Abstract
We consider the Hyers-Ulam stability problem of the generalized quadratic functional equation % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGaemyDauNaeSigI8MaemyqaeueduuDJXwAKbYu51My % VXgaiyaacqWFRaWkcqWG2bGDcqWIyiYBcqWGcbGqcqWFsislcqWFYa % GmcqWG3bWDcqWIyiYBcqWGqbaudaWgaaWcbaGae8xmaedabeaakiab % -jHiTiab-jdaYmaaBaaaleaaiyGacqGF6oWAaeqaaOGaeSigI8Maem % iuaa1aaSbaaSqaaiab-jdaYaqabaGccqWF9aqpcqWFWaamcqWFSaal % aaa!5886! $$ u \circ A + v \circ B - 2w \circ P_1 - 2_\kappa \circ P_2 = 0, $$ which is a distributional version of the classical generalized quadratic functional equation % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqipC0xg9qqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaGaemOzayweduuDJXwAKbYu51MyVXgaiyaacqWFOaak % cqWG4baEcqWFRaWkcqWG5bqEcqWFPaqkcqWFRaWkcqWGNbWzcqWFOa % akcqWG4baEcqWFsislcqWG5bqEcqWFPaqkcqWFsislcqWFYaGmcqWG % ObaAcqWFOaakcqWG4baEcqWFPaqkcqWFsislcqWFYaGmcqWGRbWAcq % WFOaakcqWG5bqEcqWFPaqkcqWF9aqpcqWFWaamcqWFUaGlaaa!5D5C! $$ f(x + y) + g(x - y) - 2h(x) - 2k(y) = 0. $$
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