Dong-Myung Lee, Jeong-Gon Lee, and Ming-Gen Cui. 1. introduction

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1 J. Kore So. Mth. Edu. Ser. B: Pure Appl. Mth. ISSN Volume 11, Number My 004), Pges REPRESENTATION OF SOLUTIONS OF FREDHOLM EQUATIONS IN W Ω) OF REPRODUCING KERNELS Dong-Myung Lee, Jeong-Gon Lee, nd Ming-Gen Cui Abstrt. In this pper we derive deomposition of the solution of Fredholm equtions of the seond kind in terms of reproduing kernels in the spe W 1 Ω). 1. introdution The solution problem in integrl equtions is to find n effetive deomposition of the solution u of the form u n s n, where the oeffiients n nd hrmonis s n hve omputble nd relevnt properties, optiml nd desired durtion properties in terms of the informtion re, respetively. e. g., Benedetto & Wlnut []). Let φx, y) be bounded on Ω [, b] [, d] in R nd let W Ω) ux, y) ux, y) is bsolutely ontinuous nd u x, u y, u } x y L Ω). It is evident from the work of Cui, Deng & Wu [4], Cui, Lee & Lee [5] tht the deomposition of the solution of Fredholm equtions depends on W 1 R), the spe of reproduing kernels. The present work ddresses the solution representtion bsed on Cui s pproh by repling R into Ω nd we formulte the struture of this deomposition in terms of the reproduing kernels in W Ω).. Spe W Ω) nd Reproduing Kernels Throughout, L Ω) denotes, s usul, the Hilbert spe of ll Lebesgue squre integrble funtions on Ω nd RA) the rnge of bounded liner opertor A on Reeived by the editors July 1, 003 nd, in revised form, April 16, Mthemtis Subjet Clssifition. 41A17, 4C15, 4C30. Key words nd phrses. Fredholm eqution, bsolutely ontinuous, reproduing kernel. This work ws supported by Won Kwng University Reserh Grnt in Kore So. Mth. Edu.

2 134 Dong-Myung Lee, Jeong-Gon Lee, nd Ming-Gen Cui W Ω). The needed fts bout reproduing kernels n be found in Lee, Lee & Kim [7], Sitoh [9]. Definition.1. Let φx, y) be bounded on Ω [, b] [, d]. If φi i ) b i d i + i i i d i b i i 0 1) for I i [ i, b i ) [ i, d i ) Ω, where [ i, b i ) nd [ i, d i ) re finite pirwise disjoint subintervls of [, b] nd [, d], respetively. Then φ is lled monotone on Ω. Definition.. Let φx, y) be bounded nd monotone on Ω [, b] [, d]. If, given ny ɛ > 0, there exists δ > 0 suh tht min b i i, d i i } < δ implies n 1 φi i) < ɛ, then φ is lled bsolutely ontinuous on Ω. We now define, for u, v W Ω), u, v Ω ux, y)vx, y) + u v x x + u v y y + u v ) dσ ) x y x y s inner produt on W Ω) nd norm u W < u, u > 1. Theorem.3. Let f W Ω). Then f sup p Ω fp) M f W, where M 1 b )d ) [ b )d ) + b ) + d ) + 1 ]. Proof. Let x 1, y 1 ), x, y ) Ω. Then x fx, y ) y f x 1 y 1 y + x tτ dτdt + y 1 f τ x 1, τ)dτ + fx 1, y 1 ). x 1 f tt, y 1 )dt 3) Let the both sides of 3) integrte over Ω. Then, Cuhy-Shwrtz, nd Hölder s inequlities llow us to ompute fx, y ) dx 1 dy 1 Ω Ω + x y x 1 y x f tτ t, τ)dτdt + f tt, y 1 )dt 4) y 1 x 1 f τ x 1, τ)dτ + fx 1, y 1 ) dx 1 dx. y 1

3 FREDHOLM EQUATIONS IN W Ω) OF REPRODUCING KERNELS 135 On the other hnd, sine fx, y ) b )d ) b )d ) f tτ t, τ) dtdτ + b ) + d ) f τ x 1, τ) dτdt + b )d ) b )d ) + b ) + fx 1, y 1 ) dx 1 dy 1 f tτ t, τ) dτdt f tt, y 1 ) ) 1 dtdy 1 + d ) fx 1, y 1 ) ) 1 dx 1 dy 1 } we hve [ b )d ) + b ) + d ) + 1 fx, y ) b )d ) M f W., ] 1 f tt, y 1 ) dtdy 1 ) 1 f τ x 1, τ) ) 1 dτdx 1 } } 1 f tτ t, τ) + f tt, y 1 ) + f τ x 1, τ) + f tτ t, τ) dx 1 dy 1 We rell, see Cui & Deng [3], Cui, Deng & Wu [4]), tht ny ux) W 1 [, b]) n be represented in terms of reproduing kernel suh s ux) u ), R x ), where R x y) 1 [ ] hy + x b) + h y x + b), y [, b]. shb ) Our next result extends this formul to the se of Ω [, b] [, d]. Theorem.4. R xy, ) is reproduing kernel of spe W Ω), where R xy, ) R x )R y ). Proof. For ny u W Ω), u, ), R xy, ) dη uξ, η)r xy ξ, η) + u ξ ξ, η)r xyξ, η) ξ + u ηξ, η)r xyξ, η) η + u ξη ξ, η)r xyξ, η) ξη }dξ uξ, η)r xy ξ, η)dξ + u ξ ξ, η)r xyξ, η) ξ

