Gomaa, Adel Mahmoud On the solution sets of four-point boundary value problems for nonconvex differential inclusions. (English) Zbl 1220.34021 Int. J. Geom. Methods Mod. Phys. 8, No. 1, 23-37 (2011). Consider the problem\[ \ddot{u}(t) \in F(t,u(t),\dot{u}(t)),\quad u(0) = x_{0},\quad u(\eta)= u(\theta)=u(T),\quad \text{a.e. on }I, \tag{1} \]where \(I= [0,T]\), \(0 < \eta < \theta < T\), \(F\) is a multifunction from \(I\times \mathbb R^{n}\times \mathbb R^{n}\) to nonempty compact subsets of \(\mathbb R^{n}\).Theorem. Let \(F\) be a multifunction satisfying the following conditions: 8mm (i) for each \((x,y) \in \mathbb R^{n}\times \mathbb R^{n}\), the multifunction \(F(\cdot,x,y)\) is measurable;(ii) for each \(t\in I\) a.e., the multifunction \((x,y)\to F(t,x,y)\) is continuous;(iii) for each \((t,x,y)\in I\times \mathbb R^{n}\times \mathbb R^{n}\), \[ \|F(t,x,y)\|=\sup\{\|v\|: v\in F(t,x,y)\}\leq a(t) + c_{1}(t)\|x\| +c_{2}(t)\|y\| \]with \(a,c_{1}, c_{2}\in L^{p}(I,\mathbb R^{+})\), \(2\leq p<\infty;\) (iv) the spectral radius \(r(L) < 1,\) where \(L:C(I,\mathbb R^{n})\times C(I,\mathbb R^{n})\to C(I,\mathbb R^{n})\times C(I,\mathbb R^{n})\), \(L(f,g)=(L_{1}(f,g), L_{2}(f,g))\) with \[ L_{1}(f,g) =\int_{0}^{T}|G(t,\tau)|(c_{1}(\tau)f(\tau) +c_{2}(\tau)g(\tau))\,d\tau, \] \[ L_{2}(f,g) =\int_{0}^{T}|\partial{G(t,\tau)}/\partial{\tau}|(c_{1}(\tau)f(\tau) +c_{2}(\tau)g(\tau))\,d\tau, \]where \(G\) is the Green function.Then problem (1) admits a solution. Reviewer: Oleg Filatov (Samara) Cited in 2 Documents MSC: 34A60 Ordinary differential inclusions 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations Keywords:differential inclusions; four-point boundary value problems; Green functions PDF BibTeX XML Cite \textit{A. M. Gomaa}, Int. J. Geom. Methods Mod. Phys. 8, No. 1, 23--37 (2011; Zbl 1220.34021) Full Text: DOI References: [1] Azzam D. L., Control Cybernet. 31 pp 659– [2] DOI: 10.1090/S0002-9939-1991-1045587-8 · doi:10.1090/S0002-9939-1991-1045587-8 [3] Bressan A., Studia Math. 90 pp 69– [4] DOI: 10.1007/BFb0087685 · doi:10.1007/BFb0087685 [5] De Blasi F. S., Topol. Methods Nonlinear Anal. 2 pp 303– [6] DOI: 10.2140/pjm.1986.123.9 · Zbl 0595.47037 · doi:10.2140/pjm.1986.123.9 [7] Gomaa A. M., J. Egypt. Math. 12 pp 97– [8] Hartman P., Ordinary Differential Equations (1967) · Zbl 0161.31301 [9] DOI: 10.1016/0047-259X(77)90037-9 · Zbl 0368.60006 · doi:10.1016/0047-259X(77)90037-9 [10] Hille E., Amer. Math. Soc. Colloq. Publ. 31 [11] DOI: 10.1512/iumj.1973.22.22058 · Zbl 0274.28008 · doi:10.1512/iumj.1973.22.22058 [12] DOI: 10.4153/CJM-1969-041-7 · Zbl 0202.33803 · doi:10.4153/CJM-1969-041-7 [13] Himmelberg C., Bull. Acad. Polon. Sci. Math. Astronom. Phys. 19 pp 911– [14] Ibrahim A. G., J. Egypt. Math. Soc. 8 pp 155– [15] Ibrahim A. G., Appl. Math. Comput. 136 pp 297– [16] Klein E., Theory of Correspondence (1984) [17] Kuratowski K., Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 pp 397– [18] Ricceri B., Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nature 81 pp 283– [19] DOI: 10.1016/j.jmaa.2006.10.081 · Zbl 1127.34033 · doi:10.1016/j.jmaa.2006.10.081 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.