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condition the associated linear operator is uninvertible. Third order differential equations arise in a variety of different areas of applied mathematics and physics, as the deflection of a curved beam having a constant or varying cross section, three layer beam and so on [20]. Recently much attention has been paid to the existence of solutions, especially for the positive solutions, of third-order multi-point boundary value problems at non-resonance (for details see [1, 21 30, 10, 7, 17, 22, 12, 27]). Anderson [1] established the existence of at least three positive solutions to

POSITIVE SOLUTION FOR NONLINEAR THIRD-ORDER MULTI-POINT BOUNDARY VALUE PROBLEM AT RESONANCE