This paper addresses the stability of linear mappings in Banach spaces. The problem was posed by S. M. Ulam, asking for conditions under which a linear mapping near an approximately linear mapping exists. The author provides a solution to this problem. The main theorem states that if a mapping $ f: E_1 \to E_2 $ between Banach spaces satisfies a certain inequality involving a parameter $ p \in [0,1) $ and a constant $ \theta \geq 0 $, then there exists a unique linear mapping $ T: E_1 \to E_2 $ that approximates $ f $ within a specific bound. The inequality is: $$ \frac{\|f(x+y)-f(x)-f(y)\|}{\|x\|^{p}+\|y\|^{p}}\leqslant\theta $$ for all $ x, y \in E_1 $. The theorem shows that $ T $ is linear and satisfies: $$ \frac{\|f(x)-T(x)\|}{\|x\|^{p}}\leqslant\frac{2\theta}{2-2^{p}} $$ for all $ x \in E_1 $. The proof involves showing that the sequence $ \frac{f(2^n x)}{2^n} $ converges to a linear mapping $ T $. It also demonstrates that $ T $ is unique. The result generalizes a previous result by D. H. Hyers for the case $ p = 0 $. The paper concludes that the solution to Ulam's problem is established, providing a generalized solution for the stability of linear mappings in Banach spaces. The author thanks D. H. Hyers for his comments on an earlier version of the manuscript.This paper addresses the stability of linear mappings in Banach spaces. The problem was posed by S. M. Ulam, asking for conditions under which a linear mapping near an approximately linear mapping exists. The author provides a solution to this problem. The main theorem states that if a mapping $ f: E_1 \to E_2 $ between Banach spaces satisfies a certain inequality involving a parameter $ p \in [0,1) $ and a constant $ \theta \geq 0 $, then there exists a unique linear mapping $ T: E_1 \to E_2 $ that approximates $ f $ within a specific bound. The inequality is: $$ \frac{\|f(x+y)-f(x)-f(y)\|}{\|x\|^{p}+\|y\|^{p}}\leqslant\theta $$ for all $ x, y \in E_1 $. The theorem shows that $ T $ is linear and satisfies: $$ \frac{\|f(x)-T(x)\|}{\|x\|^{p}}\leqslant\frac{2\theta}{2-2^{p}} $$ for all $ x \in E_1 $. The proof involves showing that the sequence $ \frac{f(2^n x)}{2^n} $ converges to a linear mapping $ T $. It also demonstrates that $ T $ is unique. The result generalizes a previous result by D. H. Hyers for the case $ p = 0 $. The paper concludes that the solution to Ulam's problem is established, providing a generalized solution for the stability of linear mappings in Banach spaces. The author thanks D. H. Hyers for his comments on an earlier version of the manuscript.