Monotone positive solution of fourth order boundary value problem with mixed integral and multi-point boundary condition
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Monotone positive solution of fourth order boundary value problem with mixed integral and multi-point boundary conditions Faouzi Haddouchi1,2 · Nourredine Houari2 Received: 31 May 2020 / Revised: 13 August 2020 / Accepted: 18 August 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper, we investigate the existence of monotone positive solutions for a fourth order boundary value problem with dependence on the derivative in nonlinearity under integral and multi-point boundary conditions. By applying the fixed point theorem in a cone, some criteria on the existence of positive solutions are acquired. These criteria are given by explicit conditions which are generally weaker than those derived by using the classical norm-type expansion and compression theorem. As applications, three examples are presented to illustrate the validity of our mains results. Keywords Existence · Monotone positive solution · Fixed point theorem · Boundary value problem · Cone Mathematics Subject Classification 34B15 · 34B18
1 Introduction Fourth order boundary value problems have been the subject of intense research in recent decades, and arise in the study of various areas of research as mechanics, chemical engineering, flow and thermo-elasticity. Several problems have been considered and different approaches and techniques have been adopted in many papers during the years.
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Faouzi Haddouchi [email protected] Nourredine Houari [email protected]
1
Department of Physics, University of Sciences and Technology of Oran-MB, Oran, Algeria
2
Laboratory of Fundamental and Applied Mathematics of Oran, Department of Mathematics, University of Oran 1, Oran, Algeria
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F. Haddouchi, N. Houari
Mixed integral and multi point boundary conditions gained much attention over the years, which change according to the requirements of the mathematical applications in many fields. The research of monotone positive solutions have received a lot of attention by many authors, and a considerable number of works have been published, see for example [2,17,19]. Recently, many people have established the existence of positive solutions of some boundary value problems of nonlinear differential equations, readers can see [1,6,8,10,12,13,15,20–27] and references cited therein. The author in [16] used a fixed point theorem of generalized concave operators to investigate the existence and uniqueness of monotone positive solutions for the following fourth order two points boundary value problem with nonlinear boundary conditions u (t) = f (t, u(t), u (t)), 0 < t < 1, u(0) = u (0) = u (1) = 0, u (1) = g(u(1)), where f ∈ C([0, 1] × R × R) and g ∈ C(R) are real functions. In [11], the authors based on a fixed point theorem in a specially constructed cone due to Krasnoselskii and Zabreiko to deal with the existence and uniqueness of positive solutions of the following fourth order m-point boundary value problem ⎧ ⎪ 0 < t < 1, ⎨u (t) +αu − βu = f (t, u), m−2 m−2 bi u(ξi ), u(
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