Existence and stability theorems in vector-valued metric spaces for fixed point and coincidence point problems governed by multi-valued weak contractions

Abstract In this work, in the setting of a vector-valued metric space X , the fixed point inclusion $$x\in G(x), x\in X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>G</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> governed by a multi-valued operator $$G:X\multimap X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo>⊸</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> satisfying a weak contraction type condition on its graph is studied. Some stability results are obtained and the coincidence point problem involving two multi-valued operators is also studied. The best proximity point problem in vector-valued metric spaces is finally discussed.