Existence of Solution for a Nonlinear Operator Inclusion Problem and Its Application to a Doubly Nonlinear Elliptic Equation

Teffera M. Asfaw

Abstract and Applied Analysis, 2026, vol. 2026, 1-10

Abstract: The main goal of the study is to establish an existence result which guarantees the solvability (in the Sobolev space W1,pΩ with p≥2) of a doubly nonlinear elliptic Dirichlet boundary value problem, with nonlinear higher order term of the form −div qx,πux,∇πux and a lower order term of the type gx,ux−fx, with x∈Ω― and u∈W1,pΩ, where f∈LqΩ, p and q are conjugate exponents to each other, Ω is a nonempty, bounded, and open subset of RN, and q, g, and π are Carathèodory functions satisfying mild conditions. In order to complete this goal, we first derive new abstract existence theorems on the solvability of operator inclusion problems of the type f∗∈MAw+Gw with Aw∈DM, where M:X⊇DM→2X∗ is a maximal monotone operator of type S+, G:X→X∗ is a compact operator and A:X→X is a continuous and β-expansive operator such that for some α∈0,2−1β, we have Aw−Av,Jw−v≥−αw−v2 for all w,v∈X2. The abstract results and their application to study problem given in Equation (1) give improvements of the existing results. The arguments of the proofs lie on fundamental properties of the Leray–Schauder degree mapping and a subjectivity result for expansive operator.MSC2020 Classification: 47H07, 47H11, 47H14, 35J15, 35J60, 35J66

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:2239090

DOI: 10.1155/aaa/2239090

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