In this paper, governed by the fundamental solutions we introduce the Green’s function of the second-order differential equations in general form with respect to boundary conditions and deal with the solvability of the infinite system of a second-order differential equations. Using the ideas of Hausdorff measure of noncompactness and Meir-Keleer condensing operator we seek the sufficient conditions to justify the existence of solutions for the aforementioned system in the Banach sequence space l_{p}.

Throughout this paper, using the p-adic wavelet basis together with the help of separation of variables and the Adomian decomposition method (as a scheme in numerical analysis) we initially investigate the solution of Cauchy problem for two classes of the first and second order of pseudo-differential equations involving the pseudo-differential operators such as Taibleson fractional operator in the setting of p-adic field.

P-adic numbers serve as the simplest ultrametric model for the tree-like structures arisingin various physical and biological phenomena. Recently p-adic dynamical equations started to beapplied to geophysics, to model propagation of fluids (oil, water, and oil-in-water and water-in-oilemulsion) in capillary networks in porous random media. In particular, a p-adic analog of theNavier–Stokes equation was derived starting with a system of differential equations respectingthe hierarchic structure of a capillary tree. In this paper, using the Schauder fixed point theoremtogether with the wavelet functions, we extend the study of the solvability of a p-adic field analogof the Navier–Stokes equation derived from a system of hierarchic equations for fluid flow in acapillary network in porous medium. This equation describes propagation of fluid’s flow throughGeo-conduits, consisting of the mixture of fractures (as well as fracture’s corridors) and capillarynetworks, detected by seismic as joint wave/mass conducts. Furthermore, applying the Adomiandecomposition method we formulate the solution of the p-adic analog of the Navier–Stokes equationin term of series in general form. This solution may help researchers to come closer and find morefacts, taking into consideration the scaling, hierarchies, and formal derivations, imprinted from theanalogous aspects of the real world phenomena.

Throughout this paper, via the Schauder fixed-point theorem, a generalization of Krasnoselskii’s fixed-point theorem in a cone, as well as some inequalities relevant to Green’s function, we study the existence of positive solutions of a nonlinear, fractional three-point boundary-value problem with a term of the first order derivative [...]

In this paper, we deal with the existence results for mild solutions of abstract fractional evolution equations with non-instantaneous impulses on an unbounded interval. We also establish the existence of S-asymptotically -periodic mild solutions. The applied techniques are supported by the concept of measure of noncompactness in conjunction with the well-known Darbo-Sadovskii and Tichonov fixed-point theorems. Furthermore, an example to the fractional initial/boundary value Cauchy problem is concerned to illustrate our main results.

Throughout this work, using the technique of measure of noncompactness together with Meir–Keeler condensing operators, we study the solvability of the following infinite system of third-order differential equations in the Banach sequence space c_{0} as a closed subspace of ℓ∞ :u′′′_{i}+au′′_{i}+bu′_{i}+cu_{i}=f_{i}(t,u1(t),u2(t),...)where fi∈C(R×R^{∞},R) is ω -periodic with respect to the first coordinate and a,b,c∈R are constant. Our approach depends on the Green's function corresponding to the aforesaid system and deduce some conclusions relevant to the existence of ω -periodic solutions in Banach sequence space c_{0} . In addition, some examples are supplied to illustrate the usefulness of the outcome.

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