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The present paper is particularly exhibits about the derive results of a third order singular multipoint boundary value problem at resonance using coindence degree arguments. G. Pushpalatha | S. K. Reka "Singular Third-Order Multipoint Boundary Value Problem at Resonance" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: https://www.ijtsrd.com/papers/ijtsrd11058.pdf Paper URL: http://www.ijtsrd.com/mathemetics/other/11058/singular-third-order-multipoint-boundary-value-problem-at-resonance/g-pushpalatha
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@ IJTSRD | Available Online @ www.ijtsrd.com
ISSN No: 2456
InternationalResearch
Singular ThirdValue Problem
G. Pushpalatha Assistant Professor, Department of Mathematics, Vivekanandha College of Arts and Sciences for
Women, Tiruchengode, Namakkal, Tamilnadu, India
ABSTRACT The present paper is particularly exhibits about the derive results of a third-order singular multipoint boundary value problem at resonance using coindence degree arguments.
Keywords: The present paper is particularly exhibits about the derive results of a third-order singular multipoint boundary value problem at resonance using coindence degree arguments.
INTRODUCTION
This paper derive the existence for the thirdsingular multipoint boundary value problem at resonance of the form
๐ขโฒโฒโฒ = ๐(๐ก), ๐ข(๐ก), ๐ขโฒ(๐ก), ๐ขโฒโฒ(๐ก)) +
๐ขโฒ(0) = 0, ๐ขโฒโฒ(0) = 0,
๐ข(1) = ๐ ๐ ๐ข ๐ ,
,
Where ๐ โถ [0,1] ร โ โ โ is caratheodoryโs function (i.e., for each (๐ข, ๐ฃ) โ โ the function measurable on [0,1]; for almost everywhere the function ๐(๐ก, . , . ) is continuous on ๐ โ (0,1), ๐, ๐ = 1,2, โฆ , ๐ โ 3, and 1, where ๐ and โ have singularity at ๐ก=1.
In [1] Gupta et al. studied the above equation when and โ have no singularity and โ ๐,
obtained existence of a ๐ถ [0,1] solution by utilizing
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume โ 2 | Issue โ 3 | Mar-Apr 2018
ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume
International Journal of Trend in Scientific Research and Development (IJTSRD)
International Open Access Journal
Third-Order Multipoint BoundaryValue Problem at Resonance
Department of Mathematics, Vivekanandha College of Arts and Sciences for
Women, Tiruchengode, Namakkal, Tamilnadu, India
S. K. RekaResearch Scholar, Department of Mathematics, Vivekanandha College of Arts and Science
Women, Tiruchengode, Namakkal, Tamilnadu, India
The present paper is particularly exhibits about the order singular multipoint
boundary value problem at resonance using coindence
e present paper is particularly exhibits order singular
multipoint boundary value problem at resonance
This paper derive the existence for the third-order multipoint boundary value problem at
) + โ(๐ก)
is caratheodoryโs function the function ๐(. , ๐ข, ๐ฃ) is
for almost everywhere ๐ก โ [0,1], is continuous on โ ). Let
and โ ๐ ๐ =,
=1.
In [1] Gupta et al. studied the above equation when ๐ ๐ โ 1. They
solution by utilizing
Letay-Schauder continuation principle. These results correspond to the nonresonance case. The scope of this article is therefore to obtained when โ ๐ ๐ = 1, (the resonance case) and when ๐ and โ have a singularity at ๐ก
Definition 1
Let ๐ and ๐ be real Banach spaces. One says that the linear operator ๐ฟ: ๐๐๐ ๐ฟ โ ๐mapping of index zero if Ker finite dimension, where ๐ผ๐ ๐ฟ denotes the image of
Note
We will require the continuous projections ๐: ๐ โ ๐ such that Im P = Ker L,Ker Q = Im L, ๐ =Ker L โจ Ker ๐, ๐๐ฟ| โ : dom L โ isomorphism.
Definition 2
Let ๐ฟ be a Fredholm mapping of index zero and bounded open subset of U such that dom ๐ฟโฮฉ โ ๐. The map M: ๐ โ ๐on ๐, if the map ๐๐(ฮฉ) is bounded and compact, where one denotes by ๐๐๐ ๐ฟ โ ๐พ๐๐ ๐ the generalized inverse of addition ๐ is ๐ฟ-completely continuous if it on every bounded ฮฉ โ ๐.
