periodic solution for the problem Thensome preliminary lemma is introduced which is valuable forthe proof of our main results
Let 1198621119879be the space of continuously differentiable 119879-
periodic functions with norm sdot given by
119909 = max119905isin[0119879]
|119909 (119905)| + max119905isin[0119879]
100381610038161003816100381610038161199091015840
(119905)
10038161003816100381610038161003816 (9)
It is well known that 1198621119879is a Banach space
We now introduce the following lemma
Lemma 6 119906(119905) and V(119905) are 119879-periodic functions If 0 ≨ V(119905)for all 119905 isin 119877 then the equation
11990910158401015840= 119906 (119905) 119909
1015840+ V (119905) 119909 (10)
has a unique 119879-periodic solution 119909(119905) equiv 0
Proof Assume there exists a 119879-periodic solution 119909(119905)119909(119905) equiv 0 Since 0 ≨ V(119905) 119909(119905) is not a constant Otherwise wewill get a contradiction by substituting 119909 = 119862 = 0 into (10)Then we claim that there exist 119905
0and 1199051 1199050lt 1199051 such that
119909 (119905) gt 0 for 119905 isin (1199050 1199051)
1199091015840(1199050) gt 0 119909
1015840(1199051) = 0
(11)
Now we prove it
Case 1 119909(119905) has zero points Assume that 119909(120579) = 0 then wehave 1199091015840(120579) = 0 because the initial value problem of (10) has aunique solution If 1199091015840(120579) gt 0 then we let 119905
0= 120579 if 1199091015840(120579) lt 0
by the periodic condition there exists 1199050 1199050gt 120579 such that
So we can find a 1199051isin (1199050 120578) such that 1199091015840(119905
1) = 0 Thus under
Case 1 we verify (11) holds
Case 2 119909(119905) has no zero point Then 119909(119905) has a constant signSuppose 119909(119905) gt 0 (if not we can consider the 119879-periodicsolution 119909(119905) = minus119909(119905) 119909(119905) gt 0) Since 119909(119905) is periodicand is not a constant there exists 119905
0such that 1199091015840(119905
0) gt 0
119909(1199050) = 119909(119905
0+ 119879) So we can find a 119905
1 1199051gt 1199050 1199091015840(1199051) = 0
Then we prove that (11) holdsMultiplying both sides of (10) by expminus int1199051
1199050
119906(119904)119889119904 andintegrating between 119905
0and 1199051we get
0 gt minus1199091015840(1199050) = int
1199051
1199050
V (119905) 119909 (119905) expminusint1199051
1199050
119906 (119904) 119889119904 119889119905 ⩾ 0
(14)
which leads to a contradiction This completes the proof ofLemma 6
Remark 7 We can also prove the following If 119906(0) = 119906(119879)V(0) = V(119879) 0 ≨ V(119905) ae 119905 isin [0 119879] then the followingperiodic boundary value problem
11990910158401015840= 119906 (119905) 119909
1015840+ V (119905) 119909 119905 isin [0 119879]
119909 (0) = 119909 (119879) 1199091015840
(0) = 1199091015840
(119879)
(15)
also has a unique solution 119909(119905) equiv 0
3 Proof of the Main Results
Proof of Theorem 1 First prove the uniqueness Assume that1199091(119905) and 119909
2(119905) are two 119879-periodic solutions of (1) Setting
Proof of Compactness For each bounded set 119878 sub 1198621119879 we claim
that 119875(119878) is bounded in 1198621119879 If not by an analogous manner
as above we will reach a contradiction For every 119909 isin 119878119910 = 119875119909 is defined by (20) Since 1199101015840 119910 119891
119909 and 119891
1199091015840
are all bounded then 11991010158401015840 lt infin Proceeding as proof ofcontinuity we conclude that 1199101015840 and 119910 are bounded andequicontinuous sequences of functions By the Ascoli-ArzelaTheorem 119875 is a compact operator
We claim that 119875(1198621119879) is bounded in 1198621
119879 If not there exist
119909119896 119896 = 1 2 such that 119875119909
119896 rarr infin (119896 rarr infin)
Let 119910119896= 119875119909
119896 Then (22) holds Take 120596
119896= 119910119896119910119896 Then
120596119896 sub 119862
1
119879 120596119896 = 1 and (23) (24) (25) and (27) hold
By the above proof we know 1205961015840
119896 and 120596
119896 are bounded
and equicontinuous sequences of functions and contain auniformly convergent subsequence respectively (also use thesame notations) such that
