Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2232 https://internationalpubls.com A New Insight with Trigonometric Coefficients of Additive-Quadratic Functional Equations and its Stability Analysis V. Vijayan 𝟏 P. Agilan πŸβˆ—, M. Sophia πŸ‘, N. Maheshkumar πŸ’ 1Department of Electronics and Instrumentation Engineering, St.Joseph’s College of Engineering, OMR, Chennai - 600 119, TamilNadu, India. E-mail:vinvpn@gmail.com 2Department of Mathematics, St.Joseph’s College of Engineering, OMR, Chennai - 600 119, TamilNadu, India. E-mail: agilram@gmail.com 3Department of Mathematics, SIMATS Engineering, Saveetha Nagar, Thandalam, Kanchipuram-Chennai Rd, Chennai- 602105, Tamil Nadu, India. E-mail: sophia.raj2005@gmail.com. 4Department of Science and Humanities, Faculty of Engineering, Karpagam Academy of Higher Education, Coimbatore-641021, TamilNadu, India. E-mail: maheshkumar.natarajan@kahedu.edu.in. βˆ—Corresponding author: agilram@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: This study introduces a novel framework for analyzing the Ulam-Hyers stability of mixed-type additive-quadratic functional equations with trigonometric constant coefficients in Banach spaces. Employing advanced analytical techniques and leveraging the unique properties of trigonometric functions, we derive sufficient conditions for the stability of these equations. The intricate relationship between additive and quadratic components is rigorously examined, emphasizing the pivotal role of trigonometric coefficients in influencing stability behavior. Our results provide fresh insights into the structural stability of functional equations and broaden the scope of existing stability theories. This work lays the groundwork for future research on mixed-type functional equations in both theoretical and applied mathematical contexts Keywords: Additive, quadratic functional equations, generalized Hyers - Ulam - Rassias stability 1. Introduction The study of functional equations and their stability has been a fundamental aspect of mathematical analysis for decades. The concept of stability in functional equations originated with StanisΕ‚aw Ulam in 1940 [1], who posed the question of whether an approximate solution to a functional equation could be approximated by an exact solution. In 1941, Donald Hyers [2] provided the first affirmative answer to Ulam’s question, establishing the stability of linear functional equations. This foundational result, now known as Ulam-Hyers stability, has since been generalized to a wide range of functional equations [3, 4, 5], including quadratic, cubic, and mixed-type equations. Mixed-type functional equations, which integrate distinct mathematical structures such as additive and quadratic components, have garnered significant attention due to their applications in fields like physics, economics, and engineering. The inclusion of trigonometric coefficients introduces additional complexity and depth to the analysis, as the inherent periodicity and symmetry of trigonometric functions play a crucial role in shaping stability properties. Despite their theoretical and practical significance, the stability of mixed-type functional equations with trigonometric coefficients remains a relatively underexplored area of research [6, 7, 8]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2233 https://internationalpubls.com The study of Ulam-Hyers stability has seen significant advancements, emerging as a crucial area of research in functional analysis and its applications [9, 10, 11, 12, 13]. Functional equations with mixed structures, such as additive-quadratic forms, present unique challenges and opportunities for mathematical investigation. These equations naturally arise in various fields, modeling systems where linear and nonlinear behaviors interact. This paper focuses on a novel class of mixed-type additive- quadratic functional equations featuring trigonometric constant coefficients. The incorporation of trigonometric terms introduces distinctive properties, making the stability