Existence of solutions of three-point boundary value problems for nonlinear fourth order differential equation

In this paper, the author uses the methods in [1, 2] to study the existence of solutions of three point boundary value problems for nonlinear fourth order differential equation.% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa% aaleqabaGaaiikaiaaisdacaGGPaaaaOGaeyypa0JaaGOKbiaacIca% caWG0bGaaiilaiaadMhacaGGSaGabmyEayaafaGaaiilaiqadMhaga% GbaiaacYcaceWG5bGbaibacaGGPaaaaa!4497![y^{(4)} = f(t,y,y',y',y')] with the boundary conditions% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe% qaaiaadEgacaGGOaGaamyEaiaacIcacaWGHbGaaiykaiaacYcaceWG% 5bGbauaacaGGOaGaamyyaiaacMcacaGGSaGabmyEayaagaGaaiikai% aadggacaGGPaGaaiilaiqadMhagaGeaiaacIcacaWGHbGaaiykaiaa% cMcacqGH9aqpcaaIWaGaaiilaiaadIgacaGGOaGaamyEaiaacIcaca% WGIbGaaiykaiaacYcaceWG5bGbayaacaGGOaGaamOyaiaacMcacaGG% PaGaeyypa0JaaGimaaqaaiqadMhagaqbaiaacIcacaWGIbGaaiykai% abg2da9iaadkgadaWgaaWcbaGaaGymaaqabaGccaGGSaGaam4Aaiaa% cIcacaWG5bGaaiikaiaadogacaGGPaGaaiilaiqadMhagaqbaiaacI% cacaWGJbGaaiykaiaacYcaceWG5bGbayaacaGGOaGaam4yaiaacMca% caGGSaGabmyEayaasaGaaiikaiaadogacaGGPaGaaiykaiabg2da9i% aaicdaaaGaayzFaaaaaa!7059![left. begin{gathered} g(y(a),y'(a),y'(a),y'(a)) = 0,h(y(b),y'(b)) = 0 hfill y'(b) = b_1 ,k(y(c),y'(c),y'(c),y'(c)) = 0 hfill end{gathered} right}] For the boundary value problems of nonlinear fourth order differential equation% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa% aaleqabaGaaiikaiaaisdacaGGPaaaaOGaeyypa0JaaGOKbiaacIca% caWG0bGaaiilaiaadMhacaGGSaGabmyEayaafaGaaiilaiqadMhaga% GbaiaacYcaceWG5bGbaibacaGGPaaaaa!4497![y^{(4)} = f(t,y,y',y',y')] many results have been given at the present time. But the existence of solutions of boundary value problem (*). (**) studied in this paper has not been involved by the above researches. Morcover, the corollary of the important theorem in this paper, i. e. existence of solutions of the boundary value problem of equation (*) with the following boundary conditions.% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb% WaaSbaaSqaaiaaicdaaeqaaOGaamyEaiaacIcacaWGHbGaaiykaiab% gUcaRiaadggadaWgaaWcbaGaaGymaaqabaGcceWG5bGbauaacaGGOa% GaamyyaiaacMcacqGHRaWkcaWGHbWaaSbaaSqaaiaaikdaaeqaaOGa% bmyEayaagaGaaiikaiaadggacaGGPaGaey4kaSIaamyyamaaBaaale% aacaaIZaaabeaakiqadMhagaGeaiaacIcacaWGHbGaaiykaiabg2da% 9iaadMhadaWgaaWcbaGaaGimaaqabaGccaGGSaGaamOyamaaBaaale% aacaaIWaaabeaakiaadMhacaGGOaGaamOyaiaacMcacqGHRaWkcaWG% IbWaaSbaaSqaaiaaikdaaeqaaOGabmyEayaagaGaaiikaiaadkgaca% GGPaGaeyypa0JaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiqadMha% gaqbaiaacIcacaWGIbGaaiykaiabg2da9iaadMhadaWgaaWcbaGaaG% OmaaqabaGccaGGSaGaam4yamaaBaaaleaacaaIWaaabeaakiaadMha% caGGOaGaam4yaiaacMcacqGHRaWkcaWGJbWaaSbaaSqaaiaaigdaae% qaaOGabmyEayaafaGaaiikaiaadogacaGGPaGaey4kaSIaam4yamaa% BaaaleaacaaIYaaabeaakiqadMhagaGbaiaacIcacaWGJbGaaiykai% abgUcaRiqadogagaGeaiaacIcacaWGJbGaaiykaiabg2da9iaadMha% daWgaaWcbaGaaG4maaqabaaaaaa!7DF7![begin{gathered} a_0 y(a) + a_1 y'(a) + a_2 y'(a) + a_3 y'(a) = y_0 ,b_0 y(b) + b_2 y'(b) = y_1 hfill y'(b) = y_2 ,c_0 y(c) + c_1 y'(c) + c_2 y'(c) + c'(c) = y_3 hfill end{gathered} ] has not been dealt with in previous works.