Apr 2018 Page: 623
6470 | www.ijtsrd.com | Volume - 2 | Issue โ 3
Scientific (IJTSRD)
International Open Access Journal
Order Multipoint Boundary
Reka Department of Mathematics,
Vivekanandha College of Arts and Sciences for Women, Tiruchengode, Namakkal, Tamilnadu, India
Schauder continuation principle. These results correspond to the nonresonance case. The scope of this article is therefore to obtained the survive results
(the resonance case) and when ๐ก = 1.
be real Banach spaces. One says that the ๐ โ ๐ is a Fredholm
if Ker ๐ฟ and ๐/๐ผ๐ ๐ฟ are of denotes the image of ๐ฟ.
We will require the continuous projections ๐: ๐ โ ๐, such that Im P = Ker L,Ker Q = Im L,
๐ =Im L โจ Im Q, ker ๐ โ Im ๐ฟ Iis an
be a Fredholm mapping of index zero and ฮฉ a bounded open subset of U such that dom
๐ is called ๐ณ-compact is bounded and ๐ (๐ผ โ ๐) is
compact, where one denotes by ๐ โถ Im ๐ฟ โthe generalized inverse of ๐ฟ. In
completely continuous if it ๐ฟ-compact
International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume โ 2 | Issue โ 3 | Mar-Apr 2018 Page: 624
Theorem 1
Let ๐ฟ be a Fredholm operator of index zero and let ๐ be ๐ฟ-compact on ฮฉ. Assume that the following conditions are satisfied :
(i) ๐ฟ๐ข โ ๐ ๐๐ข for every (๐ข, ๐ ) โ[(๐๐๐ ๐ฟ โ ๐พ๐๐ ๐ฟ) โฉ ๐๐ ] ร (0,1).
(ii) ๐๐ข โ ๐ผ๐ ๐ฟ, for every ๐ข โ ๐พ๐๐ ๐ฟ โฉ ๐๐ . (iii) deg (๐๐| โฉ , ๐ โฉ ๐พ๐๐ ๐ฟ, 0) โ 0,
with ๐: ๐ โ ๐ being a continuous projection such that ๐พ๐๐ ๐ = ๐ผ๐ ๐ฟ. then the equation ๐ฟ๐ข = ๐๐ข has at least one solution in ๐๐๐ ๐ฟ โฉ ฮฉ.
Proof :
We shall make use of the following classical spaces, ๐ถ[0,1], ๐ถ [0,1], ๐ถ [0,1], ๐ฟ [0,1], ๐ฟ [0,1],
and ๐ฟโ[0,1]. Let ๐ด๐ถ[0,1] denote the space of all absolute continuous functions on [0,1], ๐ด๐ถ [0,1] ={๐ข โ ๐ถ [0,1] โถ ๐ขโฒโฒ(๐ก) โ ๐ด๐ถ[0,1]}, ๐ฟ [0,1] =
๐ข: ๐ข|[ , ] โ ๐ฟ [0,1] for every compact interval [0, ๐] โ [0,1].
๐ด๐ถ [0,1) = ๐ข: ๐ข|[ , ] โ ๐ด๐ถ[0,1] .
Let U be the Banach space defined by
๐ = {๐ข โ ๐ฟ [0,1] โถ (1 โ ๐ก )๐ข(๐ก) โ ๐ฟ [0,1]},
With the norm
โ๐ฃโ = (1 โ ๐ก ) |๐ฃ(๐ก)|๐๐ก.
Let ๐ be the Banach space
๐ = {๐ข โ ๐ถ [0,1): ๐ข โ ๐ถ[0,1], lim โ (1 โ
๐ก ) ๐ขโฒโฒ ๐๐ฅ๐๐ ๐ก๐ } ,
With the norm
โ๐ขโ = max โ๐ขโโ , (1 โ ๐ก )๐ขโฒโฒ(๐ก)โ
.
(1)
Where โ๐ขโโ = sup โ[ , ]|๐ข(๐ก)| .
We denote the norm in ๐ฟ [0,1] by โ. โ . we define the linear operator ๐ฟ: ๐๐๐ ๐ฟ โ ๐ โ ๐ by
๐ฟ๐ข = ๐ขโฒโฒโฒ(๐ก), (2)
Where
๐๐๐ ๐ฟ = ๐ข โ ๐ โถ ๐ขโฒ(0) = 0, ๐ขโฒโฒ(0) = 0, ๐ข(1)
= ๐ ๐ ๐ข(๐)
,
And ๐: ๐ โ ๐ is defined by
๐ฟ๐ข = ๐ ๐ก, ๐ข(๐ก), ๐ขโฒ(๐ก), ๐ขโฒโฒ(๐ก) + โ(๐ก).
