ITB J.
Sci.
Vol.
43 A.
No.
1, 2011, 51-58
Existence and Uniqueness Results for Difference A-Laplacian Boundary Value Problems Bondar1.
Borkar2 & S.
Patil3 A P.
Department of Mathematics.
Science College.
India Department of Mathematics.
Yeshwant Mahavidyalaya.
India Email: klbondar_75@rediffmail.
Abstract.
This paper is devoted to study the existence and uniqueness of solutions to nonlinear difference A-Laplacian boundary value problems with mixed and Dirichlet boundary conditions.
Keywords: fixed points words.
Laplacian problems.
Introduction The study of nonlinear difference second order boundary value problems has been developed recently in many papers.
The first order difference equation with nonlinear functional boundary conditions is considered in .
where as the nth order difference equations are studied in .
The existence and uniqueness of solutions for some nonlinear boundary value problems are given .
Some of these results for second order difference equations have been generalized in .
for A-Laplacian problems.
The existence and uniqueness of solutions to difference A-Laplacian problems along with comparison results are given by A.
Cabada and V.
Espinar in .
In this paper we study the existence and uniqueness of solution for the equation AE(A(AEu.
)) = f.
,u.
) .
with different boundary conditions.
The results in this paper are generalizations of the results obtained in .
The corollaries in this paper are nothing but some of the theorems proved in .
The technique of fixed point theory of contraction mapping is utilized in this work to prove existence and uniqueness of solution of solution of Eq.
Throughout this paper we denote I = .
,1,2,C,N Ae .
P = .
,1,2,C,N}.
J =
,1,2,C,N .
By summation convention we have Eu k A n A A1 ( x k ) A 0 if m < Suppose B is a set of all real valued functions defined on P and for each Received April 17th, 2010.
Revised July 10th, 2010.
Accepted for publication December 21st, 2010.
Bondar, et al.
xEaB, define a norm on B by x A Eu k A0 x.
) the usual 1-norm in EC N A1 .
Hence B is a Banach space.
In the sequel we state the following conditions.
(H.
: A :ECC EC is a homeomorphisim and A-1 is Lipschitizian function on EC i.
there is H > 0 satisfying A A1 ( .
A A A1 ( .
C H x A y for all x,yEaEC.
(H.
: A function f : P C ECCEC is Lipschitz w.
the second variable i.
there is L > 0 such that f .
, .
A f .
, y ) C L x A y for all kEaP and x,yEaEC (H.
: A : ECCEC is nondecreasing.
Existence and Uniqueness Results This section is devoted to the existence and uniqueness of solutions to difference A-Laplacian boundary value problem AE(A(AEu.
)) = f.
,u.
) with different boundary conditions.
Theorems 2.
1, 2.
2, and 2.
3 in this paper are the generalizations of the Theorems 2.
1, 2.
2, and 2.
3 in .
Theorem 2.
Assume (H.
and (H.
If L Ea EE 0 .
E then the problem HN ( N A .
E AE AuA A AE u ( k ) AAy A f A k , u ( k A .
k Ea I ( PM 1 ) E AE u ( 0 ) A N 0 .
u ( N A .
A N 1
has a unique solution u : J C R for all N0.
N1EaEC.
Proof.
Define a mapping T : B C B by Difference A-Laplacian Boundary Value Problems k A1 Tv ( k ) A A ( N 0 ) A Eu f E l .
N 1 A Eu A ( v ( s )) E , lA0 s A l A1 for vEaB and kEaP.
We verify that u: J C EC given by u .
) A N 1 A Eu A A1 ( v ( s )) sAk is a solution of problem (PM.
provided vEaB is a fixed point of T and uniqueness of the solution follows from the existence of unique fixed point of T.
From Eq.
3 and by summation convention, it is obvious that u(N .
= N1.
Now.
AEu .
) A A Eu A A1 s A k A1 ( v ( s )) A Eu A A1 ( v ( s )) sAk ( v ( k )) Au A ( AE u ( k )) A v ( k ).
Since v is a fixed point of T, k A1 s A l A1 A ( AE u ( k )) A A ( N 0 ) A Eu f E l .
