Clear["Global`*"]
DSolve[{p'[c]==(p[c]-c)/(α(1+λ)/(2λ)-c),p[α]==v},p[c],c]//FullSimplify

Solve[((v α-((c^2)+(v-α) α) λ)/(α-2 c λ+α λ))==p,c]//FullSimplify
D[((v α-((c^2)+(v-α) α) λ)/(α-2 c λ+α λ)),c]//FullSimplify

{{c->((p λ-(Sqrt[λ ((-p+v) α+((p^2)-(p+v) α+(α^2)) λ)]))/λ)},{c->((p λ+(Sqrt[λ ((-p+v) α+((p^2)-(p+v) α+(α^2)) λ)]))/λ)}}
D[((p λ-(Sqrt[λ ((-p+v) α+((p^2)-(p+v) α+(α^2)) λ)]))/λ),p]//FullSimplify
D[(1-λ)(v-p)/(2λ(p-0)),p]//FullSimplify

Manipulate[Plot[((v α-((c^2)+(v-α) α) λ)/(α-2 c λ+α λ)),{c,0,α}],{{λ,1},0,1},{{α,1},0,1},{{v,1},1,3}];
Manipulate[Plot[{((λ+((λ (α-2 p λ+α λ))/(2 (Sqrt[λ ((-p+v) α+((p^2)-(p+v) α+(α^2)) λ)]))))/λ)*1/α,-((v (-1+λ))/(2 (p^2) λ))},{p,0,v}],{{λ,1/2},0,1},{{α,1},0,1},{{v,1},1,3}]