has a unique 119879-periodic solution because the correspondinghomogeneous equation only has trivial 119879-periodic solution119909(119905) equiv 0
We define 119875 1198621119879rarr 1198621
119879 for each given 119909 isin 1198621
119879119910(119905) =
119875[119909](119905) is the unique 119879-periodic solution of (38) Hencethe existence of the periodic solutions is equivalent to theexistence of fixed points of 119875 in the Banach space 1198621
119879
Proceeding as the proof of Theorem 1 we can prove that119875 is a compact continuous operator and 119875(1198621
119879) is a bounded
subset of 1198621119879
Then there exists a constant 1198701gt 0 such that 119875119909 ⩽ 119870
1
for all 119909 isin 1198621119879 Let119863 = 119909 isin 119862
1
119879 119909 ⩽ 119870
1+ 1 By Schauderrsquos
fixed-point theorem 119875 119863 rarr 119863 has at least one fixed pointThis completes the proof of Theorem 4
4 Another Simple Proof
Actually by the anonymous referee we know that the proofof the theorem can be much simplified if we use the theoremin [8] In fact for equation
119906(119899)= 119891 (119905 119906 119906
(119899minus1)) (39)
where 119891 R119899+1 rarr R is continuous and 119879-periodic in 119905Theorem 22 of [8] implies the following propositions
Journal of Function Spaces 5
Lemma 8 Let there exist continuous and 119879-periodic in thefirst argument functions 119891
119896 R119899+1 rarr R (119896 = 1 2 119899)
such that1003816100381610038161003816100381610038161003816100381610038161003816
are satisfied on R119899+1 where 119903 = 119888119900119899119904119905 gt 0 and 119901119894119896
R rarr R (119894 = 1 2 119896 = 1 2 119899) are continuous119879-periodic functions Let moreover for any continuous 119879-periodic functions 119901
119896 R rarr R (119896 = 1 2 119899) satisfying
1199011119896(119905) ⩽ 119901
119896(119905) ⩽ 119901
2119896(119905) (119896 = 1 2 119899) (41)
the equation
119906(119899)=
119899
sum
119896=1
119901119896(119905) 119906(119896minus1) (42)
have no nontrivial 119879-periodic solution Then equation has atleast one 119879-periodic solution
Lemma 9 Let the function 119891 in the last 119899 arguments havecontinuous partial derivatives satisfying
1199011119896(119905) ⩽
120597119891119896(119905 1199091 119909
119899)
120597119909119896
⩽ 1199012119896(119905) (119896 = 1 2 119899)
(43)
where 119901119894119896 R rarr R (119894 = 1 2 119896 = 1 2 119899) are continuous
119879-periodic functions Let moreover for any 119879-periodic 119901119896
R rarr R (119896 = 1 2 119899) satisfying (41) (42) has nonontrivial119879-periodic solutionThen the equation has a unique119879-periodic solution
On the basis of these theorems we can prove the theoremby checking the conditions in the previous lemma ByLemma 6 the conditions above can be easily proved
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper is supported by National Natural Science Founda-tion of China (Grant no 11301209) The author would like tothank the anonymous referee for the valuable comments andsuggestions on the paper especially for the considerations inSection 4 The author would like to thank Professor Yong Lifor his helpful instruction and valuable suggestions
References
[1] F Cong ldquoPeriodic solutions for second order differential equa-tionsrdquo Applied Mathematics Letters vol 18 no 8 pp 957ndash9612005
[2] J Ward ldquoPeriodic solutions for a class of ordinary differentialequationsrdquo Proceedings of the American Mathematical Societyvol 78 no 3 pp 350ndash352 1980
[3] R Kannan ldquoExistence of periodic solutions of nonlinear dif-ferential equationsrdquo Transactions of the AmericanMathematicalSociety vol 217 pp 225ndash236 1976
[4] K Schmitt ldquoA note on periodic solutions of second orderordinary differential equationsrdquo SIAM Journal onAppliedMath-ematics vol 21 pp 491ndash494 1971
[5] Q Yao ldquoPositive solutions of nonlinear second-order periodicboundary value problemsrdquoAppliedMathematics Letters vol 20no 5 pp 583ndash590 2007
[6] I Kiguradze and S Stanck ldquoOn periodic boundary valueproblem for the equation11990610158401015840 = 119891(119905 119906 1199061015840)with one-sided growthrestrictions on frdquo Nonlinear Analysis vol 48 pp 1065ndash10752002
