Semilinear evolution systems with nonlinear constraints (Q2823168)

For \(t\in [0,T]\) the following system of ODE's is considered NEWLINE\[NEWLINEx'(t)+A_1x(t)=f_1(t,x(t),y(t)),NEWLINE\]NEWLINE NEWLINE\[NEWLINEy'(t)+A_2y(t)=f_2(t,x(t),y(t)),NEWLINE\]NEWLINE NEWLINE\[NEWLINEg_1(x,y)=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINEg_2(x,y)=0,NEWLINE\]NEWLINE where \(A_i\) are linear operators in Banach spaces \(X_i\) generating the strongly continuous semigroups \(S_i\), and \(g_i\), \(i=1,2\) are nonlinear operators. The constraints defined by \(g_1\) and \(g_2\) have to define the `nonlocal initial conditions at \(t=0\)' for \(x\) and \(y\) respectively. A very important role in all considerations here plays the `\textit{support of the above mentioned nonlocal initial conditions}'.NEWLINENEWLINEThe authors are interested in the global mild solutions to the stated problem, i.e., the solution of the following integral equations NEWLINE\[NEWLINEx(t)=S_1(t)x(0)+\int_0^tS_1(t-s)f_1(s,x(s),y(s))ds,NEWLINE\]NEWLINE NEWLINE\[NEWLINEy(t)=S_2(t)y(0)+\int_0^TS_2(t-s)f_2(s,x(s),y(s))ds,NEWLINE\]NEWLINE where \(x(0)\) and \(y(0)\) have to be understand in the `nonlocal sense'.NEWLINENEWLINEThis last system may be viewed as a fixed point problem in a certain Banach space. To this end the authors introduce the generalized vector valued norm and define generalized Banach space based at this notion. Following this way and using the generalized fixed point theory for generalized contractions, the authors prove, using Perov theorems, that if \(f_1\) and \(f_2\) satisfy a kind Lipschitz conditions and if some other conditions are satisfied, then the mild solution exists and is unique. The paper contains two versions of the existence theorem. Two examples are joint.