4 136 Dong-Myung Lee, Jeong-Gon Lee, nd Ming-Gen Cui dη R y η) + u ηξ, η)r xyξ, η) η + u ξη ξ, η)r xyξ, η) ξη }dξ uξ, η)rx ξ) + u ξ ξ, η)r xξ) ξ ) dξ + R yη) η Applying the reproduing properties of R y ξ), we hve So tht, u η ξ, η)r x ξ) + u ξη R xξ) ξ ) dξ }. [uξ, η)r x ξ) + u ξ ξ, η)r xξ) ξ ] dξ u, η), R x ) ux, η). u ηξ, η)r x ξ) + ξ u η ξ, η) ) R xξ) ξ dξ u η, η), R x ) u ηx, η). u, ), R xy, ) } ux, η)r y η) + u ηx, η)r xη) dη. Finlly, we use the definition of reproduing kernels to get the onlusion. 3. Deomposition by Reproduing kernels Let us onsider the following lssil Fredholm eqution ux, y) λ nd let where I is identity opertor, λ is prmeter, nd Ku kx, y; t, τ)ut, τ)dtdτ fx, y), I λk)u f, 5) kx, y; t, τ)ut, τ)dtdτ. Then, for given sequene p i ) in Ω [, b] [, d], n elementry liner lgebr rgument shows tht φ j p i )) is linerly independent Lee, Lee, & Kim [7], where φ j p i ) R pj p i ) for p i Ω. We begin by stting the next results, whose onlusion will be needed for our purposes. Lemm 3.1. Let A be bounded liner opertor on W Ω), A be djoint of A, nd p i ) be dense in Ω. Then A φ j p i )) is omplete if nd only if A is one-to-one.

5 FREDHOLM EQUATIONS IN W Ω) OF REPRODUCING KERNELS 137 Proof. Assume A is one-to-one nd for u W 1Ω), let u, A φ j p i ) 0. Then u, A φ j p i ) Au, φ j p i ) Au)p i ) 0. Thus, the ssumption of A implies u 0. Conversely, let Au 0. For eh φ j p i ), Au, φ j p i ) u, A φ j p i ) 0. Hene we hve u 0. Lemm 3.. If RA) W Ω), then A φ j p i )) is linerly independent. Proof. Let i ) C, nd let j A φ j p i ) 0. Then, for u W Ω), Au, j φ j 0. It follows esily tht j φ j p i ) RA) o}. So tht we hve j 0. We use now Lemms 3.1 nd 3. to prove the following hrteriztion of reproduing kernels. Theorem 3.3. Let Au f from 5), where A I λk nd p p i ) be dense in Ω. Then u is deomposed by reproduing kernels. Proof. The bove Lemms show tht A φ j p)) is omplete nd linerly independent. We now let ψ j p) A φ j p)). By using Grm-Shmidt orthogonliztion proedure of ψ j ), we obtin ψ j ) suh tht, s usul, ψ i, ψ j 1 if i j nd 0 if i j. Let ψ i p) i j1 β ijψ j p). Then, we hve u u, ψ k ψ k k u, A β kj φ j p)) ψ k k1 k1 j1 k1 j1 k β kj Au, φ j ψ k p) f k ψk p), k1 k1 j1 k β kj fp) ψ k p) where f k k j1 β kjfp), nd the theorem is proved. Referenes 1. N. Aronszjn: Theory of reproduing kernels. Trns. Amer. Mth. So. 68, 1950) MR 14,479. J. Benedetto & D. Wlnut: Gbor frmes for L nd relted spes. In: John J. Benedetto & Mihel W. Frzier Eds.), Wvelets: mthemtis nd pplitions pp ). CRC Press, Bo Rton, FL, MR 94f:4048

6 138 Dong-Myung Lee, Jeong-Gon Lee, nd Ming-Gen Cui 3. M. Cui & Z. X. Deng: On the best opertor of interpoltion in W 1. Mth. Numer. Sini ), no., MR 87j: M. Cui, Z. X. Deng & B. Y. Wu: Anlyti solutions to Fredholm integrl equtions of the seond kind. Numer. Mth. J. Chinese Univ ), no. 1, MR 91: M. G. Cui, D. M. Lee & J. G. Lee: Fourier Trnsforms nd Wvelet Anlysis. Kyung Moon Press, Seoul, D. M. Lee, J. G. Lee & S. H. Yoon: A onstrution of multiresolution nlysis by integrl equtions. Pro. Amer. Mth. So ), no. 1, MR 003f: D. M. Lee, J. G. Lee & I. S. Kim: Representtion of integrl opertors on W Ω) of reproduing kernels. Submitted. 8. S. Sitoh: Theory of reproduing kernels nd its pplitions. Longmn Sientifi & Tehnil, Hrlow; opublished in the United Sttes with John Wiley & Sons, In., New York, MR 90f: : Integrl trnsforms, reproduing kernels nd their pplitions. Longmn, Hrlow, MR 98k:46041 D. M. Lee) Deprtment of Mthemtis, Won Kwng University, 344- Shinyong-dong, Ik-Sn, Chunbuk , Kore Emil ddress: dmlee@wonkwng..kr J. G. Lee) Deprtment of Mthemtis, Won Kwng University, 344- Shinyong-dong, Ik-Sn, Chunbuk , Kore M. G. Cui) Hrbin Institute of Tehnology WEI HAI brnh Institute, 6409 Wei Hi, Shndong, P. R. Chin Emil ddress: mgyfs63.net