In this paper, we give some coupled fixed point results in the framework of C * -algebra-valued b-metric spaces and in particular in the setting of C * algebra-valued metric spaces. These results, with shorter proofs, generalize and improve other theorems recently introduced. We have used a method of reducing coupled fixed point results to the respective ones for mappings with one variable in the framework of b-metric spaces. Finally, two examples are given to support our theoretical work.

We prove boundedness, Hölder continuity, Harnack inequality results for local quasiminima to elliptic double phase problems of p-Laplace type in the general context of metric measure spaces. The proofs follow a variational approach and they are based on the De Giorgi method, a careful phase analysis and estimates in the intrinsic geometries.

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We prove the existence of at least two positive homoclinic solutions for a discrete boundary value problem of equations driven by the (p, q)-Laplace operator. The properties of the nonlinearity ensure that the energy functional, corresponding to the problem, satisfies a mountain pass geometry and a Palais-Smale compactness condition.Remark 1.1. By (H1) it follows that g(z, 0) = 0 for all z ∈ N.

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