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1991 Mathematics Subject Algebra Assume AT(G automorphism Banach spaces boundary conditions bounded Cayley plane coefficients commutative compact manifold completes the proof complex numbers constant convergence convex function Corollary curvature and torsion defined Definition denote Department of Mathematics derive difference equations differential differential inequalities domain eigenvalues element eventually positive solution exists finite flag manifold formula function f geodesic given groupoid H-function Hadamard's inequalities harmonic Hence holds homotopic Hua's inequality idempotent implies inner product integral Jto Jto Key words Lemma Let f linear locally convex spaces Math Mathematics Subject Classification matrix minimal nonassociative ring nonlinear norm obtain Oscillation oscillatory p-energy p-harmonic maps p-minimizing p-SSU positive integer problem properties prove quasigroup Rg(x Riemannian manifold right loop S. S. Dragomir satisfies sectional curvature semigroup sequence spaces SpT(a subgroup Suppose Tamkang University Theorem 2.1 vector Volume 28 words and phrases wp(A Zayed