Qisuvchi akslantirishlar prinsipining integral tenglamalarga tadbiqi.
Fredholm tenglamasi.Qisuvchi akslantirishlar prinsipini ushbu
Ikkinchi tur Fredholm integral tenglamasi yechimining mavjudligi va yagonaligini isbotlash uchun qo ‘llaymiz.Bu yerda integral tenglama yadrosi, -berilgan funksiya ,-izlanayotgaan funksiya ,-esa haqiqiy parameter.
Ko ‘rsatamizki, qisuvchi akslantirishlar prinsipi parametrning yetarlicha kichik qiymatlardaa qo ‘llash mumkin.
Faraz qilamiz - kvadratda uzluksiz funksiya
bo ‘lsin.Shunday ekan musbat son mavjud bo ‘lib ,barcha
uchun tengsizlik bajariladi. To ‘la fazoni
o‘zini-o‘ziga
(6)
Formula vositasida akslantiruvchi akslantirsh berilgan bo ‘lsin. U holda
![](data:image/png;base64,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)
Yoki
![](data:image/png;base64,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)
Shunday ekan
(7)
Bo ‘lganda qisuvchi akslantirish bo ‘ladi.Qisuvchi akslantirishlar prinsipiga asoslanib xulosa qilamizki,(7)shartni qanoatlantiruvchi ixtiyoriy da (5) Fredholm tenglamasi yagona uzluksiz yechimga ega
Bu yechimga intiluvchi ketma-ket yaqinlashishlar ![](data:image/png;base64,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)
Ko ‘rinishga ega ,bu yerda sifatida ixtiyoriy uzluksiz funksiyani olish mumkin.
Chiziqlimas integral tenglamalar.Qisuvchi akslantiruvchi prinsipining
Ko ‘rinishdagi chiziqlimas integral tenglamalarga tadbiqini qaraymiz.Bu yerda va funksiyalar uzluksiz bo ‘lib ,bundan tashqari o ‘zining 3 chi funksional argument bo ‘yicha Lipshits shartini qanoatlantirsin ya’ni shunday mavjud bo ‘lib
Tengsizlik barcha va lar uchun o ‘rinli bo ‘lsin.Bu holda fazoni o ‘zini-o ‘ziga
formula vositasida akslantiruvchi akslantirish uchun
tengsizlik o ‘rinli bo ‘ladi, bu yerda , .Shunday ekan,
Shartda akslanish qisuvchi bo ‘ladi.
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