Abstract
We consider general solution and the generalized Hyers-Ulam stability of an Euler-Lagrange quadratic functional equation <svg style="vertical-align:-3.56265pt;width:333.63751px;" id="M1" height="16.6625" version="1.1" viewBox="0 0 333.63751 16.6625" width="333.63751" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D453" d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19
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q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" /></g><g transform="matrix(.017,-0,0,-.017,37.307,12.162)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" /></g><g transform="matrix(.017,-0,0,-.017,51.059,12.162)"><path id="x1D460" d="M352 391q0 -31 -27 -44q-14 -7 -24 6q-39 48 -84 48q-23 0 -39.5 -15t-16.5 -40q0 -43 73 -90q49 -32 70 -58t21 -57q0 -58 -62 -105.5t-129 -47.5q-40 0 -75.5 25t-35.5 52q0 28 32 46q7 4 15 3t11 -6q19 -31 48.5 -50.5t54.5 -19.5q34 0 54 19.5t20 42.5q0 43 -65 81
q-97 56 -97 123q0 50 51 96q19 17 58 32.5t62 15.5q37 0 61 -18t24 -39z" /></g><g transform="matrix(.017,-0,0,-.017,57.433,12.162)"><path id="x1D466" d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26
q13 28 43 28q18 0 30 -15.5t12 -39.5z" /></g><g transform="matrix(.017,-0,0,-.017,67.276,12.162)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g><g transform="matrix(.017,-0,0,-.017,76.931,12.162)"><use xlink:href="#x2B"/></g><g transform="matrix(.017,-0,0,-.017,90.683,12.162)"><use xlink:href="#x1D45F"/></g><g transform="matrix(.017,-0,0,-.017,97.856,12.162)"><use xlink:href="#x1D460"/></g><g transform="matrix(.017,-0,0,-.017,104.231,12.162)"><use xlink:href="#x1D453"/></g><g transform="matrix(.017,-0,0,-.017,115.144,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,121.026,12.162)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,134.319,12.162)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,148.071,12.162)"><use xlink:href="#x1D466"/></g><g transform="matrix(.017,-0,0,-.017,157.913,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,168.52,12.162)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,183.207,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,189.089,12.162)"><use xlink:href="#x1D45F"/></g><g transform="matrix(.017,-0,0,-.017,200.053,12.162)"><use xlink:href="#x2B"/></g><g transform="matrix(.017,-0,0,-.017,213.805,12.162)"><use xlink:href="#x1D460"/></g><g transform="matrix(.017,-0,0,-.017,220.179,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,226.061,12.162)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,231.925,12.162)"><use xlink:href="#x1D45F"/></g><g transform="matrix(.017,-0,0,-.017,239.099,12.162)"><use xlink:href="#x1D453"/></g><g transform="matrix(.017,-0,0,-.017,250.012,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,255.894,12.162)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,265.396,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,275.051,12.162)"><use xlink:href="#x2B"/></g><g transform="matrix(.017,-0,0,-.017,288.803,12.162)"><use xlink:href="#x1D460"/></g><g transform="matrix(.017,-0,0,-.017,295.178,12.162)"><use xlink:href="#x1D453"/></g><g transform="matrix(.017,-0,0,-.017,306.091,12.162)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,311.973,12.162)"><use xlink:href="#x1D466"/></g><g transform="matrix(.017,-0,0,-.017,321.815,12.162)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,327.696,12.162)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g> </svg> in fuzzy Banach spaces, where <svg style="vertical-align:-0.1638pt;width:7.3000002px;" id="M2" height="7.9499998" version="1.1" viewBox="0 0 7.3000002 7.9499998" width="7.3000002" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45F"/></g> </svg>, <svg style="vertical-align:-0.1638pt;width:6.5px;" id="M3" height="7.9499998" version="1.1" viewBox="0 0 6.5 7.9499998" width="6.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D460"/></g> </svg> are nonzero rational numbers with <svg style="vertical-align:-1.35135pt;width:124.45px;" id="M4" height="17.525" version="1.1" viewBox="0 0 124.45 17.525" width="124.45" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><use xlink:href="#x1D45F"/></g> <g transform="matrix(.012,-0,0,-.012,7.238,7.612)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g> <g transform="matrix(.017,-0,0,-.017,17.35,15.775)"><use xlink:href="#x2B"/></g><g transform="matrix(.017,-0,0,-.017,31.102,15.775)"><use xlink:href="#x1D45F"/></g><g transform="matrix(.017,-0,0,-.017,38.275,15.775)"><use xlink:href="#x1D460"/></g><g transform="matrix(.017,-0,0,-.017,48.424,15.775)"><use xlink:href="#x2B"/></g><g transform="matrix(.017,-0,0,-.017,62.176,15.775)"><use xlink:href="#x1D460"/></g> <g transform="matrix(.012,-0,0,-.012,68.562,7.612)"><use xlink:href="#x32"/></g> <g transform="matrix(.017,-0,0,-.017,78.675,15.775)"><use xlink:href="#x2212"/></g><g transform="matrix(.017,-0,0,-.017,92.41,15.775)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,103.408,15.775)"><path id="x2260" d="M535 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q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g> </svg>, <svg style="vertical-align:-1.35135pt;width:55.012501px;" id="M5" height="12.6" version="1.1" viewBox="0 0 55.012501 12.6" width="55.012501" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45F"/></g><g transform="matrix(.017,-0,0,-.017,11.01,10.862)"><use xlink:href="#x2B"/></g><g transform="matrix(.017,-0,0,-.017,24.762,10.862)"><use xlink:href="#x1D460"/></g><g transform="matrix(.017,-0,0,-.017,33.975,10.862)"><use xlink:href="#x2260"/></g><g transform="matrix(.017,-0,0,-.017,46.775,10.862)"><use xlink:href="#x30"/></g> </svg>.
Highlights
The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms
For r + s ≠ 1, there exists a real number θ ≥ 0 such that a mapping f : X → Y with f(0) = 0 satisfies the inequality
In the theorem, we are going to consider a stability problem concerning the stability of (6) by using a fixed point theorem of the alternative for contraction mappings on generalized complete metric spaces due to Margolis and Diaz [25]
Summary
The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. A mapping f : X → Y between linear spaces satisfies the functional equation f (kx + ly) + f (kx − ly) = kl [f (x + y) + f (x − y)]
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