[7] P Yan andMZhang ldquoHigher order non-resonance for differen-tial equations with singularitiesrdquo Mathematical Methods in theApplied Sciences vol 26 no 12 pp 1067ndash1074 2003
[8] I Kiguradze ldquoBoundary value problems for systems of ordinarydifferential equationsrdquo Itogi Nauki i Tekhniki Seriya Sovremen-nye Problemy Matematiki Noveishie Dostizheniya vol 30 pp3ndash103 1987 (Russian) English translation Journal of SovietMathematics vol 43 no 2 pp 2259ndash2339 1988
By Theorem 4 the equation has at least one 2120587-periodicsolution
To establish the main results we introduce some appro-priate function space By using Schauderrsquos fixed-point theo-rem the existence and uniqueness of periodic solutions areobtained We know from the anonymous referee that theproof of this paper can be simplified by using the results in[8] The proof of this paper can be seen as an application ofSchauderrsquos fixed-point theorem
2 Preliminary
In this sectionWefirst introduce the function space inwhichwe will obtain the periodic solution for the problem Thensome preliminary lemma is introduced which is valuable forthe proof of our main results
Let 1198621119879be the space of continuously differentiable 119879-
periodic functions with norm sdot given by
119909 = max119905isin[0119879]
|119909 (119905)| + max119905isin[0119879]
100381610038161003816100381610038161199091015840
(119905)
10038161003816100381610038161003816 (9)
It is well known that 1198621119879is a Banach space
We now introduce the following lemma
Lemma 6 119906(119905) and V(119905) are 119879-periodic functions If 0 ≨ V(119905)for all 119905 isin 119877 then the equation
11990910158401015840= 119906 (119905) 119909
1015840+ V (119905) 119909 (10)
has a unique 119879-periodic solution 119909(119905) equiv 0
Proof Assume there exists a 119879-periodic solution 119909(119905)119909(119905) equiv 0 Since 0 ≨ V(119905) 119909(119905) is not a constant Otherwise wewill get a contradiction by substituting 119909 = 119862 = 0 into (10)Then we claim that there exist 119905
0and 1199051 1199050lt 1199051 such that
119909 (119905) gt 0 for 119905 isin (1199050 1199051)
1199091015840(1199050) gt 0 119909
1015840(1199051) = 0
(11)
Now we prove it
Case 1 119909(119905) has zero points Assume that 119909(120579) = 0 then wehave 1199091015840(120579) = 0 because the initial value problem of (10) has aunique solution If 1199091015840(120579) gt 0 then we let 119905
0= 120579 if 1199091015840(120579) lt 0
by the periodic condition there exists 1199050 1199050gt 120579 such that
So we can find a 1199051isin (1199050 120578) such that 1199091015840(119905
1) = 0 Thus under
Case 1 we verify (11) holds
Case 2 119909(119905) has no zero point Then 119909(119905) has a constant signSuppose 119909(119905) gt 0 (if not we can consider the 119879-periodicsolution 119909(119905) = minus119909(119905) 119909(119905) gt 0) Since 119909(119905) is periodicand is not a constant there exists 119905
0such that 1199091015840(119905
0) gt 0
119909(1199050) = 119909(119905
0+ 119879) So we can find a 119905
1 1199051gt 1199050 1199091015840(1199051) = 0
Then we prove that (11) holdsMultiplying both sides of (10) by expminus int1199051
1199050
119906(119904)119889119904 andintegrating between 119905
0and 1199051we get
0 gt minus1199091015840(1199050) = int
1199051
1199050
V (119905) 119909 (119905) expminusint1199051
1199050
119906 (119904) 119889119904 119889119905 ⩾ 0
(14)
which leads to a contradiction This completes the proof ofLemma 6
Remark 7 We can also prove the following If 119906(0) = 119906(119879)V(0) = V(119879) 0 ≨ V(119905) ae 119905 isin [0 119879] then the followingperiodic boundary value problem
11990910158401015840= 119906 (119905) 119909
1015840+ V (119905) 119909 119905 isin [0 119879]
119909 (0) = 119909 (119879) 1199091015840
(0) = 1199091015840
(119879)
(15)
also has a unique solution 119909(119905) equiv 0
3 Proof of the Main Results
Proof of Theorem 1 First prove the uniqueness Assume that1199091(119905) and 119909
2(119905) are two 119879-periodic solutions of (1) Setting
