YENİ QUADRATİC FONKSİYONEL DENKLEM VE BU DENKLEMİN HYERS ULAM RASSIAS KARARLILIĞI

Diferansiyel denklemlerin kararlılığında asıl mesele bir diferansiyel denklemi yaklaşık olarak sağlayan bir dönüşümün denklemin tam çözümüne yaklaşması ne zaman gerçek olur sorusuna cevap verilmesidir. Bu nedenle diferansiyel denklemlerin Hyers-Ulam ve Hyers Ulam Rassias kararlılığı esastır. Bu günlerde araştırmacılar diferansiyel denklemlerin Hyers-Ulam ve Hyers-Ulam Rassias kararlılığını araştırmak için çeşitli metotlar (açık dönüşüm, direkt metot, integral çarpanı, sabit nokta metodu) kullanmaktadır. Direkt metot birçok farklı fonksiyonel diferansiyel denklemlerin HyersUlam Rassias kararlılığını araştırmak için başarılı bir şekilde uygulanmaktadır. Fakat bu metot bazı önemli durumlar için yeterli değildir. İkinci en popüler metot sabit nokta metodudur. Bu çalışmada direkt metot ve sabit nokta metodunu kullanarakg(    )  g(    )  4g()  g(  )  g(   2 )  g(  )  g(  ) şeklindeki yeni bir quadratic tipten fonksiyonel denklemin Hyers-Ulam Rassias kararlılığını belirlemek için girişimde bulunduk. Bu araştırmanın quadratic fonksiyonel denklemlerin Hyers Ulam kararlılığı üzerine çalışan yazarlara fayda sağlayabileceğini ve ilgili literatüre katkı sağlayacağını düşünüyoruz

NEW QUADRATIC FUNCTIONAL EQUATION AND ITS (HURS)

The primary subject in the stability of differential equations is to answer the question of when is it real that a mappingwhich roundly satisfies a differential equation must be close to an exact solution of the equation. For this reason, theHyers-Ulam and Hyers-Ulam Rassias stability of differential equations is fundemantal. Currently, researchers have usedvarious methods (open mapping, direct method, integral factor, fixed point method) to research that the Hyers-UlamRassias and Hyers-Ulam stability of differential equations. The direct method has been succesfully apllied for investigateof the Hyers-Ulam Rassias stability of many different functional differential equations. But it does not enough for someimportant cases. The second most popular method is the fixed point method.In this study, we make an attemp to establish the Hyers-Ulam Rassias stability (HURS) of a new quadratic type functionalequation (QFE) g(x+y+v+ç ) + g(x-y-v-ç ) = 4g(x) + g(y+ç ) +g(y+ç+2v ) - g(x-v ) - g(x+v )by direct method and fixed point method. We consider that this research will contribute to the related literature and itmay be useful for authors studying on the Hyers-Ulam Stability of the quadratic functional differential equations.

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[1] Skof, F. "Proprieta locali e approssimazione di operatori", Rend. Sem. Mat. Fis. Milano, 53. No., 113- 129, 1983.
[2] Cholewa, P.W. ‘‘Remarks on the stability of functional equations’’, Aequationes Math., 27., 76−86, 1984.
[3] Czerwik, S. Functional Equations and Inequalities in Several Variables, World Scientific, Singapore, 2002.
[4] Jung, S. M. ‘’On the Hyers-Ulam-Rassias stability of a quadratic functional equation’’, J. Math. Anal. Appl., 232., 384-393, 1999.
[5] Jung, S. M. ‘’On the Hyers-Ulam stability of the functional equations that have the quadratic property’’, J. Math. Anal. Appl., 222., 126-137, 1998. [6] Lee, Y. W. ‘’On the stability of a quadratic Jensen type functional equation’’, J. Math. Anal. Appl., 270., 590-601, 2002.
[7] Park, C. and Rassias, T.M. ‘’Fixed Points and generalized Hyers-Ulam Stability of Quadratic Functional Equations’’, Journal of Mathematical Inequalities, 1.4., 515-528, 2007.
[8] Aoki, T. "On the stability of the linear transformation in Banach spaces’’, J. Math. Soc. Japan, 2. 64-66, 1950.
[9] Biçer, E. and Tunç, C. ‘’On the Hyers-Ulam stability of certain partial differential equations of second order’’, Nonlinear Dyn. Syst. Theory. 17. 2., 150-157, 2017.
[10] Biçer, E. and Tunç, C. ‘’On the Hyers -Ulam stability of Laguerre and Bessel equations by Laplace transform method’’, Nonlinear Dyn. Syst. Theory, 17. 4., 340-346, 2017.
[11] Biçer, E. and Tunç, C. ‘’New Theorems for HyersUlam Stability of Lienard Equation with Variable Time Lags’’, International Journal of Mathematics and Computer Science, 13. 2., 231-242, 2018.
[12] Kenary, H. A., Rezaei, H., Gheisari, Y., Park, C. ‘’On the stability of set-valued functional equations with the fixed point alternative’’, Fixed Point Theory Appl., 17 pages 2012.
[13] Diaz, J. B. and Margolis, B. ‘’A fixed point theorem of the alternative for contractions on a generalized complete metric space’’, Bull. Amer. Math. Soc., 74. 305-309, 1968.
[14] Shen, H.Y. and Lan, Y. Y. ‘’On the general solution of a quadratic functional equation and its Ulam stability in various abstract spaces’’, J. Nonlinear Sci. Appl., 7., 368-378, 2014.
[15] Ulam, S.M. Problems in Modern Mathematics, Wiley, New York, 1964.
[16] Tunç, C. and Biçer, E. ‘’Hyers-Ulam stability of non-homogeneous Euler equations of third and fourth order’’, Scientific Research and Essays. 8.5., 220-226, 2013.
[17] Tunç, C. and Biçer, E. ‘’Hyers-Ulam-Rassias stability for a first order functional differential equation’’, J. Math. Fundam. Sci. 47. 143-153, 2015.
[18] Rassias, T. M. ‘’On the Stability of the Linear Mapping in Banach Spaces’’, Proc. Amer. Math. Soc., 72., 297-300, 1978.
[19] Hyers, D. H. ‘’On the Stability of the Linear Functional Equation’’, Proc. Nat. Acad. Sci. 27., 222- 224, 1941.
[20] Başcı, Y., O g rekçı̇, S., Mısır, A. ‘‘On Hyers-Ulam stability for fractional differential equations including the new Caputo Fabrizio fractional derivative’’, Mediterranean Journal of Mathematics 16: 130-144, 2019.
[21] Başcı, Y., Mısır, A., O g rekçı̇, S. ‘‘On the stability problem of differential equations in the sense of Ulam’’, Results in Mathematics 75, 2020.