(3)
Then boundary value problem (1) can be written as
๐ฟ๐ข = ๐๐ข.
Therefore ๐ฟ = ๐.
Lemma
If โ ๐ ๐ = 1, then
(i) ๐พ๐๐ ๐ฟ = {๐ข โ ๐๐๐ ๐ฟ: ๐ข(๐ก) = ๐, ๐ โ โ, ๐ก โ[0,1]} ;
(ii) ๐ผ๐ ๐ฟ =๐ฃ โ ๐ง:
โ ๐ ๐ โซ โซ โซ ๐ฃ(๐)๐๐๐๐ = 0,
(iii) ๐ฟ: ๐๐๐ ๐ฟ โ ๐ โ ๐ is a Fredholm operator ๐: ๐ โ ๐ can be defined by
๐๐ฃ =๐
โ๐ ๐ ๐ฃ(๐)๐๐๐๐
,
,
(4)
Where
โ = ๐ ๐
,
๐ + ๐ + ๐ โ ๐ โ ๐ โ 1
(iv) The linear operator ๐ : ๐ผ๐ ๐ฟ: โ
๐๐๐ ๐ฟ โ ๐พ๐๐ ๐ can be defined as
๐ = ๐ฃ(๐)๐๐ (5)
(v) ๐ ๐ฃ โค โ๐ฃโ for all ๐ฃ โ ๐.
Proof :
(i) It is obvious that ๐พ๐๐ ๐ฟ = {๐ข โ ๐๐๐ ๐ฟ: ๐ข(๐ก) = ๐, ๐ โ โ}.
International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume โ 2 | Issue โ 3 | Mar-Apr 2018 Page: 625
(ii) We show that
๐ผ๐ ๐ฟ =๐ฃ โ ๐:
โ ๐ ๐ โซ โซ โซ ๐ฃ(๐)๐๐๐๐ = 0,. (7)
To do this, we consider the problem
๐ค โฒโฒโฒ(๐ก) = ๐ฃ(๐ก) (8)
And we show that (5) has a solution ๐ค(๐ก) satisfying
๐ค โฒโฒ(0) = 0, ๐ค โฒ(0) = 0, ๐ค(๐ก) = ๐ ๐ ๐ค ๐ ๐
,
If and only if
๐ ๐ ๐ฃ(๐)๐๐๐๐ = 0
,
(9)
Suppose (3) has a solmution ๐ค(๐ก) satisfying
๐ค โฒโฒ(0) = 0, ๐ค โฒ(0) = 0, ๐ค(๐ก) = ๐ ๐ ๐ค ๐ ๐
,
Then we obtain from (5) that
๐ค(๐ก) = ๐ค(0) + ๐ฃ(๐)๐๐๐๐ ,
(10)
And applying the boundary conditions we get
๐ ๐ ๐ฃ(๐)๐๐๐๐
,
= ๐ฃ(๐)๐๐๐๐ , (11)
Since โ ๐ ๐ = 1, , and using (i) and we get
๐ ๐ ๐ฃ(๐)๐๐๐๐ = 0
,
On the other hand if (6) holds, let ๐ข โ โ; then
๐ค(๐ก) = ๐ค(0) + ๐ฃ(๐)๐๐๐๐ ,
Where ๐ฃ โ ๐
๐ค โฒโฒโฒ(๐ก) = ๐ฃ(๐ก) (12)
(iii) For ๐ฃ โ ๐, we define the projection ๐๐ฃ as
๐๐ฃ =๐
โ๐ ๐ ๐ฃ(๐)๐๐๐๐
,
,
๐ก โ [0,1], (13)
Where
โ = ๐ ๐
,
๐ + ๐ + ๐ โ ๐ โ ๐ โ 1 โ 0.
We show that ๐: ๐ โ ๐ is well defined and bounded.
|๐๐ฃ(๐ก)| โค|๐ |
|โ||๐ | ๐
,
(1 โ ๐ ) |๐ฃ(๐ )|๐๐
=1
|โ||๐ | ๐
,
โ๐ฃโ |๐ |
โ๐๐ฃโ โค (1 โ ๐ก) |๐๐ฃ(๐ก)|๐๐ก
โค1
|โ||๐ | ๐
,
โ๐ฃโ |๐ | (1 โ ๐ ) ๐๐
=1
|โ||๐ | ๐
,
โ๐ฃโ โ๐ โ .