Proof of Compactness For each bounded set 119878 sub 1198621119879 we claim
that 119875(119878) is bounded in 1198621119879 If not by an analogous manner
as above we will reach a contradiction For every 119909 isin 119878119910 = 119875119909 is defined by (20) Since 1199101015840 119910 119891
119909 and 119891
1199091015840
are all bounded then 11991010158401015840 lt infin Proceeding as proof ofcontinuity we conclude that 1199101015840 and 119910 are bounded andequicontinuous sequences of functions By the Ascoli-ArzelaTheorem 119875 is a compact operator
We claim that 119875(1198621119879) is bounded in 1198621
119879 If not there exist
119909119896 119896 = 1 2 such that 119875119909
119896 rarr infin (119896 rarr infin)
Let 119910119896= 119875119909
119896 Then (22) holds Take 120596
119896= 119910119896119910119896 Then
120596119896 sub 119862
1
119879 120596119896 = 1 and (23) (24) (25) and (27) hold
By the above proof we know 1205961015840
119896 and 120596
119896 are bounded
and equicontinuous sequences of functions and contain auniformly convergent subsequence respectively (also use thesame notations) such that
has a unique 119879-periodic solution because the correspondinghomogeneous equation only has trivial 119879-periodic solution119909(119905) equiv 0
We define 119875 1198621119879rarr 1198621
119879 for each given 119909 isin 1198621
119879119910(119905) =
119875[119909](119905) is the unique 119879-periodic solution of (38) Hencethe existence of the periodic solutions is equivalent to theexistence of fixed points of 119875 in the Banach space 1198621
119879
Proceeding as the proof of Theorem 1 we can prove that119875 is a compact continuous operator and 119875(1198621
119879) is a bounded
subset of 1198621119879
Then there exists a constant 1198701gt 0 such that 119875119909 ⩽ 119870
1
for all 119909 isin 1198621119879 Let119863 = 119909 isin 119862
1
119879 119909 ⩽ 119870
1+ 1 By Schauderrsquos
fixed-point theorem 119875 119863 rarr 119863 has at least one fixed pointThis completes the proof of Theorem 4
4 Another Simple Proof
Actually by the anonymous referee we know that the proofof the theorem can be much simplified if we use the theoremin [8] In fact for equation
119906(119899)= 119891 (119905 119906 119906
(119899minus1)) (39)
where 119891 R119899+1 rarr R is continuous and 119879-periodic in 119905Theorem 22 of [8] implies the following propositions
Journal of Function Spaces 5
Lemma 8 Let there exist continuous and 119879-periodic in thefirst argument functions 119891
119896 R119899+1 rarr R (119896 = 1 2 119899)
such that1003816100381610038161003816100381610038161003816100381610038161003816
are satisfied on R119899+1 where 119903 = 119888119900119899119904119905 gt 0 and 119901119894119896
R rarr R (119894 = 1 2 119896 = 1 2 119899) are continuous119879-periodic functions Let moreover for any continuous 119879-periodic functions 119901
119896 R rarr R (119896 = 1 2 119899) satisfying
1199011119896(119905) ⩽ 119901
119896(119905) ⩽ 119901
2119896(119905) (119896 = 1 2 119899) (41)
the equation
119906(119899)=
119899
sum
119896=1
119901119896(119905) 119906(119896minus1) (42)
have no nontrivial 119879-periodic solution Then equation has atleast one 119879-periodic solution
Lemma 9 Let the function 119891 in the last 119899 arguments havecontinuous partial derivatives satisfying
1199011119896(119905) ⩽
120597119891119896(119905 1199091 119909
119899)
120597119909119896
⩽ 1199012119896(119905) (119896 = 1 2 119899)
(43)
where 119901119894119896 R rarr R (119894 = 1 2 119896 = 1 2 119899) are continuous
119879-periodic functions Let moreover for any 119879-periodic 119901119896
R rarr R (119896 = 1 2 119899) satisfying (41) (42) has nonontrivial119879-periodic solutionThen the equation has a unique119879-periodic solution
On the basis of these theorems we can prove the theoremby checking the conditions in the previous lemma ByLemma 6 the conditions above can be easily proved