analysis both complex and fascinating. Trigonometric coefficients are not only mathematically significant but also hold practical relevance in modeling periodic and oscillatory phenomena, such as wave functions and signal processing. The primary objective of this study is to establish the Ulam-Hyers stability of these functional equations within the framework of Banach spaces. By employing advanced techniques in functional analysis and leveraging the inherent properties of trigonometric coefficients, this work offers new insights into the stability of such equations. Furthermore, this research enhances the broader understanding of how trigonometric factors influence the stability of mixed-type functional equations, paving the way for further theoretical developments and practical applications. Recently, Agilan et al. have explored stability results for various additive functional equations across different normed spaces, as evidenced in [14, 15, 16, 17, 18, 19, 20, 21,22]. Through these motivations, the paper aims to deepen the understanding of the stability properties of functional equations in Banach spaces and to demonstrate the effectiveness of combining direct and fixed point methods in such analyses. Authors have proved the generalized Ulam - Hyers stability of a mixed type general additive quadratic functional equation 𝒬1 (π’³π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 ) + 2𝒴𝑠𝑒𝑐ℒ ( πœ‹ 4 )) + 𝒬1 (𝒴𝑠𝑒𝑐 β„’ ( πœ‹ 4 ) βˆ’ π’³π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) + 𝒬1 (π’³π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 ) βˆ’ 𝒴𝑠𝑒𝑐ℒ ( πœ‹ 4 )) = ( π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )+3π‘π‘œπ‘ π‘’π‘2β„’( πœ‹ 4 ) 2 )𝒬1(𝒳) + ( 3π‘π‘œπ‘ π‘’π‘2β„’( πœ‹ 4 )βˆ’π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 ) 2 )𝒬1(βˆ’π’³) + (𝑠𝑒𝑐ℒ ( πœ‹ 4 ) + 3𝑠𝑒𝑐2β„’ ( πœ‹ 4 ))𝒬1(𝒴) + (3𝑠𝑒𝑐 2β„’ ( πœ‹ 4 ) βˆ’ 𝑠𝑒𝑐ℒ ( πœ‹ 4 )) 𝒬1(βˆ’π’΄) (1) in Banach spaces. let us consider 𝒳 and 𝒴 to be a normed space and a Banach space, respectively. Define a mapping 𝒬:β„‹ β†’ ℐ by 𝒬(𝒳, 𝒴) = 𝒬1 (π’³π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 ) + 2𝒴𝑠𝑒𝑐ℒ ( πœ‹ 4 )) + 𝒬1 (𝒴𝑠𝑒𝑐 β„’ ( πœ‹ 4 ) βˆ’ π’³π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) +𝒬1 (π’³π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 ) βˆ’ 𝒴𝑠𝑒𝑐ℒ ( πœ‹ 4 )) βˆ’ ( π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )+3π‘π‘œπ‘ π‘’π‘2β„’( πœ‹ 4 ) 2 )𝒬1(𝒳) βˆ’ ( 3π‘π‘œπ‘ π‘’π‘2β„’( πœ‹ 4 )βˆ’π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 ) 2 )𝒬1(βˆ’π’³) βˆ’ (𝑠𝑒𝑐ℒ ( πœ‹ 4 ) + 3𝑠𝑒𝑐2β„’ ( πœ‹ 4 )) 𝒬1(𝒴) βˆ’ (3𝑠𝑒𝑐 2β„’ ( πœ‹ 4 ) βˆ’ 𝑠𝑒𝑐ℒ ( πœ‹ 4 ))𝒬1(βˆ’π’΄) for all 𝒳,𝒴 ∈ β„‹. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2234 https://internationalpubls.com 2. Stability Results: Odd case Theorem 2.1 Let 𝒯:𝒳2 β†’ [0,∞) be a function such that βˆ‘βˆžπ’°=0 𝒯((cosecβ„’( Ο€ 4 )) 𝒰 𝒳,(cosecβ„’( Ο€ 4 )) 𝒰 𝒴) (cosecβ„’( Ο€ 4 )) 𝒰 converges in β„› and lim π’°β†’βˆž 𝒯((cosecβ„’( Ο€ 4 )) 𝒰 𝒳,(cosecβ„’( Ο€ 4 )) 𝒰 𝒴) (cosecβ„’( Ο€ 4 )) 𝒰 = 0 (1) for all 𝒳,𝒴 ∈ β„‹. Let π’¬π‘Ž:β„‹ β†’ ℐ be an odd function satisfying the inequality β€–π’¬π‘Ž(𝒳, 𝒴)β€– ≀ 𝒯(𝒳,𝒴) (2) for all 𝒳,𝒴 ∈ β„‹. Then there exists a unique additive mapping 𝐴:β„‹ β†’ ℐ such that β€–π’¬π‘Ž(𝒳) βˆ’ 𝐴(𝒳)β€– ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) βˆ‘βˆžβ„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ (3) for all 𝒳 ∈ β„‹. The mapping 𝐴(𝒳) is defined by 𝐴(𝒳) = lim π’°β†’βˆž π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝒰 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝒰 (4) for all 𝒳 ∈ β„‹. Proof. Replacing (𝒳,𝒴) by (𝒳, 0) in (2) and using oddness of π’¬π‘Ž, we get β€–π’¬π‘Ž(𝒳) βˆ’ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 ))𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β€– ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝒯(𝒳, 0) (5) for all𝒳 ∈ β„‹. Now replacing 𝒳 by (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )𝒳) and dividing by (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) in (5), we obtain β€– π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 ))𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) βˆ’ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 2 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 β€– ≀ 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 ))𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 (6) for all𝒳 ∈ β„‹. It follows from (5) and (6) that β€–π’¬π‘Ž(𝒳) βˆ’ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 2 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 β€– ≀ β€–π’¬π‘Ž(𝒳) βˆ’ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 ))𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β€– (7) + β€– π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 ))𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) βˆ’ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 2 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 β€– ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) [𝒯(𝒳, 0) + 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 ))𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) ] (8) for all𝒳 ∈ β„‹. In general for any positive integer 𝑁 , we get β€–π’¬π‘Ž(𝒳) βˆ’ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑁 β€– ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) βˆ‘π’°βˆ’1β„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2235 https://internationalpubls.com ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) βˆ‘βˆžβ„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ (9) for all𝒳 ∈ β„‹. In order to prove the convergence of the sequence { π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑁 }, replace 𝒳 by (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑀 𝒳 and divide by (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑀 in (1), for any 𝑀,𝑁 > 0 , to deduce β€– π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑀 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑀 βˆ’ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁+𝑀 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) (𝑁+𝑀) β€– = 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑀 β€–π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑀 𝒳) βˆ’ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁 β‹…(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑀 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑁 β€– ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) βˆ‘π’°βˆ’1β„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹+𝑀 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹+𝑀 ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) βˆ‘βˆžβ„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹+𝑀 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹+𝑀 β†’ 0 π‘Žπ‘  𝑀 β†’ ∞ for all 𝒳 ∈ β„‹. Hence the sequence { π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑁 } is a Cauchy sequence. Since ℐ is complete, there exists a mapping 𝐴:β„‹ β†’ ℐ such that 𝐴(𝒳) = lim π’°β†’βˆž π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑁 βˆ€ 𝒳 ∈ β„‹. Letting 𝑁 β†’ ∞ in (1) we see that (3) holds for all𝒳 ∈ β„‹. To prove 𝐴 satisfies (1), replacing (𝒳,𝒴) by ((π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒳, (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒴) and dividing by (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 in (2), we obtain 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑁 βˆ₯ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳, (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒴) βˆ₯ ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑁𝒯((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳, (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒴) for all 𝒳,𝒴 ∈ β„‹. Letting 𝑁 β†’ ∞ in the above inequality and using the definition of 𝐴(𝒳), we see that π’¬π‘Ž(𝒳,𝒴) = 0. Hence 𝐴 satisfies (1) for all 𝒳,𝒴 ∈ β„‹. To show 𝐴 is unique, let 𝐡(𝒳) be another additive mapping satisfying (1) and (3), then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2236 https://internationalpubls.com β€–A(𝒳) βˆ’ 𝐡(𝒳)β€– = 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑁‖𝐴((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳) βˆ’ 