Let (X, [[sigma].sub.b]) be a b-metric-like space with parameter s [greater than or equal to] 1, and let f, g : X [right arrow] X and [alpha] : X x X [right arrow] [0, [infinity]) be given mappings and
arbitrary constant p such that p [greater than or equal to] 2.
(b) By the first equation of (7), we easily obtain the fact that [r.sub.1]u(1-u/K)-[rho][u.sup.[alpha]]v [less than or equal to] [r.sub.1]u(1-u/K) in [0, +[infinity])x[OMEGA]; the first result follows easily from the simple comparison argument for parabolic problems, and thus there exists T [member of] (0, +[infinity]) such that u(t, x) [less than or equal to] K + [epsilon] in [T, +[infinity]) x [OMEGA] for an
arbitrary constant [epsilon] > 0.
where [c.sub.0] is an
arbitrary constant. From (4), we have
For
arbitrary constant values, [[alpha].sub.2] = [[alpha].sub.1] = l, [[alpha].sub.7] = -1, [[alpha].sub.6] = 1, [[alpha].sub.10] = 0, [[alpha].sub.11] = 1, and [[alpha].sub.8] = 5.
Proof If f is a constant function, say f(x) = c for all x [member of] G, then using the functional equation (10) we have g(x) = c(2c- 1) for any
arbitrary constant c [member of] C.
Here, we note, for rigid body displacement field, that [u.sub.r] = a + b x r, where a and b are
arbitrary constant vectors, and, for constant electric potential, [[phi].sub.r] [tau]([u.sub.r], [[phi].sub.r]) = 0.
Here we propose [f.sub.2](T) = T + [epsilon][T.sup.2], where e is an
arbitrary constant. For this model, manipulation of (13)-(15) yields
After integration we get [[lambda].sub.2]g' = [c.sup.2][phi](g') + [c.sub.1], where [c.sub.1] is an
arbitrary constant. Suppose that [lim.sub.x[right arrow][infinity]]([partial derivative]u/[partial derivative]x)(x, 0) = A.
Optical band gap is denoted as Eg for the crystal and 'A' is the
arbitrary constant. Plank's constant is denoted by 'h' and frequency of incident photons is denoted by 'Y'.
where m is an
arbitrary constant. These values of (eq.) and (eq.) are define the famous Schwarzschild spacetime.
Later, inhomogeneous ODE (17) is solved uniformly in all three cases using method of
arbitrary constant variation [5].