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper is supported by National Natural Science Founda-tion of China (Grant no 11301209) The author would like tothank the anonymous referee for the valuable comments andsuggestions on the paper especially for the considerations inSection 4 The author would like to thank Professor Yong Lifor his helpful instruction and valuable suggestions
References
[1] F Cong ldquoPeriodic solutions for second order differential equa-tionsrdquo Applied Mathematics Letters vol 18 no 8 pp 957ndash9612005
[2] J Ward ldquoPeriodic solutions for a class of ordinary differentialequationsrdquo Proceedings of the American Mathematical Societyvol 78 no 3 pp 350ndash352 1980
[3] R Kannan ldquoExistence of periodic solutions of nonlinear dif-ferential equationsrdquo Transactions of the AmericanMathematicalSociety vol 217 pp 225ndash236 1976
[4] K Schmitt ldquoA note on periodic solutions of second orderordinary differential equationsrdquo SIAM Journal onAppliedMath-ematics vol 21 pp 491ndash494 1971
[5] Q Yao ldquoPositive solutions of nonlinear second-order periodicboundary value problemsrdquoAppliedMathematics Letters vol 20no 5 pp 583ndash590 2007
[6] I Kiguradze and S Stanck ldquoOn periodic boundary valueproblem for the equation11990610158401015840 = 119891(119905 119906 1199061015840)with one-sided growthrestrictions on frdquo Nonlinear Analysis vol 48 pp 1065ndash10752002
[7] P Yan andMZhang ldquoHigher order non-resonance for differen-tial equations with singularitiesrdquo Mathematical Methods in theApplied Sciences vol 26 no 12 pp 1067ndash1074 2003
[8] I Kiguradze ldquoBoundary value problems for systems of ordinarydifferential equationsrdquo Itogi Nauki i Tekhniki Seriya Sovremen-nye Problemy Matematiki Noveishie Dostizheniya vol 30 pp3ndash103 1987 (Russian) English translation Journal of SovietMathematics vol 43 no 2 pp 2259ndash2339 1988
Proof of Compactness For each bounded set 119878 sub 1198621119879 we claim
that 119875(119878) is bounded in 1198621119879 If not by an analogous manner
as above we will reach a contradiction For every 119909 isin 119878119910 = 119875119909 is defined by (20) Since 1199101015840 119910 119891
119909 and 119891
1199091015840
are all bounded then 11991010158401015840 lt infin Proceeding as proof ofcontinuity we conclude that 1199101015840 and 119910 are bounded andequicontinuous sequences of functions By the Ascoli-ArzelaTheorem 119875 is a compact operator
We claim that 119875(1198621119879) is bounded in 1198621
119879 If not there exist
119909119896 119896 = 1 2 such that 119875119909
119896 rarr infin (119896 rarr infin)
Let 119910119896= 119875119909
119896 Then (22) holds Take 120596
119896= 119910119896119910119896 Then
120596119896 sub 119862
1
119879 120596119896 = 1 and (23) (24) (25) and (27) hold
By the above proof we know 1205961015840
119896 and 120596
119896 are bounded
and equicontinuous sequences of functions and contain auniformly convergent subsequence respectively (also use thesame notations) such that
has a unique 119879-periodic solution because the correspondinghomogeneous equation only has trivial 119879-periodic solution119909(119905) equiv 0
We define 119875 1198621119879rarr 1198621
119879 for each given 119909 isin 1198621
119879119910(119905) =
119875[119909](119905) is the unique 119879-periodic solution of (38) Hencethe existence of the periodic solutions is equivalent to theexistence of fixed points of 119875 in the Banach space 1198621
119879
Proceeding as the proof of Theorem 1 we can prove that119875 is a compact continuous operator and 119875(1198621
119879) is a bounded
subset of 1198621119879
Then there exists a constant 1198701gt 0 such that 119875119909 ⩽ 119870
1
for all 119909 isin 1198621119879 Let119863 = 119909 isin 119862
1
119879 119909 ⩽ 119870
1+ 1 By Schauderrsquos
fixed-point theorem 119875 119863 rarr 119863 has at least one fixed pointThis completes the proof of Theorem 4
4 Another Simple Proof
Actually by the anonymous referee we know that the proofof the theorem can be much simplified if we use the theoremin [8] In fact for equation
119906(119899)= 119891 (119905 119906 119906
(119899minus1)) (39)
where 119891 R119899+1 rarr R is continuous and 119879-periodic in 119905Theorem 22 of [8] implies the following propositions
Journal of Function Spaces 5