In addition it is easily verified that
๐ ๐ฃ = ๐๐ฃ, ๐ฃ โ ๐. (14)
We therefore conclude that ๐: ๐ โ ๐ is a projection. If ๐ฃ โ ๐ผ๐ ๐ฟ, then from (6) ๐๐ฃ(๐ก) = 0. Hence ๐ผ๐ ๐ฟ โ ๐พ๐๐ ๐. Let ๐ฃ = ๐ฃ โ ๐๐ฃ ; that is, ๐ฃ โ๐พ๐๐ ๐. Then
International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470
@ IJTSRD | Available Online @ www.ijtsrd.com | Volume โ 2 | Issue โ 3 | Mar-Apr 2018 Page: 626
๐ ๐ ๐ฃ (๐)๐๐๐๐
,
= ๐ ๐ ๐ฃ(๐)๐๐๐๐
,
โ1
โ๐ฃ(๐)๐๐๐๐
,
Here โ = 0 we get
๐ ๐ ๐ฃ (๐)๐๐๐๐
,
= ๐ ๐ ๐ฃ(๐)๐๐๐๐
,
Therefore ๐ฃ = ๐ฃ.
Thus ๐ฃ โ ๐ผ๐ ๐ฟ and therefore ๐พ๐๐ ๐ โ ๐ผ๐ ๐ฟ and hence
๐ = ๐ผ๐ ๐ฟ + ๐ผ๐ ๐ = ๐ผ๐ ๐ฟ + โ.
It follows that since ๐ผ๐ ๐ฟ โฉ โ = {0}, then ๐ =๐ผ๐ ๐ฟ โจ ๐ผ๐ ๐.
Therefore,
dim ๐พ๐๐ ๐ฟ = dim ๐ผ๐ ๐ = dim โ = ๐๐๐๐๐ ๐ผ๐ ๐ฟ = 1.
This implies that ๐ฟ is Fredholm mapping of index zero.
(iv) We define ๐ โถ ๐ โ ๐ by ๐๐ข = ๐ข(0), (15)
And clearly ๐ is continuous and linear and ๐ ๐ข =๐(๐๐ข) = ๐๐ข(0) = ๐ข(0) = ๐๐ข and ๐พ๐๐ ๐ ={๐ข โ ๐ โถ ๐ข(0) = 0}. We now show that the generalized inverse ๐พ = ๐ผ๐ ๐ฟ โ ๐๐๐ ๐ฟ โฉ ๐พ๐๐ ๐ of ๐ฟ is given by
๐ ๐ฃ = ๐ฃ(๐)๐๐ (16)
For ๐ฃ โ ๐ผ๐ ๐ฟ we have
(๐ฟ๐ )๐ฃ(๐ก) = ๐ ๐ฃ (๐ก)โฒโฒ
= ๐ฃ(๐ก) (17)
And for ๐ข โ ๐๐๐ ๐ฟ โฉ ๐พ๐๐ ๐ we know that
(๐ ๐ฟ) ๐ข(๐ก) = ๐ขโฒโฒ(๐)๐๐๐๐
= (๐ก โ ๐ )๐ขโฒโฒ ๐๐ (18)
= ๐ข(๐ก) โ ๐ขโฒ(0) ๐ก โ ๐ข(0) = ๐ข(๐ก)
Since ๐ข โ ๐๐๐ ๐ฟ โฉ ๐พ๐๐ ๐, ๐ข(0) = 0, and ๐๐ข = 0.
This shows that ๐ = (๐ฟ| โฉ ) .
(v) ๐ ๐ฃโ
โค max โ[ , ] โซ (๐ก โ ๐ ) |๐ฃ(๐ )|๐๐
โค (๐ก โ ๐ ) |๐ฃ(๐ )|๐๐
โค โ๐ฃโ .
We conclude that
๐ ๐ฃ โค โ๐ฃโ .
References
[1] Gupta C.P, Ntouyas S.K, and Tsamatos P.C, โSolvability of an ๐-point boundary value problem for second order ordinary differential equations,โ Journal of Mathematical Analysis and Applications, vol. 189, no. 2, pp. 575-584, 1995.
[2] Ma R and OโRegan D, โSolvability of singular second order ๐-point boundary value problem,โ Journal of Mathematical Analysis and Applications, vol. 301, no. 1, pp. 124-134, 2005.
[3] Infante G, and Zima M.A, โPositive solutions of multi-point boundary value problems at resonance,โ Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 8, pp. 2458-2465, 2008.
[4] Kosmatov N, โA singular non-local problem at resonance,โ Journal of Mathematical Analysis and Applications, vol. 394, no. 1, pp. 425-431, 2012.