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Using some ideas of the Wesson induced matter theory, we obtain a new kind of \(F(^{(4)}\!R,\varphi )\) modified gravity theory as an effective four-dimensional (4D) theory derived from \(f(^{(5)}\!R)\) gravity in five dimensions (5D). This new theory exhibits a different matter coupling than the one in BBHL theory. We show that the field equations of the Wesson induced matter theory and of some brane-world scenarios can be obtained as maximally symmetric solutions of the same \(f(^{(5)}\!R)\) theory. We found criteria for the Dolgov–Kawasaki instabilities for both the \(f(^{(5)}\!R)\) and the \(F(^{(4)}\!R,\varphi )\) theories. We demonstrate that under certain conditions imposed on the 5D geometry it is possible to interpret the \(F(^{(4)}\!R,\varphi )\) theory as a modified gravity theory with dynamical coefficients, making this new theory a viable candidate to address the present accelerating cosmic expansion issue. Matter sources in the \(F(^{(4)}\!R,\varphi )\) case appear induced by the 5D geometry without the necessity of the introduction of matter sources in 5D.

New effective coupled $$F(^{(4)}\!R,\varphi )$$ F ( ( 4 ) R , φ ) modified gravity from $$f(^{(5)}\!R)$$ f ( ( 5 ) R ) gravity in five dimensions