Lemma 8 Let there exist continuous and 119879-periodic in thefirst argument functions 119891
119896 R119899+1 rarr R (119896 = 1 2 119899)
such that1003816100381610038161003816100381610038161003816100381610038161003816
are satisfied on R119899+1 where 119903 = 119888119900119899119904119905 gt 0 and 119901119894119896
R rarr R (119894 = 1 2 119896 = 1 2 119899) are continuous119879-periodic functions Let moreover for any continuous 119879-periodic functions 119901
119896 R rarr R (119896 = 1 2 119899) satisfying
1199011119896(119905) ⩽ 119901
119896(119905) ⩽ 119901
2119896(119905) (119896 = 1 2 119899) (41)
the equation
119906(119899)=
119899
sum
119896=1
119901119896(119905) 119906(119896minus1) (42)
have no nontrivial 119879-periodic solution Then equation has atleast one 119879-periodic solution
Lemma 9 Let the function 119891 in the last 119899 arguments havecontinuous partial derivatives satisfying
1199011119896(119905) ⩽
120597119891119896(119905 1199091 119909
119899)
120597119909119896
⩽ 1199012119896(119905) (119896 = 1 2 119899)
(43)
where 119901119894119896 R rarr R (119894 = 1 2 119896 = 1 2 119899) are continuous
119879-periodic functions Let moreover for any 119879-periodic 119901119896
R rarr R (119896 = 1 2 119899) satisfying (41) (42) has nonontrivial119879-periodic solutionThen the equation has a unique119879-periodic solution
On the basis of these theorems we can prove the theoremby checking the conditions in the previous lemma ByLemma 6 the conditions above can be easily proved
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper is supported by National Natural Science Founda-tion of China (Grant no 11301209) The author would like tothank the anonymous referee for the valuable comments andsuggestions on the paper especially for the considerations inSection 4 The author would like to thank Professor Yong Lifor his helpful instruction and valuable suggestions
References
[1] F Cong ldquoPeriodic solutions for second order differential equa-tionsrdquo Applied Mathematics Letters vol 18 no 8 pp 957ndash9612005
[2] J Ward ldquoPeriodic solutions for a class of ordinary differentialequationsrdquo Proceedings of the American Mathematical Societyvol 78 no 3 pp 350ndash352 1980
[3] R Kannan ldquoExistence of periodic solutions of nonlinear dif-ferential equationsrdquo Transactions of the AmericanMathematicalSociety vol 217 pp 225ndash236 1976
[4] K Schmitt ldquoA note on periodic solutions of second orderordinary differential equationsrdquo SIAM Journal onAppliedMath-ematics vol 21 pp 491ndash494 1971
[5] Q Yao ldquoPositive solutions of nonlinear second-order periodicboundary value problemsrdquoAppliedMathematics Letters vol 20no 5 pp 583ndash590 2007
[6] I Kiguradze and S Stanck ldquoOn periodic boundary valueproblem for the equation11990610158401015840 = 119891(119905 119906 1199061015840)with one-sided growthrestrictions on frdquo Nonlinear Analysis vol 48 pp 1065ndash10752002
[7] P Yan andMZhang ldquoHigher order non-resonance for differen-tial equations with singularitiesrdquo Mathematical Methods in theApplied Sciences vol 26 no 12 pp 1067ndash1074 2003
[8] I Kiguradze ldquoBoundary value problems for systems of ordinarydifferential equationsrdquo Itogi Nauki i Tekhniki Seriya Sovremen-nye Problemy Matematiki Noveishie Dostizheniya vol 30 pp3ndash103 1987 (Russian) English translation Journal of SovietMathematics vol 43 no 2 pp 2259ndash2339 1988
Proof of Compactness For each bounded set 119878 sub 1198621119879 we claim
that 119875(119878) is bounded in 1198621119879 If not by an analogous manner
as above we will reach a contradiction For every 119909 isin 119878119910 = 119875119909 is defined by (20) Since 1199101015840 119910 119891
119909 and 119891
1199091015840
are all bounded then 11991010158401015840 lt infin Proceeding as proof ofcontinuity we conclude that 1199101015840 and 119910 are bounded andequicontinuous sequences of functions By the Ascoli-ArzelaTheorem 119875 is a compact operator
We claim that 119875(1198621119879) is bounded in 1198621
119879 If not there exist
119909119896 119896 = 1 2 such that 119875119909
119896 rarr infin (119896 rarr infin)