N 1 A Eu A lA0 ( v ( s )) E.
It follows that E AE (A ( AE u ( k )) A f E k .
N 1 A Eu A ( v ( s )) E s A k A1 A f ( k , u ( k A .
From Eq.
4 it follows that A(AEu.
) = A(N.
and injectivity of A implies AEu.
= N0.
Now we prove that T has a unique fixed point in B.
For v1,v2EaB and using conditions (H.
, (H.
we have k A1 Tv 1 ( k ) A Tv 2 ( k ) C Eu f E l .
N 1 A Eu A ( v 1 ( s )) E A f E l .
N 1 A Eu A ( v 2 ( s )) E s A l A1 s A l A1 Bondar, et al.
k A1 C LEu Eu A A1 ( v 1 ( s )) A A A1 ( v 2 ( s )) l A 0 s A l A1 C LH v 1 A v 2 k Au Tv 1 A Tv 2 C Eu LH v 1 A v 2 k k A0 C LH v 1 A v 2 N ( N A .
Since L Ea A0 .
HN .
N A1 ) A.
T is a contraction and hence has a unique fixed point which completes the proof.
Theorem 2.
Assume (H.
and (H.
If L Ea A0 .
HN .
N A 1 ) A, then the boundary value problem E AE (A ( AE u ( k ))] A f ( k , u ( k A .
) ( PM 2 ) E E u.
A N 0 .
AE u ( N ) A N 1 has a unique solution u : J C R for all N0.
N1EaR.
Proof.
Define a mapping T : B C B by N A1 Tv ( k ) A A ( N 1 ) A Eu f E l .
N 0 A Eu A ( v ( s )) E , lAk for kEaP and vEaB.
Similar to Theorem 2.
1 it can be verified that u : J C EC given by N A1 u .
) A N 0 A Eu A A1 ( v ( s )) .
sA0 is a desired unique solution of B.
(PM.
if and only if vEaB is a unique fixed point of T.
Using these theorems following corollaries can be obtained which are the Theorems 2.
1 and 2.
2 in .
Corollary 2.
Assume condition (H.
If M Ea A HN A( N2 A 1 ) , 0 A then the problem A AE (A ( AE u ( k ))) A Mu ( k A .
A A ( k ).
k Ea I Difference A-Laplacian Boundary Value Problems AE u .
A N 0 .
u ( N A .
A N 1 has a unique solution uEaB for each AEaB and all N0,N1EaEC.
Proof.
If we put f.
=Mu-A.
the above problem reduces to problem (PM.
Moreover for u, vEaB, f .
, u ) A f .
, .
A A M u A v .
Therefore f is a Lipschitz function with L = - M.
Since M Ea A HN (AN2 A .
, 0 A .
L Ea 0 .
HN .
N A 1 ) so the conclusion follows from Theorem 2.
Corollary 2.
Assume condition (H.
and (H.
If M Ea A HN A( N2 A 1 ) , 0 A then the A AE (A ( AE u ( k ))) A Mu ( k A .
A A ( k ).
k Ea I u ( 0 ) A N 0 .
AE u ( N A .
A N 1 has a unique solution uEaB for each AEaB and all N0.
N1EaEC.
Now we prove the existence and uniqueness of solution for Dirichlet Boundary value problem of A-Laplacian difference equation.
Theorem 2.
Assume (H.
, (H.
and (H.
If L Ea A0 .
HN .
N A 1 ) A, then the boundary value problem E AE (A ( AE u ( k ))] A f ( k , u ( k A .
) ( PM 3 ) E E u.
A N 0 .
u ( N A .
A N 1 has a unique solution u:J C EC for all N0,N1EaEC.
Proof.
Define a mapping T:BCB by E Tv ( k ) A C v A Eu f E l .
N 0 A Eu A ( v ( s )) E , lAk where Cv is solution of N1 A N 0 A Eu A k A0 E v Eu Eu A ( v ( s )) E E.
Bondar, et al.
