## 2 Formulation and Solution of Nonlinear VFIDEs  
First, we formulate the nonlocal boundary problem (3), (4) in a convenient operator form. Let \(X=C[a,b]\), \(a,b\in\mathbb{R}\), and let the operators \(A,K:X\to X\) defined as  
\[Au = \sum_{i=0}^{n}a_{i}u^{(n-i)}(x), \tag{5}\] \[Ku = \sum_{i=0}^{n}\int_{a}^{x}k_{i}(x,t)u^{(n-i)}(t)dt, \tag{6}\]  
be an \(n\)th-order linear differential operator and a linear Volterra-type integral operator, respectively, where \(n\in\mathbb{N}\), \(a_{0}\
eq 0\), \(a_{i}\), \(i=1,2,\ldots,n\), are real constants; \(u(x)\in C^{n}[a,b]\); and kernel functions \(k_{i}(x,t)\in X\times X\), \(i=0,1,\ldots,n\). Furthermore, let the nonlinear Fredholm.type integral operator \(\bar{K}:X\to X\) be defined as  
\[\bar{K}u=\sum_{j=1}^{m}\int_{a}^{b}q_{j}(x,t)\varphi_{j}\Big{(}t,u(t),u^{ \prime}(t),\ldots,u^{(n)}(t)\Big{)}dt, \tag{7}\]  
where \(q_{j}(x,t)\), \(\varphi_{j}(\cdot)\), \(j=1,2,\ldots,m\), are continuous functions. We assume that kernels \(q_{j}(x,t)\) are degenerate, i.e., without loss of generality of the form  
\[q_{j}(x,t)=g_{j}(x)h_{j}(t),\quad j=1,2,\ldots,m,\]  
where \(g_{j}(x)\), \(h_{j}(t)\in X\). In this case, Equation (7) becomes  
\[\bar{K}u=\sum_{j=1}^{m}g_{j}(x)\int_{a}^{b}h_{j}(t)\varphi_{j}\Big{(}t,u(t),u ^{\prime}(t),\ldots,u^{(n)}(t)\Big{)}dt. \tag{8}\]  
By defining the row vector of functions  
\[g=\big{(} \begin{array}{ccccc}g_{1}&g_{2}&\cdots&g_{m}\end{array} \big{)}, \quad g_{j}=g_{j}(x),\quad j=1,2,\ldots,m,\]  
and the column vector of functionals\[\Psi(u)=\left( \begin{array}{c}\Psi_{1}(u)\\ \Psi_{2}(u)\\ \vdots\\ \Psi_{m}(u)\end{array} \right),\quad\Psi_{j}(u)=\int_{a}^{b}h_{j}(t)\varphi_{j} \Big{(}t,u(t),u^{\prime}(t),\ldots,u^{(n)}(t)\Big{)}dt, \tag{9}\]  
where \(\Psi_{j}\), \(j=1,2,\ldots,m\), are nonlinear functionals defined on \(X\), Equation (8) is written compactly as  
\[\bar{K}u=\sum_{j=1}^{m}g_{j}\Psi_{j}(u)=g\Psi(u). \tag{10}\]  
Let the column vectors of linear bounded functionals  
\[\Phi(u)=\left( \begin{array}{c}\Phi_{1}(u)\\ \Phi_{2}(u)\\ \vdots\\ \Phi_{n}(u)\end{array} \right),\quad\Theta(u)=\left( \begin{array}{c}\Theta_ {1}(u)\\ \Theta_{2}(u)\\ \vdots\\ \Theta_{l}(u)\end{array} \right),\]  
and the \(m\times\ell\) constant matrix  
\[\mathbf{N}=\left( \begin{array}{cccc}v_{11}&v_{12}&\cdots&v_{1l}\\ v_{21}&v_{22}&\cdots&v_{2l}\\ \vdots&\vdots&\ddots&\vdots\\ v_{n1}&v_{n2}&\cdots&v_{nl}\end{array} \right),\]  
where \(v_{ij}\), \(i=1,2,\ldots,n\), \(j=1,2,\ldots,l\), are real constants, and write the nonlocal boundary conditions in the matrix form  
\[\Phi(u)-\mathbf{N}\Theta(u)=\mathbf{0}, \tag{11}\]  
where \(\mathbf{0}\) denotes the zero column vector. By employing (5), (6) and (11) define the linear Volterra-type integro-differential operator \(V:X\to X\),  
\[ \begin{array}{rcl}Vu&=&Au-Ku,\\ \mathcal{D}(V)&=&\{u:u\in C^{n}[a,b],\;\Phi(u)-\mathbf{N}\Theta(u)=\mathbf{0} \},\end{array}  \tag{12}\]  
and then, by means of (10), the nonlinear Volterra-Fredholm-type integro-differential operator \(T:X\to X\),  
\[Tu=Vu-g\Psi(u),\quad\mathcal{D}(T)=\mathcal{D}(V).\]  
Thus, the boundary value problem (3), (4) can be written in the operator form  
\[Tu=f,\quad f\in X.\]  
We now state the following theorem. Theorem 1: _If linear operator \(V\) is bijective on \(X\) and its inverse is denoted by \(V^{-1}\), then the exact solution to the problem \(Tu=f\) for all \(f\in X\) is given by_  
\[u=V^{-1}f+V^{-1}g\mathbf{c}^{*}, \tag{13}\]  
_for every vector \(\mathbf{c}^{*}=\Psi(u)\) that is a solution of the nonlinear algebraic (transcendental) system_  
\[\mathbf{c}-\Psi\Big{(}V^{-1}f+V^{-1}g\mathbf{c}\Big{)}=\mathbf{0}. \tag{14}\]Proof.