Let 119910119896= 119875119909
119896 Then (22) holds Take 120596
119896= 119910119896119910119896 Then
120596119896 sub 119862
1
119879 120596119896 = 1 and (23) (24) (25) and (27) hold
By the above proof we know 1205961015840
119896 and 120596
119896 are bounded
and equicontinuous sequences of functions and contain auniformly convergent subsequence respectively (also use thesame notations) such that
has a unique 119879-periodic solution because the correspondinghomogeneous equation only has trivial 119879-periodic solution119909(119905) equiv 0
We define 119875 1198621119879rarr 1198621
119879 for each given 119909 isin 1198621
119879119910(119905) =
119875[119909](119905) is the unique 119879-periodic solution of (38) Hencethe existence of the periodic solutions is equivalent to theexistence of fixed points of 119875 in the Banach space 1198621
119879
Proceeding as the proof of Theorem 1 we can prove that119875 is a compact continuous operator and 119875(1198621
119879) is a bounded
subset of 1198621119879
Then there exists a constant 1198701gt 0 such that 119875119909 ⩽ 119870
1
for all 119909 isin 1198621119879 Let119863 = 119909 isin 119862
1
119879 119909 ⩽ 119870
1+ 1 By Schauderrsquos
fixed-point theorem 119875 119863 rarr 119863 has at least one fixed pointThis completes the proof of Theorem 4
4 Another Simple Proof
Actually by the anonymous referee we know that the proofof the theorem can be much simplified if we use the theoremin [8] In fact for equation
119906(119899)= 119891 (119905 119906 119906
(119899minus1)) (39)
where 119891 R119899+1 rarr R is continuous and 119879-periodic in 119905Theorem 22 of [8] implies the following propositions
Journal of Function Spaces 5
Lemma 8 Let there exist continuous and 119879-periodic in thefirst argument functions 119891
119896 R119899+1 rarr R (119896 = 1 2 119899)
such that1003816100381610038161003816100381610038161003816100381610038161003816
are satisfied on R119899+1 where 119903 = 119888119900119899119904119905 gt 0 and 119901119894119896
R rarr R (119894 = 1 2 119896 = 1 2 119899) are continuous119879-periodic functions Let moreover for any continuous 119879-periodic functions 119901
119896 R rarr R (119896 = 1 2 119899) satisfying
1199011119896(119905) ⩽ 119901
119896(119905) ⩽ 119901
2119896(119905) (119896 = 1 2 119899) (41)
the equation
119906(119899)=
119899
sum
119896=1
119901119896(119905) 119906(119896minus1) (42)
have no nontrivial 119879-periodic solution Then equation has atleast one 119879-periodic solution
Lemma 9 Let the function 119891 in the last 119899 arguments havecontinuous partial derivatives satisfying
1199011119896(119905) ⩽
120597119891119896(119905 1199091 119909
119899)
120597119909119896
⩽ 1199012119896(119905) (119896 = 1 2 119899)
(43)
where 119901119894119896 R rarr R (119894 = 1 2 119896 = 1 2 119899) are continuous
119879-periodic functions Let moreover for any 119879-periodic 119901119896
R rarr R (119896 = 1 2 119899) satisfying (41) (42) has nonontrivial119879-periodic solutionThen the equation has a unique119879-periodic solution
On the basis of these theorems we can prove the theoremby checking the conditions in the previous lemma ByLemma 6 the conditions above can be easily proved
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper is supported by National Natural Science Founda-tion of China (Grant no 11301209) The author would like tothank the anonymous referee for the valuable comments andsuggestions on the paper especially for the considerations inSection 4 The author would like to thank Professor Yong Lifor his helpful instruction and valuable suggestions
References
[1] F Cong ldquoPeriodic solutions for second order differential equa-tionsrdquo Applied Mathematics Letters vol 18 no 8 pp 957ndash9612005
[2] J Ward ldquoPeriodic solutions for a class of ordinary differentialequationsrdquo Proceedings of the American Mathematical Societyvol 78 no 3 pp 350ndash352 1980
[3] R Kannan ldquoExistence of periodic solutions of nonlinear dif-ferential equationsrdquo Transactions of the AmericanMathematicalSociety vol 217 pp 225ndash236 1976
