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Nonlinear Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces. (English) Zbl 1297.34103
The paper establishes the existence of solutions of the nonlinear (dynamic) Sturm-Liouville boundary value problem with mixed derivatives on \([0, \infty)\) with a real-valued nonlinear term. The mixed derivatives used are the \(\Delta-\) and \(\nabla-\) derivatives. After a brief review of time scale calculus, the authors present a result on the measure of noncompactness for bounded subsets \(A\) of the Banach space \(C_1^{\Delta\nabla}(\mathbb{T}, E)\) where \(\mathbb{T}\) is a timescale and \(E\) is a Banach space. The main existence result is obtained by using Mönch’s fixed point theorem. As a special case, the results obtained are valid for Sturm-Liouville \(q-\) difference equations.

MSC:
34N05 Dynamic equations on time scales or measure chains
34G20 Nonlinear differential equations in abstract spaces
34A40 Differential inequalities involving functions of a single real variable
34B24 Sturm-Liouville theory
47N20 Applications of operator theory to differential and integral equations
39A13 Difference equations, scaling (\(q\)-differences)
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Full Text: Euclid