𝐡((π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒳)β€– ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑁 {‖𝐴((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳) βˆ’ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳)β€– + β€–π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳) βˆ’ 𝐡((π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒳)β€–} ≀ 2 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) βˆ‘βˆžβ„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹+𝑁 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) (β„‹+𝑁) β†’ 0 π‘Žπ‘  𝑁 β†’ ∞ for all 𝒳 ∈ β„‹. Hence 𝐴 is unique. Corollary 2.2 Let 𝒯 and P be non negative real numbers. Let an odd function 𝒬a:β„‹ β†’ ℐ satisfy the inequality ‖𝒬a(𝒳, 𝒴)β€– ≀ { 𝒯, 𝒯{||𝒳||P + ||𝒴||P}, P β‰  1; 𝒯{||𝒳||P||𝒴||P + {||𝒳||2P + ||𝒴||2P}}, P β‰  1 2 ; (10) for all 𝒳,𝒴 ∈ β„‹. Then there exists a unique additive function A:β„‹ β†’ ℐ such that ‖𝒬1(𝒳) βˆ’ A(𝒳)β€– ≀ { 𝒯 (cosecβ„’( Ο€ 4 ))βˆ’1 , 𝒯||𝒳||P |(cosecβ„’( Ο€ 4 ))βˆ’(cosecβ„’( Ο€ 4 )) P | , 𝒯||𝒳||2P |(cosecβ„’( Ο€ 4 ))βˆ’(cosecβ„’( Ο€ 4 )) 2P | , (11) for all 𝒳 ∈ β„‹ 3.Stability Results: Even Case Theorem 3.1 Let 𝒯: X2 β†’ [0,∞) be a function such that βˆ‘βˆžπ’°=0 𝒯((cosecβ„’( Ο€ 4 )) 𝒰 𝒳,(cosecβ„’( Ο€ 4 )) 𝒰 𝒴) (cosecβ„’( Ο€ 4 )) 2N converges in β„› and lim π’°β†’βˆž 𝒯((cosecβ„’( Ο€ 4 )) 𝒰 𝒳,(cosecβ„’( Ο€ 4 )) 𝒰 𝒴) (cosecβ„’( Ο€ 4 )) 2N = 0 (12) for all 𝒳,𝒴 ∈ β„‹. Let π’¬π‘ž:β„‹ β†’ ℐ be an even function satisfying the inequality ‖𝑄(𝒳,𝒴)β€– ≀ 𝒯(𝒳,𝒴) (13) for all 𝒳,𝒴 ∈ β„‹. Then there exists a unique Quadratic mapping 𝑄:β„‹ β†’ ℐ such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2237 https://internationalpubls.com β€–π’¬π‘ž(𝒳) βˆ’ 𝑄(𝒳)β€– ≀ 1 3(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 βˆ‘ ∞ β„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2β„‹ (14) for all 𝒳 ∈ β„‹. The mapping 𝐴(𝒳) is defined by 𝑄(𝒳) = lim π’°β†’βˆž π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝒰 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑁 (15) for all 𝒳 ∈ β„‹. Proof. Replacing (𝒳,𝒴) by (𝒳, 0) in (13) and using evenness of π’¬π‘ž, we get β€–π’¬π‘ž(𝒳) βˆ’ π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 ))𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 β€– ≀ 1 3(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝒯(𝒳, 0) (16) for all𝒳 ∈ β„‹. Now replacing 𝒳 by (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 ))𝒳 and dividing by (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 2 in (16), we obtain β€– π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 ))𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 βˆ’ π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 2 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 4 β€– ≀ 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 ))𝒳,0) 3(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 4 (17) for all𝒳 ∈ β„‹. It follows from (16) and (17) that β€–π’¬π‘ž(𝒳) βˆ’ π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 2 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 4 β€– ≀ β€–π’¬π‘ž(𝒳) βˆ’ π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 ))𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 β€– + β€– π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 ))𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 βˆ’ π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 2 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 4 β€– ≀ 1 3(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 [𝒯(𝒳, 0) + 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 ))𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 ] (18) for all𝒳 ∈ β„‹. In general for any positive integer 𝑁 , we get β€–π’¬π‘ž(𝒳) βˆ’ π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑁 β€– ≀ 1 3(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2βˆ‘ π’°βˆ’1 β„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2β„‹ ≀ 1 3(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2βˆ‘ ∞ β„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2β„‹ (19) for all𝒳 ∈ β„‹. In order to prove the convergence of the sequence Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2238 https://internationalpubls.com { π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑁 }, replace 𝒳 by (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 2𝑀 𝒳 and divide by (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 2𝑀 in (1), for any 𝑀,𝑁 > 0 , to deduce β€– π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑀 π‘₯) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑀 βˆ’ π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁+𝑀 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2(𝑁+𝑀) β€– = 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑀 β€–π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑀 𝒳) βˆ’ π’¬π‘Ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁 β‹…(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑀 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑁 β€– ≀ 1 3(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2βˆ‘ π’°βˆ’1 β„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹+𝑀 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2(β„‹+𝑀) ≀ 1 3(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2βˆ‘ ∞ β„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹+𝑀 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2(β„‹+𝑀) β†’ 0 π‘Žπ‘  𝑀 β†’ ∞ for all 𝒳 ∈ β„‹. Hence the sequence { π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑁 } is a Cauchy sequence. Since ℐ is complete, there exists a mapping 𝑄:β„‹ β†’ ℐ such that π’¬π‘ž(𝒳) = lim π’°β†’βˆž π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’( πœ‹ 4 )) 𝑁 𝒳) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑁 βˆ€ 𝒳 ∈ β„‹. Letting 𝑁 β†’ ∞ in (1) we see that (14) holds for all𝒳 ∈ β„‹. To prove 𝑄 satisfies (1), replacing (𝒳,𝒴) by ((π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒳, (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒴) and dividing by (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 2𝑁 in (13), we obtain 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑁 βˆ₯ π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳, (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒴) βˆ₯ ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑁𝒯((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳, (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒴) for all 𝒳,𝒴 ∈ β„‹. Letting 𝑁 β†’ ∞ in the above inequality and using the definition of 𝑄(𝒳), we see that π’¬π‘ž(𝒳,𝒴) = 0. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2239 https://internationalpubls.com Hence 𝑄 satisfies (1) for all 𝒳,𝒴 ∈ β„‹. To show 𝑄 is unique, let 𝐡(𝒳) be another quadratic mapping satisfying (1) and (14), then ‖𝐴(𝒳) βˆ’ 𝐡(𝒳)β€– = 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑁‖𝐴((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳) βˆ’ 𝐡((π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒳)β€– ≀ 1 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑁 {‖𝐴((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳) βˆ’ π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳)β€– + β€–π’¬π‘ž((π‘π‘œπ‘ π‘’π‘ β„’ ( πœ‹ 4 )) 𝑁 𝒳) βˆ’ 𝐡((π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) 𝑁 𝒳)β€–} ≀ 2 3(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2βˆ‘ ∞ β„‹=0 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹+𝑁 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2(β„‹+𝑁) β†’ 0 π‘Žπ‘  𝑁 β†’ ∞ for all 𝒳 ∈ β„‹. Hence 𝑄 is unique. Corollary 3.2 Let 𝒯 and P be non negative real numbers. Let an even function 𝒬q:β„‹ β†’ ℐ satisfy the inequality ‖𝑄(𝒳,𝒴)β€– ≀ { 𝒯, 𝒯{||𝒳||P + ||𝒴||P}, P β‰  2; 𝒯{||𝒳||P||𝒴||P + {||𝒳||2P + ||𝒴||2P}}, P β‰  1; (20) for all 𝒳,𝒴 ∈ β„‹. Then there exists a unique quadratic function Q:β„‹ β†’ ℐ such that ‖𝒬1(𝒳) βˆ’ Q(𝒳)β€– ≀ { 𝒯 3((cosecβ„’( Ο€ 4 )) 2 βˆ’1) , 𝒯||𝒳||P 3|(cosecβ„’( Ο€ 4 )) 2 βˆ’(cosecβ„’( Ο€ 4 )) P | , 𝒯||𝒳||2P 3|(cosecβ„’( Ο€ 4 )) 2 βˆ’(cosecβ„’( Ο€ 4 )) 2P | , (21) for all 𝒳 ∈ β„‹. 