[4] K Schmitt ldquoA note on periodic solutions of second orderordinary differential equationsrdquo SIAM Journal onAppliedMath-ematics vol 21 pp 491ndash494 1971
[5] Q Yao ldquoPositive solutions of nonlinear second-order periodicboundary value problemsrdquoAppliedMathematics Letters vol 20no 5 pp 583ndash590 2007
[6] I Kiguradze and S Stanck ldquoOn periodic boundary valueproblem for the equation11990610158401015840 = 119891(119905 119906 1199061015840)with one-sided growthrestrictions on frdquo Nonlinear Analysis vol 48 pp 1065ndash10752002
[7] P Yan andMZhang ldquoHigher order non-resonance for differen-tial equations with singularitiesrdquo Mathematical Methods in theApplied Sciences vol 26 no 12 pp 1067ndash1074 2003
[8] I Kiguradze ldquoBoundary value problems for systems of ordinarydifferential equationsrdquo Itogi Nauki i Tekhniki Seriya Sovremen-nye Problemy Matematiki Noveishie Dostizheniya vol 30 pp3ndash103 1987 (Russian) English translation Journal of SovietMathematics vol 43 no 2 pp 2259ndash2339 1988
are satisfied on R119899+1 where 119903 = 119888119900119899119904119905 gt 0 and 119901119894119896
R rarr R (119894 = 1 2 119896 = 1 2 119899) are continuous119879-periodic functions Let moreover for any continuous 119879-periodic functions 119901
119896 R rarr R (119896 = 1 2 119899) satisfying
1199011119896(119905) ⩽ 119901
119896(119905) ⩽ 119901
2119896(119905) (119896 = 1 2 119899) (41)
the equation
119906(119899)=
119899
sum
119896=1
119901119896(119905) 119906(119896minus1) (42)
have no nontrivial 119879-periodic solution Then equation has atleast one 119879-periodic solution
Lemma 9 Let the function 119891 in the last 119899 arguments havecontinuous partial derivatives satisfying
1199011119896(119905) ⩽
120597119891119896(119905 1199091 119909
119899)
120597119909119896
⩽ 1199012119896(119905) (119896 = 1 2 119899)
(43)
where 119901119894119896 R rarr R (119894 = 1 2 119896 = 1 2 119899) are continuous
119879-periodic functions Let moreover for any 119879-periodic 119901119896
R rarr R (119896 = 1 2 119899) satisfying (41) (42) has nonontrivial119879-periodic solutionThen the equation has a unique119879-periodic solution
On the basis of these theorems we can prove the theoremby checking the conditions in the previous lemma ByLemma 6 the conditions above can be easily proved
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper is supported by National Natural Science Founda-tion of China (Grant no 11301209) The author would like tothank the anonymous referee for the valuable comments andsuggestions on the paper especially for the considerations inSection 4 The author would like to thank Professor Yong Lifor his helpful instruction and valuable suggestions
References
[1] F Cong ldquoPeriodic solutions for second order differential equa-tionsrdquo Applied Mathematics Letters vol 18 no 8 pp 957ndash9612005
[2] J Ward ldquoPeriodic solutions for a class of ordinary differentialequationsrdquo Proceedings of the American Mathematical Societyvol 78 no 3 pp 350ndash352 1980
[3] R Kannan ldquoExistence of periodic solutions of nonlinear dif-ferential equationsrdquo Transactions of the AmericanMathematicalSociety vol 217 pp 225ndash236 1976
[4] K Schmitt ldquoA note on periodic solutions of second orderordinary differential equationsrdquo SIAM Journal onAppliedMath-ematics vol 21 pp 491ndash494 1971
[5] Q Yao ldquoPositive solutions of nonlinear second-order periodicboundary value problemsrdquoAppliedMathematics Letters vol 20no 5 pp 583ndash590 2007
[6] I Kiguradze and S Stanck ldquoOn periodic boundary valueproblem for the equation11990610158401015840 = 119891(119905 119906 1199061015840)with one-sided growthrestrictions on frdquo Nonlinear Analysis vol 48 pp 1065ndash10752002
[7] P Yan andMZhang ldquoHigher order non-resonance for differen-tial equations with singularitiesrdquo Mathematical Methods in theApplied Sciences vol 26 no 12 pp 1067ndash1074 2003
[8] I Kiguradze ldquoBoundary value problems for systems of ordinarydifferential equationsrdquo Itogi Nauki i Tekhniki Seriya Sovremen-nye Problemy Matematiki Noveishie Dostizheniya vol 30 pp3ndash103 1987 (Russian) English translation Journal of SovietMathematics vol 43 no 2 pp 2259ndash2339 1988