Using Kakutani type fixed point theorems and monotone operator theory in new ways, the authors obtain existence theorems for abstract operator equations which include quasi-linear evolution equations and optimal control problems involving Cauchy-Dirichlet and Cauchy-Neumann equations. Specifically, the authors obtain existence of optimal solutions without requiring seminormality for Mayer type and Lagrange type problems monitored by (1) \(Ex(t)+(\Lambda x)(t)=g(t,(Mx)(t),u(t))\), u(t)\(\in \omega (t)\), \(t\in J=[0,T]\), \(x\in X=L_ p(J,V)\) where V is a separable reflexive Banach space with dual \(V^*\). Assuming that \(\geq \alpha \| v\|\) and \(\| g(t,z,u(t))\| \leq \ell (t)\) for measurable selections u(t) of \(\omega\) (t), the authors show sequential weak continuity of \(N_ u(x)=\Lambda x-g(\cdot,Mx,u)\) and show that \(\geq 0\) for \(\| x\| \leq (\| \ell \| /\alpha)^{1/(p-1)}.\) An existence theorem of Hou for solutions of \(Ex+N_ ux=0\) yields the existence of solutions for (1) in \(K=\{x| \| x\| \leq R\}\) where \(R\geq (\| \ell \| /\alpha)^{1/(p-1)}\). Using a variant of the McShane-Warfield implicit function theorem, the class \(\Phi\) of all solutions of (1) for all possible measurable selections u of \(\omega\), is shown to be sequentially weakly compact. These results are then applied to the case where \(E=(d/dt)\) with domain \(D=\{x\in X\), \(dx/dt\in X^*\), \(x(0)=0\}.\) Using the maximal monotonicity of E on \(X=L_ 2(J,V)\) and Hou's recent result that maximal monotone operators on Fréchet spaces have Cesari's property (Q), the authors obtain existence of weak solutions of the abstract Cauchy problem dx/dt \(\Lambda\) x\(=f\) with \(x(0)=0\), \(f\in X^*=X=L_ 2(J,V)\). Several special cases are then considered. These existence theorems are then applied to obtain optimal solutions for Mayer and Lagrange type control problems. Using a fixed point theorem of Hou for multifunctions \(\Gamma\) : \(K\to K\) with closed graph, controllability is proved for problems monitored by \(Ex(t)=g(t,Mx(t),u(t))\), u(t)\(\in \omega (t,Mx(t))\) for \(t\in [0,T]\). Several illustrative examples are given.