Since A-1 is one-one, onto and continuous form EC to itself, for given vEaB there exists unique CvEaEC satisfying Eq.
We verify that u:JCEC given by k A1 u .
) A N 0 A Eu A A1 ( v ( s )) .
sA0 is a desired unique solution of B.
(PM.
if and only if vEaB is a unique fixed point of T.
From .
it is obvious that u.
=N0.
Now AEu .
) A A A1
( v ( k )) E Au A ( AE u ( k )) A C v A Eu f E l .
N 0 A Eu A ( v ( s )) E Au AE [A ( AE u ( k ))] A f E k .
N 0 A Eu A ( v ( s )) E A f ( k .
u ( k A .
From Eq.
9 and Eq.
8 it follows that u(N .
=N1.
Now we prove that T has a unique fixed point.
For each kEaP and v1,v2EaB, we E E Tv 1 ( k ) A Tv 2 ( k ) A C v1 A C v 2 A Eu E f E l .
N 0 A Eu A ( v 1 ( s )) E A f E l .
N 0 A Eu A ( v 2 ( s )) E E lAk E where C V and C V satisfy EuA k A0 E C v1 A Eu f ( l .
N 0 A Eu A ( v 1 ( s ))) E A Eu A E C v 2 A Eu f ( l .
N 0 A Eu A ( v 2 ( s ))) E.
A k E k A0 There exists k0,k1EaP such that A A1 E C v1 A Eu f E l .
N 0 A Eu A ( v 1 ( s )) E E C A E C v 2 A Eu f E l .
N 0 A Eu A ( v 2 ( s )) E E l Ak0 lAk 0 E C v1 A Eu f E l .
N 0 A Eu A ( v 1 ( s )) E E C A E C v 2 A Eu f E l .
N 0 A Eu A ( v 2 ( s )) E E.
l A k1 l A k1 Difference A-Laplacian Boundary Value Problems Condition (H.
implies that E C v1 A Eu f E l .
N 0 A Eu A ( v 1 ( s )) E C C v 2 A Eu f E l .
N 0 A Eu A ( v 2 ( s )) E lAk 0 lAk 0 C v1 A Eu f E l .
N 0 A Eu A ( v 1 ( s )) E C C v 2 A Eu f E l .
N 0 A Eu A ( v 2 ( s )) E
l A k1
l A k1
E E
Au Eu E f E l .
N 0 A Eu A ( v 1 ( s )) E E A f E l .
N 0 A Eu A ( v 2 ( s )) E C C v1 A C v 2
l A k1 E
E E
C Eu E f E l .
N 0 A Eu A ( v 1 ( s )) E A f E l .
N 0 A Eu A ( v 2 ( s )) E E.
lAk 0 E If we suppose k < k0, then we obtain Tv 1 ( k ) A Tv 2 ( k ) C LH v 1 A v 2 ( k 0 A k ), and k >k0, then we obtain Tv 1 ( k ) A Tv 2 ( k ) C LH v 1 A v 2 ( k A k 0 ).
Au Tv 1 ( k ) A Tv 2 ( k ) C LH v 1 A v 2 A k A k 0 A.
Similarly we deduce that Au Tv 2 ( k ) A Tv 1 ( k ) C LH v 1 A v 2 A k A k 1 A.
Hence from Eq.
10 and Eq.
11 we conclude that, for all kEaP Tv 1 ( k ) A Tv 2 ( k ) C LH v 1 A v 2 max A k A k 0 , k A k 1 A C LH v 1 A v 2 max Ak .
N A k A.
As a consequence, we obtain Tv 1 A Tv 2 C LH N ( N A .
v1 A v 2 .
This shows that T is a contraction and hence result holds.
Bondar, et al.
Remark.
Similar to Corollaries 2.
1 and 2.
2, we can obtain a result which is Theorem 2.
3 in .
, that the equation A AE (A ( AE u ( k )) A Mu ( k A .
A A ( k ) .
A Ea I has a unique solution for Dirichlet boundary condition u.
=N0, u(N .
= N1.
Acknowledgement Authors are very much thankful to the referees for their valuable suggestions to improve this work.
References