## 1 Introduction  
Let \(F:D\longrightarrow\mathbb{R}^{i}\) be a continuously differentiable operator; \(D\subseteq\mathbb{R}^{i}\) an open set, \(K\subset D,K\) a closed convex set, and \(H:D\rightrightarrows\mathbb{R}^{i}\) be a set-valued operator equipped with a nonempty closed graph. We shall study the generalized equation  
\[x\in K,0\in F(x)+H(x). \tag{1}\]  
We are interested in finding a solution \(x^{*}\) of the generalized equation, since many problems in nonlinear programming and other disciplines can be reduced to this equation using mathematical modeling.  
Wy utilize the Newton Inexact Projection Algorithm (NIPA) formally defined in [1] by:  
**Step a.**: Choose \(x_{0}\in K\) and let \(\{\tau_{m}\}\subset[0,\infty)\) be given, and set \(n=0\).
**Step b.**: If \(0\in F(x_{n})+H(x_{n})\), then terminate; otherwise, compute \(v_{n}\in\mathbb{R}^{i}\) so that  
\[0\in F(x_{n})+F^{\prime}(x_{n})(v_{n}-x_{n})+H(v_{n}).\]
**Step c.**: If \(v_{n}\in K\), let \(x_{n+1}=v_{n}\); otherwise choose any \(x_{n+1}\in K\) so that  
\[x_{n+1}\in P_{K}(v_{n},x_{n};\tau_{n}).\]
**Step d.**: Let \(n\gets n+1\), and repeat **Step a**.  
Here \(P_{K}(.,x,\tau):\mathbb{R}^{i}\rightrightarrows K\) is the operator given as  
\[P:=P_{K}(y_{1},x,\tau):=\{y_{2}\in K:\langle y_{1}-y_{2},y_{3}-y_{2}\rangle\leq \tau\|y_{1}-x\|^{2}\text{ for all }y_{3}\in K\}.\]  
Operator \(P\) is called a feasible inexact projection.  
The study of generalized equations was inaugurated by [PERSON] back in the 1970s [1, 2]. A plethora of results was produced ever since utilizing iterative algorithms since solutions in closed form are rarely attainable [1, 2, 3, 4, 5, 6, 7, 8]. It is known that if \(H\equiv\{0\}\), then (1) reduces to a constrained generalized equation [2, 3, 4, 7, 8, 9]. In these studies the superlinear and/or quadratic local convergence has been established under the condition of the metric regularity or strong metric regularity of the partial linearization of the function that defines a generalized equation. There is a surge in studies of this type since they provide an abstract model for several applications such as equilibrium, linear and nonlinear complementary problems, and variational inequality problems [1, 2, 5, 9]. A common feature in these studies is the realization that region of accessibility or more commonly called convergence ball is not large in general. Hence, the choice of initial points for the iterative algorithm that guarantee its convergence is limited. Moreover, the upper error estimations on the distances are pessimistic in most cases, so more than necessary iterations are carried out to achieve a certain error tolerance. Extending the uniqueness of the solution ball is also an important issue. Motivated by the elegant work by [PERSON] et al. [6], which generalized the earlier ones [2, 3, 4, 7], as well as the aforementioned problems, we develop a technique leading to smaller Lipschitz constants than before [6, 7, 8], and consequently to a finer ball convergence. It is important to notice that these extensions are obtained without additional conditions. In particular, our technique is given for the NIPA. But it can be used with the same advantages on other iterative algorithms.  
In Section 2 the ball convergence of NIPA is given, whereas the examples appears in Section 3.