4 Stability Results: Mixed Case Theorem 4.1 Let 𝒯: X2 β†’ [0,∞) be a function satisfying (2.1) and (3.1) for all 𝒳,𝒴 ∈ β„‹. Let 𝒬:β„‹ β†’ ℐ be a function satisfying the inequality ‖𝒬(𝒳, 𝒴)β€– ≀ 𝒯(𝒳,𝒴) (22) for all 𝒳,𝒴 ∈ β„‹. Then there exists a unique additive mapping A:β„‹ β†’ ℐ and a unique quadratic mapping Q:β„‹ β†’ ℐ such that Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2240 https://internationalpubls.com ‖𝒬1(𝒳) βˆ’ 𝐴(𝒳) βˆ’ 𝑄(𝒳)β€– ≀ 1 2 [ 1 (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4) ) βˆ‘ ∞ β„‹=0 ( 𝒯((π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4) ) β„‹ 𝒳, 0 (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) β„‹ + 𝒯(βˆ’(π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4) ) β„‹ 𝒳, 0) (π‘π‘œπ‘ π‘’π‘β„’ ( πœ‹ 4 )) β„‹ ) + 1 3(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2βˆ‘ ∞ β„‹=0 ( 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2β„‹ + 𝒯(βˆ’(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2β„‹ )] (23) for all 𝒳 ∈ β„‹. The mapping 𝐴(𝒳) and 𝑄(𝒳) are defined in (4) and (15) respectively for all 𝒳 ∈ β„‹. Proof. Let π’¬π‘œ(𝒳) = π’¬π‘Ž(𝒳)βˆ’π’¬π‘Ž(βˆ’π’³) 2 for all𝒳 ∈ β„‹. Then π’¬π‘œ(0) = 0 and π’¬π‘œ(βˆ’π’³) = βˆ’π’¬π‘œ(𝒳) for all𝒳 ∈ β„‹. Hence β€–π’¬π‘œ(𝒳,𝒴)β€– ≀ 𝒯(𝒳,𝒴) 2 + 𝒯(βˆ’π’³,βˆ’π’΄) 2 (24) for all 𝒳,𝒳 ∈ β„‹. By Theorem 2.1, we have β€–π’¬π‘œ(𝒳) βˆ’ 𝐴(𝒳)β€– ≀ 1 2(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) βˆ‘βˆžβ„‹=0 ( 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ + 𝒯(βˆ’(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ ) (25) for all 𝒳 ∈ β„‹. Also, let 𝒬𝑒(𝒳) = π’¬π‘ž(𝒳)+π’¬π‘ž(βˆ’π’³) 2 for all𝒳 ∈ β„‹. Then 𝒬𝑒(0) = 0 and 𝒬𝑒(βˆ’π’³) = 𝒬𝑒(𝒳) for all𝒳 ∈ β„‹. Hence ‖𝒬𝑒(𝒳,𝒴)β€– ≀ 𝒯(𝒳,𝒴) 2 + 𝒯(βˆ’π’³,βˆ’π’΄) 2 (26) for all 𝒳,𝒴 ∈ β„‹. By Theorem 3.1, we have ‖𝒬𝑒(𝒳) βˆ’ 𝑄(𝒳)β€– ≀ 1 6(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2βˆ‘ ∞ β„‹=0 ( 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) ((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2β„‹ + 𝒯(βˆ’(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2β„‹ ) (27) for all 𝒳 ∈ β„‹. Define 𝒬(𝒳) = 𝒬𝑒(𝒳) + π’¬π‘œ(𝒳) (28) for all 𝒳 ∈ β„‹. From (25),(27) and (28), we arrive ‖𝒬1(𝒳) βˆ’ 𝐴(𝒳) βˆ’ 𝑄(𝒳)β€– = ‖𝒬𝑒(𝒳) + π’¬π‘œ(𝒳) βˆ’ 𝐴(𝒳) βˆ’ 𝑄(𝒳)β€– ≀ β€–π’¬π‘œ(𝒳) βˆ’ 𝐴(𝒳)β€– + ‖𝒬𝑒(𝒳) βˆ’ 𝑄(𝒳)β€– ≀ 1 2(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) βˆ‘βˆžβ„‹=0 ( 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ + 𝒯(βˆ’(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2241 https://internationalpubls.com + 1 6(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2βˆ‘ ∞ β„‹=0 ( 𝒯((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2β„‹ + 𝒯(βˆ’(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) β„‹ 𝒳,0) (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2β„‹ ) for all 𝒳 ∈ β„‹ Corollary 4.2 Let 𝒯 and P be non negative real numbers. Let a function 𝒬:β„‹ β†’ ℐ satisfy the inequality ‖𝒬(𝒳, 𝒴)β€– ≀ { 𝒯, 𝒯{||𝒳||P + ||𝒴||P}, P β‰  1,2; 𝒯{||𝒳||P||𝒴||P + {||𝒳||2P + ||𝒴||2P}}, P β‰  1 2 , 1; (29) for all 𝒳,𝒴 ∈ β„‹. Then there exists a unique additive function A:β„‹ β†’ ℐ and a unique quadratic function Q:β„‹ β†’ ℐ such that ‖𝒬1(𝒳) βˆ’ 𝐴(𝒳) βˆ’ 𝒬(𝒳)β€– ≀ { ( 𝒯 (π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 ))βˆ’1 ) + ( 𝒯 ((π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 βˆ’1) ) , [ 1 |(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 ))βˆ’(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑃 | + 1 3|(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 βˆ’(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 𝑃 | ] ||𝒳||𝑃, [ 1 |(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 ))βˆ’(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑃 | + 1 3|(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2 βˆ’(π‘π‘œπ‘ π‘’π‘β„’( πœ‹ 4 )) 2𝑃 | ] ||𝒳||2𝑃 (30) for all 𝒳 ∈ β„‹ Refrences [1] S.M. Ulam, Problems in Modern Mathematics, Science Editions,Wiley, NewYork, 1964 (Chapter VI, Some Questions in Analysis: 1, Stability). [2] D.H. Hyers, On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A.,27 (1941) 222-224. [3] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. 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