Let be two operators such that.
Let be two operators such that (i) whenever, (ii) is compact and continuous, (iii) is a contraction mapping. .
Let A, B be two operators such that: (i) A x + B y ∈ M whenever x, y ∈ M ; (ii) A is compact and continuous; (iii) B is a contraction mapping. .
Let A and B be two operators such that: (H1): (Ax+Byin {M}), wherever (x,yin {M}); (H2): A is compact and continuous; and (H3): B is a contraction mapping.
Let (mathcal{A}), (mathcal{B}) be two operators such that (i) (mathcal{A}x+mathcal{B}yinmathcal{M}) whenever (x, yinmathcal{M}), (ii) (mathcal{A}) is compact and continuous, (iii) (mathcal{B}) is a contraction mapping.
Let G, H be two operators such that (i) (G x)+H y)in W) for (x,yin W); (ii) G is compact and continuous; (iii) H is a contraction mapping.
(2) More generally, if (T,S OmegarightarrowOmega) are two operators such that both T and S are γ-condensing, then the operator (A OmegatimesOmegarightarrowOmega) defined by A ( x,y ) =frac{1}{2} bigl( T ( x ) +S ( y ) bigr) satisfies the γ-condensing condition ((C_{1})).
More generally, if (T,S OmegarightarrowOmega) are two operators such that both T and S are γ-condensing, then the operator (A OmegatimesOmegarightarrowOmega) defined by A ( x,y ) =frac{1}{2} bigl( T ( x ) +S ( y ) bigr) satisfies the γ-condensing condition ((C_{1})).
Suppose that (G Imrightarrow Im ) and (F SrightarrowIm ) are two operators such that (a) there exists (varphi_{C} inPhi ) such that for all (theta,varthetainIm ), we have Vert Gtheta-Gvartheta Vert leqdeltavarphi_{G}bigl Vert theta-vartheta Vert bigr), for some constant (delta>0), (b) F is completely continuous, (c) (theta=Gtheta+BvarthetaRightarrowthetain S) for all (varthetain S).
suppose that (A:Xrightarrow X) and (B Srightarrow X) are two operators such that (C1) there exists (varphi_{A} inPhi) such that for all (x,y in X), we have |Ax-Ay|leqsigmavarphi_{A}bigl Vert x-yVert bigr), for some constant (sigma>0), (C2) B is completely continuous, (C3) (x=Ax+By Rightarrow x in S) for all (y in S).
Suppose that A : M → X and B : X → X are two operators such that (i) A is weakly-strongly continuous, and A ( M ) is relatively weakly compact, (ii) B is nonexpansive and ω-condensing, (iii) I − B is demiclosed, (iv) if λ ∈ ( 0, 1 ) and x = λ B x + A y for some y ∈ M, then x ∈ M. .
