Some new existence theorems for nonlinear inclusion with an application (Q2778296)

This article deals with the problem of finding \(u\in X\), \(x\in Su\) and \(y\in Tu\) such that \(u=A(y,x)\), where \(X\) is a nonempty set in an ordered (by a cone \(P)\) Banach space \(X\), \(A:X\times X\to E\) a nonlinear mapping, \(S,T:X\to 2^X\) two set-valued mappings. Under the assumptionsNEWLINENEWLINE \((A_1)\) \(A(x_0,y_0)\leq u_0(x_0\in Su_0)\), \(v_0\leq A(y_0,x_0)y_0\in Tv_0)\);NEWLINENEWLINE \((A_2)\) there exist two constants \(M\geq 0\) and \(0\leq N<\tfrac 12\) such that, for any \(u^{(1)}, u^{(2)}, v^{(1)}, v^{(2)}\in D\), \(u^{(1)}\leq u^{(2)},v^{(2)}\leq v^{(1)}\) implies that NEWLINE\[NEWLINEA\bigl(y^{(2)}, x^{(2)}\bigr) -A\bigl(y^{(1)} ,x^{(1)}\bigr)\geq-M \bigl(u^{(2)} -u^{(1)}\bigr) -N\bigl(v^{(1)} -v^{(2)}\bigr)NEWLINE\]NEWLINE NEWLINE\[NEWLINEA\bigl (y^{(1)}, x^{(1)} \bigr)-A \bigl(y^{(2)},x^{(2)}\bigr)\geq-M\bigl( v^{(1)} -v^{(2)}\bigr) -N\bigl(v^{(2)}-v^{(1)}\bigr)NEWLINE\]NEWLINE for all \(x^{(1)} \in Su^{(1)}\), for all \(x^{(2)}\in Su^{(2)}\), \(y^{(1)}\in Tv^{(1)}\), \(y^{(2)}\in Tv^{(2)}\);NEWLINENEWLINE \((A_3)\) there exists a constant \(\beta\in(0,1-2N)\) such that, for any \(u,v,u\leq v\) implies that NEWLINE\[NEWLINE-(M+N) (v-u)\leq A(x,y)- A(y,x)\leq\beta (v-u)\;x\in Su,\;y\in Tv;NEWLINE\]NEWLINE \((A_4)\) for any \(u,w,w,u\leq w\leq v\) implies that NEWLINE\[NEWLINEA(y',x')-A(x,y)\leq M(v-w)+N(w-u)\;x\in Su,\;y\in Tv,\;x'\in Sw, \;y'\in Tw,NEWLINE\]NEWLINE the authors prove the solvability of the problem above and the convergence of the corresponding successive approximations. Some modifications are also presented.